functionally graded materials: design, processing and applications

FUNCTIONALLY GRADED MATERIALS

Design, Processing and Applications

MATERIALS TECHNOLOGY SERIES

Series editor: Renee G. Ford

The Materials Technology series is dedicated to state-of-the-art areas of materials synthesis and processing as related to the applications of the technology. By thorough presentation of the science underlying the technology, it is anticipated that these books will be of practical value both for materials scientists and engineers in industry and for engineering students to acquaint them with developments at the forefront of materials technology that have potential commercial significance.

Ceramic Injection Molding Beebhas C. Mutsuddy and Renee G. Ford Hardbound (0412 53810 5)

Cryochemical Technology of Advanced Materials Yu. D. Tretyakov, N.N. Oleynikov and O.A. Shkyajhtin Hardbound (0 412 63980 7)

Modelling of Materials Processing Gregory C. Stangle Hardbound (041253810 5)

Porous Materials Kozo Ishizaki, Sridhar Komameni, Makota Nanko Hardbound (0412711109)

FUNCTIONALLY GRADED MATERIALS

Design, Processing and Applications

edited by

Y. Miyamoto Professor, Joining and Welding Research Institute

Osaka University

W. A. Kaysser Director, Institute for Materials Research

German Aerospace Center

B.H. Rabin President, GA Powders, Inc.

A. Kawasaki Professor, F aculty of Engineering

Tohoku University

Renee G. Ford President, Renford Communications, Ltd.

SPRINGER SCIENCE+ BUSINESS MEDIA. llC

Library of Congress Cataloging-in-Publication Data

Functionally graded materials : design, processing, and applications I edited by Y. Miyamoto.

p. em. --(Materials technology series)

I. Functionally gradient materials. I. Miyamoto, Yoshinari. II. Series: Materials technology series (Springer Science+ Business Media, LLC) TA418.9.F85F86 1999 620.1'1--dc21 99-40751

Copyright ® 1999 by Springer Science+ Business Media New York Originally published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1st edition1999

CIP

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+ Business Media, LLC

Printed on acid-free paper.

ISBN 978-0-412-60760-8 ISBN 978-1-4615-5301-4 ( eBook)DOI 10.1007/978-1-4615-5301-4

ABOUT THE SCULPTURE SHOWN ON THE COVER

The cover image of "Winged Torso", a sculpture which is composed of a cast polyester resin substrate that has been flame-sprayed fIrst with zinc and then with bronze. After polishing, the patina of the fmal piece is similar to foundry bronze and the high fIdelity of the substrate's features is maintained. Throughout history, sculpture has been the focus of interaction between mankind's need for creative expression and the available materials technology. The Fusion Bronze™ process used to create this sculpture by Boston sculptor Barbara Rubin-Katz l unites flame sprayed functionally graded metals with the sculptor's art.

FUSION BRONZE™

The Fusion Bronze™ process utilizes molten metal spray technology in which an electric current melts a bronze wire. The molten bronze (at about 1 OOO°C) is entrained in a flow of compressed air in a confmed space. This creates a jet of very small liquid bronze particles that can coat a sculptural substrate (plaster, resin, Fiberglas™, or even paper, wax, or wood) with a bronze overlayer. If the sculptural substrate is a material which might be deformed or destroyed by contact with molten bronze, a layer of zinc, which melts at about 400°C, is applied fIrst and then the bronze is deposited. The zinc acts as a thermal plane dissipating the heat as the bronze is applied. Thus the fusion bronze overlayer is truly a Functionally Graded Material (FGM).

Substrate - Cast Polyester Resin

/ Zinc Layer o

0.10 inch

Schematic of the Microstructure of a FGM, Fusion Bronze sculpture.

Surface Oxide or Sulfide Patina

IBaroara Rubin-Katz trained at the Philadelphia College of Art and the Pennsylvania Academy of Fine Arts, and also studied classical figurative sculpture with Evangelos Frudakis in Philadelphia. On moving to Boston she continued her studies under the guidance of the late Peter Abate, a noted New England sculptor and teacher. She started showing her work publicly in the late 1980's and from the outset received recognition and awards. Since the early 1990's her work has been seen in many exhibitions at the Copley Society of Boston, the Federal Reserve Bank of Boston, Fanueil Hall, Hellenic College, Montserrat College of Art, Bradford College, Worcester Polytechnic Institute and many regional galleries. In 1992 the Copley Society of Boston awarded Barbara Rubin-Katz the highly esteemed designation of "Copley Artist".

Contents

Contributors IX

Preface Xlll

Acknowledgements xv

INTRODUCTION 1

LESSONS FROM NATURE 7

GRADED MICROSTRUCTURES 29

MODELING AND DESIGN 63

THE CHARACTERIZATION OF PROPERTIES 89

PROCESSING AND FABRICATION 161

APPLICATIONS 247

SUMMARY AND OUTLOOK 315

Index 319

Contributors

Prof. N. Arakt(5.3.2, 5.3.4)* Department of Energy and Mechanical Engineering, Shizuoka

University, Hamamatsu 432-8011, Japan Dr. L. Chen (1) Institute for Materials Research, Tohoku University, Katahira 2-1-1,

Aoba-ku, Sendai 980-0812, Japan Dr. N. Cherradi_(5.3.3) 38 Avenue de Montoie CH-I007 Lausanne, Switzerland Prof. M. J. Cima_( 6.7.1) Massachusetts Institute of Technology, 77 Mass. Ave., Room 12-011,

Cambridge, MA 02139, U.S.A. Prof. F. Erdogan (5.4.3a, 5.4.3b) Department of Mechanical Engineering and Mechanics, Lehigh

University, Bethlehem PA, 18015, U.S.A. Dr. R. G. Ford (Ch.1 - 8) Editor-in-chief, Materials Technology, P.O. Box 72, Harrison, NY

10582-0072, U.S.A. Prof. M. Gasik (7.5) Laboratory of Materials Processing and Powder Metallurgy, Helsinki

University of Technology, Vuorimiehentie 2A, FIN-02150 Espoo, Finland Prof. A. M. Glaeser_(6.6.2) Department of Materials Science and Mineral Engineering, Hearst

Mining Building, University of Cali fomi a Berkley, CA 94720 U.S.A. Prof. T. Hirai (6.3.5, 7.2.1b) Institute for Materials Research, Tohoku University, Katahira 2-1-1,

Aoba-ku, Sendai 980-0812, Japan

x Contributors

Dr. K. Hirano (5.4.3c) Department of Materials Science and Bioengineering, Mechanical

Engineering Laboratory, AIST, MIT!, Namiki, 1-2, Tsukuba, 305-0044, Japan

Mr. T. Hirano (4.3.2, 7.3.2c) Electronic Engineering Laboratory, Daikin Industries, Ltd., Aza-Ohtani,

Okamoto, Kusatsu-City, 525-0044, Japan Prof. B. I1schner (6.2.1, 6.2.3a) Swiss Federal Institute of Technology, MX-D, EPFL Ecublens, CH-1015

Lausanne, Switzerland Dr. H. Imai (7.3.3) Research Association for Nuclear Facility Decommissioning, 821-100,

Funaishikawa, Tokai, Ibaraki, 319-1111, Japan Dr. Y. Itoh (7.3.3) Power and Industrial Systems Research and Development Center,

Toshiba Corporation, Ukijima 2-1, Kawasaki, Kawsaki-City, 210-0862, Japan

Dr. C. Kawai (5.5.4) Itami Research Laboratories, Sumitomo Electric Industries, Ltd., 1-1-1,

Koya-Kita, Itami, Hyogo, 664-0016, Japan Prof. A. Kawasaki (Ch.5, 5.1,5.3.1,5.3.5,5.4.1,5.4.2,5.4.4,5.4.5,5.5) Department of Materials Processing, Faculty of Engineering, Tohoku

University, Sendai 980-8579, Japan Prof. Dr. W. A. Kaysser (Ch.6 and 7, 6.1, 6.2, 6.3.1, 6.3.2, 6. 3. 3a,

6.3.4, 6.3.6, 6.3.7a, 6.4, 6.5, 6.6.1, 6.6.3, 6. 7.2, 7.1, 7.2.1c, 7.3.1, 7.3.2a, 7.3.2b, 7.3.4, 7.4)

Director, Institute of Materials Research, German Aerospace Center 51140 Cologne, Germany

Prof. Y. Koike (7.7) Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi,

Kohoku-ku, Yokohama, 223-0061, Japan Prof. M. KoizumU 1) REC, Ryukoku University, Otsu, 520-2123, Japan Mr. K. Kurihara (6.3.5) Fujitsu Laboratories Ltd., 10-1, Wakamiya Morinosato, Atsugi-shi,

Kanagawa, 243-0122, Japan Prof. J. J. LannuttU6.2.1a) Department of Materials Science and Engineering, Ohio State University,

477 Watts Hall, 2041 College Road, Columbus, OH 43210-1179 U.S.A. Prof. M. I. MendelsonJ6.3.3b, 7.2.1b, 7.2.2, 7.2.3) College of Science and Engineering, Loyola-Marymount University,

Loyola Blvd. at 80th West St., Los Angeles, CA 90045-2699 U.S.A.

Contributors Xl

Prof. Y. Miyamoto (Ch.l- 8, 1, 5.2, 6.2.3f, 6.7.1, 7.7, 7.8.2, 8) Joining and Welding Research Institute, Osaka University, Ibaraki,

Osaka 567-0047, Japan Dr. T. Nagano (6.7.3) Japan Science and Technology Corporation, Ceramic Superplasticity

Project, Fine Ceramic Center, Atsuta, Rokuno 2-4-1, Nagoya 456-8587, Japan

Dr. M. Niino (1) Kakuda Research Center, National Aerospace Laboratory, STA Kakuda,

Miyagi 981-1525, Japan Prof. I. Nishida (7.3.2c) National Research Institute for Metals, 1-2-1, Sengen, Tsukuba-shi,

Ibaraki, 305-0047, Japan Prof. F. Nogata (2) Department of Mechanical System Engineering, Faculty of Engineering,

Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan Dr. H. Ohnishi (7.8.1) Department of Orthopaedic Surgery, Artificial Joint Section and

Biomaterial Research Laboratory, Osaka Minami National Hospital, 677-2, Kido-Cho, Wachinagano-Shi, Osaka, 586-0001, Japan

Dr. B. H. Rabin (Ch. 3 and 4, 3, 4) President, GAPowders Inc., 2300 N. Yellowstone Idaho Falls, ID 83404,

U.S.A. Mr. K. S. Ramesh (4.3.4) Battelle Pacific Northwest Laboratory, Richland, WA 99352-9668,

U.S.A. Dr. P. Sarkar (6.3. 7b) Advanced Industrial Materials and Processes Group, Alberta Research

Council, 250 Karl Clark Road, Edmonton, Alberta, Canada, T6N 1E4 Prof. M. Sasaki (6.3.5, 7.2.1b) Department of Materials Science, Faculty of Engineering, Muroran

Industrial University, Muzumoto, 27-1, Muroran, 050-0071, Japan Mr. N. Shimoda (6.3.3b) Steel Research Laboratories, Nippon Steel Corporation, 20-1, Shintomi

Futtsu, Chiba, 293-8511, Japan Prof. I. Shiota_(7.4.1) Department of Chemical Engineering, Kogakuin University, 2665-1,

Nakano, Hachioji, Tokyo, 192-0015, Japan Dr. Y. Tada_(7.2.1a, 7.2.1b) Foundation for Promotion of Japanese Aerospace Technology, 1-16-6

Izumi-Chuo, Izumi-ku, Sendai, Miyagi, 981-3133, Japan Prof. M. Tamura_(6.2.2)

xu Contributors

Department of Materials Science and Engineering, National Defense Academy, 1-10-20 Hashirimizu Yokosuka-City, 239-8686, Japan

Dr. T. Tateishi (7.8.3) Bio-Group, National Institute for Advanced Interdisciplinary Research,

AIST, MITI, Higashi 1-1-4, Tsukuba 305-0046, Japan Dr. R. L. Williamson (3, 4)

Idaho National Engineering and Environmental Laboratory, P.O. Box 1625, Idaho Falls, ID 83404, U.S.A.

Dr. J. Yoo (6. 7.1) Massachusetts Institute of Technology, 77 Mass. Ave., Room 12-011

Cambridge, MA 02139, U.S.A. Prof. J. Yoshino (7.6) Department of Physics, Tokyo Institute of Technology, Ohokayama,

Meguro, Tokyo 152-0033, Japan

* The numbers in parentheses are the contributed sections and edited chapters.

Preface

Seven years have elapsed since Dr. Renee Ford, editor-in-chief of Materials Technology, first suggested to me to publish a book on Functionally Graded Materials (FGMs). She said that the FGM concept, then largely unknown outside of Japan and a relatively few laboratories elsewhere, would be of great interest to everyone working in the materials field because of its potentially universal applicability. There was no book about FGMs in English at that time, although the number of research papers, review articles, and FGM conference proceedings had been increasing yearly. We discussed what the book should cover, and decided it should present a comprehensive description from basic theory to the most recent applications of FGMs. This would make it useful both as an introduction to FGMs for those simply curious about what this new materials field was all about, and also as a textbook for researchers, engineers, and graduate students in various material fields. The FGM Forum in Japan generously offered to support this publication program.

Because it is very difficult for an individual author to write a book that covers such a wide range of various aspects of many different materials, I invited more than 30 eminent materials scientists throughout the world, who were associated with FGM research, to contribute selected topics. I also asked several leading researchers in this field to edit selected chapters: Dr. Barry H. Rabin, then at the U.S. Department of Energy's Idaho National Engineering and Environmental Laboratory and now President of GA Powders, Inc.; Dr. Wolfgang A. Kaysser, Director of the Institute for Materials Research at the German Aerospace Center in Cologne; and Dr. Akira Kawasaki, Professor of Materials Science at Tohoku University in Japan. Dr. Ford reviewed each edited chapter and rewrote the book in a

XIV Preface

uniform style. This approach proved to be very time consuming, and is the major reason for the time it has taken to complete this book for publication. However, it has been constantly updated to reflect the latest developments.

A unique feature of this book is that its writing and production has been accomplished completely electronically. Nothing was printed on paper until the book's actual pUblication. Internet communication has developed rapidly worldwide since we started the actual writing and editing in 1994. We decided to use e-mail for communicating and for transmitting the files to each other. However, at the beginning we encountered some confusion and incompatibility when exchanging files. Part of the problem was due initially to inexperience with using the Internet and part due to our different computer systems - some of us are Macintosh users and others use pes. However, we soon overcame our hardware and software incompatibilities, developed a system for file identification so we could keep track of the most recent versions of each chapter, and became quite adept at electronic editing. Older manuscripts were updated quickly and the latest research results were added easily using e-mail. Many of the figures and tables were reformed or newly prepared using computer graphics by my colleagues, Dr. J. S. Lin and Miss K. Agu assisted by some of my students. Finally, Dr. Rabin formatted the entire manuscript for submission to the publisher electronically. We are proud that this is the first book written and edited completely electronically spanning three continents.

The sculpture on the book's cover is by the contemporary artist, Barbara Rubin-Katz. Throughout human history, sculpture has been the focus of interaction between the need for artistic expression and the materials technology available. The fusion bronze process used in creating this sculpture, which unites flame sprayed functionally graded metals with the sculptor's art, exemplifies this tradition. This abstract sculpture of a torso symbolizes that optimized FGMs can be used to create beauty as well as function.

Dr. Yoshinari Miyamoto, Editor

Professor of Materials Science, Joining and Welding Research Institute, Osaka University. Japan

Acknowledgements

The authors would like to gratefully acknowledge the sponsorship of FGM FORUM, The Society for Non-Traditional Technology, Kotohira Kaikan Bldg. 1-2-8, Toranomon, Minato-ku, Tokyo 105-0001, Japan. We also appreciate the tremendous patience and support shown by all the contributors.

Chapter 1

INTRODUCTION

Keywords: Element, material ingredient

1. DEFINITIONS AND HISTORICAL PERSPECTIVE

In a Functionally Graded Material (FGM) both the composition and the structure gradually change over the volume, resulting in corresponding changes in the properties of the material. The structural unit of an FGM is referred to as an element [1] or a material ingredient [2, 3]. It is a conceptual unit for constructing an FGM that includes various aspects of its chemical composition, physical state, and geometrical configuration. The term, material ingredient, probably expresses the overall concept best. Typical examples are listed in Table 1.1. Material ingredients can resemble biological units such as cells and tissues. For example, bamboo, shell, tooth, and bone all have graded structures consisting of biological material ingredients. Graded structures and functions in nature are discussed in Chapter 2 on Lessons from Nature.

T, hIll FGM a e , , scan b C e ompose d fV' M. 0 anous ateria 11 d' ngre lents, Chemical inorganic, organic, ceramic, metal, polymer Physical electronic state, ionic state

crystalline state, dipole moment, magnetic moment band gap, potential well, barrier

Geometrical granule, rod, needle, fiber, platelet, sheet pore, texture, orientation

Biological complex macromolecule, organelle, cell, tissue

2 Chapter 1

In the simplest FGMs, two different material ingredients change gradually from one to the other as illustrated in Figure 1.1 (a). The material ingredients can also change in a discontinuous way such as the stepwise gradation illustrated in Figure 1.1 (b). This type of structure can also be considered an FGM. The most familiar FGM is compositionally graded from a refractory ceramic to a metal. It can incorporate incompatible functions such as the heat, wear, and oxidation resistance of ceramics with the high toughness, high strength, machinability, and bonding capability of metals without severe internal thermal stress.

(a) (b)

Figure 1.1 . (a) Continuous and (b) stepwise graded structures.

Pores also are important material ingredients of FGMs. A gradual increase in the pore distribution from the interior to the surface can impart many properties such as mechanical shock resistance, thermal insulation, catalytic efficiency, and the relaxation of thermal stress. Even if the gradation of the material ingredients is limited to a specific location in the material such as the interface, a joint, or a surface as shown in Figure 1.2, the material can be considered to be an FGM because it incorporates the FGM concept. Although this concept can be extended to materials with functions that are designed to change gradually over time or with environmental conditions, such as a drug delivery system, these time dependent functions are actually produced by tailoring the spatial gradation of the material ingredients. The creation of multiple or new functions with graded structures, rather than the graded material itself, is the basis for the FGM concept as reflected in this book.

The general idea of structural gradients first was advanced for composites and polymeric materials in 1972 [4, 5]. Various models were suggested for gradients in composition, in filament concentration, and in polymerization along with possible applications for the resulting graded structures. However, there was no actual investigation about how to design, fabricate, and evaluate graded structures until the 1980s.

Introduction 3

Figure 1.2. Local gradients at the joint (a) and surface (b).

In 1985, the use of continuous texture control was proposed in order to increase the adhesion strength and minimize the thermal stress in the ceramic coatings and joints being developed for the reusable rocket engine [6]. The developers realized that this continuous control of a property could be extended to a more general concept that could be applied to impart new properties and functions to any material by gradually changing its texture or composition. At this time, the concept of the material ingredient was introduced for designing such materials [1].

In 1986, these types of materials were termed functionally gradient materials, which soon became abbreviated to the now familiar, FGM. In 1995, as a consequence of a discussion at the Third International Symposium on FGMs held in Lausanne in 1994, it was decided to change the full name to functionally graded materials because it is more accurate both descriptively and grammatically.

Since FGMs are not homogeneous materials, it was recognized that in order to create them, comprehensive studies would need to be done in design methodology and theoretical modeling as well as in processing and evaluation. Therefore, in 1987 a 5-year research program, "Fundamental Studies on the Relaxation of Thermal Stress by Tailoring Graded Structures," was initiated in Japan [7]. The program's focus was to develop FGMs for high temperature use with the objective of using them for the hypersonic space plane. Since 1989, the results of these research and development programs have been disseminated worldwide via papers, articles, international meetings, and exchange programs. An international symposium on FGMs has been held every 2 years starting with the first one in 1990 in Sendai followed by San Francisco, Lausanne, Tsukuba, and

4 Chapter 1

Dresden. Today, the FGM concept extends over a variety of materials fields worldwide.

2. OVERVIEW OF APPLICATIONS

The FGM concept is applicable to many fields, as illustrated in Figure 1.3. In the engineering applications to cutting tools, machine parts, and engine components, incompatible functions such as heat, wear, and corrosion resistance plus toughness, and machinability are incorporated into a single part. For example, throwaway chips for cutting tools made of graded tungsten carbide/cobalt (WC/Co) and titanium carbonitride (TiCN)-WC/Co have been developed and commercialized that incorporate the desirable properties of high machining speed, high feed rates, and long life [8]. Various combinations of these ordinarily incompatible functions can be applied to create new materials for aerospace, chemical plants, and nuclear energy reactors.

ENGINEERING

NUCLEAR ENERGY

Nudear reactor ~enls

Figure 1.3. Potentially applicable fields for FGMs.

Introduction 5

The FGM concept is also applicable to functional materials. A graded potential for electrons or ions can be tailored by using techniques such as molecular beam epitaxy (MBE), metal-organic chemical vapor deposition (MOCVD), and scanning probe microscopy (SPM) to create graded structures on the atomic scale. This can produce unique quantum effects that could result in new applications for semiconductor and sensor materials. Graded band theory and its application to semiconductor materials are discussed in Chapter 7.

The application of FGMs to biomaterials is growing in importance. Over 2500 surgical operations to incorporate graded hip prostheses have been successfully performed in Japan over the past twelve years. These graded hip implants enable a strong bond to develop between the titanium implant, bone cement, hydroxyapatite (HAp), and bone. The bone tissue penetrates HAp granules inserted between the implant and the bone forming a graded structure. The application of the FGM concept to hip implants is also discussed in Chapter 7.

There are many potentially useful applications of the FGM concept. For example, structural walls that combine the two functions of thermal and sound insulation with good specific strength could be made by the gradation of both the porosity and the composition. Also, it might be possible to design attractive interference colors for automobiles by dispersing graded coated particles such as titanium dioxide/mica (Ti02/mica) in the body coating. Graded combinations of flexibility or elasticity and rigidity could enhance sports equipment such as golf clubs, tennis rackets, and skis. The sculpture on the book's cover shows the use of gradation in the creation of a work of art.

By exploiting the myriad possibilities inherent in the FGM concept, it is anticipated that the properties of materials will be optimized and new functions for them created. A comprehensive discussion of design, modeling, processing, and evaluation of FGMs as well as their applications is covered in this book.

REFERENCES

1. Hirai, T. (1996) Functional gradient material, in Processing o/Ceramics, Part 2, (ed. R.J. Brook), Materials Science and Technology, 17B, 293-341, VCH Publishers, Weinheim.

2. Mortensen, A. and Suresh, S. (1995) Functionally graded metals and metal-ceramic composites: Part I "Processing", International Materials Reviews, 40(6), 239-265.

3. Mortensen, A. and Suresh, S. (1998) Fundamentals o/Functionally Graded Materials, 10M Communications Ltd., London.

4. Bever, M.B. and Duwez, P.E. (1972) Gradients in composite materials, Mater. Sci. Eng., 10, 1-8.

6 Chapter 1

5. Shen, M. and Bever, M.B. (1972) Gradients in polymeric materials, J. Mater. Sci., 7, 741-746.

6. Niino, M. et. al. (1984) Fabrication of a high pressure thrust chamber by the eIP forming method, AIAA Paper No. 84-1227.

7. Koizumi, M. and Niino, M. (1995) Overview ofFGM research in Japan, in Functionally Gradient Materials, (eds. B.H. Rabin and I. Shiota), MRS Bull. XX, 19-21.

8. Miyamoto, Y. (1997) Applications ofFGM in Japan, in Functionally Graded Materials: Manufacture, Properties, and Applications, (eds. A. Ghosh et al.) Ceramic Transactions, Am. Ceram. Soc. 76, 171-189.

Chapter 2

LESSONS FROM NATURE

Keywords: Biological structure, composite material, multifunctionality, optimum design, mechanosensor, intelligent material system, adaptability, self-optimization, self-adaptive modeling, structural efficiency, weight-cost, fiber reinforcement, microstructure, cellular structure, bamboo, mollusk shell, palm tree, bone

1. INTRODUCTION

In examining biological load carriers such as the stems of plants and the trunks of trees, animal bones, mollusk shells, and other biological hard tissues, it can be seen that their geometry changes to accommodate to their physical environment. This implies that they are highly adapted to all boundary and loading conditions defined by their environment. Only the most economical construction is able to survive the intense competition for energy as well as the external physical conditions with the minimal amount of materials available to them in their limited living space. For example, the interior structure (architecture) of a bone has an optimized shape with respect to the direction of principal stress and the magnitude of the shear stress [1]. This has been explained to be due to an optimized mechanical design that is characterized by uniform stress distribution with no localized stress peaks [2]. This suggests that both bone and other biological tissues are managed by a self-optimizing system with sensing mechanisms that can detect external mechanical stimuli in order to control the modeling and remodeling of the skeletal system [3]. It can be inferred, therefore, that the shape and ingenious construction of biological hard tissues are the result of a continuous process of intelligent optimization.

8 Chapter 2

The basic characteristics of biological hard tissue, such as microstructure, function, and modeling systems both fascinate and inspire the designers of engineering structure. The basic difference between biological and artificial structures is that the former has living organisms that can be characterized by multi functionality, hierarchical organization, and adaptability [4, 5]. Consequently, biological structures are complicated and nonuniform. This suggests that an intelligent and functional combination of elements, materials, and components with different strengths in the same structure can result in hybrid systems whose properties are designated for specific purposes.

This chapter describes the ingenious construction, strength, and natural process of optimization by a cell-based mechanosensing system in certain biological tissues. Bamboo, is used as an example for the development of new material and structural concepts, such as composites of multiphased and functionally graded materials.

2. GENERAL MORPHOLOGICAL CHARACTERISTICS OF BIOLOGICAL TISSUE

A structure can be defined as any assemblage of materials intended for sustaining loads [6]. Although it is difficult to distinguish between structures and materials in biological structural systems, from a macroscopic viewpoint there are three ways to sustain external loads as shown in Figures 2.1, 2.2, and 2.3. 1. By changing the microstructure (that is by moderating the thickness

and/or the shape), e.g., bamboo, mollusk shell. 2. By changing the size and/or shape of a body, e.g., the trunk of a tree and

the stem of a plant, the shaft of a leaf or a feather. 3. By combining the first two types, e.g., human and animal bone, the

spicule of a sea urchin, a rat tooth. Graded structures of the first type (1) are more common in nature than

the other two types.

Lesson from Nature

bamboo

• fibe r plaooment

• volume densuty

lamellar

foliated nacre

mol usc shell

Purplish Washington clarY

(Saxidomus purpuratus)

Figure 2.1. A way to accommodate or sustain external loads by changing the microstructure (1) in biological systems.

sunnyG) ® ® 0 ® side shaft

4 4A ~~

, I 100 -..I 500 .I

900 13)0

1700 L:2700 nm

The shape of a ClOSS section of the leaf

CD ® ® (3) ® ~~~~~ tree trurk

Figure 2.2. A way to accommodate or sustain external loads by changing the size and/or shape of a body (2) in biological systems.

9

10 Chapter 2

Femur head

• shape and size • para sity

animal bone tooth

Figure 2.3. A way to accommodate or sustain external loads by combining types (1) and (2) in biological systems.

3. A NATURAL PROCESS OF OPTIMIZATION: ADAPTIVE MODELING WITH A CELL-BASED MECHANOSENSOR

3.1 Bamboo

Bamboo belongs to the genus Bambuseae in the grass family, Gramineae. Like other grasses, bamboo grows in thick clumps, but like trees it often attains great heights. In some species in Asian monsoon forests, the woody, hollow aerial stems of bamboo often grow to heights of about 30 m and to diameters of about 25 cm.

When the transverse section of a bamboo stem is cut, many beautiful brown spots can be seen (Figure 2.4 (a». These small spots on the outside and the inside of the cross section vary widely in shape. Figure 2.4 (b) is an enlarged photograph of one of these spots located near the inside of the cross section. It is shaped like a flower. In the center, there are two large holes (these are vessels in the xylem -- the supporting and water conducting tissue of vascular plants), and a few small holes (these are sieve tubes in the phloem -- the food conducting tissue of vascular plants). If both of these holes are replaced by one big hole, the significance of the flower shape can be understood by comparing it with the stress distribution around a single

Lesson from Nature 11

hole in an infinite plate (see Figure 2.5 (a)) and around three holes in an actual plate, when both are subjected to uniaxial tension (see Figure 2.5 (b) -- a photo of elastic stress patterns). The optimum way to reinforce these holes is to insert fiber bundles in accordance with the stress distribution. Therefore, the shape of the fiber bundles (the black areas in Figure 2.4 (b) that consist of four or five hundred fine fibers per bundle) suggests the presence of stress around the vascular structures in the xylem and phloem.

L....-___ .....I' Smm

(a) Transverse section through the stem of Moso bamboo

L..--...J O.1mm

(b) Enlargement of a vascular bundle

Figure 2.4. A transverse section showing the placement of fiber bundles. (a) Transverse section through the stem of Moso bamboo. (b) Enlargement of a vascular bundle.

The contours of biological structures such as tree stems, red deer antlers, human tibia, and tiger claws have been shown to be highly optimized with respect to their mechanical strength and minimum weight [2, 6]. This implies that biological structures may possess mechanical sensing capability. Voltage signal curves obtained from a bamboo stem SUbjected to an external bending moment, measured using an electrocardiograph, are shown in Figures 2.6 and 2.7 (the box shows a typical biological electric signal) [7]. The curves indicate the presence of a spike on loading and unloading. The higher voltage signal was recorded on the compression side rather than the tension side of the bamboo stem. This implies that the sensitivity of a bamboo cell is greater for detecting compression (stress/strain) than tension. These signals could be used to trigger adaptive growth related to the direction of the stress. Data, obtained from other plants (see Figure 2.8), show that the characteristic features of the signals depend on the kinds of

12 Chapter 2

plants. In addition, no voltage signals were induced from a specimen boiled in hot water for 1 hour or from a dried specimen that lost 50% of its weight. Because boiling or drying specimens kills the plant cells, the voltage signals recorded must have been produced by live cells in the stressed materials [8]. This indicates that live bamboo cells have the ability to sense at least some of the information induced by external mechanical stimuli.

(a) Stress distribution around a hole in an infinite plate

cr

L...J 1 mm

cr

(b) Photoelastic stress pattern around three holes

Figure 2.5. Stress distribution around holes in the structure of bamboo. (a) Stress distribution around a hole in an infinite plate. (b) Photoelastic stress pattern around three holes.

Moso bamboo

(Phyllostachys pubescens Mazel)

voltmeter

@ CD ~r

:';'"

@ W=19.6 N

Figure 2.6. Experimental set up to measure the voltage signals of bamboo.

Lesson from Nature

Tension

CD-Compression

@-> E

stress duration

o~IC==========~~ stress duration

@=CD-(g) 20min .

@- t--J,...J"',,----__ ----1-.L~

~I '--l--===========----I stress duration

G)® : unipolar lead system

® bipolar lead system

gj ~I--...L.....-"""'T'"----r-en

Typical bone signal [Williams,et aI. ~9751

13

Figure 2.7. A typical example of the voltage signals induced by the bending moment of a bamboo stem. The induced signals were recorded using unipolar and bipolar lead systems that make possible recording both tension and compression points.

Piezoelectric properties have been found in bone subjected to stress [9]. Several experimental observations have been reported that bone demonstrates a piezoelectric effect [10, 11, 12, 13]. This is used to explain the concept of stress- or strain-induced bone remodeling referred to as Wolfs law [4]. It appears that bone converts mechanical stress to an electrical potential that influences the activity of osteoclasts (large cells in growing bone) and osteoblasts (bone-forming cells) [14]. As described above, the interior structure of bone (trabecular architecture) is arranged in compressive and tensile systems corresponding to the direction of the principal stress [1].

The properties of the voltage signals induced in bamboo may be similar to the piezoelectric effect in bone. Therefore, the electrical properties of both bone and bamboo probably play an important role in the modeling and remodeling of the skeletal system in biological hard tissue. As evidence, bamboo that was grown on steep ground has an enlarged vascular bundle [8]. The deformed contours of the bamboo stem and the asymmetric shape of the fiber bundles (deformed petal shape) reflect the biased loading conditions of their environment. The recorded electric signals and the location of the fiber

14

i 1.1. 1 v-t curve

recorded signals

bipolar lead system

stress dll"a tion , i

restingpo1ential ~

II ~[~ '-----'

"1"1 ~[E9 20 min.

~~[~

rubber plant (Firus elastica Roxb)

palm tree (Butia ya tay)

fig tree (Firus carica Unn)

bamboo (Mosobamboo)

Chapter 2

Figure 2.B. Typical examples of the voltage signals induced by the bending moment of certain plants including bamboo.

bundles suggest that bamboo has a stress/strain-induced adaptive modeling system.

The data suggest that bamboo cells may possess a mechanosensor that determines the shapes in which they grow, such as the thickness of the stem and the volume density of the fiber, thereby compensating for the applied external load in order to obtain a homogeneous distribution of the stress. Bamboo could be regarded as an "intelligent material system" that has a mechanosensor. The volume density of the fiber and its distribution in the bamboo provide important information about its mechanical properties.

In bone remodeling, it is known that a bone that is bent by a mechanical load adapts by depositing new bone on the concave (compression) side, and by resorbing the bone on the convex (tension) side [4]. In this way the bone becomes adapted to an imposed mechanical stress. A similar adaptation (growing) mechanism can be seen with plants. Specifically, the compression side of a softwood tree grows faster than the tension side (called


compression wood), and the tension side of a hardwood tree grows faster than the compression side (called tension wood).

Figure 2.9 shows the transverse section of two tree stems: a Japanese cedar (softwood) and an ash (hardwood). Figure 2.10 shows the voltage signals induced in them by the bending moment of these two types of plants. The sensitivity of the electrical signals induced from the tension and compression sides is consistent with the growth direction of the tree trunks. Thus the growing mechanism of plants is governed by the ability of the cells to detect stresses. The evidence indicates that electrical signals control the growth of plants subjected to a load. Figure 2.11 is a scanning electron micrograph of the fractured fibers that form the backbone of bamboo. The distribution of fiber density at two different transverse sections (the lower position is specimen A and the upper is specimen B) in a bamboo stem is shown in Figure 2.12. This graph indicates that in a bamboo stem, the fiber density gradually increases from the inside to the outside surface as well as from the lower part to the upper part. These graded structures produce a uniform internal stress distribution in both the radial and axial directions. To examine the mechanical properties of bamboo, tension tests were performed on very small specimens with cross-sectional areas of about 0.25 mm2• The specimens were taken from nine areas arranged as shown in Figure 2.4.

A tree growng on steep ground

Softwood HaJdwood

Figure 2.9. The growing sections of two types of trees: Japanese cedar and ash.

Figure 2.13 shows the tensile strength and Young's modulus for specimens A and B along the transverse section of a bamboo stem. It can be seen that the strength of the bamboo stem gradually increases from the inside to the outside, and also that specimen B is stronger than specimen A. This is the same variation in the volume density of the fibers that was previously noted in Figure 2.12. The strength of an inner specimen (No.9 in Figure 2.4

16

stressed duration ~===========::::!.----1 tension side

(j) y \' CD

~ ~ n-r--~-----'y

<D@: unipolar lead system """1Driin.

> E o

coI11l ression side <2.>

[

stressed duration

r (

'----' 10mn.

Typical examples ofthe voltage sign als in duced for two types of trees: softwood and haltlwood

Chapter 2

Figure 2. J O. The voltage signals of a Japanese cedar and an ash tree induced by stress, tension, and compression. Note the portion grown to sustain extemalloads and the sensitivity of the tree cells.

(a)) made of pure ground tissue was found to be about 25 MPa. Using the rule of mixtures, the strength of a pure fiber was estimated to be about 810 MPa, which is equivalent to steel (600-1000 MPa). Furthermore, the Young's modulus of a pure fiber was found to be 55 GPa, which is about 25% that of steel (200 GPa). These data show that bamboo has high strength and flexibility with low rigidity.

Figure 2. J J. A scanning electron micrograph of fractured fibers in a vascular bundle.

The model selected to examine graded structures and strength in terms of the stress analysis of the transverse section of a composite beam, is a cantilevered beam carrying a uniformly distributed load between points I and

Lesson from Nature

100~--------~Q~~------------~

Epidermis

-ffffi-III-&-i-+-+- t:7.9mmktr 0=106.3

i=84.5 t:10.9mm

O(inside) 0.5 1.0

Distance from inner surface, rlt

Figure 2.12. The distribution of fiber density along the transverse section of a bamboo stem.

17

II (see Figure 2.12). This could be regarded as corresponding to pressure exerted by the wind on branches and leaves. The results show that the ratio of the measured strength (au) to the calculated strength (aj), au/aj, has almost the same value, about 25-30, for all test points on the composite beam, in both specimens A and B. This is believed to be one of the safety factors in this plant. It implies that a bamboo structure is designed to have uniform strength everywhere, both in the radial direction on the transverse section and in the axial (lengthwise) direction. Therefore it has been concluded that bamboo has a self-optimizing graded structure constructed with a system for sensing external mechanical stimuli. This could be considered a model of an intelligent adaptive system.

3.2 The leaf shaft of a palm tree

The conventional image of a palm tree is a tall tree with a single trunk and large feather- or fan-shaped leaves. Furthermore, palms, which are both tropical and subtropical trees, can withstand very strong winds that might blow from almost any direction. Figure 2.2 shows the change in shape of the cross section of a palm tree perpendicular to the leaf shaft. Figure 2.14 plots

18

Ii Q.

~ ::l

tl

s: .... C)

5i ... 1ii ~ 1/1

5i ...

1ij' Q.

~ W

Ii ::l

:a 0 E 1/1 -C)

I:

~ >

100% fiber-tensile slren~h estimated by the rule of mixtures

0 600

500

400

100% fiber- Young's modJlus estimated • by the rule of mixture e . \. • o _

60 55 GPa -e-. e 0 0 0 0 50 0

0 40 o CB t=10.9mm

30 • @t=7.9 ~ Modulus of elasticity of a rTiniature ,

sp3cirnen. @ 20

10

o (inside)

O~ ~O (outside)

Distance from inner surface, r/t

Chapter 2

Figure 2. J 3. The distribution of tensile strength and Young's modulus along the transverse section of a bamboo stem.

the torsional rigidity (Glp, where G is the shear modulus and Ip is the polar moment of inertia of an area) along the shaft. The data indicate that this plant is protected from strong winds because it has low torsional rigidity (X/L= more than 0.7) with minimal resistance, which allows the leaves to move freely.

Figure 2.15 shows the experimental set up to measure voltage signals induced by tension and compression stresses on the leaf shaft of a palm tree (a different specimen than the one in Figure 2.14). The data show the detection capability of the cell as seen in Figure 2.16. It also appears that the


-(\J

E 10 • (.)

Z -'<t

0 or-X

0-

CJ >. 5 :!::! :2 0> ·c

ctS c: 0

·00 ~

0 I- 0

0 0.5 1

XlL

XlL=0.44

I---~x

L=2700mm

Figure 2. J 4. The torsional rigidity of the leaf shaft of the palm tree, Butia yatay.

electrical signal controls the growth activity of the leaf shaft to conform to the physical environment. This exemplifies a typical design for a biological structural system of type (2).

20

Butia yatay

(J)(2): unipolar lead system ® bipolar lead system

Chapter 2

Figure 2. J 5. The experimental set up to measure the voltage signals of the palm tree, Butia yatay. Half-size diagnostic EKG electrodes are pasted on the top and bottom sides of the leaf s shaft.

stressed duration

tension side -tt)

> E 0

a) stressed duration 10min

> E 0

-G)

b) stressed duration 10min

Figure 2. J 6. Typical voltage signals induced by the bending moment of the palm tree, Butia yatay. Upper graph: voltage signals oftension(<D) and compression(~) points. Lower graph: voltage signal of the difference (@) between the tension and the compression stress.

3.3 Bone

Bone consists of a protein, collagen, and an inorganic mineral phase, hydroxyapatite [Cas(P04)J(OH)]. Figure 2.17 shows the tensile strength and Young's modulus along the transverse section of the humerus bone (a section


of compact bone) of a young rhinoceros [16]. It indicates high strength at the outside that gradually become lower toward the inside. Because there are many miniature blood vessels inside the compact bone, a variation in porosity is observed along the transverse section of this compact bone [17]. It is known that the degree and distribution of porosity affect the tensile strength and the Young's modulus [18]. Therefore, bone has a graded structure created by altering the porosity of the bone.

Rhinoceros humerus (8.5 kN, weight)

m ~ ~ 200 ~40 ~--

<Ii .2 ::J

~20 '" § >-

a) ~ ______ ~5~2~m~m~ __ ~~

1 mm

b)

Figure 2.17. The mechanical properties of the humerus bone of a rhinoceros [11].

Bone also exhibits a piezoelectric effect, which would lead one to believe that this effect is used both for detecting an external stress and to remodel bone structures so that no peak stress occurs at any point [8]. Therefore, bone structure and strength (tensile strength and Young's modulus) can be shown to result from a continuous process of self-optimization by an intelligent, adaptive modeling system.

22 Chapter 2

3.4 Mollusk shells

Mollusk shells consist of one or more ceramic phases embedded in a proteinaceous matrix. The mineral in these ceramic phases is mainly calcium carbonate (CaC03), which is unsuitable as a structural material because of its brittleness. Nevertheless, shells are by nature remarkably strong and tough structures. In order to examine their mechanical properties, a relatively large shell, the Purplish Washington clam was chosen. Figure 2.18 shows a clam shell of the Purplish Washington clam and its structure. Figure 2.19 shows the tensile strength and Young's modulus in the crosswise (BB') direction of this shell. As can be seen, the strength gradually increases toward the inside from the outside. This is also true for the microhardness.

How do shells construct a structure with extremely high strength by using the same material in three different phases? Figures 2.19 and 2.20 show that the microstructure of a shell is different in each of its three phases. The outside section, a lamellar structure, consists of 40 x 5 x 0.5 J..lm crystal plates; the midsection, a foliated structure, consists of 0.1 x 10 J..lm needlelike crystals; and the inside section, a nacreous structure, consists of 5 x 5 x 0.2 J..lm crystal plates.

It is known that the mechanical properties of shells depend on the combination of different structural types, and that a nacreous structure is considerably stronger than other types. There are several structural features of nacre that contribute to its strength: the thickness of the sheets (~ 0.5 J..lm); the uniform thickness of the plates; and the staggered arrangement of the blocks [19]. In addition, a softer area is located at the phase boundaries (Figure 2.18) that toughens the structure by allowing flexible deformation rather than rigid phase boundaries. These examples of ingenious construction show that shell structures have limited strength to protect them from external stresses such as point loading and hydraulic pressure. The stress analysis along the thickness (YY') direction indicates that the structure of the shell also has uniform strength ((Ju/(Jj is about 4.5). Therefore, one can conclude that mollusk shells form graded structures by combining different microstructures to achieve uniform strength.

The structural efficiency in terms of weight-cost for some performance indices based on loading-bearing modes for selected materials is summarized in Table 2.1 [6,20,21,22]. Some of the data plotted in Figure 2.21 show the relationship between the specific modulus, E/p, and the specific strength, (J/p. It can be seen that pure bamboo fiber, the shaft of a feather, and wood all have excellent specific strengths comparable to engineering alloys and ceramics, and that bone and antler have almost the same strength as engineering alloys. Also, in terms of -.J Ej p, which governs the weight-cost of overall deformability, bamboo fiber is superior to steel.

Lesson from Nature

78mm

A

Figure 2.18. A clam shell of the Purplish Washington clam and its structure.

Microhardness Hv (300g)

~~4~ __ ~~~~~1~ __ ~N N

50 40 30 20 10 Young's modulus E(GPa)

St rength (fu (M P a)

o

",E ..:.s. Ul Ul Q)

co .x:

<D .U o~

Distribution of tensile strength, Young's modulus and microhardness

~--

T T

)~ c:i~ ;<-,-t- I<>i - r-

IO. C:;:G) 0

--' -lJ

Purplish Washington clam J.Q..mm (Saxdomus purpura/us)

Figure 2.19. The mechanical properties of a clam shell.

23

24 Chapter 2

Table 2.1. The structural efficiency in terms of weight-cost for selected materials and the performance index for a mode of loading.

Materials E(GPa)*l (E/p)*('<'E/p)* au(MPa)*2 (aulp)* (a/3/p)* p*3 Ref.

_ ~Qd_s!e.:l ___ ~O_O _____ (~5..:6) __ (1.:.82 __ 4..02-.?~0 ___ J~I.:~42 ___ iI!;9.:~·~t __ 7·~ _____ _

Carbon 200 (25.6) (1.8) 450-700 (58-90) (7.5-10) 7.8 steel

_ ~r~.? !t.:~ __ .?~o _____ (~5..:6) __ (1.:.82 __ 8_02-1~.o.9 ___ (!~2:1.?±t _ il}:ljj2.. __ 7.~ _____ _

_ ~I_a~<ry~ ___ 73 ______ (~6) ___ (3.:.02 __ 1_42-.?~0 ____ (~0.:~9.?2.. __ i9J~.-~3.:.6] __ ~.~ _____ _ Ceramics:

_ ~i~ ______ ~~O _____ (~I] ___ (.5.:.02 __ 4..°2 ______ (!2)2 ____ i~6J2 ____ 2'~ _____ _

_ ~~2 _____ !7_0 _____ (!4..61 __ (3.:.52 __ 1_62 ______ (~3) _____ 17..:9) _____ 2·7 _____ _

Ah03 390 (100) (5.0) 500 (128) (63.0) 3.9

Bamboo: _ ~'!!~ _____ ~I ______ (! 7..:8] __ (3J2 __ 2}7 ______ (~3_42 ____ i3_6.:.02 ___ _ 1·1 ~ __ ~ __ _ !!1~t~~ ____ ~ ______ (!.~7) __ (1.:.02 __ 2) _______ (!8..:4) ____ i6..:3) ____ _ 1'2~ __ ~ __ _ ~l!.f~ f!b_e~ __ ~5 _____ J~2)) __ Q .:.02 __ 8}2 ______ (~7_02 ____ i8}}2 ___ _ l·Q~ __ ~ __

Wood II (27.5) (8.3) 100 (250) (53.8) 0.4 20

(pine)

Mollusc shell:

Purplish Wa.

clam 32.4 (11.8) (2.0) 44.1 (16) (4.5) 2.75 8

~nacre2

Bone:

rhino 13 (6.5) (1.7) 15O (75) (14.0) 2.01 16 J~l!~e!"!t _________________________________________________ _

antler 14 (7.3) (1.92 200 (105) (18.02 1.9 20

Leaf shaft:

palm tree (Butia

yatay)

Feather

shaft:

2 (1.8) (1.2) 4 (3.6) (2.3) 1.1 8

Stanley 7.2 (10.7) (4.0) 310 (462) (68.3) 0.67 8 ____________________________________________________ _

duck 12 (16.2) (4.6) 245 (331) (52.9) 0.74 8

*1: E=Young's modulus, *2: au: Tensile strength, *3: p: Density (Mg, megagram) *Performance index for a mode ofloading[21,22]: E/p=GPaI(Mg/m\ au/p=MPaI(Mg/m3) for tension. ('<'E/p)=" GPaI(Mg/m3), (au 2131 p )=MPa2l3/(Mg/m3) for bending.

4. WHAT CAN BE LEARNED

The ultimate goals of this work are to understand the principles of design and the processes found in biological materials, and to apply these findings to develop new and superior structural concepts for materials. For example,

Lesson from Nature

Direction of

al{t.5 G) Lamellar 5

@ Foiated

II G) Nacre

5~·2Ilm 5

Size of one crystal

Figure 2.20. The microstructure of a clam shell.

---. cf) .g 0> ~

~ a.. ~

a. -... W (If ::;, '5 "0 0 E t.l

~ Q. (j)

103

102

10

10-1

10-2

10-1 10

crane

Engineering polymers

Specific strength, 0u/p (MPaI(Mg/m3))

Figure 2.21. The specific modulus, E/p, as a function of the specific strength, alp, for some selected biological hard tissues.

25

26 Chapter 2

composites of multi phased and functionally graded materials could be created by using or modifying models found in living organisms. Thus rather than developing new materials with high stiffness (see Figure 2.22), it might be more advantageous to create structures using the optimal placement of fibers, various microstructures, or porous or cellular structures. In addition, the self-optimizing, cell-based sensing system for analyzing external physical stimuli adapted by plants could serve as a useful model for constructing new, intelligent, multiphased materials systems.

REFERENCES

I. Koch, I.C. (1917) The laws of bone architecture, Am. J Anat., 21, 177-198. 2. Mattheck, C. and Burkhardt, S. (1990) A new method of structural shape optimization

based on biological growth, Internat/. J Fat., 12 (3), 185-190. 3. Mattheck, C. (1990) Engineering components grow like trees, MaterialwissenschaJt und

WerkstofJtechnik, 21, 143-168. 4. Wolff, I. (1870) XXII: Uber die innere Architecture der Knochen und ihre Bedeutung fUr

die Frage von Knochenwachsthum (Concerning the internal architecture of bone and its implication for the study of bone growth), Archiv for Pathologisch Anatomie, 50 (3), 389-450.

5. Srinivasan, A.V., Haritos, G.K., and Hedberg, F.L. (1991) Biomimetics: Advancing manmade materials through guidance from nature, App. Mech. Rev., 44 (11), 463-481.

6. Gordon, I.E. (1978) Structures or Why Things Don't Fall Down, Penguin Books, London, 17.

7. Williams, W.S. and Breger, L. (1975) Piezoelectricity in tendon and bone, J Biomech., 8, 407-413.

8. Nogata, F. and Takahashi, H. (1995) Intelligent functionally graded material: bamboo, J Compos. Eng., 5 (7), 743-751.

9. Fukada, E. and Yasuda, I. (1957) On the piezoelectric effect of bone, J Phys. Soc. Jap. 12 (10), 1158-1162.

10. Martin, R.B. (1972) The effects of geometric feedback in the development of osteoporosis, J Biomech., 5, 447-455.

11. Gjelsvik, A. (1973) Bone remodeling and piezoelectricity-I, ibid., 6,69-77. 12. Cowin, S.c. and Hegedus, D.H. (1976) Bone remodeling I: Theory of adaptive

elasticity, J Elastic. 6 , 313-326. 13. Cowin, S.c. and Van Buskirk, W.C. (1978) Internal bone remodeling induced by a

medullary pin, J Biomech., 11,269-275. 14. Hayes, W.C. et al. (1982) Stress-morphology relationships in trabecular bone of the

patella, in Finite Elements in Biomechanics, (eds. R.H. Gallagher, et al.), John Wiley & Sons, New York, 223-268.

15. Japan Society of Wood (1991) World of Beautiful Wood (in Japanese), Ohtsu, Japan, Kaiseisha, 25.

16. Currey, I.D. and Nogata, F. (1991) unpublished data, investigation performed at University of York, U.K.

17. NHK Project team (1989) Human Body (in Japanese), NHK publications, Tokyo, 50.

Lesson from Nature

18. Alexander, R.M. (1984) Animal Mechanics (second edition), Blackwell Scientific Publications, London, 123.

27

19. Currey, J.D. (1980) Mechanical properties of mollusk shell in The Mechanical Properties 0/ Biological Materials, 34th Symposium of the Society for Experimental Biology, (eds. J.F.V. Vincent and J.D. Currey), Cambridge University Press, London, 75-97.

20. Gordon, J.E. (1988) The Science o/Structures and Materials, Scientific American Library, New York, 25 and 176.

21. Ashby, M.F. (1992) Materials Selection in Mechanical Design, Pergamon Press, Oxford, 35.

22. Ashby, M.F. et al. (1995) The mechanical properties of natural materials I, Material property charts in Proceedings o/the Royal Society London, Series A, Mathematical and Physical Sciences, 450, 123-140.

Chapter 3

GRADED MICROSTRUCTURES

Keywords: Structure, microstructure, microstructure characterization, microstructural analysis, nonuniform materials, characteristic dimensions, spatial variation, volume fraction, connectivity, field parameter.

1. INTRODUCTION

It is well known that microstructure plays a predominant role in determining material behavior. Materials engineers therefore seek to control microstructure through processing. Processing studies have traditionally focused on optimizing microstructural characteristics with the intent of producing a uniform microstructure throughout the material. Increasing microstructural uniformity has long been considered a fruitful means of improving properties. In contrast, FGMs are produced containing deliberate spatial nonuniformities in their microstructures. By treating microstructure as a variable that is dependent on position, different material characteristics can be incorporated in a single component. Such a component can be considered a materials system integrated at the microstructural level to achieve optimum performance in a specific application. This is what distinguishes FGMs from other materials.

Quantitative microstructural analysis methods aid in understanding the fundamental relationships between structure and properties needed for effective utilization of this new class of materials. For example, certain aspects of the microstructure must be specified as a function of position so that local material properties can be estimated for modeling purposes. Alternatively, an FGM designed using modeling techniques must be

30 Chapter 3

described in terms of the spatial distribution of its key microstructural features that define goals for synthesis and processing.

The most widely studied implementation of the FGM concept involves the joining of dissimilar materials (e.g. ceramic-to-metal bonding), where residual stresses develop during cooling as a result of the property mismatch across the bonded interface. These FGMs typically consist of an interiayer between the ceramic and metal that exhibits a gradual transition in the relative amounts (volume fractions) of the ceramic- and metal-rich phases. The spatial variation in volume fraction is optimized to minimize the thermal stresses that can cause thejoint to fail during fabrication or while in service. Although this is one of the most important applications for FGMs, the FGM concept can extend far beyond this. It has been suggested that, in principle, a spatial variation in any microstructural feature can be produced through appropriate manipulation of material and processing variables [1]. Countless material combinations and microstructural arrangements are therefore possible. It is reasonable to expect that all types of microstructures known from the study of spatially uniform materials are relevant to the field of FGMs.

FGMs can be produced by numerous processing methods that are described in detail in Chapter 6 [1, 2]. Undoubtedly, FGMs produced by different processing methods will exhibit vastly different microstructures. To illustrate the wide range of possibilities, Figure 3.1 shows several examples of graded materials. Differences occur due to other factors besides processing, since within a given graded material, different microstructural types exist at different locations. This level of complexity presents significant challenges for quantitative analysis. For this reason it is important to establish a common framework to describe and analyze FGM microstructures.

The goals of this chapter are to present a brief overview of the relevant principles of microstructural analysis, and to discuss spatially nonuniform microstructures with particular attention to differences between these materials and conventional (spatially uniform) materials.

Graded Microstructures

2. RELEVANT PRINCIPLES OF MICROSTRUCTURE CHARACTERIZATION

2.1 Background

31

The goal of microstructural analysis is to develop a quantitative description of microstructure that can be used to establish its relationship to properties and processing. The level of effort required to fully characterize a given material, as well as the choice of appropriate analysis methods, will depend to a large extent on the kind of information desired and the accuracy needed. Excellent reviews of microstructural analysis principles and applications can be found in the literature [3-5], therefore only a brief introduction is presented here.

Figure 3.1. Examples of different graded microstructures produced by various processing methods (a) powder processed alumina-nickel (AI20 3-Ni), (b) thermal sprayed AhOrNi, (c) chemical vapor deposited silicon carbide-carbon (CVD SiC-C), (d) vapor deposited aluminum-silicon (AI-Si)

32 Chapter 3

In general, microstructures consist of three different types of elements -phases, interfaces, and defects, all of which are relevant to the study of FGMs. Within this framework, numerous features can be identified. To arrive at a useful description of microstructure, certain characteristics of the microstructural features of interest must be quantified. Table 3.1 lists the basic elements that compose all microstructures, along with some common features of interest and examples of characteristics used to describe them.

Characterization typically involves specifying a combination of crystallographic parameters (e.g., crystal structure and chemical composition) and geometrical parameters (e.g., volume fraction, size, shape, orientation, and spatial arrangement) for the various microstructural elements. The primary focus of most analyses is on specifying the chemical composition and geometrical parameters for the phases present. However, other information is often important, depending on the material and the application. For example, point defect concentrations influence the electrical properties in semiconductors, and dislocation density affects the strength of cold-worked metals.

Table 3 1 The Basic Elements of Microstructure .. Typical Features

Elements of Interest Example Characteristics

phases grains, particles, fibers, voids state of matter, chemical composition, volume fraction, crystallinity, size, shape, orientation

interfaces surfaces, grain boundaries, specific boundary area, contiguity, interphase boundaries degree of order (coherency), impurity

segregation

defects vacancies, impurities, number density, arrangements, size dislocations, cracks

Real microstructures are three-dimensional, yet most analyses are performed on two-dimensional metallographic cross sections. There are various options available for conducting two-dimensional analyses. In the simplest case, two-dimensional parameters are used to describe the microstructure directly. The fraction of an area and the mean length of an intercept are examples. More complex two-dimensional parameters, such as shape factors, can be defined and used to describe phase morphology. Threedimensional descriptions using stereological parameters such as volume fraction or specific surface area can often be derived from two-dimensional measurements. However, for certain parameters (e.g., size distributions) simplifying assumptions must be made regarding the shape and arrangement of the features. Direct evaluation of three-dimensional geometry can only be made by laborious serial sectioning methods.

Graded Microstructures 33

Analysis of two-dimensional cross sections involves numerous steps. Foremost, careful specimen preparation is required to obtain an image that is representative of the material and suitable for quantitative analysis. This can be a challenge with FGMs because of the vastly different material characteristics that can be encountered within a single specimen. In addition, the instrument used to generate the image must have adequate resolution to distinguish the features of interest. Furthermore, the image itself must have enough contrast to allow the features of interest to be identified.

Once an image is obtained, it can be analyzed using point counting, linear analysis or areal analysis methods. Point counting and linear analyses can provide highly accurate results and are simple to perform. Recently, fully automated image analysis systems and powerful software packages have become commercially available. These instruments make it relatively easy to obtain large amounts of data rapidly. However, problems related to the acquisition and interpretation of digital images are common. In automated instruments, the measurements performed by the computer involve manipulating and analyzing the characteristics of a very finely spaced grid of points (pixels) that compose the digitized image.

Typically, average values are sought for the geometrical parameters. But individual microstructural constituents rarely need to be described. This forms the basis for the statistical nature of most microstructural measurements. Quantities measured on individual microstructural constituents are called feature parameters, whereas those that are representative of many features are called field parameters. Field parameters are usually obtained and reported assuming that the microstructure is uniform (i.e. only random errors are present) within the area of measurement. This assumption can have significant implications with respect to the analysis of graded microstructures, as will be discussed in this chapter.

Many different parameters can be used to characterize microstructures. Their selection depends on the nature of the material being examined and the kind of information being sought. Several key parameters in the characterization ofFGM microstructures are discussed below.

2.2 Crystal Parameters

2.2.1 Composition

Phase composition must be specified for any multicomponent system. Composition is typically expressed in either weight or atomic (mole) percent. In the case of single-phase materials (i.e. solid solutions), the

34 Chapter 3

volume average or bulk composition of the material is specified. In the case of multi phase materials the composition of each phase must be specified in addition to the bulk composition. Under equilibrium conditions, phase diagrams can provide useful information about the composition of the phases present. However, quantitative chemical analysis must be employed to accurately measure actual compositions. Such analyses can be performed using a wide variety of techniques, each having certain advantages and disadvantages. The choice of method depends on the material and the degree of accuracy required. Common techniques include x-ray diffraction, x-ray fluorescence, energy dispersive x-ray analysis, and wet chemical analysis, among others. Only a few methods are capable of determining composition locally at the microscopic scale, a particularly useful piece of information in the study ofFGMs.

2.2.2 Crystal structure

The structure of an atomic or molecular condensed phase can be described by the degree of short-range or long-range order present. Amorphous phases exhibit only short-range order, whereas completely crystalline phases exhibit long-range order. For each crystalline phase present in a material, the crystal structure must be specified. Under equilibrium conditions, phase diagrams can also provide useful information about the crystallographic structure of the phases present. Experimentally, conventional x-ray diffraction or electron diffraction techniques provide most of the desired crystallographic information. In certain cases, more sophisticated techniques are needed to obtain information about phase defect structures, the extent of ordering, and the like.

2.3 Geometrical Parameters

2.3.1 Nomenclature

In the analysis of microstructures it is necessary to use a common nomenclature to avoid confusion. Following conventional notation, in the following discussion volumes are denoted V, surfaces are denoted S, areas are denoted A, lines are denoted L, points are denoted P, and numbers are denoted N. Subscripts are used to refer to the test region, bars are used to represent mean values, and Greek letters are used to represent individual phases.


2.3.2 Volume fraction

Volume fraction is perhaps the single most important microstructural parameter of interest in multiphase materials. The volume fraction of phase b, expressed as VyCf)), is given by:

V~ VyCf)) = Vo

C3.1)

where V ~ is the volume of the b phase within the test region and Vo is the volume of the test region. Volume fraction can be determined from linear, areal, or point counting analysis since, within the limits of statistical errors, these yield equivalent results. Thus,

(3.2)

where AACf)), LLCf)), and PpCf)) represent the area, length, and number of points of the b phase per unit test area, per unit test length, and per number of test points, respectively. For a material containing n distinct phases the following relationship holds:

VyCI) + Vy(2) + ... VyCn) = 1 (3.3)

Thus, in a two-phase material it is sufficient to analyze the volume fraction of only one of the phases. Care must be taken when using this formula, since in some cases phases may be present that are either not anticipated or not readily observed. An example would be the presence of residual porosity in a powder processed two-phase material, where for the purposes of Equation 3.3 porosity should be treated as a third phase.

2.3.3 Boundary area density and specific boundary area

The surface or interfacial area of a phase per unit volume is known as the boundary (or interface) area density. The boundary area density of the b phase determined relative to the total volume of the test region, expressed as SyCf)), is given by:

(3.4)

36 Chapter 3

where S(~) is the surface area of the b phase, PL(~) is the number of intersection points with the b phase boundary per unit length of a test line, and LA(~) is the length of the b phase boundary per unit test area. Note that in applying Equation 3.4, grain boundary intersections must be counted twice because the boundary is shared by two surfaces.

The surface or interfacial area of a phase can also be related to the volume of the phase itself. In this case, the specific boundary ( or interface) area is obtained, expressed as:

Sv(~') = S(~) = Sv(~) = 2PL(~) = 4. LA(~) V(~) Vv(~) Pp(~) 1t AA(~) (3.5)

It has been pointed out that, other than volume fraction, the boundary area density and specific boundary area are probably the most important microstructural parameters [5]. This is because they relate directly to many microstructural phenomena and material properties, they are threedimensional quantities obtained from two-dimensional analyses without assumptions regarding phase shape or arrangement, and they are extremely useful for defining other microstructural parameters.

2.3.4 Size

The most commonly used method for expressing the size of grains or second phases involves the mean intercept length, L, expressed as:

- Lo 1 L=-=-

N NL (3.6)

where N is the number of grain boundaries intersected along a test line of length Lo. NL is simply the number of grains (or grain boundaries counted once) per unit length of test line. This method of specifying dimensions is also the most general, since no assumptions about grain shape are required. This measure can also be applied to determine the mean linear size of particles or agglomerates of a second phase embedded in a matrix, using the formula:

[(~) = Lo· Vv(~) = Vv(~) N(~) NL(~) (3.7)


where N(~) is the number of b particles intersected along a test line of length Lo. Similarly, the mean linear spacing between particles can be expressed as:

L(~-~) = Lo'( 1-Vv(~)) = 1 - Vv(~) N(~) NL(~) (3.8)

A distribution of grain or particle sizes generally exists and is often of interest. A complete discussion of size distribution analysis is beyond the scope of this review. In general, data on the size of microstructural features is obtained in the form of intercept (linear chord length) distributions obtained from cross sections, and analyzed in the form of frequency plots (histograms) or cumulative frequency plots. The most commonly used distributions are the Gaussian and log-normal functions. Distributions are described by parameters such as the mean, mode, median, and standard deviation.

2.3.5 Shape

The most common descriptions of phase shape are those based on simple geometrical objects, such as spheres and cubes (or other regular polyhedra) for which straightforward relations exist between the characteristic dimensions and such measurable quantities as surface area and volume. Other (irregular) shapes can be described using qualitative descriptors such as flakes, ligaments, ellipsoids, and whiskers. Quantitative descriptions for these features can be quite simple or complex, depending on the information required. The shape of individual features is most easily described based on planar shape parameters obtained from cross sections or projections. For example, the aspect ratio is defined as the maximum particle dimension divided by the minimum particle dimension, and is equal to 1 for a sphere, 3 - 10 for an ellipsoid or ligament, and as high as 200 for needles or whiskers.

Another commonly used shape parameter compares the length of the perimeter of a planar object with the circumference of a circle having the same area. In three dimensions, there is considerable difficulty in capturing all aspects of shape characterization of spatially distributed features using a single parameter. Typically, shape parameters defined for three-dimensional objects attempt to describe the deviation from spherical shape by combining both volume and surface area measurements into a dimensionless factor. However, the presence of a distribution in the size of features often complicates the interpretation of these results.

38 Chapter 3

2.3.6 Connectivity

As the content of the second phase of a two-phase material is increased, there is a critical volume fraction at which the transition from a dispersed phase to an interconnected microstructure takes place. A phase is said to be connected in three dimensions when a continuous path can be found through the material within that phase without intersecting the other phase. A quantitative measure of connectivity is the mean number of contacts that a particle or grain makes with particles or grains of the same phase. Unfortunately, the number of contacts per particle measured in two dimensions cannot be directly related to the three-dimensional connectivity. More frequently, the critical volume fraction (percolation threshold) at which the transition from a dispersed phase to an interconnected microstructure takes place is specified. While this value is difficult to measure metallographically, applicable methods have been described [3]. More typically, transport-based methods, such as electrical or thermal conductivity, are used to determine the percolation threshold, particularly when the two phases have markedly different transport properties. Whether or not a phase is connected strongly influences the properties of a two-phase material. This microstructural transition is of particular importance in FGMs.

2.3.7 Contiguity

Contiguity is a measure of the relative amount of a phase in contact with itself within a two-phase mixture. This parameter has also been shown to correlate with many properties of two-phase mixtures. Contiguity can be defined from the specific boundary areas as [6]:

Sv(~~) C=------

Sv(~~) + Sv(a~) (3.9)

where Sv(~~) represents the specific interface area for contact between b grains, Sv(a~) represents the specific interface area for contact between a and b grains, and NL(~~) and NL(a~) represent the number of intersections with b-b and a-b boundaries per unit length of the test line, respectively. The contiguity characterizes the skeletal nature of the microstructure, but does not specify whether or not the phase is connected in three dimensions.


2.3.8 Other features

A number of other characteristics of microstructural features can be quantified if they are useful for establishing relationships with processing or properties. Examples of topics that may be of interest include among others: the degree and direction of preferred orientation (texture); issues related to arrangement or packing of microstructural features (e.g., coordination number, particle density, nearest neighbor spacing); surface curvature; and angles between interfaces (e.g., contact angle, dihedral angle) [1,2].

2.4 Useful Mathematical Relationships

There are many mathematical relationships between the microstructural parameters that are useful. For example, in a single-phase material the grain boundary surface area is inversely proportional to the mean linear grain size:

- 4 L=

Sy (3.10)

In a two-phase material, the mean linear particle size of the second phase L(I3) is related to the contiguity C, volume fraction Vy(l3) and the mean distance between particles LCI3-I3) by the relation '

[(13) = [(13-13 ). (I-C). VyCI3) 1-VyCI3) (3.11 )

Therefore, only three independent parameters must be determined from the experimental data in order to calculate the value for the other microstructural parameter in Equation 3.11.

In actual microstructures, the distribution of the shape, spatial arrangement, and size features is complicated. The stereological parameters described above are useful for characterizing these microstructures, but are often unsuitable for use in modeling the behavior of the material. Therefore, simplified geometrical models for which well-defined mathematical relationships exist are often employed. For example, the specific surface area of spheres of radius r is given by:

4nr2 3 Sy=---

4/3.nr3 r (3.12)

40 Chapter 3

and from the relationship between the mean linear particle size and the surface area:

3L r=-

4 (3.13)

This expression can be used to calculate the radius of equally sized spheres of a second phase from the mean linear particle size measured on the cross section. Other commonly used geometrical models involve the assumption of equally sized cubes or spheres arranged in space according to a periodic lattice. Using such models, along with the relationships for the surface areas and volumes of these objects, many useful relationships can be readily derived that are extremely useful for comparing experimental measurements with theoretical predictions, such as the distance between particle centers or the spacing between particle surfaces.

3. UNIQUE ASPECTS OF SPATIALLY NONUNIFORM MATERIALS

3.1 General Comments

In the broadest sense, a graded material can be defined as any material in which the microstructure exhibits a spatial variation in at least one dimension. FGMs refer to a special class of graded materials in which the microstructural variation (and associated variation in properties or functions) has been designed and intentionally introduced into the material to optimize its performance in a particular application, or under a specific set of functional requirements.

Many materials contain gradients in composition; some of these (e.g., carburized steel) are produced by transport-based processes that have been utilized for many years. Although the composition gradient often also gives rise to gradients in other microstructural features, these spatial distributions are constrained by the processes used to produce them (e.g., solid state diffusion). With the emergence of the FGM concept, the idea of spatial variations in microstructure has been broadened beyond composition to include essentially all microstructural parameters. Now, for example, in addition to composition (or volume fraction), it is reasonable to discuss gradients in the size, shape, and arrangement of phases.


In addition, many new fabrication routes have been developed that allow controlled spatial distributions of these microstructural features to be introduced into a material. Thus, in principle, microstructural gradients can be designed a priori, and spatial distributions within the material can be established that are very different from those previously accessible via transport-based processes. This is an important distinction between the FGMs being produced today and the graded materials of the past. It is also the reason that microstructural characterization issues are so important in this field.

Some materials are referred to as "compositionally graded" to reflect that chemical composition is their primary variable of interest. However, it should be recognized that chemical gradients are not necessary for a material to have a graded microstructure. Neither does this term imply that composition is the only microstructural parameter varied in such a material. For example, a material's resistance to fatigue crack initiation might be improved simply by altering the grain size as a function of position near the material's surface. This type of material could be considered an FGM even in the absence of any compositional gradient. Similarly, in order to improve the corrosion resistance of a single-phase solid solution, the composition of an alloying element could be varied as a function of distance near the material's surface. Such a material could be considered an FGM, even though no changes in the geometrical features of the microstructure are evident.

These examples illustrate that a variation in any of the microstructural features described in the previous section can be considered within the context of FGMs. Furthermore, within a single graded material, a spatial variation often exists in more than one microstructural feature. A material described as being "compositionally graded" may therefore also contain gradients in the amount, size, shape, or arrangement of a second phase. The different microstructural gradients present within a material can be related to one another through conservation laws or fundamental geometrical relationships. Alternatively, it is possible that the different gradients have been introduced completely independently of each other in order to impart multiple functions to the material. This is often the case with FGMs.

The level of complexity possible in microstructures containing multiple graded features is illustrated in Figure 3.2 using a hypothetical graded material as an example. Without noting how or why this material might have been formed, the following qualitative observations can be made. First, the volume fraction of b increases with increasing distance from left to right. The size of the b particles also increases in the same direction. In addition, the b particles become more angular, and there is more contact between them as the volume fraction of b increases. At the far left, the microstructure

42 Chapter 3

consists of isolated b particles distributed uniformly throughout a matrix of a, whereas at the far right the b phase forms an interconnected network with islands of a existing along grain boundaries of b. The porosity present within this structure is located only within certain localized regions in the structure. For those parts of the material that contain nearly equal amounts of a and b, the porosity is located entirely within the a phase. At high contents of b, the size of the porosity is smaller and located entirely within the b phase.

The purpose of this example is solely to illustrate the kinds of variations that may be present within a given graded material. It should be emphasized that some features within the material may not have been introduced intentionally, i.e. they may simply be present as a result of processing. Such features mayor may not be of interest to the FGM designer, depending on their degree of influence on the material's performance. Obviously, in cases where the features are not relevant, less effort needs to be spent for their characterization. However, features that are critical for achieving the desired performance generally warrant a greater characterization effort. Close collaboration between the design, materials processing, and microstructural characterization efforts therefore is required.

In most cases, microstructural gradients discussed in the literature have been one-dimensional (I-D). This kind of graded material is the easiest to visualize, fabricate, and model. However, two-dimensional (2-D) and threedimensional (3-D) gradients are not only possible, but may be desirable and necessary for effectively implementing the gradient materials concept in practical applications. For example, consider the simple application of minimizing the thermal mismatch stresses that occur between two joined dissimilar materials. If edge effects are ignored while designing the

Figure 3.2. Diagram ofa hypothetical graded structure that has gradients in several different microstructural features.


microstructure of the component, then the stress distribution will be found to be purely a function of axial distance along the length of the specimen. Under such circumstances, a i-D microstructural gradient will be sufficient for reducing the peak thermal stress. However, in actual parts, edges and comers invariably result in 2-D and 3-D stress distributions that are highly geometry dependent. In these cases, 2-D and 3-D microstructural gradients may be necessary to minimize residual stresses. A diagram of differences between i-D, 2-D, and 3-D graded materials is shown in Figure 3.3. Little work has been conducted to date on the fabrication of materials containing 2-D and 3-D microstructural gradients. This is an important area for future research.

1-0

2-D

3-D

Figure 3.3. Diagram illustrating the difference between materials graded in 1-,2-, and 3-dimensions.

Because any of the common microstructural features can be considered as variables within a graded material, characterization of a graded structure typically involves many of the same analysis methods used for

44 Chapter 3

characterizing spatially homogeneous materials. However, since the graded material is spatially nonuniform, there are special considerations with respect to quantitative analysis of the spatial distribution of the characteristics of features.

3.2 Characteristic Dimensions

Just as microstructural features can have sizes from the atomic to the macroscopic, gradients can also exist on any size scale. Examples of the common size scales are listed in Table 3.2, along with the associated ranges of dimensions for the size of the relevant microstructural features plus examples of graded structures for each of the size scales.

Table 3.2. Characteristic Microstructural Length Scales Typical Feature

Size Scale Dimensions Examples of Graded Structures

atomic < lOOrun epitaxial films, optical thin films

microscopic lOO run - lOO Ilm diffusion bonding, interfaces in composites meso scopic lOO !lm - 1.0 mm surface hardening, thermal barrier coatings

macroscopic > 1.0mm graded ceramic-metal joints

In order to apply the description and analysis of graded materials across the wide range of size scales covered by this field, it is important to understand the relationships between the different dimensions involved in their analysis. At least three fundamental characteristic length scales can be identified, as shown schematically in Figure 3.4. The first, designated D1,

involves the dimensions of the particular microstructural feature of interest; the second, designated D2, involves the dimensions over which the microstructure can be considered to be "locally homogeneous," and the third, D3, involves the dimensions over which the various locally homogeneous regions are combined to produce the spatial variation in microstructure.

The first characteristic dimension is relatively straightforward, and in the case of grains or second phase particles, can be represented by the mean intercept length, as defined above. In graded multi phase materials, compositional variations often are discussed in terms of the volume fraction of the phases, for which the mean intercept length provides the appropriate characteristic dimension. However, these materials are also sometimes referred to as being "compositionally graded." Since volume fraction can be correlated with chemical composition, use of this term is valid. Confusion can result, however, because composition gradients can also exist within


single-phase materials. In such cases the more appropriate characteristic length scale is on the order of atomic dimensions.

To avoid ambiguity about the appropriate length scale, the term "compositionally graded" probably should be restricted to single-phase materials where composition is the primary microstructural variable of interest. Multiphase materials should be referred to more appropriately as being "phase graded" or "volume fraction graded". The dimension DJ can change at different locations, and in a multi phase material each phase can have its own characteristic dimension.

To understand the second characteristic length scale, D2, it is necessary to introduce the concept of a "locally uniform" microstructure, which is of particular importance in the field of FGMs. The first reason, discussed in Chapter 4, is the need to estimate local thermomechanical properties in order to model, design, and predict FGM behavior. The second reason relates to the statistical nature of the field parameter measurements performed during microstructural analysis.

Within the field of composite materials, mechanical and physical properties can be modeled at either the macroscopic or microscopic levels. In general, macroscopic models predict the average or global response of a composite using only the volume fraction and the properties of the individual phases. Since all composites are heterogeneous at the microscopic scale, these models are simplistic in that microstructural details such as

• , a •

Figure 3.4. Diagram illustrating the relationship between the three characteristic dimensions within a graded material.

46 Chapter 3

reinforcement size, shape, arrangement, and orientation are often ignored. Whether empirically based or derived from continuum assumptions, these models usually give rise to the well known rule-of-mixtures formulations. By accounting in various ways for the interaction between the phases, the rule-of-mixtures estimates can be modified to include some microstructural details (e.g., arrangement of the phases or degree of contact between the phases). In all cases, however, the validity of rule-of-mixtures approaches relies on the implicit assumption that the material is homogeneous (spatially uniform in terms of properties) on a macroscopic scale, i.e. the dimensions of the discrete microstructural constituents are small compared to the size scale of the material at which properties are measured. When the dimensions of the material are comparable to the dimensions of the constituents, this assumption is violated and local stress and strain fields must be accounted for.

The application of macroscopic models for estimating material behavior in FGMs therefore implicitly involves the "locally uniform" assumption. The dimensions of the "locally uniform" region within the FGM must remain sufficiently large compared with the dimensions of the microstructural constituents to justify use of these models. Thus, for the purposes of this discussion, D2 » D\.

Microscopic models typically employ mean-field theories to estimate the overall deformation response of a composite by considering the behavior of the constituent phases [1, 7-10]. The most common method involves computing the average stress and strain fields for discrete particles isolated within a matrix phase using various forms of Eshelby's equivalent inclusion method [11]. In these models, microstructural details such as the amount, size, shape, and orientation of the reinforcement can be taken into consideration. However, only the microstructures of dispersed particles (and by implication limited volume fractions) can be treated. The finite element method (FEM) has also been used extensively to investigate the effects of microstructural details on deformation response. In this way, localized nonuniformities in the deformation fields can be studied.

Using either micromechanics or FEM approaches more realistic, yet still relatively simple, microstructures can be modeled using various unit cell arrangements. In general, however, because these models do not account for large-scale nonuniformities in microstructure, their application in estimating material behavior in FGMs involves treating the spatial distribution of the microstructure as a series of discrete, "locally uniform" layers, each having a different microstructure.

The deformation response of a graded material also has been modeled without recourse to the "locally uniform" assumption, by using random unit cell arrangements that take microstructure nonuniformities into account


explicitly [12]. The point of this discussion is to emphasize that in almost all FGM modeling studies the concept of a "locally uniform" material is employed. From a microstructural analysis standpoint, it is necessary to recognize its importance.

The third characteristic dimension, 0 3, relates to the overall dimensions of the graded region of the material, regardless of the microstructural details or their spatial distribution. This dimension has a significant effect on the macroscopic behavior of the structure, and is important in the modeling and design of graded materials (discussed in Chapter 4). A key point is that the length scale, 0 3, must be larger than O2 to be effective in significantly altering the local material properties as a function of position.

With respect to microstructural analysis, the determination of an appropriate size for the test region must be based on knowledge of all three length scales. Recall that the measurement of field parameters representing specific microstructural features is performed assuming that the material is spatially homogeneous within the region of measurement. Typically, a number of features are measured within a given test region. Several test regions are then analyzed, and an estimate (mean) and standard deviation (error) for the measurement are subsequently determined. Errors arising from random scatter within a particular test field can be reduced by making the test region large compared with the feature size (i.e. by ensuring a sufficiently large number of features are counted within the field). Thus, the characteristic dimension of the particular microstructural feature, 0], is useful for determining an appropriate (minimum) size for the test region.

Within practical limits, larger test regions are more desirable for minimizing errors. In contrast, errors arising from the random scatter between different fields measured on the same specimen are minimized by examining many fields. Random spatial inhomogeneities in the microstructure contribute to an increased error in the field-to-field measurement. However, if a systematic spatial variation in microstructure exists, and information about the spatial distribution is sought (as is true in the analysis of graded materials), the field-to-field variations cannot be treated as errors.

Alternatively, a series of systematically arranged test regions are needed and within each test region the material must be assumed to be homogeneous. The appropriate (maximum) size of the test region is then limited by the characteristic dimension of the microstructural variation, 0 3,

and smaller test regions are more desirable for minimizing position-toposition errors. Ideally, the dimensions of the test region should be comparable to or less than the dimension O2, since at each analysis location along the graded direction, several representative test regions must be analyzed.

48 Chapter 3

It can be seen from this discussion that serious measurement errors can occur when the test region is on the same size scale as any of the characteristic dimensions. Tradeoffs will therefore be likely to exist with respect to the accuracy with which either the microstructural feature itself or its spatial variation can be determined from a single set of measurements. To illustrate, consider the hypothetical graded material shown in Figure 3.4. The results of performing image analyses to determine volume fraction, using test regions of different sizes, are shown in Figure 3.5. In Figure 3.5 (a), the analysis region was chosen smaller than the characteristic dimensions of the b particles (i.e. < DJ)' Unreliable results are obtained near the left side of Figure 3.4 where these particles are widely spaced. In comparison, because the analysis region is larger than the smallest characteristic dimension of the a phase, it could detect the locally high concentration of b that occurs about two-thirds of the way across the structure. In Figure 3.5 (b), the analysis region is approximately the same size as the characteristic dimension D2, and all the trends in volume fraction are correctly observed. In Figure 3.5 (c) the analysis region is large compared with D3• This analysis was unsuccessful at detecting the local variations in volume fraction.

These problems can be further exacerbated in the more general case when spatial variations exist for several different microsrtuctural features, and when the different features each have different characteristic dimensions, as in Figure 3.2. Similar to the case shown in Figure 3.4, both DJ and D2 can vary as a function of position within the material thereby causing further complications for analysis. In order to characterize such variations, several measurements along the length of the material may be required using different analysis techniques and test regions of different sIzes.

3.3 Diffusion and Phase Diagram Considerations

The evolution of microstructure is controlled by the thermodynamic and kinetic characteristics of the materials system, its thermal history, synthesis method, and thermomechanical processing. Diffusion and phase equilibria considerations are therefore expected to play a significant role in establishing and altering microstructural gradients. Although a complete understanding of these topics is lacking, several important points can be emphasized.

It is an important law of nature that gradients are unstable and provide a driving force for mechanisms for their elimination. Therefore, all materials containing microstructural gradients are in a nonequilibrium state. As such they are subject to processes such as diffusion or viscous flow that can significantly alter the original microstructure. In the case of a


compositionally graded material, when there is solubility for one component in the other, diffusion fluxes can become significant at elevated temperatures, thereby allowing chemical homogenization to occur. Both solubilities and diffusion coefficients are strongly temperature dependent. Therefore, temperature plays a significant role in the stability of compositional gradients.

Size of Test Region: Image Analysis Results

1.0

0.8

c 0.6 0

A- '" " ~ '" ~

0.2 0.4 0.6 0.8

normalizectiistance

1.0

0.8

B 0.6 '" B " • ~ ~ '" ~ 0.4

0.2

O. .0 0.2 0.4 0.6 0.8

normalizedistance

1.0

0.8

B 0.6

C .. ~ ~ '" ~ 0.4

0.4 0.6 0.8

norma/ized:lstance

Figure 3.5. Results of image analysis using different sized test regions to determine the volume fraction distribution in the hypothetical microstructure shown in Figure 3.4.

1.0

1.0

1.0

50 Chapter 3

Equilibrium is achieved when chemical potential gradients have been eliminated, which usually, although not always, corresponds to a compositionally homogeneous state. Even after chemical homogenization is attained, diffusion processes can continue to cause significant microstructural changes by reducing the free energy of the system. Most significant among these are microstructural coarsening phenomena, such as grain growth and Ostwald ripening, which involve a reduction in the free energy through the elimination of interfacial surface area. A major effort has been undertaken to understand and model coarsening in spatially homogeneous materials, but little work has been done to examine these phenomena within materials that initially contain microstructural gradients. This remains an important area for future research.

The importance of diffusion and phase equilibria considerations is illustrated in the simple case of a compositionally graded material produced by interdiffusion between two solids, as shown in Figure 3.6. In this case, composition is the primary variable of interest. Chemical composition is determined as a function of position, temperature, and time by the governing differential equations describing mass transport. Using appropriate initial and boundary conditions, the composition profile within the material at any point in time (shown schematically in Figure 3.6) can be predicted provided the diffusion coefficients are known. Both analytical and numerical techniques are widely used to solve the diffusion equations for many technologically important problems. Once fabricated, the composition profile of an FGM represents the initial conditions needed for obtaining solutions to subsequent diffusion problem formulations.

The extent to which diffusion influences a graded microstructure can be qualitatively predicted based on phase diagram features such as solubilities, the presence of intermediate phases, and phase transformations. For example, possessing information about the variation of composition with position, and employing the assumption of local equilibrium at phase boundaries, important microstructural information can be predicted for any given temperature. Figure 3.6 illustrates schematically the relationship between the phase variations in the composition, the phase diagram, and the expected equilibrium microstructure for hypothetical composition gradients created between two pure materials A and B. Three different cases involving very simple binary phase diagrams are shown.

This example illustrates in a simple way the important role on microstructure that can be played by diffusion and phase relationships. More complicated phase diagrams are often encountered in real systems. Consequently, interdiffusion can also cause the formation of new phases, dissolution, gas evolution, and similar effects that can be detrimental to the material's performance. Many graded materials and FGMs are intended for


Q) u C «l

.~ "0

~ :::l

1§ ~ a ~ T---------------- --- ---

I!! :::l

A %8

8

~ ,~--~~----~ ~ T -'- - - -- - -- -- - -- -- - --E a (3 2 a+(3

A %8

8

, ---: -~-:.~: :::~

A %8

8

pure 8

pure A

Figure 3.6. Diagram showing the relationship between a hypothetical composition profile, phase diagram, and the expected equilibrium microstructure at temperature T for graded materials in some simple binary systems.

51

52 Chapter 3

use in elevated temperature applications, therefore the influence of diffusion on microstructure must be considered. As is the case for composite materials, suitable combinations of materials with good chemical compatibility can be selected in order to minimize changes in the microstructure due to diffusion. In such cases, by designing and fabricating arbitrary compositions and complex graded microstructures reasonable stability of the original microstructure can be expected. Knowledge of phase stability, interdiffusion rates, and mutual solubilities is therefore of critical importance for producing materials with adequate stability. This is particularly important in systems where extensive chemical reactions can have deleterious effects on the mechanical integrity of the material. Because there is a considerable lack of data for many systems of interest, more fundamental research is needed.

Even with availability of the relevant thermodynamic and kinetic data, details of the microstructure are difficult to predict since processing often results in highly nonequilibrium structures. It should also be emphasized that the phase diagram only specifies the composition of the phases present and their relative amounts but does not provide any information on microstructural geometry. Modeling the details of the evolution of microstructural geometry in two-phase materials is an important area of study in the field of materials science. Significant progress is being made using a variety of modeling techniques to predict the size, shape, and arrangement of second phases for special cases involving spatially uniform materials. Technologically important examples include grain growth, coarsening of second phase particles, and the evolution of pore structures during sintering. Much more work is needed in order to apply these modeling techniques to the complicated, nonuniform microstructures found in FGMs.

3.4 Dispersed versus Connected Microstructures

Much of the work on FGMs relates to materials consisting of two-phase mixtures. Figure 3.7 shows a schematic representation of the common microstructural changes that occur in a material as the second phase content is increased [13]. At low volume fractions of the second phase it exists as isolated particles dispersed within a matrix phase. The dispersed particles can be either single crystal or polycrystalline. The particles can be randomly distributed within the matrix, or they can exist preferentially at sites such as grain boundaries or grain junctions. The volume fraction, size, and spacing of dispersed particles can be interrelated through the use of simple geometrical models similar to those described in the previous section. Such models have been used extensively, for example, to predict the yield stress


of dispersion strengthened alloys or to model coarsening of second phase particles.

o •

o

• • • •

Figure 3. 7. Schematic of microstructural changes that occur in a two-phase material as the volume fraction of the second phase increases.

As the content of the second phase of a two-phase material increases, the particles begin to interact through the formation of particle-particle contacts. This leads to the formation of an agglomerated microstructure in which the second phase remains dispersed, but often consists of polycrystalline aggregates containing many particles or grains. The contiguity parameter discussed above is extremely useful for characterizing the degree of particle contact in these materials [6].

c: .Q -(/) o a. E o u

A

distance

B

c: o ~ (/)

o a. E o u

A

distance

Figure 3.8. Diagram showing the difference between a layered microstructure and a continuously graded microstructure.

As the volume fraction is increased further, a critical microstructural transition takes place in which the second phase is no longer dispersed but

B

54 Chapter 3

rather becomes interconnected over long distances. The transition from a dispersed to an interconnected second phase structure can have a profound effect on the behavior of the material [14]. Transport based properties such as electrical or thermal conductivity, or gas permeability in porous solids, are particularly sensitive methods for determining when an interconnected structure is formed. In addition, certain mechanical properties, such as elastic modulus and fracture toughness, are also strongly affected by long-range connectivity of the second phase.

Connectivity in microstructures has been studied extensively using percolation models [13-15]. These models treat the microstructure as a network of nodes and interconnections in which composition is measured by the number of completed connections or occupied nodes (lattice sites for orderly packed structures). At a critical composition, the network becomes connected over long-range distances. Typically, there is a sharp transition in properties over a very narrow range of composition near the transition point.

In actual microstructures, the critical composition for the development of an interconnected second phase often must be measured indirectly, since metallographic methods underestimate the degree of particle or grain contact that exists in three dimensions. Transport-based methods, such as electrical or thermal conductivity, are extremely useful, particularly when the two phases have markedly different transport properties. Many studies on connectivity and percolation have been carried out, generally on mixtures of conducting and nonconducting phases. The critical composition for second phase connectivity has been found to depend on several factors, including the packing arrangement (coordination number) and shape of the second phase particles, and the relative sizes of the constituents (e.g., particle or grain size ratio) [16].

In three-dimensional structures, both phases typically will be connected at intermediate compositions, as shown in Figure 3.7. For example, in a fully dense, two-phase mixture composed of equal sized grains, both phases will be connected between approximately 18 and 82 volume % [17]. Maximum interconnectivity will always occur between 40 and 60 volume %. FGMs involving two-phase mixtures that vary from 100% of phase a to 100% of phase b will always contain both kinds of dispersed phase microstructures (i.e. a particles within a b matrix, and b particles within an a matrix), as well as a wide compositional range in which the microstructures are characterized by two interconnected phases.


3.5 Smoothly Varying versus Discretely Layered Structures

55

Graded microstructures can be characterized by the nature of the spatial variation in the microstructure. There are two general types of spatial distributions, as shown schematically in Figure 3.8. In the first type, the change in composition (or any other microstructural feature) occurs continuously with position. In the second type, the microstructural feature changes in a stepwise fashion, giving rise to a multilayered structure with interfaces existing between discrete layers. Whether a material's microstructure has continuous or stepwise spatial variation, it can be represented using a continuous mathematical function.

The difference between these two fundamental types of microstructures is important and can have implications with respect to the processing, modeling, and behavior of FGMs. For example, certain processing methods are only capable of producing one type of structure or the other [2]. As already mentioned, a multilayered structure can conveniently be represented for modeling purposes using a continuous mathematical function. Although modeling may yield misleading results if the interfaces significantly affect the behavior of the material. Certain material properties are known to be influenced by the presence of internal interfaces, depending on the extent of the discontinuity that exists. Therefore it is important to distinguish between what actually exists at the microscopic level, what is represented mathematically, and what is measured experimentally.

A multilayered structure can be made to approach a continuously varying structure if the layer thickness is made smaller and smaller. But as previously noted, for all practical purposes the thickness of an individual layer need only be reduced below the characteristic size scale of the microstructural features before the effects of the stepwise structure become ambiguous. Many recently developed fabrication methods involving discrete layer formation can be adapted to produce continuously graded structures. Whether the effort and expense needed to accomplish this are practical or even desirable must be evaluated on a case-by-case basis.

The representation of a multilayered structure by a continuous mathematical function is usually accomplished by specifying the average layer composition at some position within the individual layers. The most obvious location to choose is the midpoint of the layers. However the starting or ending interface can also be used.

56 Chapter 3

3.6 Mathematical Descriptions of Gradients

Mathematical functions are frequently used to represent the spatial variation in microstructure within graded materials. A number of mathematical functions can be used, depending on the nature of the material, the particular microstructural feature being described, and the goals of mathematical modeling.

Linear gradients, which are common in steady state transport problems, represent the simplest mathematical form. Perhaps the earliest and most widely known nonlinear mathematical functions used for describing gradients involve the error function, erf(x). This function appears frequently in solutions to the differential equation describing unsteady-state transport (Fick's second law) which, in the case of solid state diffusion, can be written generally as:

(3.14)

where C is concentration, t is time, and D is the interdiffusion coefficient. In the one-dimensional case, this equation takes the familiar form:

dC =~5dC) dt dx\ dx (3.15)

Depending on the initial and boundary conditions, the solution to this equation can vary considerably. However, solutions for many practical problems of interest are readily available [18]. As an example, Figure 3.9 shows schematically a problem involving one-dimensional diffusion between two semi-infinite solids having compositions C1 and C2. For the special case in which D is independent of composition, the solution to this diffusion problem for all x>O takes the form:

,

C-Co _ 2 12fu 2 _ (X) C -C - I exp( -x )dx - erf r;:::: 2 0 'V 1t 0 2"1 Dt (3.16)

where Co is the average concentration of C1 and C2• Since erf(x) = -erf(-x) the composition profile is symmetrical about x=O. The general form of this function is shown in Figure 3.9.


C-C 2

C1-C2

C C1 o ~ en o c.. E o u

1.0

0.8

0.6

0.4

0.2

0 -2

/ original interface

Y

X=O~,

/inilial di

I

stribution

distance, x

-1 0 1 2 X/(4Dt)1/2

Figure 3.9. Diagram showing the composition profile (top) of a diffusion couple, and the profile at a later time t (bottom) for a system that exhibits complete solid solubility. The solution to the diffusion equation is the error function, as shown in the bottom diagram.

57

In contrast to diffusion problems in which composition is of interest, most research on FGMs has involved analysis of the distribution of second phases as the primary microstructural variable of concern. However, at least for simple systems the volume fraction of the second phase can be predicted from compositional information using phase diagrams. Therefore, for FGMs fabricated by transport-based methods, the diffusion equations are extremely useful.

A number of non-transport-based methods (also called constructive methods) can be used to fabricate FGMs [2]. In these materials, the initial distribution of second phases (or more generally any microstructural parameter) can be created without the constraints imposed by Equation 3.16. There is thus a need for other, more general mathematical functions, for

58 Chapter 3

describing microstructural gradients. The most common of these, initially suggested by Japanese researchers, involves a power law function, that when written in terms of the volume fraction of the b phase takes the form:

v v(J3) = (X)P (3.17)

where X is the dimensionless distance across the graded material, shown in Figure 3.10 and defined as:

(3.18)

where x 1 and x2 define the end positions of the graded material, x is the distance into the graded layer from the x 1 interface, and p is the variable that determines the shape of the curve representing the spatial distribution of the volume fraction. The advantages of this function are that it is simple, and by selecting values of p either greater or less than one, a wide variety of distribution shapes can be represented, as shown in Figure 3.10. When p=1.0, a linear spatial distribution is obtained.

The disadvantage of this function is that it does not allow for the possibility of including inflection points or local maxima and minima in the spatial distribution, as would be required for representing the material shown in Figure 3.4. Although little or no work to date has focused on producing materials containing nonmonatonic microstructural variations, they certainly could be desirable in certain circumstances. To allow for the possibility of describing arbitrary spatial distributions, other functions would be more appropriate. The simplest of these would be the polynomial function:

(3.19)

Other functional possibilities would be equally valid, although the need for more elaborate functions has not been justified yet.


~~

1

1 1 1

X 1 • I

1 - I 2 1 1 1 1 1 ..

X=x, X=~ distance

1.0r---~--r---~--r---~--r---~--~--~~

0.8

c 0 ~ 0.6 .... -c ~ c 8

0.4 ~ ~ Q) ....

0.2

0.0 ~""";:;;;;"'_======~---''------'-_"'"----1.._....o.----I 0.0 0.2 0.4 0.6 0.8 1.0

relative distance, X

Figure 3.10. Diagram showing various profiles for phase distribution obtained by varying the exponent in the power law equation.

....

60 Chapter 3

4. EXAMPLES

In most of the literature about FGMs, the characterization of the microstructure has been limited to describing the distribution of the volume fractions within FGM specimens, and its relation to various properties. There have been relatively few studies focusing on microstructural characterization.

In one study on powder processed, partially stabilized zirconia/stainless steel (PSZ/SS) FGMs, the transition from a dispersed to an interconnected microstructure was investigated from 2-D cross sections using fractal geometry concepts [19]. An interconnected phase was assumed to correspond to a percolation cluster for which a mass fractal dimension was determined, and a surface fractal dimension was assumed to characterize a dispersed phase microstructure. It was shown that at the critical volume fraction corresponding to the transition from a dispersed to an interconnected microstructure, the mass and surface fractal dimensions were equal. Beyond the critical volume fraction, the analysis based on the assumed microstructural type broke down. In this way, it was shown that the critical volume fraction could be determined by analyzing metallographic cross sections. The critical volume fraction determined using fractal analysis was in agreement with the value determined using conductivity measurements. In a similar study on nickel/magnesium oxide (NilMgO) FGMs, density scaling functions were used to obtain the mass fractal dimension and to identify the ranges over which percolation clusters and interpenetrating phases exist [20].

In another detailed study on PSZ/SS composites used in the fabrication of FGMs, phase connectivity and percolation were investigated using metallographic examination, serial sectioning, and idealized packing simulations [21]. Two dimensional measurements of the Betti number were made to quantify the range of volume fractions for both dispersed and interconnected microstructures. For the particle size ratio used to produce these materials, the phases were interconnected between about 15 and 45 volume % PSZ. Outside of this range, the materials exhibited dispersed phase microstructures. At a volume fraction of 32 volume % PSZ, the mean curvature of the metallceramic interface was shown to be zero, presumably corresponding to the situation where the matrix and dispersed phases are indistinguishable.

The percolation probability for the stainless steel phase determined by serial sectioning was shown to be in reasonable agreement with that calculated using a simple packing model. The metal phase appears to undergo a percolation transition over the volume fraction range from 30 to 80 volume % PSZ. From 60 to 80 volume % PSZ the metal phase was


presumed to exist as tree-like clusters extending over relatively long distances but with very few interconnecting branches. The conclusion is that sufficient metal connectivity is required to obtain good electrical conductivity and fracture toughness in these composites, and that the 2-D measurements appear to provide a convenient means of determining the degree of metal connectivity.

REFERENCES

1. Hirai, T. (1996) Functional gradient materials in Materials Science and Technology, Vol.17B, Processing o/Ceramics, Part 2, (ed. RJ. Brook), VCH Verlagsgesellschaft mbH, Weinheim, Germany, 293-341.

2. Mortensen, A. and Suresh, S. (1995) Functionally graded metals and metal-ceramic composites, Part 1 Processing, Int. Mater. Reviews, 40 (6), 239-265.

3. DeHoff, R.T. and Rhines, F.N. (ed.) (1968) Quantitative Microscopy, McGraw-Hill, New York.

4. Underwood, E.E. (1970) Quantitative Stereo logy, Addison-Wesley, Reading, MA. 5. Exner, H.E. and Hougardy, H.P. (1988) Quantitative Image Analysis o/Microstructures,

DGM Informationsgesellschaft mbH., Oberursel, Germany. 6. Gurland, J. (1966) An estimate of contact and continuity of dispersions in opaque samples,

Trans. AIME, 236, 642-646. 7. Hirano, T. et al. (1990) On the design offunctionally gradient materials, in Proc. First Int.

Symp. on FGM'90, (eds. M. Yamanouchi et al..), The Society of Non-Traditional Technology, 5-10.

8. Wakashima, K. and Tsukamoto, H. (1990) Micromechanical approach to the thermomechanics of ceramic-metal gradient materials, ibid., 19-26.

9. Markworth, A.J. et al. (1995) Modeling Studies Applied to Functionally Graded Materials, J Mater. Sci., 30, 2183-2193.

10. Suresh, S. and Mortensen, A. (1997) Functionally graded metals and metal-ceramic composites, Part II, Thermomechanical properties, International Materials Reviews, 45, 85-116.

11. Eshelby, J.D. (1957) The Determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Royal Society, Series A, A241, 376-396.

12. Weissenbek, E. et al. (1997) Elasto-plastic deformation of compositionally graded metalceramic composites, Acta Materialia, 45 (8), 3401-3417.

13. Nan, c.-W. (1993) Physics of inhomogeneous inorganic materials, Progress in Materials Science, 37,1-116.

14. Deutscher, G. et al. (ed.) (1983) Percolation Structures and Processes, Adam Hilger, Briston, UK.

15. Stauffer, D. (1985) Introduction to Percolation Theory, Taylor and Francis, London. 16. German, R.M. (1989) Particle Packing Characteristics, Metal Powder Industries

Federation, Princeton, NI, 253-274. 17. Cahn, I.W. (1966) A model for connectivity in multiphase structures, Acta Metall, 14,

477-480. 18. Geiger, G.H. and Poirier, D.R. (1973) Transport Phenomena in Metallurgy, Addison

Wesley, Reading, MA, 473-513.

62 Chapter 3

19. Muramatsu, K. et al. (1990) Fractal analysis of the microstructural transition in PIM functionally graded materials, inProc. First Int. Symp. on FGM'90, (eds. M. Yamanouchi et a/.), The Society of Non-Traditional Technology, 53-58.

20. Nan, c.-W. et al. (1993) The physics of meta II ceramic functionally graded materials in Ceramic Trans., 34, Proc. Second Int'/. Symp. on FGM'92, (eds. J.B. Holt et al.), The American Ceramic Society, Westerville, OH, 75-82.

2l. Watanabe, R. et at. (1995) Microstructural characterization ofmetaVceramic functionally gradient materials, in Proc. a/The Third Int '/. Symp. on Structural and Functional Gradient Materials, (eds. B. Ilschner and N. Cherradi), Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, 3-8.

Chapter 4

MODELING AND DESIGN

Keywords: Modeling, finite element method, residual thermal stress, rule of mixtures, elastic modulus, shear modulus, bulk modulus, micromechanical approach, representative volume element, Poisson's ratio, Young's modulus, isostrain, isostress, flow stress, strain hardening, fuzzy logic technique, yttria-stabilized zirconia, partially stabilized zirconia, elastic-plastic deformation, creep, power-law creep

1. INTRODUCTION

With the advent of powerful computers and robust software, computational modeling has emerged as a very informative and cost effective tool for materials design and analysis. Modeling often can both eliminate costly experiments and provide more information than can be obtained experimentally. Computational modeling has clearly played an important role in FGM research to date, and because of the considerable complexity involved, is expected to play an even greater role in future developments. This chapter introduces some of the common approaches used in modeling FGMs, identifies the major difficulties involved, and, it is hoped, provides useful guidance for future simulation efforts. It focuses mainly on continuum models of the bulk response of FGMs due to thermal or mechanical loading.

Since the term "modeling" is quite general, the discussion first centers on a more specific definition of modeling as used in this chapter. A hypothetical FGM application is introduced and a variety of models are discussed, each designed to address a particular issue or concern. Typical modeling approaches are then described, leading to the conclusion that the most significant difficulty in FGM modeling is the accurate determination of the

64 Chapter 4

material properties of the interlayers. The main body of the chapter then provides a detailed discussion on the estimation of FGM properties, including numerous comparisons with experimental data. Where possible, specific recommendations are made.

2. BACKGROUND

Models are commonly used by scientists. However, modeling is a general term that has a variety of meanings. Essentially, models are some representation of reality, either actual physical objects such as a model airplane or simply mathematical entities. Mathematical models are developed in a variety of length scales, ranging from the atomic level (e.g., molecular models) to fully integrated systems. As used in this chapter, modeling refers to the use of either analytical or numerical mathematical solutions of the basic laws of physics to simulate the response of real materials. More specifically, these solutions are designed to predict the bulk thermomechanical response (e.g., temperature, displacement, stress, and strain fields) of FGMs subjected to varying thermal or mechanical loading conditions. The ability to accurately predict the temperature, stress, and strain fields is important, since such parameters can strongly affect the performance and structural integrity of FGM components.

2.1 A Hypothetical Example

A hypothetical FGM application illustrates some of the types of mathematical models that can be used with these materials, and how such models can assist in improving a component's design, lifetime, and performance. Consider an exhaust valve for an internal combustion diesel engine, shown schematically in Figure 4.1, which controls the flow of exhaust gases from the combustion chamber. Under normal operating conditions, the valve is subjected to high temperatures (combustion gases), thermal cycling (engine start-up and shut-down), and mechanical loading (forces from rapid opening and closing). To permit higher operating temperatures and improve engine efficiency, an FGM has been proposed as a protective layer (i.e., a thermal barrier coating) on the valve face [1]. A consideration of the fabrication and application of the valve illustrates various ways in which mathematical modeling can be advantageous.

face FGM layer

seat

stem

Figure 4.1. Schematic of an exhaust valve for an internal combustion diesel engine shown as a hypothetical FGM application.

65

Thermal barrier coatings are typically applied via spray deposition or other deposition processes at high temperatures. On cooling, thermal residual stresses develop due to differences in the properties of the materials, particularly thermal expansion coefficients. Such stresses exist even if the cooling process is sufficiently slow for the temperatures to remain spatially uniform, and can be large enough to cause the component to fail. A numerical solution of the basic equations of mechanics (Le., a finite element model) can provide estimates of the residual thermal stress and strain levels in both the valve body and the FGM coating.

During operation, the valve face is exposed to high temperature combustion gases, yet the valve seat remains relatively cool due to intermittent contact (while closed) with the engine head. Thus a large thermal gradient exists across the valve. The temperature field can be approximated by solving the energy equation either analytically or numerically. Unfortunately, large thermal gradients can lead to large thermal stresses. The thermal gradients can be the result of normal operating conditions, as described above, or can be cyclic, such as occur during startup or shut-down.

By combining a thermal analysis (solution of the energy equation) with a mechanics analysis, the effect of the temperature field on component stresses and thus component integrity can be predicted. Almost always, these

66 Chapter 4

analyses can be decoupled since the mechanical analysis has little effect on the temperature field. During operation, the valve also undergoes mechanical loading/unloading cycles as it rapidly opens and closes. Further analyses can be used to determine the resulting stress and strain fields from this applied loading. For all the models described, parametric studies, in which changes in geometry, material properties, or loading are considered, can provide added insight and lead to optimized designs.

2.2 Modeling Approaches

For each type of model discussed in the above example, the basic partial differential equations from continuum mechanics are applicable and are well-established and understood [2]. Since the basic conservation equations are generally applicable to any continuum, they are not affected by the addition of an FGM layer. For simple geometries and reasonably simple material properties (e.g., elastic behavior) analytical solutions are often available. For more complex geometries or constitutive behavior, robust numerical solution techniques (e.g., finite element methods) can be used to provide accurate approximate solutions. A wide variety of software is commercially available [3].

With respect to numerical solutions that employ finite size computational cells, an FGM layer typically is approximated as a series of perfectly bonded interlayers, with each layer having slightly different material properties than its neighbors. Material properties are thus assumed to change continuously in the direction of the gradient. Numerical studies using continuum models have been designed recently to investigate localized microstructural effects [4, 5]. Essentially, these studies use very fine computational meshing to simulate individual grains of ceramics or metals (in either periodic or random arrangements), and also to simulate the evolution of local stress and strain fields. Results from these models reveal detailed information about microstructural behavior at the grain size level. It is anticipated they will provide insight for FGM optimization.

For the macrolevel continuum models, clearly the most significant difficulty encountered is the accurate determination of material properties for each interlayer. This complication is the main factor that sets FGM models apart from standard analyses. Because of its importance to accurate simulations, the estimation of material properties is the crux of this chapter.

67

3. ESTIMATION OF MATERIAL PROPERTIES

3.1 General Comments

Realistic predictions of the thermomechanical behavior of FGMs require appropriate constitutive relations containing accurate thermophysical property data. Since a tailored spatial variation in microstructure is intentionally introduced in an FGM, a variety of different microstructures can exist within a graded region. Thermophysical properties, which are dependent both on individual phase properties and on microstructural details, such as volume fraction; size, shape, orientation, and spatial distribution of the phases; and phase connectivity, similarly can vary strongly with position. Property estimates are further complicated because often both the production processes and the intended applications for FGMs involve significant temperature variations. Because of the vast number of measurements required, experimental data are limited, often including information only for the individual phases. Clearly, the ability to provide reasonable estimates of material properties based on complex microstructures and limited experimental data is a challenging but important component of the modeling process.

An FGM is a heterogeneous material that consists of regions of different materials or even of the same material in different states. In developing estimates of thermophysical properties, a basic assumption is that the microscopic length scale for the material is much larger than molecular dimensions, but much smaller than the characteristic length of the macroscopic sample. A heterogeneous material can then be regarded as a continuum on the microscopic scale, and thus macroscopic or "effective" properties can be used.

It is important to recognize that FGMs are basically composite materials. Efforts to determine analytically the effective thermophysical properties of composites were initiated more than a century ago by such famous scientists as J. C. Maxwell, Lord Rayleigh, and Albert Einstein [6]. Because of increased interest recently in composites for industrial applications, the subject of composite properties has been thoroughly developed, and a large literature base exists. Several extensive review articles are available, that provide both good overviews of the subject and insight into the significant complexities involved [6-8]. This chapter examines some of the more practical techniques that have been applied to FGMs, and provides background information and typical examples for commonly encountered thermomechanical properties. The overall goal is to provide sufficient information to guide the selection of appropriate techniques for estimating material properties.

68 Chapter 4

This section begins by identifying and briefly describing some of the common techniques used to estimate thermophysical properties in FGMs, starting with the simplest and progressing to the more complex. Specific thermal and mechanical material properties are considered next, including comparisons of the various estimation techniques and, where available, comparisons with measured data. It should be noted that the experimental data selected for comparison was restricted to the FGM literature, principally the first five international symposia (1990, 1992, 1994, 1996, and 1998 ). This was done both to keep the comparisons focused on FGM materials and to demonstrate the limited set of property data that exists for these materials.

3.2 Approaches

3.2.1 Rules of mixtures

A common approach for estimating the material properties of FGMs is to apply a rule of mixtures. Although actually not physical or mathematical rules, these relationships can be used to approximate thermal or mechanical properties of a composite material in terms of the individual properties and relative amounts of the constituents.

The simplest is the classical linear rule of mixtures (Voight estimate) for two constituent materials [9]:

(4.1)

where P is a typical property, V is the volume fraction, and the subscripts a and B are used to distinguish the two constituents. The Voight estimate is simply a volume based arithmetic mean. Another well-known mixture rule is the harmonic mean (Reuss estimate) [10]:

(4.2)

In their most basic form, the above rules of mixtures are employed using bulk constituent properties, assuming no interactions between phases. Because of their simplicity, they are often used for FGMs, since a single relationship can be used for all volume fractions and microstructures. However, also because of their simplicity, their validity is limited.

In an effort to include interaction effects between constituent materials, the rules of mixtures also are described and used in a modified form, either by utilizing in-situ constituent properties [11] or by including empirical data.

69

Complex stress-strain data have been approximated using this empirical approach.

Recently, a generalized rule of mixtures has been described for application to FGMs that includes the effects of grain shape and phase distribution [12]. Since this approach is derived solely from continuum mechanics, it is free of empirical considerations. Excellent agreement is claimed between the generalized rule of mixtures and experimental results drawn from the literature.

3.2.2 Variational approach

This approach involves the application of variational principles of thermomechanics (e.g., elasticity theory for elastic moduli, diffusion theory for thermal conductivity) to derive bounds for effective thermophysical properties. Assuming homogeneous, isotropic materials, and applying variational methods to linear elasticity theory, relations have been obtained (Hashin and Shtrikman or HS) for the upper (U) and lower (L) bounds on the effective bulk modulus (K) and shear modulus (G) [13]:

v~ KL = Ka + -----'-----

1 + 3Va K~ - Ka 3Ka + 4Ga

(4.3)

Ku = K~ + ____ V.;...la.:io..-~--1 + 3V~

Ka - K~ 3K~ + 4G~

(4.4)

(4.5)

(4.6)

70 Chapter 4

Because of the isotropic assumption, all other elastic properties can be calculated directly from the bulk and shear moduli.

A similar approach has been applied to electromagnetic theory to obtain bounds on the effective magnetic permeability [14]. By direct analogy, these results can be used to compute effective bounds on thermal conductivity (k):

(4.7)

(4.8)

For both the elastic and thermal properties, the above relations are the best possible bounds that can be derived based only on the constituent properties and volume fractions. To improve on these bounds, additional information, such as the spatial distribution of the phases (i.e. the microstructure) is needed [13, 14].

3.2.3 Micromechanical approaches

3.2.3.1 Standard Methods In contrast to the variational techniques, micromechanical approaches

attempt to include information about the spatial distribution of the constituents in the material. The standard micromechanical approach is based on the concept of a unit cell or representative volume element (RVE) to represent the microstructure of the composite. A large variety of thermophysical property relationships can be formulated based on various unit volume geometries, bonding assumptions, loading conditions, and boundary conditions.

One of the earliest and simplest micromechanical approaches (Kerner) assumes a spherical reinforcement particle in a uniform isotropic medium, with perfect bonding between the phases [15]. Considering both elastic [15] and thermal [16] problems for two phases, relations are obtained for the effective bulk (K) and shear moduli (G), the thermal conductivity (k), and the coefficient of thermal expansion (a):

71

V K V K [3Ka + 4Ga] K = a a + [3 [3 3K[3 + 4Ga

V + V [3Ka + 4Ga] a [3 3K[3 + 4Ga

(4.9)

(4.1 0)

(4.11 )

(4.12)

where v is Poisson's ratio. A more recent effort (Wakashima and Tsukamoto) employed a "mean

field" approach characterized by a random dispersion of misfitting ellipsoidal inhomogeneities, induding an interaction effect to account for the typically large number of ellipsoids that interact [17]. This approach was based on earlier work involving a single ellipsoid in a uniform elastic domain [18]. The interaction is accounted for as an approximation using the "average stress in matrix" concept [19]. Relations were derived for the effective bulk (K) and shear moduli (G), thermal conductivity (k), volumetric specific heat (C), and coefficient ofthermal expansion (a):

(4.13)

72 Chapter 4

(4.14)

(4.15)

(4.16)

u= (4.17)

For both the Kerner [15] and the Wakashima and Tsukamoto [17] approaches, the assumption of which phase is the matrix and which the reinforcement provides two unique functions analogous to the HS bounds.

Even more recently, microstructural effects have been incorporated using a periodic arrangement of perfectly bonded cubic inclusions uniformly distributed in a continuous matrix (Ravichandran) [20]. Upper and lower bounds for the elastic properties of a unit cell are then derived by considering both parallel and series arrangements of the two phases respectively, loaded in isostrain and isostress configurations. Relations are provided in terms of the Young's modulus (E).

(4.18)

(4.19)

73

(4.20)

In developing Equations 4.18-4.20, it was assumed that the Poisson's ratios of the constituent materials are equivalent, and it was also noted that a similar development provides identical relations for the shear moduli. Since the composite materials are assumed isotropic, the Young's modulus can be related easily to the bulk modulus provided by the approaches discussed above. Although the rules of mixtures (Equations 4.1-4.2) are usually considered simple averages, for the elastic moduli they can also be formally derived using this micromechanical approach.

3.2.3.2 Higher Order Methods In applying micromechanical approaches to FGMs, a need for higher

order methods has been identified. Cases have been considered in which the size of the inclusion phase is large relative to the overall dimension of the composite, and approaches have been developed that apparently eliminate the difficult and still largely unresolved issue of what constitutes a representative volume element or unit cell in an FGM [21]. The theory explicitly couples local and global effects. This higher order approach has been implemented in a computer program, MAC (Micromechanics Analysis Code) that apparently can predict the responses of both continuous and discontinuous multiphased composites with arbitrary internal microstructures and reinforcement shapes [22]. Similarly, conventional micromechanical theory has been generalized to allow linear variations in local and applied stress and strain fields [23]. This method makes it possible to estimate the significance of the reinforcement gradients on the effective properties of an FGM.

3.2.3.3 Numerical Solutions There is now a considerable effort to combine conventional

micromechanical approaches with numerical solution techniques in order to estimate the effective properties of composite materials. Since analytical solutions are only available for simple cell geometries (typically spheres and bricks) and patterns, numerical techniques permit a look at more complex representative volume elements, boundary and loading conditions, and interface properties. To date, efforts have included the effects of localized damage (plasticity, cracking) that potentially could have a significant impact on effective properties. A recent review provides an excellent summary of the large body of literature being developed in this area [24].

74 Chapter 4

3.2.4 Empirical approaches

In the absence of complete experimental data, it is possible to combine theoretical approaches (e.g., rule of mixtures) with limited empirical data to predict material properties. One example is the modified rule of mixtures that has been used to predict the non-linear, stress-strain behavior of FGMs illustrated schematically in Figure 4.2 [25]. Stress-strain curves for the composite material are constructed by dividing both the stress and the strain between the constituents using the linear rule of mixtures, as follows:

(4.21)

(4.22)

where O'n , O'~ and En, E~ are the true stress and strain of the ceramic and metal, respectively, and O'c and Ec are the composite flow stress and strain. Since these equations do not specify the absolute amount of stress and strain transfer, an additional equation,

(4.23)

that defines the ratio of stress to strain transfer is required. As illustrated in Figure 4.2, q is the slope of a correspondence line on the stress-strain curve, with large slopes approaching the isostrain condition (En = E~ = cc) and small slopes the isostress condition (O'n = O'~ = O'c). Values for q are obtained empirically and, in general, depend on many factors, including composition, flow stress ratio and strain hardening of the constituent phases, their microstructural arrangement, and the applied strain. A constant q value obtained from experiments with systems involving coarse two-phase microstructures comprised of hard and soft constituents has been assumed as a first approximation, [25, 26].

3.2.5 Fuzzy logic techniques

In each of the micromechanical techniques described above [15, 17, 20], the analytical relations provide two unique functions based on the assumption of which phase is the matrix and which the reinforcement. In an FGM, the deliberate variation in its microstructure can result in separate

75

regions where a given material acts as either the matrix or the reinforcement, or something intermediate. Thus one function might adequately describe one microstructure within the FGM and the second another microstructure, with a connecting region that is not well described by either function.

To address this issue, Kerner's micromechanical method has been combined with fuzzy logic techniques to provide a smooth transition between bounding curves [27]. In effect, this approach provides an averaging equation that uses one of the two bounding functions at extreme volume fractions, whereas at intermediate volume fractions the effective properties are taken as some weighted average of the bounds. Additional information, which is usually obtained empirically, is required to define the interpolation functions and the volume fractions over which averaging occurs.

15' -

(l - ceramic phase (l ~ - metal phase

c - composite V - volume fraction q - tie line slope

£c

Strain (£)

Figure 4.2. Illustration of the modified rule of mixtures used to approximate stress-strain curves for FGMs.

3.3 Thermophysical Properties

3.3.1 Thermal expansion coefficient

With respect to temperature changes in FGMs (either during their processing or their application), probably the most important thermal property is the thermal expansion coefficient. For a given temperature

76 Chapter 4

change, thermal stresses are directly related to the thermal expansion difference between the two constituent materials.

Figure 4.3 shows the effective thermal expansion coefficient (normalized between zero and one) as a function of volume fraction as predicted by the simple rule of mixtures (Equations 4.1-4.2) and the Kerner (Equation 4.12) and Wakashima (Equation 4.17) micromechanical theories. Experimental data for six metal/ceramic and two ceramic/ceramic FGMs are included for comparison. Since the micromechanical relations are functions of the mechanical properties (e.g., the bulk and shear moduli) they are shown for an actual metal/ceramic composite. The Ni-Ah03 system was chosen for this comparison since it has the largest thermal expansion difference of all the material systems considered, and thus maximizes the spread between the upper and lower bounds.

2.50

2.25

2.00

1 .75

1 .50 ... tS I 1 .25 tr

"-... 1 .00 tS

I tS

0.75

0.50

0.25

0.00

-0.25 0.0

-- Voight ROM

- - - Reuss ROM ••••••••• Kerner. Wakashima

o Ni/MgO [4.281

o 304SS/PSZ [4.281

6 Ni3AI/TiC [4.281

v Ni/TiN [4.29]

~ Ni/ AI203 [4.301

~ Ni/Zr02 [4.311

X TiN/Si3N. [4.291

~ TiC/SiC (4.321

0.2 0.4 0.6 0.8 1 . 0

V,

Figure 4.3. The effective thennal expansion coefficient as a function of volume fraction as predicted by the rule of mixtures and the Kerner and Wakashima micromechanical theories. Experimental data for six metal/ceramic and two ceramic/ceramic FGMs are included for comparison. The subscripts (1 and 2) in the labels of the axes correspond to the first and second materials listed in the legend.

77

The Kerner and Wakashima theories provide identical functions, well within the bounds provided by the rule of mixtures. The spread between the two is quite small, even for the large thermal expansion difference between Ni and Ah03. Clearly there is significant scatter in the available experimental thermal expansion data for FGMs. Based on a general comparison of the predictions and experimental data, there appears to be no justification for using other than a simple linear rule of mixtures for estimating the effective thermal expansion coefficients of FGM materials. With respect to the effects of porosity, experiments with porous materials have concluded that there is no influence of porosity on the effective coefficient of thermal expansion [28, 31].

3.3.2 Thermal conductivity

Figure 4.4 compares the effective thermal conductivity (normalized between zero and one) as a function of volume fraction for a two-constituent composite, as predicted by the rule of mixtures (Equations 4.1-4.2), the HS variational technique (Equations 4.7-4.8), and the Kerner (Equation 4.11) and Wakashima (Equation 4.15) micromechanical approaches. Three separate plots are provided corresponding to conductivity ratios (kl/k2) of 32, 12.5, and 3.5. Experimental data for three metal/ceramic FGMs with corresponding conductivity ratios are included for comparison.

The variational technique and the two micromechanical theories produce identical results for thermal conductivity, providing upper and lower curves that lie well within the rule of mixtures approximations. These bounding functions are quite close for moderate thermal conductivity ratios (e.g., k1/k2

= 3.5 in Figure 4.3(c)) but diverge considerably as this ratio increases. The comparison between theoretical predictions and experimental data provides interesting and valuable insight for the estimation of thermal conductivity in FGMs. Compared with the thermal expansion coefficient, the linear rule of mixtures is clearly inadequate, which indicates the need for the kind of additional information included in the micromechanical theories.

A clear example is shown in Figure 4.4(a), where experimental data for the same material system (Ni/Zr02) from two different sources are plotted. The data sets are widely apart, with the Takemura [33] data nicely fit by the lower micromechanical theory (appropriate for metal particles in a ceramic matrix) and the Matsumura [34] by the upper relation (appropriate for ceramic particles in a metal matrix). The Takemura material was produced using layered powders and sintering, resulting in a ceramic matrix, and the Matsumura material by an electroplating technique, resulting in a metal matrix. The comparison with micromechanical theory is thus very good.

78 Chapter 4

At a reduced conductivity ratio (k]ik2=12.5, Figure 4A(b», the data for hot pressed 304SSIYSZ (304 stainless steeVyttria-stabilized zirconia) are reasonably well represented by an average of the upper and lower HS or micromechanical bounds. Data for the thermal sprayed material are not well characterized by any of the relations, which may be due to porosity. A threematerial theory (e.g., metal/ceramic/void) is probably needed for materials with considerable porosity. At the lowest conductivity ratio considered (k]/k2=3.5, Figure 4.4(c)), the upper and lower HS and micromechanical relations are quite close and the data are reasonably well represented by either function.

1.1

1. a 0.9

0.8

0.7

~ 0.6 1-... 0.5 ';; ... I 0.4 ...

0.3

0.2

0.1

0.0

-0.1 0.0

-- Voight ROM

- - - Reuss ROM ••••••.•• HS, Kerner, and Wakashimo

o Ni/Zr02 [4.33]

C Ni/Zr02 [4.34]

0.2 0.4 0.6

VI

(a)

IiId '

.. 0 ....

....... [I

: I ! I

: I ! I

P I ... I

0."" I .' I

... ~ .......... ",/'

'"

0.8 1.0

Figure 4.4. The effective thennal conductivity as a function of volume fraction as predicted by the rule of mixtures, the HS variational technique, and the Kerner and Wakashima micromechanical theories. Experimental data for three metal/ceramic FGMs with conductivity ratios (k/k2) of32 (Figure 4.4(a», 12.5 (Figure 4.4(b», and 3.5 (Figure 4.4(c» are show for comparison. The subscripts (1 and 2) in the labels of the axes correspond to the first and second materials listed in the legend.

1.2

1.1

1.0

0.9

0.8

0.7 N ... '- 0.6 ...

';:; 0.5 ... , ... 0.4

0.3

0.2

0.1

0.0

-0.1 0.0

1.1

1.0

0.9

0.8

0.7

~ 0.6 }

0.5 ';:; ... , 0.4 ... 0.3

0.2

0.1

0.0

-0.1 0.0

-- Voight ROil

- - - Reuss ROil ...•.•..• HS, Kerner, and Wakashima

o 304SS/YSZ - hot pre .. (4.351

o 304SS/YSZ - spray [4.351

0.2 0.4 0.6

V,

(b)

-- Voight ROW

- - - Reuss ROil ......... HS, Kerner, and Wakashima

o IN100/ZrOz (4.361

0.2 0.4 0.6

V,

(c)

0.8 1.0

.... :.: .• ) ...... :::: ... ::···~II

0.8 1.0

79

Figure 4.4. The effective thennal conductivity as a function of volume fraction as predicted by the rule of mixtures, the HS variational technique, and the Kerner and Wakashima micromechanical theories. Experimental data for three metal/ceramic FGMs with conductivity ratios (k1ik2) of32 (Figure 4.4(a)), 12.5 (Figure 4.4(b)), and 3.5 (Figure 4.4(c)) are show for comparison. The subscripts (1 and 2) in the labels of the axes correspond to the first and second materials listed in the legend.

80 Chapter 4

3.3.3 Heat capacity

Of the approaches reviewed above, only the Wakashima micromechanical theory and the simple rules of mixtures can be used to estimate the effective volumetric heat capacity. Experimental data for this property are very limited in the FGM literature, probably because it is less important than the thermal expansion coefficient or conductivity (heat capacity is only needed for transient analyses and has no effect on the steady state). The available data are insufficient to either support or refute the linear or nonlinear rule of mixtures. Because the micromechanical approach (Equation 4.16) results in a simple linear relation that is the same as the linear rule of mixtures (Equation 4.1), a linear approximation is recommended.

3.4 Mechanical Properties

3.4.1 Elastic modulus

For an isotropic material (which is assumed here), all elasticity properties can be expressed in terms of just two independent constants. As shown above, the HS, Kerner, and Wakashima relations are all derived in terms of the bulk and shear moduli, whereas Ravichandran's approach assumes that the constituents have equivalent Poisson's ratios and provides a relationship for the Young's modulus. Since the Young's modulus can be computed easily from the bulk and shear moduli, it can be used to make comparisons between the various approaches.

Figure 4.5 compares the effective Young's modulus (normalized between zero and one) as a function of the volume fraction for a two component composite as predicted by the rule of mixtures (Equations 4.1-4.2), the HS variational method (Equations 4.3-4.6), and the Kerner (Equations 4.9-4.10), Wakashima (Equations 4.13-4.14), and Ravichandran (Equations 4.18- 4.20) micromechanical approaches. Three separate plots are shown corresponding to modulus ratios (E lfE2) of 3.7, 1.9, and 1.2. Experimental data for three FGMs with corresponding modulus ratios are included for comparison. During the process of generating data for the plots it was discovered that the seemingly very different analytical relations for the HS and Kerner approximations produced identical numerical results. Therefore, they were combined into a single set of curves in each of the Figures.

Figures 4.5(b) and 4.5(c) indicate that for moderate mismatch ratios the HS, Wakashima, and Ravichandran bounds yield almost identical results. Furthermore, the upper and lower bounds are quite close, and probably within the experimental error for single-phase properties. As the mismatch

81

ratio increases, shown in Figure 4.S(a), the upper and lower bounds increasingly separate. Of the various methods considered here, the Ravichandran approach produces the closest upper and lower bounds, followed by the Wakashima bounds, then the HS bounds, and finally the rule of mixtures. The Ravichandran bounds are reasonably close, even for relatively large mismatch ratios. Again, the comparison of theory with the experimental data provides interesting and important insights into the prediction of properties.

For the metal/metal (W/Cu) FGM shown in Figure 4.S(a), in general the experimental data appear to fall between the linear rule of mixtures relation and any of the upper bounds from the variational or micromechanical approaches, since they are all similar. Taking into consideration experimental accuracy, it is expected that either the linear or anyone of the non-linear upper relations should provide a reasonable prediction of the effective modulus. However, because data were available for only one metal-metal FGM, the generalizability of this suggestion is uncertain.

1.1

1.0

0.9

0.8

0.7

~ 0.6 '-~ 0.5 <oj ,

0.4 <oj

0.3

0.2

0.1

0.0

-0.1 0.0

-- Voight/Reuss ROW

- - HS, Kerner ----- Wakashima ......... Rovichandran

o W/Cu [4.37J

0.2 0.4

(a)

0.6 0.8 1.0

Figure 4.5. The effective Young's modulus as a function of volume fraction as predicted by the rule of mixtures, the HS variational technique, and the Kerner, Wakashima, and Ravichandran micro mechanical theories. Experimental data for three FGMs with modulus ratios (E/E2) of3.7 (Figure 4.5(a», 1.9 (Figure 4.5(b», and 1.2 (Figure 4.5(c» are shown for comparison. The subscripts (I and 2) in the labels of the axes correspond to the first and second materials listed in the legend.

82 Chapter 4

, .6

-- Voight/Reuss ROW , .4

- - H5, Kerner

----- Walca.hima , .2

......... Rovichandron

, .0 o AIZ03/Ni - part. reinforced [4.381

o AI203/Ni - interpenetrating [4.381 N ...

0.8 L ... ";:; ... 0.6 I ...

0.4 0

0.2 C

0.0 0 0

-0.2 0.0 0.2 0.4 0.6 0.8 f .0

V,

(b)

, .2

,. , -- Voight/Reuss ROW

, . a - - H5. K,rner ----- Wakashima

0.9 ......... Ravlchandran

0.8 o PSZ/304SS [4.281

~ 0.7

L 0.6 ... ";:; 0.5 ...

I ... 0.4 o

0.3

0.2 o o O. ,

0.0

-0. , 0.0 0.2 0.4 0.6 O.B , . a

V,

(c)

Figure 4.5. The effective Young's modulus as a function of volume fraction as predicted by the rule of mixtures, the HS variational technique, and the Kerner, Wakashima, and Ravichandran micromechanical theories. Experimental data for three FGMs with modulus ratios (E/E2) of3.7 (Figure 4.5(a», 1.9 (Figure 4.5(b», and 1.2 (Figure 4.5(c» are shown for comparison. The subscripts (I and 2) in the labels of the axes correspond to the first and second materials listed in the legend.

83

For the interpenetrating phase ceramic/metal (AhOiNi) FGM shown in Figure 4.5(b), the comparison of experimental data with any of the variational or micromechanical theories is excellent for ceramic/metal volume fractions of 0.2 and 0.4, but very poor at 0.6. For the particlereinforced material, the 0.2 volume fraction is in reasonable agreement, but at larger volume fractions, none of the predictions fits the experimental data well. This discrepancy is attributed to damage effects from debonding between phases, poor sinter bonding between contiguous ceramic particles, or particle fracture. These possible damage effects are not included in the theories. The fit has been improved by including damage effects in the Ravichandran micromechanical approach [38]. Even when the moduli of the two phases are very similar, such as for the 304SS/PSZ FGM shown in Figure 4.4 c, the comparison is not good for ceramic/metal volume fractions above about 0.2. Again, this is believed to be due to damage effects.

3.4.2 Plasticity

Although clearly important for many FGM applications, few simulations to date have included plasticity effects. In most cases, elastic behavior has been assumed, sometimes resulting in significant errors. The major reason plasticity effects are neglected is because of the difficulty and uncertainty involved in predicting effective properties. Examples of stress-strain curves for FGMs exemplify this point.

Figure 4.6(a) shows experimentally obtained stress-strain curves for a PSZINi (partially stabilized zirconia/nickel) FGM produced by hot-pressing [39]. Although not shown in this Figure, pure nickel exhibits classic elasticplastic behavior, while pure PSZ is linear-elastic to failure. With only 20 volume % PSZ, the composite displays typical elastic-plastic deformation similar to pure nickel, while at 40 and 80 volume % PSZ, the response is linear elastic. At 60 volume % PSZ, however, the response becomes nonlinear again. It is suggested that this nonlinear behavior is due to a microstructure transition at this composition, resulting in loose connectivity ofthe two phases [39].

Figure 4.6(b) shows experimentally obtained stress-strain curves for a YSZ/SS304 FGM produced by sintering [35]. The pure stainless steel shows typical stress-strain behavior for a metallic material, with a failure strain of approximately 11 %. Each of the composite compositions displays slightly nonlinear behavior, with failure strains of approximately 0.5% or less. Interestingly, the 40 volume % YSZ material appears to be more nonlinear than the 20 volume % YSZ.

The two sets of data described above vary considerably, even in basic trends. Although not shown, additional data are available for an Ah03INi

84 Chapter 4

FGM [38] and indicate even different behavior. The data clearly show the difficulty of accurately describing this complex behavior with simple theories such as the modified rule of mixtures (Equations4.21-23). Such relationships can provide rough approximations of stress-strain behavior, but are apparently too simplistic for most ceramic/metal FGMs because of microstructural complexities and local damage.

3.4.3 Creep

As with plasticity, creep behavior in FGMs has not been well researched, and little is known about the estimation of effective properties. The use of simple interpolation schemes like the linear (Equation 4.1) or modified rule of mixtures (Equations 4.21- 23) is questionable because of the extreme nonlinearity in typical creep constitutive relations (e.g., powerlaw creep with powers on the order of 5 or more). In very recent work, the fracture behavior in a creeping FGM has been modeled [40]. However, the creep properties of the single FGM layer considered in this study are simply assumed.

4. COMMENTS ABOUT FGM DESIGN

The modeling of material properties and component behavior, the focus of this chapter, is a very useful tool for the design engineer. In general, the design of an FGM is expressed in terms of its constituent materials, its geometry, and the spatial distribution of its composition, microstructural features, and properties. These specifications can then be used during processing to create the desired FGM microstructure, or they can be used to predict the behavior of a larger system containing an FGM component.

During any design study, the materials modeler and the design engineer must communicate effectively about the geometry, loading conditions, performance requirements, and failure criteria for the FGM component under consideration. The modeler needs to know specifically what kind of information (e.g., stresses, temperatures) is desired, what level of accuracy is required, and what kind of property data are available for input into the model. The designer needs to know specifically what materials properties (e.g., mechanical, thermal) are most important for obtaining accurate modeling results, and also should be made aware of what boundary conditions and other assumptions are used in the model, and how these can affect the modeling results. Only when all of these issues have been adequately addressed can a modeling study be conducted that will be useful for design purposes.

600

550 I

500 I I

450 I

I

400 I

I

'0' 350 I ... I

~ I .. 300 I .. . ,

J: 250 , '" , ,

.... ,1

, 200 :, : , 150 /,'

.~' 100 .>

.y ~.

-- 20PSZ/BONi

- - - 40PSZ/60NI

----- 60PSZ/40Ni 50

, ~ ......•.. BOPSZ/20Ni

0 0.0 0.2 0.4 0.6 O. B 1.0

Slrain (%)

450

400

350

300

'0' ... 250 ~

: 200 J:

'" 150

-- OYSZ/l00SS

100 - - 20YSZ/BOSS

- - - 40YSZ/60SS

50 ----- 60YSZ/40SS .......•. BOYSZ/20SS

0.6 O.B 1.0

Slrain (I)

Figure 4.6. Experimental stress-strain data for (a) a partially stabilized zirconia/nickel FGM [39] and (b) an yttria-stabilized zirconia/stainless stee1304 FGM [35].

85

Previous modeling studies concerned with residual stresses in ceramicmetal structures bonded with an FGM interlayer can be used to illustrate some of the difficulties associated with modeling and design issues. For example, consider the simple calculation of residual stresses induced during the cooling of a thin disk versus a long, bar-shaped specimen, where the same materials properties are used in the calculations. In a long bar that has

86 Chapter 4

an FGM sandwiched between a ceramic at one end and a metal at the other, the modeling results show there is a large tensile stress acting on the ceramic near the interface at the edge of the specimen. Therefore, cracks would be expected to initiate in this region and propagate parallel to the interface.

By comparison, it has been shown that a thin disk composed of an FGM sandwiched between a ceramic surface and a metal surface undergoes significant bending. This results in large tensile stresses on the surface of the ceramic, which act parallel to the interface. Therefore cracks would be expected to be initiated on the ceramic surface and propagate perpendicular to the interface. Consequently, two different stress components and resulting failure modes are associated with these different geometries. Each would require a different FGM design solution.

All FGM design problems rely heavily on modeling, and highly accurate modeling results are dependent on the availability of reliable material property data and detailed constitutive models. Examples of the ways in which modeling studies of FGMs can be applied for design purposes are given in Chapters 5 and 7.

REFERENCES

1. Beardsley, M.B. (I 997) Functionally graded thermal barrier coatings for diesel engines, Symposium on Functionally Graded Materials, Fall Meeting of the Materials Research Society Meeting, December 1-5, 1997, Boston, MA.

2. Malvern, L.E. (1969) Introduction to the mechanics of a continuous medium, Prentice Hall, Inc., Englewood Cliffs, N.J.

3. ABAQUS Computer Program (1997) Hibbitt, Karlsson & Sorenson Inc., Pawtucket, RI. 4. Dao, M. et al. (1997) Acta Mater, 45, 3265. 5. Weissenbek, E., Pettermann, H.E., and Suresh, S. (1997) Acta Mater, 45,3401. 6. Hashin, Z. (1983) J. Appl. Mech., 50, 481. 7. Torquato, S. (1991) Appl. Mech. Rev., 44 (2), 37. 8. Nan, C.W. (1993) Progress in Materials SCience, 37, 1. 9. Voight, W. (1889) Wied. Ann., 38,573. 10. Reuss, A. (1929) ZAMM, 9,49. II. Cho, K. and Gurland, 1. (1988) Met. Trans. A, 19A, 2027. 12. Fan, Z., Tsakiropoulos, P., and Miodownik, A.P. (1994) J. Mater. Sci., 29, 141. 13. Hashin, Z. and Shtrikman, S. (1963) 1 Mech. Phys. Solids, 11, 127. 14. Hashin, Z. and Shtrikman, S. (1962)1 Appl. Phys., 33, 3125. 15. Kerner, E.H. (1956) Proc. Phys. Soc., B69, 808. 16. Taki, M. et al. (1990) A fundamental study on the application ofFGM to high

temperature rotating members, in Proc. of The First In! 'I. Symp. on FGM'90, (eds. M. Yamanouchi et al.), Functionally Gradient Materials Forum, Toranomom, Minato-ku, Tokyo, Japan, 353-358.

17. Wakashima, K. and Tsukamoto, H. (1992) ISIJ International, 32, 883. 18. Eshelby, J.D. (1957) Proc. Royal Soc. London, 241, 376. 19. Mori, T. and Tanaka, K. (1973) Acta Metall., 21, 571.

20. Ravichandran, K. (1994) JAm. Ceram. Soc., 77[5], 1178. 21. Aboudi, J., Pindera, M., and Arnold, S. (1996) Int. J Solids Structures, 33[7],931. 22. Wilt, T.E. and Arnold, S.M. (October 1994) Micromechanics Analysis Code, NASA

Technical Memorandum I 06706. 23. Zuiker, J. and Dvorak, G. (1994) Composites Engineering, 4, 19. 24. Suresh, S. and Mortensen, A. (1998) Fundamentals o/Functionally Graded Materials,

10M Communications Ltd., London. 25. Williamson, R.L., Rabin, B.H., and Drake, 1.T. (l993) J Appl. Phys., 74, 1310. 26. Fischmeister, H. and Karlsson, B. (1977) Z. Metallkde, 69,311. 27. Hirano, T., Teraki, J., and Yamada, T. (1990) On the design of functionally gradient

materials, in Proc. o/The First Int'l. Symp. on FGM'90, (eds. M. Yamanouchi et al.), Functionally Gradient Materials Forum, Toranomom, Minato-ku, Tokyo, Japan, 5-10.

28. Zhai, P.C., Jiang, C.R., and Zhang, Q.J. (1993) Application of three-phase micromechanical theories to ceramic/metal functionally gradient materials, in Ceramic Transactions 34, Proc. Second Int 'I. Symp. on FGM '92, (eds. J.B. Holt et al.), The American Ceramics Society, Westerville, OH, 449-456.

87

29. Larker, R. and Beckman, T. (1995) Compositional gradation between silicon nitride and superalloys using ShN4/TiN CMC and TiNlNi MMC layers, in Proc. Third Int 'I. Symp. on Structural and Functional Gradient Materials, (eds. B. Ilschner and N. Cherradi), Presses poly techniques et universitaires romandes, Lausanne, Switzerland, 495-501.

30. Bruck, H.A. and Rabin, B.H. (1999) An Evaluation of Rule-of-Mixtures Predictions of Thermal Expansion in Powder Processed Ni-A1203 Composites, J Amer. Ceramic Soc., in press.

31. Takemuma, M. et al. (1990) Proceedings, The First International Symposium on FGM, (eds. M. Yamanouchi et al.), Functionally Gradient Materials Forum, Toranomom, Minato-ku, Tokyo, Japan, 97-100.

32. Kawai, C. et al. (1990) Oxidation resistant coating with TiC-SiC gradient composition on carbon fiber reinforced composites by CVD, in Proc. o/The First Int 'I. Symp. on FGM'90, (eds. M. Yamanouchi et at.), Functionally Gradient Materials Forum, Toranomom, Minato-ku, Tokyo, Japan, 77- 82.

33. Takemuma, M. et al .. (1993) Evaluation of thermal and mechanical properties of functionally gradient material of ZrOz-Ni system, in Ceramic Transactions 34, Proc. Second Int 'I Symp. on FGM'92, (eds. 1.B. Holt et at.), The American Ceramics Society, Westerville, OH, 271-278.

34. Matsumura, S. et at. (1993) A technology to form FGMs by composite electroforming, ibid., 331-338.

35. Igari, T. et al. (1990) Mechanical properties offunctionally gradient material for fast breeder reactor, in Proc. o/The First Int'l. Symp. on FGM'90, (eds. M. Yamanouchi et al.), Functionally Gradient Materials Forum, Toranomom, Minato-ku, Tokyo, Japan, 209-213.

36. Akama, S. (1997) Mechanical and thermal properties ofPSZlNi-base superalloy composite, in Proc. o/The Fourth Int 'I. Symp. on FGM '96 , (eds. I. Shiota and Y. Miyamoto), Elsevier Science B.V., Amsterdam, The Netherlands, 451- 456.

37. Jedamzik, R. and Neubrand, A. (l997) Ceramics Group, TU-Darmstadt, Germany, private communication.

38. Bruck, H.A. and Rabin, B.H. (1998) Evaluating Microstructural and Damage Effects in Rule-of-Mixtures Predictions of the Mechanical Properties ofNi-AIz03 Composites for Use in Modeling Functionally Graded Materials, J Mater. Sci., 33, I-II.

88 Chapter 4

39. Zhu, J.C. e! a/. (1997) Mechanical performance of Zr02-Ni functionally graded material by powder metallurgy, in Proc. Fourth In! 'I. Symp. on FGM'96, (eds. I. Shiota and Y. Miyamoto), Elsevier Science B.V., Amsterdam, The Netherlands, 203-208.

40. Biner, S.B. (1997) Engineering Fracture Mechanics, 56, 657.

Chapter 5

THE CHARACTERIZATION OF PROPERTIES

Keywords: Electrical conductivity, Betti number, percolation, fractal, quasi-electric field, graded band semiconductor, dielectric permittivity, dielectric constant, capacitance, ceramic actuator, piezoelectricity, thermal conductivity, thermal diffusivity, thermal expansion, thermal stress, effective thermal conductivity, apparent thermal diffusivity, Fourier number, PSZlNiCrAIY, ~-TiffiC, VN2C, Young's modulus, AIIAhNi, Poisson's ratio, acoustic microscope, PSZ/SS, NiAIIAI, residual stress, TiBlTi, CulNi/Cu, strain hardening, fracture mechanics, fracture toughness, stress intensity factor, fatigue, creep, toughening ratio, TiC-Ni, Cr3C21Ni/Cr3C2, thermal stress, finite element method, partially stabilized zirconia/stainless steel, acoustic emission, PSZlNi, TiB2/Cu, TiClNi, thermal fatigue, SiCITiCICC.

1. INTRODUCTION

The technology of Functionally Graded Materials (FGMs) enables the realization of innovative and multiple functions that cannot be achieved with conventional homogeneous materials. Predetermined chemical composition profiles (the spatial distribution of their components) as well as predetermined transitions in their microstructure, are intentionally introduced to perform desired functions. Therefore, in order to use FGMs in practical applications, it is important to characterize their properties.

The characterization of properties includes evaluating the local microstructure and properties of the FGM to determine the potential performance of the designed structure and the distribution of its properties. It also includes the evaluation of the overall performance of the material's properties. In the microstructural evaluation, it is necessary to quantitatively determine the size, configuration, orientation, and contiguity of phases. This can be accomplished using ordinary image analysis. In some cases,

90 Chapter 5

conventional techniques can be used such as microscale chemical analysis and microhardness tests. But in others, some modifications in the techniques are necessary because the continuous change in the properties in a local region make measurement difficult. However, in evaluating the overall performance of properties such as electrical, magnetic, thermal, and mechanical, if conventional methods are not applicable, it may be necessary to modifY them or even to develop new techniques.

Furthermore, fracture behavior in a ceramic/metal FGM can change from brittle to ductile fracture due to the gradual change in the contiguity of the ductile, metal phase. Consequently, the overall mechanical behavior of an FGM has to be evaluated not only on the macroscopic scale but also on the microscopic scale, such as for damage growth, microcrack initiation at interfaces, and crack propagation. The evaluation of thermal stress is also important, because thermal stresses are generated during an FGM's fabrication and heat treatment as well as by differences in the coefficients of thermal expansion (CTE) of its components.

In many applications, an FGM is exposed to regular or alternating thermal loading with a high temperature gradient between its two ends. Therefore, its thermal stability must be evaluated, because microstructural changes may have occurred due to Ostwald ripening (the reorganization of many small particles into fewer larger particles), and also because the compositional distribution may be unstable due to the diffusion of component elements. In addition, if the FGM will be subjected to rapid heating and cooling at elevated temperatures, its thermal shock resistance, thermal fatigue characteristics, temperature profile, and the overall heat flow must be evaluated.

Some of the methods now used for evaluating the performance of FGMs are described in this chapter. However, many of these still need to be standardized, and new techniques need to be developed.

2. ELECTRICAL PROPERTIES

2.1 Electrical Conductivity in Graded Materials

In homogeneous materials electrical conductivity (or resistivity) is assumed to be constant. In the case of FGMs, however, it changes spatially corresponding to the graded composition and structure. The electrical conductivity of an FGM should be expressed as a function of position. For example, c(x) expresses the conductivity profile of an FGM with a onedimensional gradation as illustrated in Figure 5.1. In an ideal case, the curve is completely continuous, which indicates the absence of a sharp interface.

91

The conductance between positions Xl and X2 in an FGM with a cross sectional area, S, is expressed by integrating the conductivity profile as follows:

as:

(5.1 )

Position x

Figure 5.1. Conductivity profiles of an FGM.

The electrical current flowing through an FGM with a length .e. is written

]= sv J:dx/O"(x)

where V is the applied voltage.

(5.2)

The profile of the voltage drop between positions Xl and x 2 is given by:

92

Vf2 dx/a(x) v( x) = _--,~I=------

fodx/a(x)

Chapter 5

(5.3)

The Joule heat (e) between Xl and x 2 ' and the total heat (E) generated by a current through an FGM are derived as follows:

SV2 f;2 dx/a(x)

e(x, - x,) ~ U:dx/O"(X)r

SV2 E= --,----

foldx1a(x)

(5.4)

(5.5)

When the electrical conductivity is measured along the direction perpendicular to the gradation as indicated in Figure 5.2, the conductance along the length, (Y2 - Yl)' and the width, (Xl - x2), of the gradation can be described as follows:

fhf~ f~ s = a(x)dxdy = (Y2 - Yl) a(x)dx ~ ~ ~

(5.6)

The relation between electrical conductivity and a graded structure has been studied for an FGM of stainless steellzirconia (SS/Zr02) [1, 2]. Figure 5.2 shows the dependence of the electrical conductivity on the composition of homogeneous composites of stainless steel and Zr02. Each polycrystalline composite (A and B) was made by vacuum sintering mixtures of two different size powders of stainless steel and zr02. (A: 70 J..lm SS /2-lO J..lm Zr02 and B: 180 J..lm SS/3 J..lm Zr02). The large change in conductivity between 20 and 60 volume % Zr02 reflects the change in connectivity of the matrix phase. The region of high conductivity below 20 volume % Zr02 contains dispersed nonconducting Zr02 in the stainless steel matrix. In the zone of high to low conductivity from 20 to 60 volume % Zr02, the composite has a skeletal structure of both phases. In the insulating zone, the matrix changes to the Zr02 phase and the stainless steel phase is isolated.

The phase connectivity in a two-phase mixture is characterized by the Betti number, which is a topological parameter for the connectivity of a network [3]. Figure 5.3 shows the Betti number derived by analyzing the

93

connectivity of the stainless steel phase as a function of the volume fraction of Zr02. The similarity of the change in both the Betti number and the electrical conductivity as a function of the volume fraction of Zr02 suggests a direct relation between the electrical resistivity and the connectivity of the phases.

1.2 ~--""T"""--...... ---.....---.....,....-----. ----.t*H;- (A)

1.0 -------:----------------.-:------"(B)"----

0.8 ---- ---- ~--- -----... -- _.- --_ .. ~- .. -------- -_. _.

!! ;

0.6 ............ : ............... ~m~mlmmm ••••

0.4

, "

.......... " ........... 1 ................. 1 .........•.......

: : ,J: , , ,

0.2

0.0 L...---"""----........ --......;10;..;;;::.. ...... """""'iI---..

0.0 0.2 0.4 0.6 0.8 1.0

Volume fraction of Zr02

Figure 5.2. The electrical conductivity as a function of the volume fraction of zr02 for the stainless steel/ Zr02 composites. Curves (A) and (B) represent the data for composites sintered by combining two different particle sizes: A: 70 /lm SS/2-IO /l 2 and B: 180 /lm SSt 3 /lm zr02.

The change in conductivity in a two-phase mixture of an insulator and a conductor can be regarded as a percolation problem [4]. Such a complex structure can be analyzed using fractal theory [5]. Figure 5.4 shows the relation of the fractal dimension, D, as a function of the volume fraction of Zr02 in SS/Zr02 composites. From percolation theory, the D value is known to be equal to 1.896 when clusters are connected infinitely in two dimensions. If the fractal dimension for the stainless steel phase is above this critical value, a continuous path is formed resulting in metallic conductivity.

94

Q) (/) co

400~----~----~------~----~----~

-----Jlk-- (A)

--it--- (6)

-a 3001-····················:\ Q) Q)

til (/) (/)

J!2 c: .(ij 200 1-.................. .,. .. \ til

1 00 /-................... ; ...... \ ............ ; ..... \ ............... ; ...... .

oL---~----~--~~~~~----~

Chapter 5

0.0 0.2 0.4 0.6 0.8 1.0


Figure 5.3. The Betti number as a function of the volume fraction of ZrOz for the stainless steel phase in the stainless steel/ZrOz composites. Curves (A) and (B) refer to Figure 5.2.

In designing and analyzing the electrical properties of FGMs, it is important to know the relation of the conductivity profile to the change in structure.

2.2 Quasi-Electric Fields in Graded Semiconductors

Graded band engineering of semiconductors was initiated in the 1980s with the development of molecular beam epitaxy (MBE). This technique enables the preparation of compositional gradation as well as superlattices on an atomic scale. Figure 5.5 shows energy band diagrams for compositionally graded semiconductors [6]. In the case of an intrinsic semiconductor (a), the graded conduction and valence band edges, Ec (z)

dEc dEv F =-- F. =+-

and Ev (z), produce" quasi-electric" fields, e dz and h dz , that can accelerate the velocity of carriers. For extrinsic semiconductors, the band diagrams change somewhat as shown in Figure 5.5 (b) and (c).

95

2.1

2.0

0 1.9 c 0 '(ij c Q)

1.8 - --- --. --;--. ------- ---- ------;-' .- ------------ _ .. _. ~.------ .. _. _ .... _.

E '6 Cii 1.7 -u ctS

: : , , ----------1--·------------·····-r·--·-------------

... u. 1.6 ... -~--- ---- "----- .. ----.. ~

1.5

1.4 L...-__ ....I-__ ....... ___ ...I..-__ -'-__ --'

0.0 0.2 0.4 0.6 0.8 1.0


Figure 5.4, The fractal dimension as a function of the volume fraction of zr02 in the stainless steellZr02 composites.

e~ ~EC(Z) ~EV(Z)

@~ Fh

(a)

--------------- Ef

(b) (c)

Figure 5.5. Energy band diagrams of graded bandgap semiconductors: (a) intrinsic, (b) ptype, (c) n-type. Fe is the quasi-electric field for electrons, Fh is the quasi-electric field for holes, EJz) is the conduction band edge, Elz) is the valence band edge, and n f is the Fermi level.

96 Chapter 5

In a p-type graded-gap material, there is no effective field acting on holes dE

while the effective field for electrons is F = - --g . This is because of e dz

cancellation between the quasi-electric field, ~, due to the drifting and

accumulation of holes and the resultant space charge field. This effective field can be significantly greater than in the intrinsic case. For an n-type material the effective field can act on holes.

The concept of a graded band material is expected to modify the carrier transport and optical properties of semiconductors and to be applicable to various devices such as high speed transistors, efficient photodiodes, lasers.

2.3 Dielectric Properties in Graded Materials

Dielectric materials are widely used for capacitors, piezoelectric devices, surface acoustic devices, electronic memories, and the like. The dielectric property is represented by the dielectric permittivity, E, which depends on the frequency of the applied electromagnetic wave. If E changes spatially in a graded material along its width like the conductivity profile, a(x), the electrical capacitance through the cross sectional area S and the width f is expressed as follows:

S C== ---fdx

o E(X)

(5.7)

where e(X) is the profile of the dielectric permittivity for an FGM. Ceramic capacitors are produced presently by introducing various

additives into barium titanate (BaTi03) in order to lower the temperature coefficient of the dielectric constant. If the Curie point of a capacitor (the transition temperature between the ferroelectric and paraelectric phases) can be changed as a function of position by grading its composition, the transition from the ferroelectric to the paraelectric phase would be broadened with respect to the temperature. Consequently, the temperature coefficient of the dielectric constant could be lowered. Figure 5.6 shows the temperature dependence of the capacitance for a Bal _ xSrx Ti03 FGM compared with that of BaTi03 [7]. In the FGM, five different layers with compositions (x == 1.00,0.70,0.54,0.47,0) are stacked together. This type of graded capacitor is expected to have lower temperature coefficient and higher dielectric permittivity for a wide range of temperatures.

In ceramic actuators a piezoelectric plate is sandwiched between two metal electrodes. Usually these plates are bonded with an adhesive, which

~ 1kHz (J)

1.00 ~ .. :\ Co) c •• res ~ -'u ... / /':, .. res 0.75 r0-c.. res ~ . Co)

-0 Ba1 ' XSrJi03 FGM ~ /':, • (J) 0.50 ~ /':, /':, 0 .~ l'> \ /':, /':, (ij

/':,t::,/':,/':,/':,/':, E /':, ... 0.25 ~ BaTi03 0 Z

0.00 • I I I I I I I • I •

0 40 80 120 160

Temperature CC)

Figure 5.6. A comparison of the temperature dependence of the capacitance for a Bal. xSr, Ti03 FGM and BaTi03.

97

can result in several problems (peeling off, softening at high temperatures, cracking at low temperatures). By fabricating an actuator with a sandwiched structure consisting of a piezoelectric layer, an intermediate compositionally graded layer, and a ceramic layer with a high dielectric constant, as illustrated in Figure 5.7, the adhesive problems can be eliminated.

Piezoelectric maerial

High dielectric constant maerial

Electrode

} Intermediate

layer

Figure 5.7. The structure of an FGM piezoelectric actuator.

The FGM with a periodic structure composed of a piezoelectric material and a high dielectric constant material can be used for an ultrasonic motor that is driven by a sinusoidal progressive wave [8, 9]. Figure 5.8 is a diagram

98 Chapter 5

of such a graded piezoelectric actuator. The composition of the piezoelectric material, A, and the high dielectric constant material, B, are Pb(Ni1/3Nb2l3)o.5(Tio.7ZroJ)o. 5 0 3 and Pb(Ni1/3Nb2l3)o7(Tio.7Zro.3)o303, respectively. The symmetric structure of the FGM elements can prevent warping during fabrication.

-Figure 5.8. A schematic of an ultrasonic motor with graded piezoelectric actuator showing the progressive motion it produces.

An optical fiber with a graded index is another example of a graded dielectric material. If a material is not magnetic, the following relation exists between the dielectric constant and the refractive index:

(5.8)

where n is the refractive index and Eo is the dielectric constant of vacuum. The spatial change of the dielectric permittivity produces a similar spatial change in the refractive index. An optical fiber with a graded index is a material tailored to have a higher index in its core than in its outer layer in order to transmit light with total refraction.

3. THERMAL PROPERTIES

Some general methods for analyzing the thermophysicaJ properties of FGMs, mainly thermal conductivity and thermal diffusivity, are described in this section, and thermophysical properties under both steady state and transient heating conditions are evaluated. The analysis of transient characteristics such as temperature response is important for optimizing

99

graded structures that can withstand thermal cycling conditions at high temperatures.

3.1 Thermal Expansion

The thermal expansion of FGMs is an important characteristic that affects their mechanical behavior in severe thermal environments such as for engine components and space structures. The thermal expansion coefficient (CTE) of a material, defined as the linear expansion of the strain per unit of temperature change, is measured with a dilatometer [10]. The thermal expansion of metal-ceramic composites with different compositions has been studied extensively in order to optimize their graded compositions through the relaxation of thermal stress. However, no model exists that is valid for the full range of compositions. To precisely measure geometric change of an FGM, or the distribution of local thermal expansion, two- or threedimensional measurement techniques must be used, such as the moire interferometry [11], laser interferometry [12], or digital image correlation [13, 14].

3.2 Thermal Conductivity

For homogeneous materials, the heat shielding ability is fairly well understood, and the heat transport is described by the thermal conductivity and the thermal diffusivity. These properties can be readily measured. However, most conventional methods cannot predict the distribution of thermophysical properties in FGMs directly. The measurable temperature is limited to the surface of a material under heat flux. For the total heat resistance of an FGM, the steady state method can be used to determine thermal conductivity.

If the thermal conductivity of a component in an FGM is Ai, the effective thermal conductivity Ae can be expressed as a total property by the following equation:

(5.9)

where L is the total thickness of the FGM, Ii is the thickness of the FG M component with a thermal conductivity of Ai, a is the thermal diffusivity, c is the heat capacity, p is the density, and n ---t 00 denotes that the material is composed of components with continuous properties. The effective thermal conductivity is obtained from the thermal resistance LIA", which is

100 Chapter 5

determined by dividing the mean temperature gradient between the front and back surfaces by the rate of heat flow through the FGM.

Therefore, a set of thermal conductivities for the components of an FGM can give its total heat resistance. However, this set of thermal conductivities does not correspond to the thermal conductivity ofthe FGM.

Thermal conductivity is calculated using the following formula:

A=acp (5.10)

This formula must be modified for heterogeneous materials such as FGMs, or the results will be inaccurate.

There are steady state and transient methods for measuring the thermal conductivity of materials. Thermal diffusivity is measured directly by the laser flash method, which can measure temperatures as high as 700·C. When the front surface of an FGM is heated by a laser pulse and the temperature response at the back surface is measured, the effective thermal diffusivity, ae , is obtained from the half-time to reach the maximum temperature. Combining Equation 5.10 with the mean values for the specific heat capacity, cm, and the density, Pm, gives an apparent value for the thermal conductivity, Aa , of an FGM. However, as shown below, it cannot describe the thermal conduction process. In general, the thermal conduction equation for a material is expressed as follows [15]: (See also Appendix 5.3 A for the derivation of the equation for the heat conduction of a material with no heat source and sink.)

ar a ar pC-=-(A-) at (}z (}z

For an FGM, this equation can be replaced by:

where

1 L

Pm = - J p(z'}dz, Lo

1 L Cm = -J p(z)c(z)dz

Pm L 0

(5.11)

(5.12)

(5.13)

For a two-layered composite in which each layer has different thermophysical properties, the apparent thermal diffusivity, aa, is obtained

101

from the temperature response at the composite's back surface using the relation expressed in Equation 5.12. The apparent thermal conductivity, Aa, can then be calculated by combining the mean specific heat capacity, em, and the density, rm, using Equation 5.10. The result of this calculation is shown by the dashed line in Figure 5.9 [16, 17]. The solid line shows the effective thermal conductivity, Ae, calculated from the following relation:

103

a/a;F5.36

p/r2=7.64xlO3

...... 102 c,/c2=0.377 I

::.:: )'./1;F1.55xlO4 ......

I

E ~ 10' '-'

~ til

c....: 100

10"

.4=t,+e,=3mm

o 2 3

it (mm)

Figure 5.9. A comparison of the thennal conductivities for an FGM. The apparent thermal conductivity Aa and the effective thennal conductivity Ae were obtained from equation (5.3.4) and (5.3.6), respectively.

(5.14)

In this calculation, the thickness of the composite is assumed to be L = {£ 1 + £ 2) = 3 mm. The thermal conductivity is plotted as a function of the thickness of the first layer. There is a large difference between the

102 Chapter 5

apparent thermal conductivity, Aa , and the effective thermal conductivity, Ae, in this simple two-layered model. This is discussed further below.

3.3 Temperature Distribution under Steady State Heating

Under steady state conditions, heat flow is independent of time. Therefore, Equation 5.11 becomes:

V(AVT) = 0 (5.15)

It is assumed that the sample is thermally insulated and that the heat flows along the z axis as shown in Figure 5.10. For the sample of length L (z = 0 to z = L), the boundary conditions can then be written as T(O) = T sand T(L) = Tb.

Q

t t t t t

Material B

z

Figure 5.10. Schematic of a graded sample under steady state heating.

The equation for unidirectional heat flow for a plate-shaped sample is:

~(A(Z) dT) = 0 dz dz

and its solution is:

Z dz T(z)=AJ-+B

o Il(z)

103

(5.16)

(5.17)

Applying the boundary conditions, the constants A and B can be defined. The temperature distribution is then expressed as:

T(z)=T - 1', -~ f~ s J~ oll(z)

o Il(z)

(5.18)

This equation does not take into consideration the temperature dependence of the material's properties. The variation of thermal conductivity with temperature can be neglected when the temperature range is not too large [18].

For example, a finite element modeling program based on Equation 5.18 can be used to estimate the temperature distribution in a tungsten carbide cobalt (WC/Co) FGM. The analytical model is based on a rectangular body with a thickness L in the z-direction, in which the composition gradually changes from one side to the other according to the following relation [19]:

(5.19)

Where z is the distance from the top surface of the FGM, L is the thickness, and p is an exponent that controls the distribution function.

The graded region is treated as a series of bonded composite interlayers with compositions ranging from 5 to 25 weight % cobalt. The thermal conductivity for different compositions is computed by the linear interpolation of data compiled from the literature [20, 21].

Figure 5.11 shows the estimated temperature distribution for various compositions of a 25 x 14 x 2 mm plate shaped WC/Co FGM prepared by centrifugal powder metallurgy and attached to a substrate [22]. The computation used a heat flux of 1 MW/m2 (top surface temperature of 727°C) with fixed boundary conditions (a set temperature and a perfect contact). The analysis shows that the temperature difference through the thickness of the graded sample is relatively small, and can only be varied within about 20·C by controlling the composition.

104 Chapter 5

Figure 5.12 shows the actual temperature distribution for two homogeneous WC/Co composites and a graded sample that were welded to an iron block and heated on their top surface with a burner rig [23]. The surface temperature was measured with an infrared pyrometer and a welded thermocouple. The average heat flux was determined from the axial temperature distribution in the iron block, which was measured using five thermocouples installed along the axis of the block at 5 mm intervals starting from its top surface.

(95 wt.%WC) (75 wt.%WC

1000

Q' -- 990 @

~ L... Q) 0. E ~

980

970 L.......J.--L-L.---L.---'---L.---'---L.---'----J

0.0 0.2 0.4 0.6 0.8 1.0

Normalized distance (z/d)

0.2 0.4 0.6

P=1 2 4 6

Figure 5.11. Estimated temperature distributions for tungsten carbide/cobalt (WC/Co) FGMs with various compositions.

Because of tungsten carbide's good thermal conductivity, the temperature throughout the sample decreases with increasing WC content. In the graded sample, the heat flux behavior is average between 5 and 15 weight % Co. The temperature at the interface between the sample and the iron block was estimated by linear extrapolation of the measured values. This temperature is in agreement with that obtained from the finite element analysis, and the comparisons between the numerical simulation and the values deduced from experiments also are in good agreement.

105

3.4 Temperature Response under Transient Heating

The development of an analytical solution for the transient temperature response in FGMs is needed both for practical applications such as thermal analysis of the space shuttle during re-entry and for evaluation of thermal conductivity by a transient method. It is a complicated problem because a set of nonlinear partial differential equations has to be solved. There are two approaches to solving the transient temperature response. The first considers a multilayered model with each layer having constant thermophysical properties as shown in Figure 5.13. The second considers a simplified situation where the nonlinear equation can be made linear by regulating the distribution of thermophysical properties. Although there are a number of transient heating methods, such as pulsewise, stepwise, and periodic, the pulsewise heat flux corresponding to the laser flash method is shown here as a practical example [24, 25].

1200

• WC/Co85/15 .. WC/Co95/5 1000

.-800 ~ --Q) ....

::J - 600 co .... Q) c.. E Q) 400 ~

200

o o 5 10 15 20 25

Thickness (mm)

Figure 5.12. Measured temperature distributions for homogeneous and graded tungsten carbide/cobalt (WC/Co) composites.

106 Chapter 5

3.4.1 An approximate solution for the temperature response using a multilayered material model

As shown in Figure 5.13, the heat conduction in a multilayered material with n-Iayers can be described by a set of one-dimensional heat conduction equations. Under the boundary condition of pulsewise heat flux, the analytical solution for the transient temperature response can be obtained by using a Laplace transformation [24]. If the infinitesimal thickness of each layer is considered, the solution for multilayered materials can be extended to FGMs, which are continuous with respect to composition and thermophysical properties. Using the perturbation method, the solution for the approximate temperature response is derived as follows [25]:

Heat flux

'. " " , , ,

" , , ,

layer 1 al PI cl 11

Layer 2 a2 P2 c2 12

Layer 3 a3 P3 c3 'J , , , , , , , , , , , ,

layer n-1 a n-l P n-l c n-l In-1

Layer n an P n c n In

n Temperature detector

o

z

Figure 5.13. A model of a multilayered material in which the thermophysical properties of each layer are constant (lin = thermal diffusivity, rn = density, Cn = heat capacity, and .en = the thickness of the nth layer).

107

e = 1 + 2 f (_I)k cos(knr)e -(kn)2 Fo (1 + lP ) p ~ ~

k=l (5.20)

with

1 0 d 0 dz r=-f-Z, f ~ r:r:\) 'h = _L~a(z) 17L z -ya(z) (5.21)

(5.22)

where 8 p is the temperature response normalized with respect to the maximum temperature rise; 1h is the square root of the total thermal diffusion time, that is the duration for thermal diffusion through the total thickness; , is the ratio of the square root of the thermal diffusion time to

11L, which is actually a normalized diffusion time; ,= 1 signifies the front

surface and' = 0 the back surface; lPp,k is the correction term (when the

properties of two neighboring layers are fairly close, lP p,k can be ignored, i.e.

lPp,k = 0); Fa is the Fourier number, a nondimensional time, which is

expressed as at I L2 for a homogeneous material [24]. When the correction term is ignored, the form of the approximate

temperature response solution (Equation 5.20) is the same as that for a homogeneous material under pulsewise heating, except for the meaning of , and Fa. Therefore, an FGM can be treated as a homogeneous material in which the apparent thermal diffusivity, a;, is expressed as follows:

(5.23)

The dimensionless temperature response at the back surface (' = 0) will correspond to that of a single-layered material when the Fourier number is expressed as:

(5.24)

where a(z) in Equation 5.23 describes the distribution of the thermal diffusivity along the z direction. Therefore, a; can be referred to as the

108

::>

1.5 ,-,----.,...----.,-----..,.-----, , " ...

........... , ....... ' ..... '------------_ .... _------

1.0

0.5

0.0 ......... O""::"_---'~ __ __L ___ ---'-___ __'

0.0 0.1 0.2 Fo

0.3 0.4

Chapter 5

Figure 5.14. The normalized temperature response in an FGM as a function of the Fourier number with position. The solid curves are obtained from the exact analytical solution, the dashed curves from the approximate solution without correction terms, and the longlshortdashed curves from the approximate solution with correction terms.

effective thermal diffusivity. However, as the value of the correction term, f/Jp,k' increases, the "effective" component of a; is diminished.

3.4.2 Comparison of the approximate solution with the analytical solution

For certain situations, there are exact but complex analytical solutions for the temperature response under transient heating conditions [26]. These can be used to evaluate the validity of the approximate solutions. In Figure 5.14, the solid curves represent the analytical solutions, the dashed curves - the approximate solutions without correction terms (Equation 5.20, when f/J p,k = U), and the long/short-dashed curves - the approximate solutions with the correction terms. When' = 0 and 0.25, the approximate solutions and the analytical solutions agree reasonably well. Furthermore, the approximate solutions without the correction terms are closer to the analytical solutions than with the correction terms. However, when ,= 0.5 or greater, the approximate solutions do not correctly describe the temperature response in anFGM.

It can be concluded that Equation 5.20 without the correction term:

109

(5.25)

can be used to evaluate the temperature response of an FGM's back surface in most actual situations such as when using the laser flash method. Therefore, from an engineering standpoint, it is significant that the apparent thermal diffusivity, a;, as expressed in Equation 5.23, can be considered an effective parameter for describing thermal diffusion in an FGM.

While this evaluation method does not deal with thermal conductivity directly but rather with thermal diffusivity, it avoids the error of deriving the thermal conductivity from the relation Aa = a;cmPm (Equation 5.10). However, if the distribution of the heat capacity and density (c, p) is known and that of the thermal diffusivity is measured, the thermal conductivity can be estimated. The effective thermal conductivity also can be obtained from Equation 5.9.

3.5 Thermal Stability

Because a graded composition and structure are not usually in a state of thermodynamic equilibrium, they may not be stable at high temperatures. Thermally activated atoms diffuse along the temperature and composition gradients and disrupt the graded structures. Therefore it is essential to evaluate the thermal stability of FGMs as a function of service temperature and time, particularly for applications at high temperatures.

In the case of plasma spray coatings of FGMs of partially stabilized zirconia/nickel, chromium, aluminum, yttrium (PSZINiCrAIY), the PSZ is re-sintered when heated at 1200°C, and vertical cracks appears due to inhomogeneous shrinkage. However, no change occurs in the graded structure between 600°C-1200°C [27].

The thermal stability of linearly graded ~-titanium/titanium carbide (~TilTiC) and vanadium/vanadium carbide (VN2C) FGMs has been evaluated numerically [28]. In general, carbon atoms in metals diffuse much faster than metal atoms in metals or carbon atoms in carbides. Therefore, changes in the graded composition of these FGMs can be predicted by analyzing the carbon diffusion with temperature.

When these FGMs are exposed to a temperature gradient of 1427°C on the carbide side and 927°C on the metal side, the carbide tends to be condensed at the high temperature side for the ~-Ti/TiC system and at the low temperature side for the VN2C system, as shown in Figures 5.15 and 5.16, respectively. Carbon diffusion in metals is governed by the heat of

110 Chapter 5

transportI, Q*, and the partial molar enthalpy of solution, l1Hs, with respect to the carbon atom solute.

40 r---------------....,

TiC

30

ii :e ~Ti + TiC c 0

~ .... C 20 Q) u c 0 u c 0 .0 .... <1l 0

10

~Ti o~~~~~~~==~===c o 0.1 0.2 0.3 0.4 0.5

Distance (mm)

Figure 5.15. The change in the graded structure of a ~ titanium/titanium carbide (~-Ti/TiC) FGM with time when exposed to a temperature range of927°C"-1427°C.

For the ~-Ti/TiC system, the estimated value ofQ* + l1Hs is negative (-4.2 kJ/mol ), so the carbon atoms diffuse toward the high temperature side and TiCx is formed. While for the VIV2C system, the estimated value of Q* + l1Hs is positive ( 44.5 kJ/mol), and a V2C rich phase is formed at the low temperature side. The increase of the carbide layers of TiCx and V 2C and the

I It is defined as the heat transported by the diffusion of solute atoms. When a part of an homogeneous alloy is heated, the thermal diffusion occurs along the temperature gradient produced and the solute atoms diffuse. If the heat of transport is negative, its direction is opposite to that of atom diffusion. In this case, solute atoms diffuse toward the higher temperature side. While it is positive, the directions both of heat and of atom transport coincide.

111

metal layers of Ti and V respectively on both sides gradually results in disrupting the graded structures. Therefore, before using an FGM in a high temperature application, the temperature range and time duration it can safely sustain must be investigated. However, in order to extend the service temperature and lifetime of FGMs, materials that inhibit the diffusion of atoms sometimes need to be applied.

c o 'iij ... E ~ c 8 c .8 Qj o

Temperature (K)

1500 1400 1300 1200 1700 40~--~~--~~--~----~-----'

Distance (mm)

Figure 5.16. The change in the graded structure of a vanadium/vanadium carbide (V N 2C) FGM with time when exposed to a temperature range from 927°C-1427 DC.

112 Chapter 5

4. MECHANICAL PROPERTIES

An FGM is a material in which elastic properties such as the Young's modulus, Poisson's ratio, the thermal expansion coefficient and thus the internal stress, as well as the microstructure, change spatially. Therefore, an FGM's toughness and strength cannot be evaluated simply by applying conventional fracture mechanics. This section addresses the analysis and measurement of spatially changing elastic properties; the influence of internal or residual stress on strength; fracture mechanics as applied to FGMs; and the evaluation of fracture toughness, creep, and fatigue.

4.1 Elastic Properties

The modulus of elasticity, E, also known as the Young's modulus, is the slope of a straight line when a material's stress is plotted as a function of its elastic strain. It relates to stiffness. The higher the value of E, the stiffer the material. In general, the Young's modulus is constant for a given material at a given temperature. In the case ofFGMs, however, it changes spatially.

For determining the overall Young's modulus of an FGM, the stressstrain curve can be measured by attaching a strain gauge to the sample's surface during a conventional four-point-bending test. The dependence of the Young's modulus on an FGM's composition can be estimated by measuring the flexural resonant frequencies of a rectangular bar specimen using a forced-resonance technique [29]. The Young's modulus of an aluminum/nickel aluminide (AIIAI3Ni) FGM estimated by this method varies from 80 to 102 GPa between the Al and the AhNi surfaces compared with 187 GPa for AhNi [30).

The distribution of the Young's modulus and Poisson's ratio (the ratio of the transverse or lateral strain to the axial strain) can be measured simultaneously by using a line-focus-beam (LFB) acoustic microscope, shown schematically in Figure 5.17 [31, 32, 33, 34, 35]. The relative intensity of the received surface-reflected wave is measured as a function of the distance between the acoustic lens and the specimen. The measurement frequency of 215 MHz, which is the frequency at the center of the LFB acoustic lens, is used for the analysis, and distilled water is used as an acoustic coupler. Velocity measurements of two types of surface wave modes: leaky surface acoustic waves (LSA W) and leaky surface-skimming compressive waves (LSSCW) are derived from the Young's modulus, E, and Poisson's ratio, Y, according to a modification of elasticity theory [36]:

113

O.87+1.12v ~ CLSAW = 1 + v ~2ci+ v)p

(5.26)

CLSSCW = E(l- v)

(5.27) (1 + v)(l- 2v)p

These equations are valid if two assumptions are made: (1) CLSAW and CLSSCW are nearly equal to the velocity of the Rayleigh wave and the longitudinal wave, respectively [33, 34]; and (2) the samples are isotropic and acoustically homogeneous. This depends on the relation between the microstructural size and the ultrasonic wavelength. That is, the measurement with the acoustic microscope of the velocity at the higher frequency (shorter wave length) has higher resolution for the localized velocity measurement. However, if the estimated elastic constant is too localized, it could deviate somewhat from the macroscopic elastic constant in Equations 5.26 and 5.27.

Pulse mode Digital wave measurement memorizer system

Computer system controller analyzer

tic lens Acous (LFB 0 r PFB)

Transducer

Digital

Coup (wate

voltmeter

lant ~ Thermocouple r)

\/ 1'-..'-..'-..'-..'-..'-..'-..'-.. ...................... Specimen

Mechanical stage Stepping motor system

Figure 5.17. Schematic of an acoustic microscope.

r---

114 Chapter 5

The estimated elastic constants as a function of composition for an FGM of partially stabilized zirconia/stainless steel (PSZ/SS) along with acoustic data for its homogeneous components are shown in Figure 5.18. The elastic constant distribution is in good agreement with the data for the components. The marked change in the elastic constants in the region of the 60 volume % PSZ layer could be attributed to debonding at the metal/ceramic interfaces.

250

co Q..

~ 200 I/)

::::J '5 "C 0 E I/) 150 -0)

c: --0-- FGM ::::J 0 >- --e- Homogeneous compo site

100 0 20 40 60 80 100

0.5

0.4 0

~ 0.3 ... I/) -c: 0 I/) 0.2 I/)

·0 Q.. -<>- FGM

0.1 --e- Homogeneous composite

0.0 0 20 40 60 80 100

Volume fraction of PSZ (%)

Figure 5.18. The estimated Young's modulus and Poisson's ratio for PSZ/SS FGMs.

4.2 Deformation and Strength

Both the strength and the deformation behavior of materials are generally evaluated by tensile tests and three- or four-point bending tests. The bending strength of an AIlNiAl FGM is 3-4 times greater than NiAl itself [37]. At

115

temperatures up to 727°C, the fracture is not catastrophic because the propagation of a crack that is initiated in the aluminum side is retarded in the NiAI phase.

The residual stress strongly influences the tensile behavior of FGMs by shifting the stress state to tension or compression, and by strain hardening when the elastic limit is exceeded. The strain to failure in FGMs is determined by that of the lowest ductile component. The relation between the true stress and the applied load can be deduced using laminate theory.

In a titanium/titanium boride (TilTiB) laminate (three layers: 80 mol % TiB, 40 mol % TiB, and Ti, prepared by pulsed electric current sintering), the residual tensile stress below 300 MPa appears in the 80 mol % TiB layer. Almost the same amount of stress was reduced in the bending strength of the TiITiB FGM as in the 80 mol % TiB monolayer (400 MPa). The variation in the bending strength of this TiB layer with relative thickness agrees qualitatively with the calculation of the residual stress generated during cooling from the processing temperature [38].

Figure 5.19 shows the measured and calculated residual stresses in CulNi/Cu FGMs prepared by cold compaction followed by annealing at 800°C [39]. The regions between the pure copper and pure nickel layers consist of Cu-Ni alloys with different compositions. These alloys are known to have a solid solution strengthening effect. For example, Cu-20 weight % Ni and Cu-80 weight % Ni have higher yield stresses and ultimate strengths than pure copper and nickel, respectively. Therefore, the behavior of these FGMs cannot be predicted by applying a simple rule of mixtures to the properties of their components, pure copper and pure nickel.

The residual stress was measured by etching off one side electrochemically and monitoring the deflection of the graded alloy part [40]. Numerical data were obtained by calculating the stress produced during cooling from the stress free temperature, using a finite element code (ABAQUS) [41]. In the annealed FGMs, the calculated residual stresses were as high as 100 MPa. However, they were negligible in the cold compacted ones.

The calculated residual stress can be used to determine the stress-strain behavior of an annealed FGM. For example, at x/d values of 0.1 for Cu-20Ni and 0.9 for Cu-80Ni, where x is the distance from one end of a graded layer with thickness d, after processing, these layers are under a tensile stress of 80 MPa. Therefore they yield at a lower strain than samples with the same composition of layers in a stress free state. On the other hand, a layer that is initially under compression will be unloaded elastically first before starting to be loaded in tension.

Numerical analysis has shown that some parts of an FGM are subject to plastic yielding during cooling after processing [41]. This plastic

116 Chapter 5

defonnation is accompanied by strain hardening. The highest plastic strain encountered in a CulNi/Cu FGM is about 0.1 % for both the pure copper and the pure nickel layers. A correction can be introduced for the strain hardening of individual layers by horizontally shifting their stress-strain curves, if the following assumptions are made: (1) plastic deformations in tension and compression have the same strain hardening effect, (2) plastic deformations created at any temperature between room temperature and 400°C produce a similar strengthening effect, and (3) the strain hardened layers of an FGM are deformed elastically under the applied tensile load up to the plastic strain already encountered during cooling.

150 Tension

100

'@" CL 50 ~ I/) I/) Q) 0 .... \ , iii " iii

... ... ,

:l -50 "0 'iii Q)

a: -100

Compression

-150 0.0 1.0

Thickness (mm)

Figure 5.19. Residual stresses in CulNilCu FGMs after cold compaction and annealing at 800·C. The solid curves with and without closed squares are the results obtained from the finite element analysis and the experiment for annealed samples. The dashed curve is the experimental result for cold pressed samples.

In Figure 5.20, the stress-strain curves for a CulNi/Cu FGM and a virtual FGM, which is composed of data for six layers of homogeneous material, are compared with a curve that takes into account the effects of both residual stresses and plastic strain hardening. The good agreement between this curve and the stress-strain behavior of an actual FGM confirms the validity of the analysis and the assumptions.

C? a.. ~ (/) (/) CJ.) ... en

150r-----~------~------~----_,

100

50

0.1 0.2 Strain (%)

0.3 0.4

117

Figure 5.20. Stress-strain data for the CulNi/Cu FGMs corrected for residual stresses as well as for strain hardening. The curve with open circles was determined experimentally for annealed samples; the curve with open rhombuses was determined numerically for virtual CulNi/Cu FGMs composed of six homogeneous layers; and the curve with crosses is the curve for the virtual FGMs corrected for residual stresses and plastic hardening.

4.3 Toughness

4.3.1 Fracture mechanics

In developing FGMs, research on the mechanics, particularly on the fracture mechanics of these new classes of inhomogeneous materials is needed mainly to provide technical support to materials scientists, and design and manufacturing engineers. Fracture mechanics has been used quite successfully both as a screening tool during material processing and as a design and maintenance tool for assessing service life. In a broad sense fracture is the creation of new surfaces in solids. Fracture mechanics deals with studying the effects of the applied load, the geometry of the component or the flaw, and the environmental conditions on the failure of engineering materials and structures. The macroscopic theories of fracture are based on the principles of continuum solid mechanics and classical thermodynamics. (See Appendix 5.4A for the derivation of the fundamental criteria for fracture initiation and propagation.)

118 Chapter 5

4.3.2 Stress singularities in FGMs

In applying fracture mechanics to FGMs the basic principles and techniques with respect to fracture instability, the nonlinear fracture and the subcritical crack growth remain unchanged. However, because of the inhomogeneity in thermophysical and strength related properties, difficulties arise in characterizing the material and in solving the actual crack problems. From the standpoint of the asymptotic behavior of the stress state near a crack tip in FGMs, the significant problem appears to be the investigation of the influence of the parameters related to the material's inhomogeneity on the power of stress singularity and on the angular distribution of stresses.

The stress singularity is the magnitude of the exponent in the equation that describes the asymptotic stress field at the crack tip. This exponent determines how rapidly the stress increases as the crack tip is approached. The problem has been considered for FGMs with smoothly varying elastic properties [43, 44] and for inhomogeneous materials having a kink in the distribution of elastic parameters [45, 47]. For the plane strain problem of a crack in an inhomogeneous material the leading terms in the asymptotic expansion of stresses near the crack tip are shown to be [43]:

~ hij (8)] exp[rg(8)] , (i,j = r, 8) Ot/2r

(5.28)

where f1ij and f2ij are identical to that given in Equation 5.4.A2 (Appendix 5.4A) for homogeneous materials [42], and gee) is a known bounded function representing the inhomogeneity of the material. A similar expression is found for the antiplane shear problem [45]. From Equation 5.28 and from similar results in the literature [45, 46, 47], as r~O, the inhomogeneity of the material seems to have no effect on the asymptotic behavior of the stress state. The expressions in Equation 5.4.A2 remain valid provided that the elasticity parameters E and v are continuous at and near the crack tip, but are not necessarily differentiable functions of the space coordinates.

If E and v are discontinuous, that is, if the medium consists of bonded (homogeneous or inhomogeneous) dissimilar materials, the stress state around the tip of an interface crack or a crack intersecting the interface is known to have certain anomalous behavior [48, 49]. The stresses and crack opening displacements very near the crack tip have the asymptotic form:

119

K K uy + iux == -* .Jr exp{jw logr), a», + ia xy == l2rexp{jwlogr)

f.11 "'i £r

for an interface crack, and have the form:

k1 a =- O<a<1 »' a' r

(5.29)

(5.30)

for a crack perpendicular to the interface, where K is a complex and k1

is a real constant representing the stress intensity factors, and f.11, f.12, and

(0 are known bimaterial constants2 [48, 49]. The stress and displacement oscillations given by Equation 5.29 are physically unacceptable, whereas the main shortcoming of the nonsquare root singularity in Equation 5.30 is that the standard fracture criterion based on the energy balance concept and the self-similarity of crack tip stress and displacement behavior is no longer directly applicable. In the case of FGMs, with or without a kink or slope discontinuity in the material property distribution, (0 is zero, a is 1/2, and the

asymptotic analysis shows that not only the .Jr singularity but also the angular distribution of stresses and displacements around the crack tip turn out to be identical to those found for cracks in homogeneous materials [45, 46]. The significance of this result is that in applying computational fracture mechanics to FGMs, such as the method of finite element analysis, the techniques developed for the treatment of crack tip singularities in homogeneous materials can be used without any modification.

The above assertions about the asymptotic behavior of stresses are restricted to the leading terms only. The inhomogeneity of the material would indeed influence the higher order terms as shown in Equation 5.28. In general, the size of the so-called K-controlled region around the crack tip in FGMs would be expected to be smaller than that in homogeneous materials. For example, in a recent asymptotic study of a mode I problem for a crack perpendicular to the kink line3 of the property distribution, as r~O the crack surface displacement was shown to be [50]:

(5.31 )

2 The two constants (known also as Dundurs parameters) that are certain functions of the shear moduli and Poisson's ratios of two bonded elastic solids.

3 the line of slope discontinuity in the thermomechanical parameters of FGMs.

120 Chapter 5

where A1, A2, and A3 are constants, and A1 is a measure of the mode I stress intensity factor. The constant A3 becomes zero for homogeneous materials.

The plane strain crack problem for an infinite inhomogeneous medium is first considered in which the elastic properties vary in the x' direction only, the crack is located along the x axis, and 8 is the angle between the x and x' axes (0 < 8 < nI2). It is assumed that Poisson's ratio is constant (v = 0.3), and the inhomogeneity of the material is represented by a dimensionless parameter al3 through E(x' ) = Eo exp(,Bx'), where a is the half-crack

length. The solution of the mixed mode problem plus extensive results can be found in reference [43].

Some typical results for constant strain loading,

C~(X',+oo) = Co' perpendicular to x' are given in Table 5.1 The term

8=0 corresponds to a mode I problem for which k2 is zero and k1 is the

maximum. However, for 8 = nl2 the loading is parallel to the crack and all stress intensity factors are zero. The stress intensity factors at the crack tip x = a, on the stiffer side of the medium, are always greater than at x = - a.

Table 5.2 shows some limited results comparing the plane strain [43] and the penny-shaped crack [44] solutions for FGMs under uniform tension, ao perpendicular to the plane of the crack, where E (x,y) = Eo exp (13 y) and E (r, z) = Eo exp (13 z), v = 0,3 and 13 = 0 corresponds to a homogeneous medium. In both cases the stress intensity factors in FGMs are greater than that in homogeneous materials, and the influence of the inhomogeneity of the material on k) and k2 is more pronounced for a plane strain crack than for a penny-shaped crack.

Table 5.3 shows some results for the stress intensity factor in an FGM plane under remote bending through fixed grips, namely,

c ~ (Xl ,+00) = c1 XI . Figure 5.21 shows the normalized stress intensity

factors for the basic surface crack problem in an FGM plate. The geometry of the medium is shown in Figure 5.21 a. It is assumed that Poisson's ratio, v = 0.3, E(x) = E) exp(l3x), and I3h = log (E21E), where E)= E(O) and E2 = E(h). Figure 5.21 shows the results for various values of the material inhomogeneity parameter E21E), and for three primary loading conditions: loading by a fixed grip, C.w (x ,+00) = co; membrane or pin loading4 N

along the x = hl2 axis; and bending, M. The stresses used to normalize the results are defined by:

4 the in-plane (as opposed to bending or transverse shear) component of the extemalloads applied to the boundaries of plates, shells, or layered materials.

121

Table 5.1. The Effect of a~ and 8 on the Stress Intensity Factors; v = 0.3; Loading: Uniform

Strain, co, Away from the Crack Region, ko = Eo £0 -Ja. a~ 8ht kl{a}/ko kl{-a}1ko kzia}/ko kz{-a}1ko

0 1.196 0.825 0 0 0.1 1.081 0.750 -0.321 -0.254 0.2 0.781 0.548 -0.514 -0.422

0.25 0.3 0.414 0.290 -0.504 -0.437 0.4 0.121 0.075 -0.304 -0.282 0.5 0 0 0 0

0 1.424 0.674 0 0 0.1 1.285 0.617 -0.344 -0.213 0.2 0.925 0.460 -0.548 -0.365

0.5 0.3 0.490 0.247 -0.532 -0.397

0.4 0.146 0.059 -0.314 -0.269 0.5 0 0 0 0

0 6.317 0.115 0 0 0.1 5.376 0.117 -0.867 -0.037 0.2 3.315 0.115 -1.l55 -0.090

2.5 0.3 1.441 0.082 -0.900 -0.158 0.4 0.369 0.004 -0.429 -0.179 0.5 0 0 0 0

(5.32)

The stress intensity factors, k1' such as those given by Figure 5.21 are needed for the subcritical crack growth characterization of FGMs. The influence of Poisson's ratio, v, on the stress intensity factors is relatively insignificant [43, 44]. Therefore in problems associated with cracks, assuming a constant v would not be a serious limitation. (For further studies on the fracture mechanics ofFGMs see [51, 52, 53, 54, 55].)

Table 5.2. Stress Intensity Factors for a Plane Strain or a Penny-Shaped Crack in an FGM Under Uniform Tension, 0 o, Perpendicular to the Plane of the Crack; v = 0.3.

a~ 0 0.1 0.25 0.5 1.0 2.5 5.0 Plane Strain Crack

k1/oo.J(i 1.008 0.036 1.l01 1.258 1.808 2.869

kzloo..{a 0 0.026 0.065 0.129 0.263 0.697 1.567

Penn~-ShapedCrack

k1/( 2a o.J(i In) 1.002 1.012 1.038 1.118 1.442 2.083

k2/(2aO.J(i In) 0 0.017 0.041 0.083 0.168 0.440 0.960

122

y

o b h

(a)

5...---...---...------.

4

E IE =0.1 2 J 0.2

1. 5.

10.

O~--L---L--~

o 0.2 0.4 0.6

blh (c)

x

Chapter 5

5 r--~--""'T'"-~n

1

E21 EJ = 10. 5. 1.

0.2 0.1

O'----'----'---~ o 0.2

blh

(b)

0.4 0.6

4...--"T"'""-....,.----r---.~

3

1

o L..-_"--_-'--_....L-_ ....

o 0.2 0.4 0.6 0.8

blh

(d)

Figure 5.21. The nonnalized mode I stress intensity factor in an FGM plate with a surface crack: (a) the part-crack geometry, (b) loading by a constant strain eo or fixed grips, (c) membrane loading N, (d) bending moment M.

123

Table 5.3. Normalized Stress Intensity Factors in an FGM Plate Under Bending Away From

the Crack Region; v=O.3, kb = C 1 Eo ..ra . a@ Sin kJ(a)~ kJ(-I)/kb k2(a)~

0.5

o 0.809 -0.304 0 0.1 0.683 -0.268 0.2 0.397 -0.178 0.3 0.4 0.5

0.139 0.018 o

-0.076 -0.013 o

-0.214 -0.278 -0.184 -0.053 o

o 0.087 0.128 0.105 0.039 o

4.3.3 Crack tip blunting toughening model for ceramic/metal composites and FGMs

This section discusses the effect of a ductile metallic phase on the fracture toughness of ceramic/metal composites and FGMs. Figure 5.22 illustrates the ductile phase toughening mechanism in ceramics. The presence of a ductile metallic phase plastically blunts the tip of an initial sharp crack as the load is increased. As a result, the crack opens up, thereby increasing the fracture toughness. The toughening ratio of a composite material consisting of a matrix and toughening phases, can be formulated applying the rule of mixtures to the HRR5 field analysis at a crack tip [56, 57, 58]. The yield stress (j y and the effective fracture strain C f can then be expressed as follows:

(5.33)

(5.34)

where V_ and V_ are the volume fractions of the matrix phase, (a) and toughening phase, {I

Using HRR field theory, the toughening ratio of the composite to the matrix can be derived as follows:

1 Kc [ ( )]<n-l)/2n [ ( )]<n+l)/2n E [ ]

(n+l)/2n

/\,=-= I+V,B L-I x I+V,B A-I _c

Km Em (5.35)

5 Hutchinson-Rice-Rosengreen analysis of small deformation plasticity around a crack tip.

124 Chapter 5

where Kc and Km are the stress intensity factors, Ee and Em are the Young's modulus of the composite and matrix, a; and a; are the yield stress, cj and cj are the fracture strain of the matrix and toughening phases, and n is the reciprocal of the strain hardening exponent N. The matrix and composite materials are assumed to have equal Poisson's ratios. The toughening ratio, A, can eventually be expressed as a function of the volume fraction of the toughening phase.

Ceramic phase

Blunted crack tip Cleavage fracture

Figure 5.22. Schematic illustration of the ductile-metallic-phase toughening mechanism.

Figure 5.23 shows the toughening ratio, A, of the composites titanium carbide-nickel (TiC-Ni) and chromium carbide nickel (Cr3C2-Ni) as a function of the Ni content, prepared by self-propagating high temperature synthesis plus hot isostatic pressing (SHSIHIP) [59]. At a Ni fraction of 50 weight %, the fracture toughness is increased by a factor of 4 or 5. The small difference between the Cr3C2-Ni and TiC-Ni composites could reflect the inherent differences in the mechanical properties of Cr3C2 and TiC ceramics.

8 r-----------------------------,

"'" 6 o

-..:::;

~ 0>4 C '2 Q) ..c 0> ::J 2 ~

o o 20

TiC/Ni NonFGMs - Calculated o Experimental

CI'3C2INi NonFGMs - Calculated o Ex erimental

40

Ni content wr'/o 60

125

Figure 5.23. A comparison of the calculated and the experimentally determined A for TiC-Ni and Cr3C2-Ni composites.

Figure 5.24 shows the profiles of local fracture toughness and hardness in the symmetrically graded FGM, Cr3C2/Ni/Cr3C2, [60]. The numbers in the center of the figure indicate the weight fraction of Ni in the green body for a laminate configuration with a stepped gradient composition. The solid and open circles show whether cracking occurred during the hardness test (solid circle = cracked). The hardness curve shows the presence of a smooth gradient rather than a stepped one. X-ray microanalysis indicates the occurrence ofNi migration during the SHSfHIP process.

The fracture toughness estimated using the relation between K1C and the Ni fractions in homogeneous composites of Cr3CrNi is shown as a solid line in Figure 5.24. This agrees with the fracture toughness, 12.7 MPa.m 112,

measured in the thickness direction at the center of the Cr3C2/Ni/Cr3C2 FGM by a four-point bend test for chevron-notched specimens [61]. However, the estimated toughness is larger than the actual one. This may be due to the residual stress effect.

126

16

C13 Cr3C2fNi FGMs Cl. CJ

14

> :c 12

10

• Experimental - Estimated from Final Comp.Ratio.

20

10 -----

01....-----'-----'------''------1 o 2.65

Specimen width 5.3

mm

Chapter 5

30

~

l 20 -c:

Q) .... c: 0 ()

10 Z

Figure 5.24. Profiles of the hardness and fracture toughness of a CrJCzINi FGM.

4.4 Creep and Fatigue

The evaluation of the creep and fatigue properties of FGMs is necessary to ensure the durability and integrity of structural components. A great deal of effort has been devoted to studying crack initiation, the rate of crack propagation, and overall life prediction for conventional homogeneous materials. However, relatively little has been published on these subjects for FGMs.

127

Because of the difference in the thermal expansion coefficient of materials composing an FGM, the residual stress field that is generated during cooling from the processing temperature plays an important role in crack propagation behavior. In addition, the variation of composition within the plastic zone around the crack tip, and the long range elastic fields, can change the mechanical response compared with homogeneous materials. Alloying can cause a change in the lattice constants, which produces a bending moment in a beam with a monotonic gradient. In a beam with a symmetrical gradient, it produces a central zone under tensile stress and two outer layers under compressive stress. These effects are magnified in an FGM that is exposed to temperature changes, because the coefficient of thermal expansion depends on the localized composition.

The elastic modulus is also a function of the composition and its related elastic residual stresses. Recently, it has been reported that creep in an FGM layer can reduce the residual stresses during both fabrication and actual thermal loading [62, 63]. It is important to be aware that the reduction in thermal residual stresses during thermal loading can transform a stress state into its opposite state, for example from compression to tension. In an FGM this can result in cracking [58].

The stress amplitude required to maintain a constant rate of fatigue crack propagation has been demonstrated for sintered binary Cu-Ni alloys with graded compositions [64, 65]. Fatigue crack propagation tests were carried out on bending test specimens with both homogeneous and graded compositions using frequencies of about 100 kHz. The cracks were propagated parallel and/or perpendicular to the gradient.

For homogeneous test specimens, the rate of fatigue crack propagation follows the nth power of the stress intensity factor, which is called Paris' law. It is not clear whether the linear regime of the Paris' plot is valid over the entire concentration range. In the test mode used, the crack propagation rate was constant. When the crack propagation direction was parallel to the gradient, the range of the stress intensity factor, ~K, increased with increasing nickel content, maintaining a constant crack propagation rate (da/dN), as shown in Figure 5.25.

In a specimen with a symmetric gradient and pure copper in the center, the crack path was determined by the pattern of the concentration. There is no clear explanation for the wavy shape of the crack propagation, shown in Figure 5.26. As can be seen in Figure 5.27, in a CU/Cu50Ni sample with a symmetric gradient, the crack deviates into the copper rich side. This deviation is thought to be due to the specific material properties characteristic of this composition, and also to the long range elastic field, the residual stress distribution, and the compositional changes within the plastic zone around the crack tip. The distribution of the residual stress in graded

128 Chapter 5

copper-nickel alloys is a consequence of the difference between the thermal expansion coefficients of copper and nickel. This leads to tension in the copper rich part and compression in the nickel rich part, which can increase as well as decrease the rate of crack propagation. The creep and fatigue properties of an FGM would be improved if the distribution of the internal stress could be tailored by controlling its compositional gradients.

Fraction of Ni [wt .%]

25 50 75 100 6 ~----~----~----~----~----~

5

<I 4

3

o 2 Crad< length a [mm]

Figure 5.25. Crack propagation in a graded CulNi FGM (daldN = a constant). The crack propagation direction is parallel to the gradient.

4.5 Nondestructive Evaluation

For FGMs to gain acceptability in industrial applications, quality control using nondestructive evaluation (NDE) is required for each processing and assembly step. Although numerous NDE methods have been developed for testing a variety of engineering materials, their application to FGMs is still limited.

Plasma sprayed coatings of a Zr02INiCrAIY FGM on stainless steel substrates, which were damaged due to the thermal shock from irradiation with a CO2 laser, were inspected using a water-immersion ultrasonic method [66]. Because of the roughness of the plasma sprayed coating's surface, detecting surface cracks from the echo of ultrasonic waves injected at the surface proved to be difficult. However, the transmission echo from an ultrasonic pulse injected from the back surface and focused through the

129

coating layer at both the surface and the coating interface, contained a great deal of information about the thermal shock damage. A scanning acoustic microscope used to characterize the same coatings could detect the difference between the velocity of the surface waves from damaged and undamaged regions at the polished surface of metal-ceramic composites with various mixing ratios [67].

Q) u >. () ..... E oS z -c III -c

3

2

2 Crad< length a [mm]

8

7

6

5

4

3

2

Figure 5.26. Crack propagation in a graded Ni/CulNi FGM (da/dN = a constant). The crack propagation direction is perpendicular to the gradient.

~Cu.- 50Ni' . , "7

gradient

o

500 j,.IlTI • Cu .

Figure 5.27. Micrograph of a crack in a graded CulCu50Ni FGM perpendicular to the gradient.

130 Chapter 5

Invisible microcracks can be detected with an optical microscope. As previously noted, a line-focus beam acoustic microscope has been used successfully to determine the elastic properties and their distribution in PSZ/SS FGMs [68]. Acoustic emission, which can analyze the onset of a crack, its propagation, and fracture modes can be used to determine the microscopic fracture mechanism for metal - ceramic composites [69]. For the successful application of FGMs, their design, processing, and nondestructive evaluation must be closely linked [70].

5. COMPLEX BEHAVIOR

FGMs are expected to be used for applications such as engine components, the airframes of reusable rockets, and space transportation vehicles, which encounter extremely high cyclic heat loads in severe oxidation and corrosive environments. Under actual heat loading conditions, both microscopic and macroscopic stresses are generated by large drops in temperature during the transition from startup to shutdown, and also by differences between the thermal expansion of the ceramic-rich and the metal-rich layers. Therefore, it is important to analyze the internal thermal stresses in FGMs and to evaluate their resistance at high temperatures to thermal shock, thermal fatigue, oxidation, and corrosion.

5.1 Thermal Stress

Thermal stress is generated because of constraints in the deformation of components that are subjected to certain thermomechanical boundary conditions such as inhomogeneous temperature distribution in a homogeneous component; uniform heating of an inhomogeneous component, for example in a metal/ceramic joint [71, 72]; and the rapid heating and cooling of a component's surface [73, 74]. In FGMs, some or all of the following conditions are accompanied by thermal stress: cooling from the processing temperature, a steady state temperature gradient field, and transient temperature distribution due to thermal shock and cyclic thermal loading during operation [75, 76]. Therefore, as the first step in developing an FGM, one must know the thermal stress present in each of these conditions. The relation between the estimated thermal stress and the observed fracture mode can be used in defining the design criteria.

Two problems that must be addressed when designing FGMs for fabrication by powder metallurgy are: minimizing the thermal stress generated during cooling from the sintering temperature in order to prevent bending or cracking [77, 78], and minimizing the thermal stress generated by

131

a temperature gradient [79]. The first problem can be analyzed using experimentally determined thermal expansion coefficients and elastic constants. The second problem involves taking into consideration the properties of materials that are temperature dependent in designing the optimum composition for an FGM exposed to a temperature gradient. For thermal barrier applications, no self-consistent design scheme for minimizing the thermal stress in a temperature gradient has been developed that can be used to optimize the composition for both process and service conditions. However it has been found that the optimum composition for process conditions is also relatively effective for service conditions.

Estimating the optimum composition for an object with a given shape and size under a particular thermal loading condition, requires knowing the dependence on the composition of properties such as thermal expansion and thermal conductivity, and the elastic and plastic constants. As a first approximation, a linear law can be assumed for the dependence of these properties on a material's composition [80]. However, because a material's properties depend strongly on its microstructure, the relationship between properties and microstructure has been also investigated over a wide range of compositions [81, 82, 83].

5.1.1 A thermal stress analysis case study

For the thermal stress analysis of a partially stabilized zirconia (Zr02) Istainless steel (PSZ/SS) FGM two types of models are used: a cylinder and a disk sintered at 1450°C and cooled to room temperature. For the stress calculation, the stepwise variation of composition in the interlayer is assumed. Since the model is symmetrical, only half of the cross section is considered. The finite element mesh, which is divided into 900 rectangular elements, is shown in Figure 5.28. The elastic constants and thermal expansion coefficients of PSZ and SS are given in Table 5.4. The properties of the interlayer material are estimated assuming their linear dependencies on composition.

Table 5 4 Thennal and Mechanical Properties of the Model Materials ..

Material E (MPa) G(MPa) u .aLK"ll PSZ 1.75 x 105 0.673 X 105 0.3 10.0 x 10<;

ShN4 3.0 1.18 0.27 3.6 W 4.0 1.539 0.3 6.0 SS 1.96 0.784 0.26 14.0

As shown in Figure 5.29, the composition of the interlayer is represented by the following power law equation:

132 Chapter 5

C = (xld)P (5.36)

where C is the volume fraction of the partially stabilized zirconia (PSZ) phase, d is the thickness of the interlayer, x is the distance from the stainless steel phase, and p is a numerical constant related to the phase distribution or compositional gradient. The plastic deformation and the temperature dependence of the physical constants are neglected. However, since the plastic deformation can reduce the stress concentration, this calculation may overestimate the thermal stress.

"0

I

.-

FGM sample )--

eu sample holder

1 __

Part A

~~§§§ -NODAL POINTS : 1010 -ELEMENTS : 900

08

(mm)

030

Figure 5.28. Mesh division for the finite element analysis of cylindrical and disk-shaped FGMs. Number of cylinder elements, 1125 and nodal points, 1216. Number of disk elements, 900 and nodal points, 10 10

d

C=~x/d}P

1.0

N en a.. '0 c 0 :u ca 0.5 .... u. Q)

E :J 0 > U

0.0 0.0 0.5 1.0

88304 x/d P8Z

Figure 5.29. Compositional gradients in the interlayer as a function of the normalized distance. x is the distance from the metal end and d is the thickness of the interlayer.

5.1.2 Thermal stress during fabrication

133

As shown in Figure 5.30, the maximum tensile stress is near the surface of the ceramic side of a PSZ/SS FGM. The effect of the composition of the interlayer on the axial component of the thermal stress is shown in Figure 5.31. As the constant p decreases, the position of the maximum stress shifts toward the metal side, thereby reducing the stress in the ceramic side. From the variation of the stress distribution curve, there seems to be an optimum composition for the reduction of the maximum stress.

Figure 5.32 shows the effect of the thickness of the PSZ layer on the maximum tensile stress. As the thickness decreases from 5 mm to 1 mm, the maximum axial stress is reduced sharply by about 75 %. However, as Figure 5.33 indicates there is no evident effect of the sample's diameter on the maximum tensile stress.

134

Center line

PSZ

-2 0 2

SS304

(a)

Center line

2

4

6 8

10 -2

0

-10 2 40%

-8 -6

-4

(Unit: 1 11 OOMPa) 10mm ~

{b)p=1

Chapter 5

Figure 5.30. Contour maps of axial thermal stress: (a) directly bonded PSZ/SS and (b) compositionally graded PSZ/SS.

When joining a large area with a thin interlayer, the circumferential stress should be taken into account. As shown in Figure 5.34, there is an inverse relation between the circumferential stress on the outer periphery of the PSZ and the axial stress.

Figure 5.35 shows the distribution of axial, circumferential, and shear stresses in a disk-shaped sample of PSZ/SS cooled from 1450°C. The axial stress is highest at the surface on the side of the ceramic rich region, while the circumferential stress is highest at the center of the top surface. The maximum values for the axial and circumferential stresses are about the same, and the distribution of the radial stress and the circumferential stress is similar. Because the shear stress is small compared with the axial and circumferential stresses it can be ignored in this case. The maximum values for the axial, radial, and circumferential stresses, which are all dependent on

1000 (11 a.. :E ..... rn ! 0 10 'ii ~ -1000

88

~ ~ 4 ~ 0 2 468

Distance I mm

.! ';; is ... ~ .~ Q.

S o

CJ

Figure 5.31. The distribution of the axial stress for different compositional profiles (p): A = 0.23, B = 0.7, C = 1.0, D = 3.0.

C/) C/)

~ -C/)

(ij .~

E :J E .~

E ~ ~ 0.5 Q) c:

8

(Linear composition-control)

o~----~--~~--~~--~~--~ o 2 3 4 5

Thickness of the P8Z layer (mm)

135

Figure 5.32. The dependence of the maximum axial stress on the thickness of the interlayer.

136

In In

~ ti "iii .~

E ::J

2.0 ..... ---...,..----"1""""----,

E 1.0 .~

E ~ ~ Qi a:

PSZ ~~~.--l4

(Linear composition-control)

o~------~--------~------~ o 10 20 30

Diameter of the PSZ layer (mm)

Chapter 5

Figure 5.33. The dependence of the maximum axial stress on the diameter of the graded PSZ/SS.

Diameter of the PSZ (mm)

10 20 30

¢

t 300 I" -I Cij" i3J 0...

~ (t=5mm) In In

~ 200 ti ~ E ~ Q)

E ::J

100 ~ 6 (¢=8mm)

o~--~--~----~--~--~--~ o 2 3 4 5

Thickness of the PSZ layer (mm)

Figure 5.34. The dependence of the circumferential stress at the top periphery ofPSZ on the diameter and the thickness of the PSZ.

( a )

0

( b )6

2 -2 -6 -4 ·2 ·2

·2 6 4 2 0

( c )

0

0

6

4

2

0

(Unit 1/100 MPa ) 4 --~

0

~ 0

0 ....... :..--

(Unit 1/100 MPa)

o o o

r-------~------------~~~2

o o o (Unit 1/100 MPa)

Figure 5.35. Stress distribution in a disk shaped PSZ/SS FGM. (a) axial stress, (b) circumferential stress, (c) shear stress

137

the thickness of the sample, are shown in Figure 5.36. The axial stress predominates at the surface on the side of a disk-shaped sample, when its thickness is 2 mm or larger.

Cracks often form in samples that do not have optimized compositional gradients. The optimum compositional gradient for reducing axial stress in the PSZ/SS specimens that are about 8 mm in diameter and 4 mm thick with

138 Chapter 5

sintered joints, is shown in Figure 5.37. In the sample with the optimum composition in which the axial stress is at a minimum, the exponent pis 0.7. No cracks are formed in this specimen and the bending strength is about 250 MPa. But in the samples that do not have optimized compositions, for example, if p equals 1, cracks occur in the PSZ near the interlayer; and if p equals 0.3, cracks occur in the 70 volume % PSZ phase.

1500 • ~ axial stress

t:Jc ___ radial

- -0 - - circumferential

--fr-- shear ca a.. I 10 6

D+~D Ul 1000 Ul

~ Cij (ij E ---------0 Qi :5 E ::J E ·x 500 III

::E

o~----~----~----~----~----------~ 0.0 1.0 2.0 3.0 4.0 5.0

Thickness of SS and PSZ layers (mm)

Figure 5.36. The relation between the maximum thermal stresses and the thickness of the PSZ and stainless steel layers.

5.1.3 The thermal stress state under actual thermal loading

In thermal shock tests, numerous cracks are produced in the center of an FGM part where the combustion flame impinges. These vertical cracks are initiated at the surface as shown in Figure 5.38. Some are deflected parallel to the surface, but there is no delamination. Figure 5.39, a fracture mode map for PSZ/SS FGMs, indicates the presence of a damaged region and a crackfree region [78]. The critical surface temperature, which is defined as the

139

temperature at which the first crack is formed (about l027°C for a PSZ/SS FGM), is almost constant, and is independent of the sample's size and phase

UJ UJ Q) 10lnterlayers '- PSZ - "0, UJ 1 . !!! • C=(x/d)P )( SS (CJ 1.0 E :::l E )( 0.5 (CJ

E Q)

.~ -(CJ 0.0 1) 0.0 0.5 1.0 a: x/d

0 0.1 0.5 1.0 5.0 10

Exponentp

Figure 5.37. The maximum axial stress in an FGM nonnalized with respect to the stress in direct bonding, as a function of the exponent p (x is the distance from the PSZ phase). The minimum in each curve indicates the optimum phase distribution for effective thennal stress relaxation in the compositionally graded model.

Smm

Figure 5.38. Typical damage on the surface of the PSZ after a burner heating test.

140 Chapter 5

distribution. Crack generation can be monitored by acoustic emission (AE). During heating there are no AE signals. However, many are detected after the onset of cooling, indicating the formation of cracks.

Surface crack formation and its extended behavior are closely associated with the thermal stress fields created in FGMs during heating and cooling. Figure 5.40 shows the stress distribution in the radial and circumferential directions, on the surface of a disk-shaped sample during steady state heating at the maximum thermal output. Both the radial and the circumferential stresses in the central region are compressive (about 1.3 GPa), and the stress decreases towards the edge inversely with the radial distance.

2000

L::,. PSZlSS FGM

0 PSZ spray

1600 0 coating (non-FGM)

p Vertical crack

(\) L::,. (J

~ C1l 't: :J 1200 II) 0 0. .8 (\)

£ 0 '0 (\) 800 ....

Crack free :J iii .... (\) 0. E (\) I- 400

o~~~~~~--~~~~~~~

o 200 400 600 800 1000 1200

Temperature of the bottom surface ( ·C )

Figure 5.39. A fracture mode map for PSZ/stainless steel samples that includes regions with vertical cracks and regions without cracks.

The distribution of radial and axial stresses along the center axis are shown in Figure 5.41. The compressive stress, which is at a maximum on the top surface, decreases towards the bottom surface along the center axis. The

141

radial stress changes to a tensile stress within the stainless steel layer, and the axial stress, which is associated with delamination, is relatively small. Therefore, during heating the stresses at the center of the top surface are in biaxial compression that decreases inversely with the radial distance from the periphery.

0 ~ a.. C

~ -0.5 b "0

ta -1.0 ... b 1/1 1/1 (l) "-iii -1.5 iii E "-(l)

F -2.0 0 5 10 15

Radius r (mm)

Figure 5.40. The distribution of the thermal stresses a r and as at the heated surface of a diskshaped PSZ/SS FGM with linear composition control.

2.0 ""' m ...... a.. SS / PSZ FGM Stainless steel Ni C

N 1.0 0 "0 c: m ... 0 0

l:!

~ (l)

~ -1.0 (Jr

8 ..., iii

L1S __ .rJ' E "-(l)

.s:: -2.0 f-

0 2 3 4 5 Depthd (mm)

Figure 5.41. The distribution of the thermal stresses, a r and a z along the center axis of a diskshaped PSZ/SS FGM from the top surface to the bottom surface.

142 Chapter 5

Figure 5.42 shows typical distributions for the transient thennal stress in the radial direction at the top surface of an FGM during heating and subsequent cooling, taking into account non-linear deformation of the components and temperature dependent material properties. As can be seen, heating generates a large compressive stress at 0.1 s, but after 300 s the stress is relaxed at the center. This is because the surface temperature reaches the brittle-to-ductile transition temperature thereby allowing the nonlinear defonnation of PSZ to occur. During cooling, however, the resulting inelastic strain rapidly converts the stress into tensile stress. From the residual radial stress after cooling, it has been determined that the high tensile stress is limited to the layer close to the surface. This is because the temperature decreases abruptly toward the inside, and non-linear defonnation is limited to a shallow depth from the surface.

center periphery center hery or-------------~~ ~.~------------~~.

_2500'----.L..-..... --'--~--' o 5 10 15

.2500 L.-~_.L..-..... _-'----'_-' o 5 10 15

Radial distance (mm) Radial distance (mm)

(a) heating (b) cooling

Figure 5.42. The transient thermal stress in the radial direction at the top surface of an FGM during heating and subsequent cooling.

Based on this thermal stress analysis, a mechanism for the fonnation of vertical cracks has been proposed. During heating as shown in Figure 5.43, the top surface of the FGM is in a high biaxial compressive stress state. The stresses induce non-linear deformation when the top surface is heated above the transition temperature. During the cooling cycle, the stress converts into a tensile stress that is high enough to exceed the fracture strength of PSZ, thus causing vertical cracking. Figure 5.44 shows the dependence on the phase distribution of the maximum compressive stress at the top surface. compositional gradient.

143

Vertical crack initiation , " ,

~,' Fracture stress

Compressive stress . strain

Figure 5.43. A conceptual model for the formation of vertical cracks during a burner heating test on a ceramic/metal FGM.

Iii' 2.0 I

a.. ~ (/) 1.5 r0- O 0 -(/)

~ iii 0 Q)

0 > 1.0 r- -'(jj (/)

~ 0 c. E 0.5 0 r- -U

.!I<: ctl Q) I a.. 0.0

0.1 10

Gradient exponent, p

Figure 5.44. The peak compressive stress at the top surface of an FGM as a function of its compositional gradient.

5.2 Thermal Shock

Figure 5.45 shows tests for evaluating the development of FGMs as thermal barrier coatings [84]. The first step includes several fundamental tests for determining basic thermomechanical properties of the homogeneous components. Various heating methods such as laser, plasma-arc, burner, and

144 Chapter 5

electron beam are used for testing thermal shock and thermal fatigue. For example, the thermal shock resistance of a plasma sprayed FGM coating of PSZlNiCrAIY on stainless steel (SS) was evaluated by using a CO2 laser to irradiate the coating's surface, as shown in Figure 5.46 [84]. NiCrAIY or NiCoCrAlY is used as a bond coat to improve the adhesion of the ceramic coating to the metal components. The thermal shock resistance is characterized by determining the critical power density, Pc (the critical laser power divided by the area of the spot), when the first vertical crack appears. The effectiveness of the graded coating is evidenced by a threefold increase in its thermal shock resistance compared with a homogeneous composite coating.

Service performance

Test of actual device

Simulated service test

Component test

Single parameter test (Special)

Single parameter test (Standard)

Tests Sample size

Large scale thermal loading test 100 ....... 300 mm

Small scale thermal loading test in a simulated environment 50 ....... 100 mm

Thermal shock / fatigue test Thermal barrier performance test Erosion / corrosion test

Fracture strength test Fracture toughness test Thermal property measurement

30mm

Figure 5.45. Evaluative tests for developing FGMs with thermal barrier properties.

Specimen

Chopper

Tranalatlon Stage

Figure 5.46. Schematic of the laser beam thermal shock test. (AE = acoustic emission)

145

Heating with a plasma arc has been used to evaluate the oxidation and thermal shock resistance of the PSZINiCrAIY FGM coating on SS [85]. To induce a temperature gradient, the top surface was exposed to an argon CAr) plasma containing a controlled amount of oxygen, while the bottom surface was cooled by flowing Ar. The oxidation rate of the coating, which is independent of its graded composition, follows the parabolic law. The oxidation behavior is largely controlled by the formation of an alumina film (Alz0 3) on the NiCrAlY.

8 ]~ , ./ ..-6

,./,./

./ ..-..-2 -

'l'· 1

control unit )

3 8

liz Oz

12

Cu holder

Figure 5.47. Schematic of the test system for the burner heating test, and the configuration for holding the sample. (I) test sample, (2) torch, (3) cooling chamber, (4) shutter, (5) protective plate, (6) acoustic emission (AE) sensor, (7) pyrometer, (8) thermocouples, (9) AE apparatus, (10) monitor, (11) regulatory valve, (12) cooling water supply.

A burner heating test, shown schematically in Figure 5.47, was used to evaluate the thermal shock and fatigue resistance of FGM coatings for applications such as the thrust chamber of a rocket engine [86, 87] and the leading edge in advanced gas turbine blades [88]. The test samples were prepared by powder metallurgy and plasma spray coating. The temperature at which the first crack forms was used as a measure of the coating's performance as a thermal barrier. This temperature was almost constant for

146 Chapter 5

various samples and heating conditions, indicating the dependence of the thermal barrier property on the material. The fracture mechanism map, shown in Figure 5.48, for correlating the temperatu~e with coating damage, indicates the higher spalling resistance of the FGM coating compared with a conventional Zr02 single-layer coating.

2000..------1 0 No crack IB Vertical crack • Delamination

1600

HI HI

P 1200 ....... co . , I- 800 o .. .' q, 0 0 ' .. ".' o

400

o "If o ,fY'

,

"",,/

o ~0~2~00:--:4:1:-00=-=6l::00=-=80:-::0:--1:-:000 Tb ("C)

(a) Homogeneous component

20 Ou

16 00

DO

co I- 8 O()

4 00

0

o No crack IB Vertical crack • Delamination

• • • IB QTarget 0 o 0 IB ,4

00 , {j 0 ,0

0 , , , 00 , , 0

0 o " 0' , , , , , , , , , o 200 400 600 800 1000

Tb (OC) (b) FGM

Figure 5.48. A fracture mechanism map for (a) a homogeneous component and (b) an FGM. (T. = the temperature of the top surface. Tb = the temperature of the bottom surface.)

5.3 Thermal Fatigue

Thermal fatigue has been tested for a number of FGMs: titanium diboride/copper (TiB2/Cu) [89], titanium carbide/nickel (TiCINi) [90], partially stabilized zirconia/nickel (PSZlNi) [91], and silicon carbide/carboncarbon (SiCICC) [92] prepared by self-propagating high temperature synthesis (SHS), self-propagating high temperature synthesis plus hot isostatic pressing (SHS/HIP), low pressure plasma spray (LPPS), and chemical vapor deposition and chemical vapor infiltration (CVD/CVI), respectively. The test involved estimating the variation of the effective thermal conductivity with thermal cycling.

A diagram of the test apparatus is shown in Figure 5.49 [93]. The sample is exposed to large temperature differences by heating the top surface with the beam of a 30 kW xenon (Xe) arc lamp, and cooling the bottom surface

147

with liquid nitrogen. The heating and cooling cycles are effected by opening and closing a shutter between two reflectors.

Ifthe effective thermal conductivity (Ie) of a material is defined as:

(5.37)

where q is the heat flux loaded on the sample, Ts is the average temperature of the top surface, T b is the temperature of the bottom surface, and L is the thickness of the sample.

To stack

DC power supply

Video Radiation

Vacuum vessel

pyrometer

Figure 5.49. Diagram of the test apparatus for cyclic heating using a Xe arc lamp.

Then, the normalized effective thermal conductivity, ite ' is expressed as:

(5.38)

where I (n)e is the effective thermal conductivity after n heating cycles and I (l)e is the effective thermal conductivity at initial heating under steady state conditions.

148 Chapter 5

Results of thermal fatigue tests for TiB2/Cu FGMs are shown in Figure 5.50. The temperature difference between the top surface (~ 427°C) and the bottom surface of each sample is - 300°C. The normalized effective thermal conductivity of the 8 layer sample (GT -48) is almost constant up to 30 cycles, but decreases significantly at 40 cycles. While with the 13 layer sample (GT-51) it decreases somewhat with increasing thermal cycling. The large vertical cracks generated during thermal cycling decrease the effective thermal conductivity due to the thermal resistance of cracks and pores. The larger number of cracks in the 8 layer sample than in the 13 layer one is attributed to higher thermal stress.

1.5

~ .> U

--0-- GT-48 (8 layers)

- -6- - GT-51 (13 layers)

:::J "0 c: 0 0

Iii E ... Q)

-= Q)

.~ t5 Q) :t: 0.5 Q)

"0 Q)

.!:::! Iii E 0 z

0 0 10 20 30 40 50

Number of cycles

Figure 5.50. Effect of cyclic thermal exposure on the degradation of the normalized effective thermal conductivity of the TiB2/Cu FGM samples: GT -48 and GT -51.

The results of thermal fatigue tests for TiClNi FGMs are shown in Figure 5.51. The temperature difference in each of the samples is 350°C-650°C between the top surface (727°C-1127°C) and the bottom surface. In the OM-62 sample, which was tailored so that the phase distribution parameter for a mole fraction of Ni was p=0.3, spalling occurred during the 9th heating cycle. In the OM-63 (p=1.0), a large vertical crack appeared on the surface at the start of the 20th heating cycle. This crack probably occurred during the previous cooling step. The normalized effective thermal conductivity of

149

samples OM-61 and OM-64 decreased with an increase in thermal cycling. These results lead to the conclusion that for reducing thermal stress and extending a TiClNi FGM's life, the optimum exponent of the phase distribution function should be 0.5 - 0.7.

---*- OM-62 (n=0.3) --0-- OM-61 (n=0.5) --fr-- OM-64 (n=0.7) -{}- OM-63 (n=1.0)

o ~----~------~----~------~ o 10 20 30 40

Number of cycles

Figure 5.51. The effect of cyclic thennal exposure on the degradation of the nonnalized effective thennal conductivity of the TiClNi FGM samples: OM-61, 62, 63, and 64.

Figure 5.52 shows test results for a PSZINi FGM. The temperature of the bottom surface was maintained at 700°C during thermal exposure. During initial heating the maximum temperature of the top surface was 827°C, reaching 1227°C after 45 cycles. The normalized effective thermal conductivity decreases to a minimum at 25 cycles and remains at 60 % of the initial value. There are large vertical macro cracks and small delaminations. However, macrocracks (a large decrease in effective thermal conductivity) do not occur when the temperature of the bottom surface is maintained at -127°C [93]. The vertical cracks are apparently generated in earlier cycles due to the higher temperature of the bottom surface. This can cause inelastic deformation of the metal phases, which can act to relax the thermal stresses.

150 Chapter 5

1.5 r----r---,---.,.---r--.....,

---f:r- NS-55

10 20 30 40 50

Number of cycles

Figure 5.52. The effect of cyclic thermal exposure on the degradation of the normalized effective thermal conductivity of the PSZINi FGM sample: NS-55.

The variation in the normalized effective thermal conductivity of SiCICC FGMs is shown in Figure 5.53. The maximum temperature of the top surface, 1627°C-I727°C, is during the initial cycle, and the temperature of the bottom surface is 727°C-I027°C. The normalized effective thermal conductivity of each sample decreases with an increase in the number of cycles. When a heat flux of - 0.5 MW/m2 is loaded on SiCICC FGM samples (JS-57 and JS-58) and a non-FGM sample coated with a single layer of SiC (JS-59), the normalized effective thermal conductivity of the nonFGM (JS-59) degrades the fastest, and spalling at its surface occurs at the 20th thermal cycle. In addition, the degradation of the normalized effective thermal conductivity of the FGM sample (JS-57) is half that of the non-FGM US-59). This indicates that a SiCICC graded layer can improve thermal fatigue resistance and prevent spalling. When a higher heat flux of 1.1 MW/m2 is loaded on the FGM (JS-58), the decrease of the normalized effective thermal conductivity is larger than that of the FGM(JS-57). This implies that the higher thermal stress associated with a higher heat flux would accelerate the propagation of flaws in some FGMs.

These experimental results indicate that the exposure of FGMs to thermal cycling generates flaws such as cracks and spalling, which results in a decrease in normalized effective thermal conductivity. The change in the normalized effective thermal conductivity can be used to investigate thermal fatigue.

.?:-.s: 1.0 n :::J '0 C a C) 0.8 'iii E a; -:= 0.6 Q)

> U Q)

== 0.4 Q)

'0 Q)

.!::! 'iii 0.2 E 0 z

0 0

--0- JS-57 SiC/CC FGM ~JS-58

--0-- JS-59 SiC/CC non-FGM

10

Spalling (heating phase)

20

Number of cycles

151

30

Figure 5.53. The effect of cyclic thennal exposure on the degradation of the nonnalized effective thennal conductivity of the SiC/CC FGM samples: JS-57 and JS-58 compared with the non-FGM sample, JS-59.

5.4 Oxidation and Corrosion

Graded SiC coatings adhere well to graphite components and prevent their oxidation. SiC/C FGM coatings made by chemical vapor deposition (CVD) have good oxidation resistance up to 800°C in air [94], and those made by a thermal reaction between graphite and silicon powders up to 1400°C [95]. FGMs composed of a graded layer of 30 /lm thick SiC/C in chemical vapor infiltrated (CVI) woven carbon fiber are tough and oxidation resistant [96]. When subsequently coated by CVD with a 100 /lm thick layer of SiC, they are highly oxidation resistant to cyclic heating between 350°C-1525°C.

Although FGMs can be protectively coated against oxidation and corrosion, surface cracks are sometimes produced as a result of the tensile stress created during heating and cooling. These cracks then can act as conduits for oxidation. An increase in the volume of one or more of an FGM's components due to oxidation can open the cracks further, or produce

152 Chapter 5

new cracks, ultimately causing spalling. Introducing a compressive stress in the surface layer can inhibit cracking.

A silicon carbide/titanium carbide/carbon-carbon FGM (SiC/TiC/CC) coating deposited by CVD on a CC composite induces a strong compressive residual stress (500 MPa to 1000 MPa) in the surface SiC layer because of the higher thermal expansion rate of TiC. It has good oxidation resistance at 1300°C with few surface cracks [97]. However, as the TiC layer becomes oxidized through the cracks, the increased volume of the titanium dioxide (Ti02) seals the cracks, as shown in Figure 5.54. It is expected that this selfhealing effect will be incorporated in the design ofFGMs.

Figure 5.54. Crack sealing with Ti02 formed along the thermal crack in the coated layer of SiCffiC on the C/C composite, after heating in air at IIOO'C for Ih.

A graded SiC/TiC layer has been deposited by CVD on stainless steel (SS304) as a potential material for a system to produce hydrogen by the thermochemical dissolution of water [98]. A material for this system must be resistant up to 727°C to corrosion by the highly corrosive bromine-oxygenhydrogen bromide (Br2-02-HBr) gases produced in this process. The graded coating showed superior corrosion resistance in both isothermal and cyclic tests compared with stainless steel coated with a monolayer of TiC or SiC. This indicates that the high temperature corrosion resistance of FGMs could be further improved by optimizing their graded composition and microstructure.

An understanding about the properties of FGMs as well as their characterization is fundamental both to their successful processing, discussed in Chapter 6, and to their applications, described in Chapter 7.

IS3

APPENDIX 5.3 A

The derivation of the heat conduction equation for a material when there is no heat source and sink.

Consider a small region R in a body and its boundary surface S. Then the amount of heat escaping from R per unit time is:

ffQ·ndA (S.3A.l)

where Q. n is the component of Q in the direction of the external unit normal vector n of S. From the divergence theorem, one obtains:

ff Q·ndA = -fffV(XVT)dxdydz R

The total amount of the heat Q contained in R is:

Q = f f f cpTdxdydz R

where c is the specific heat of the material and p is the density. The rate ofthe decrease ofQ is:

JQ ar --= - If J cp-dxdydz at R at

which must equal the quantity of heat escaping from R. Thus:

ar f f f cp-:;:dxdydz = f f fV(}'VT)dxdydz

R at R

Since this holds for any region R in the body, one can obtain:

(S.3A.2)

(S.3A.3)

(5.3A.4)

(S.3A.S)

(S.3A.6)

154 Chapter 5

APPENDIX 5.4 A

Derivation of the fundamental criteria for fracture initiation and propagation

The fundamental criteria for fracture initiation and propagation can be derived by using the concept of energy balance at the crack front, which for an equilibrium crack can be expressed as [42]:

d -(U - V) = G ::::; Gc dA

(5.4A.l)

where U is the work done by the external loads, V is the strain energy, and A is the crack surface area. G (the left hand side of Equation 5.4A.l) is the energy available, and Gc is the energy required to create a unit area of new fracture surface. These terms are referred to as the crack driving force and the fracture toughness, respectively.

By using the mechanism of crack closure, it can be shown that the crack driving force can be evaluated from the asymptotic stresses and displacements near the crack tip. In homogeneous materials the asymptotic stresses, in turn, are obtained from the elasticity solution as follows [42]:

(lij (r,O) = k1 k2 (liz (r, 0) = k3

$/1(0) + 5 121 (0) , $ hi (0) , 2r IJ 2r ~

(i,j = r,O) (5.4A.2)

where f1ij , f2ij , f3i are known functions,

Iwe(O) = 12re(O) = 13e(O) = 1, and the constants kl , k2 and k3 represent

the magnitude of the applied loads and severity of the part/flaw geometry and are known respectively, as the mode I, II, and III stress intensity factors. Thus, by evaluating the local energy release, it can be shown that:

(5.4A.3)

where E, v, and !-l are respectively, the Young's modulus, Poisson's ratio,

and the shear modulus, and K] = Jiik l , Ku =.J1ik2' KDJ =..fiik3. For

155

the important case of a mode I condition, for example, the fracture criterion Equation 5.4A.l) becomes:

(5.4A.4)

The criterion, as expressed by (5.4A.l) or (5.4A.4), is very useful to study the question of fracture stability. However, many of the fracture failures such as fatigue and creep crack growth and stress corrosion cracking involve a period of subcritical crack propagation. In modeling these failure processes, the stress intensity factor, kj , or the strain energy release rate, G j ,

(I = 1,2,3), are known to be very effective.

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156 Chapter 5

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24. Araki, N. et al. (1992) An analytical solution of temperature response in multilayered materials for transient methods, International Journal of Thermo physics, 13(3),515-538.

25. Ishiguro, T. et al. (1993) Transient temperature response in functionally gradient materials, International Journal of Thermo physics, 14(1), 101-121.

26. Makino, A. and Araki, N. (1993) Evaluation of the thermal diffusivity of functionally gradient materials, High Temperatures-High Pressures, 25,545-551.

27. Shinohara, Y. et al. Thermal stability of plasma sprayed Ni-Cr-AI-YIPSZ FGM in uniform and gradient temperature fields, ibid, 255-262.

28. Aihara, T., Kaji, M., and Igarashi, T. (1990) Numerical analysis of compositional distribution change in functionally gradient materials, J Japan Inst. Metals, 54, 758-763.

29. Fukui, Y. et al. (1994) Young's modulus determination offunctionally gradient AI-AI3Ni material by a resonance method, J Mater. Sci., 29, 2281-2288.

30. Wakashima, K. (1976) Macroscopic mechanical properties of composite materials Part II, Elastic moduli and thermal expansion coefficient, Trans. Jpn. Soc. Compo Mater., 2-7.

31. Mihara, T. et al. (1992) Elastic constants of zirconia/stainless steel sintered composite materials measured by a line-focus-beam acoustic microscope, JJapan Inst. Metals, 56, 321-326.

32. Atalar, A. (1979) A physical model for acoustic signature, J Appl. Phys., 50, 8237-8239. 33. Kushibiki, J., Ohkubo, A., and Chubachi, N. (1981) Linearly focused acoustic beams for

acoustic microscopy, Electron Lett., 17,520-522. 34. Kushibiki, J. and Chubachi, N. (1985) Material characterization by line-focus-beam

acoustic microscope, IEEE Trans. Sonics. Ultrason., SU-32, 189-212. 35. Obata, M., Shimada, H., and Mihara, T. (1990) Stress dependence ofleaky surface wave

on PMMA by line-focus-beam acoustic microscope, Experimental Mechanics, 30, 32-37. 36. Viktorov, I. A. (1967) Rayleigh and Lamb Waves, Plenum Press New York, 46.

37. Lannutti, J. 1. (1994) Functionally graded materials, properties, potential and design guidelines, Composites Engineering, 4, 81-94.

38. Nakasa, K., Kato, M., and Matsuyoshi, H. (1997) Fabrication ofTiB-Ti functionally graded material by spark and resistance sintering and evaluation of the mechanical properties,J Jpn. Inst. Metals, 61, 311-318.

157

39. Delfosse, D., Cherradi, N., and I1schner, B. (1995) Influence of residual stresses on the tensile behavior of a Cu-Ni FGM, in Proc. of The Third Int 'I. Symp. on Structural and Functional Gradient Materials, (eds. B. I1schner and N. Cherradi), Presses Poly techniques et Universitaires Romandes, Lausanne, 307-313.

40. Delfosse, D., Kunzi, U. H., and I1schner, B. (1992) Experimental determination of residual stresses in materials with a one-dimensional gradient of composition, Acta Metallurgica et Materialia, 40, 2219-2224.

41. Delfosse, D. (1990) Structural part with a concentration gradient produced by powder metallurgy and properties (Bauteile mit einem konzentrationsgradienten pulvermetallurgische herstellung und eigenschaften), These No. 868, Swiss Federal Institute of Technology, Lausanne.

42. Liebowitz, H. (ed.) (1968) Fracture-An Advanced Treatise, Vol. II, Mathematical Fundamentals, Pergamon Press, Oxford.

43. Konda, N. and Erdogan, F. (1990) The mixed mode crack problem in a nonhomogeneous elastic plane, Engineering Fracture Mechanics, 47,533-545.

44. Ozturk, M. and Erdogan, F. (1993) The axisymmetric crack problem in a nonhomogeneous medium. ASME J Appl. Mech., 60, 406-413.

45. Erdogan, F. (1985) The crack problem for bonded nonhomogeneous materials under antiplane shear loading, ASME J Appl. Mech., 52, 823-828.

46. Delale, F. and Erdogan, F. (1988) Interface crack in a nonhomogeneous elastic medium, Int. J Engng. Sci., 28, 559-568.

47. Erdogan, F., Kaya, A. C. and Joseph, P. F. (1991) The crack problem in bonded nonhomogeneous materials, ASME J Appl. Mech., 58, 410-418.

48. Erdogan, F. and Gupta, G. D. (1971) Layered composites with an interface flaw, Int. J Solids Structures, 7, 1089-1107.

49. Cook, T. S. and Erdogan, F. (1972) Stresses in bonded materials with a crack perpendicular to the interface, Int. J Engng. Sci., 10,667-696.

50. Martin, P. A. (1992) Tip behavior for cracks in bonded inhomogeneous materials. J Engineering Mathematics, 26, 467-480.

51. Erdogan, F. and Wu, B.H. (1996); Crack problems in FGM layers under thermal stresses, Journal of Thermal stresses, 19,237-266.

52. Erdogan, F. (1995) Fracture mechanics of functionally graded materials, J Composites Engineering, 5, 753-770.

53. Erdogan, F. and Ozturk, M. (1995) Periodic cracking of functionally graded coatings, Int. J Engng. Sci., 33, 2179-2195.

54. Ozturk, M. and Erdogan, F. (1996) Axisymmetric crack problem in bonded materials with a graded interfacial region. Int. J Solids Structures, 33, 193-217.

55. Chen, Y.F. and Erdogan, F. (1996) The interface crack problem for a nonhomogeneous coating bonded to a homogeneous substrate, J Mech. Phys. Solids, 44, 771-787.

56. Hirano, K. and Suzuki, T. (1990) Fracture mechanics of functionally gradient materials at an ultra-high temperature, in Proc. of The First Int 'I. Symp. on FGM'90, (eds. M. Yamanouchi, M. Koizumi, T. Hirai, and I. Shiota), Sendai, Japan, FGM Forum, Society for Non-traditional Technology, Tokyo, 313-319.

158 Chapter 5

57. Hirano, K. (1994) Toughening mechanism for ceramics by a ductile metallic phase, J Materials Science Letters, 13, 1219-1221.

58. Chan, K. S. (1992) Influence of microstructure on intrinsic and extrinsic toughening in an alpha-two titanium aluminide alloy, Metal, Trans., 23A, 183-199.

59. Miyamoto, Y. et at. (1992) HIP in SHS technology, Int. J Self-Propagating HighTemperature Synthesis 1,147-154.

60. Tanihata, K. et at. (1993) Fabrication ofCr3C2INi functionally gradient materials by gaspressure combustion sintering, Ceramic Transactions, 34, in Proc.ofThe Second Int 'I. Symp. on FGM'92, (eds. 1. B. Holt, M. Koizumi, T Hirai, and Z. A. Munir), American Ceramic Society, Westerville, 361-368.

61. Munz. D. G., Shanon, 1. L., Jr., and Budsey, R. T. (1980) Fracture toughness calculation from maximum load in four point bend tests of chevron notch specimens, Int. J Fract., 16, R137-RI41.

62. Williamson, L. R., Rabin, H. B., and Byerly, G. (1993) Residual stress in joint ceramicmetal structures: FGM studies of interlayer and creep effects, ibid., 3 I 5-322.

63. Kokini, K. and Takeuchi, R. Y. (1994) Initiation of surface cracks in multilayer ceramic thermal barrier coatings under thermal loads, Material Science and Engineering, A189, 301-309.

64. Blumm, M., Dollmeier, K., and Ilschner, B. (1995) Experimental investigation of fatigue crack propagation in room temperature bending tests with Cu-Ni graded alloys, in Proc. The Third Int 'I. Symp. on Structural and Functional Gradient Materials, (eds. B. Ilschner and N. Cherradi), Presses Polytechniques et Universitaires Romandes, Lausanne, 315-320.

65. Delfosse, D. and Ilschner, B. (1992) Pulvermetallurgische herstellung von gradientenwerkstoffen, Mat. Wiss und Werkstofflech., 23, 235-240.

66. Shimada, M., Yoshii, T. and Akiyama, S. (1991) Ultrasonic Testing ofThermal Shock Damage on Plasma Sprayed FGM Coating, in Proc. of 4th Domestic Symposium on FGMs, FGM Forum Japan, 31-35.

67. Nishimori, H., Hayakawa, Y., and Nonaka, T (1991) Nondestructive evaluation of functionally gradient materials using ultrasound (2), ibid., 53-60.

68. Makino, A., Araki, N., and Ishiguro, T. (1993) Transient temperature response in FGM for step-or pulsewise heating, Ceramic Transactions, 34, in Proc.ofThe Second Int 'I. Symp. on FGM'92, (eds. J.B. Holt, M. Koizumi, T Hirai, and Z.A. Munir), American Ceramic Society, Westerville, 305-312.

69. Saito, M. et at. (1990) Acoustic emission study on stress-induced transformation and micro fracture in zirconia and metal fiber/zirconia composites, in Proc. of The 10th International Acoustic Emission Symposium, Sendai, 97-104.

70. Takahashi, H. and Hashida, T. (1990) Development of an evaluation method of functionally gradient materials, Japan Soc. Mech. Eng., Intern. J, Series I, 33, 281-287.

71. Kawasaki, A. and Watanabe, R. (1988) Powder metallurgical fabrication of the thermal stress relief-type of functionally gradient materials, in Proc. of Int. Con! on Sintering '87 Tokyo, (eds. S. Somiya et al .. ), Elsevier, London, 2, 1197-1202.

72. Nicholas, G.M. and Crispin, M.R. (1982) Diffusion bonding stainless steel to alumina using alminium interface, J Mat. Sci., 17,3347-3360.

73. Faber, TK. Huang, D.M., and Evans, G.A. (1981) Quantitative studies of thermal shock in ceramics based on a novel test technique. JAm. Ceram. Soc., 64, 296-301.

74. Schneider, A.G. and Petzow, G. (1991) Thermal shock testing of ceramics- a new testing method, JAm. Ceram. Soc., 74, 98-102.

159

75. Watanabe, R. and Kawasaki, A. (1991) Recent development offunctionally gradient materials for special application to space plane, in Proc. ofSymp. on Composite Materials ofICAM91, Strasbourg, France, 197-208.

76. Niino, M., Hirai, T., and Watanabe, R. (1987) Functionally gradient material- high temperature use for space rocket, J Japan Society of Composite Materials, 13 (6) 257-264.

77. Kawasaki, A. and Watanabe, R. (1987) Finite element analysis of thermal stress of the metaVceramic multi-layer composites with controlled composition gradients, J Japan Inst. Metals, 51, 525-529.

78. Kawasaki, A. and Watanabe, R. (1993) Fabrication of disk-shaped functionally gradient materials by hot pressing and their thermomechanical performance, in Ceramic Transactions, 34, Proc. of The Second Int'l. Symp. on FGM'92, (eds. lB. Holt, M. Koizumi, T. Hirai, and Z.A. Munir), Westerville, Am. Ceram. Soc., 157-164.

79. Kawasaki, A. and Watanabe, R. (1992) Thermal shock fracture mechanism of metaVceramic functionally gradient materials, in Proc. Intern. Workshop on Thermal Shock and Thermal Fatigue Behavior of Advanced Ceramic, Schloss Ringberg, (eds. G. Petzow and G.A. Schneider), Kluwer Academic Publishers, Netherlands, (1993), 509-520.

80. Watanabe, R., Kawasaki, A., and Takahashi, H. (1991) Mechanics and mechanisms of damage in composites and multi-materials, ESISII (ed. D. Baptiste), Mechanical Engineering Publications, London, 285-289.

81. Kawasaki, A. and Watanabe, R. (1990) Microstructural designing and fabrication of disk shaped functionally gradient material by powder metallurgy, J Japan Soc. Powder and Powder Metallurgy, 37, 253-256.

82. Watanabe, R. and Kawasaki, A. (1989) The relation between microstructural transition and material properties in sintered functionally gradient material, in Proc. 3rd Domestic Symposium on FGMs, Tokyo, Sept., FGM Forum Japan, 35-48.

83. Kawasaki, A. and Watanabe, R. (1997) Concept and PIM fabrication offunctionally gradient materials, Ceramic International, 23, 73-83.

84. Takahashi, H. and Hashida, T. (1990) Development of an evaluation method for functionally gradient materials. Japan Soc. Mech. Eng., Intern. J., Series J, 33, 281-287.

85. Takahashi, H. et al. (1994) Oxidation/thermal shock fracture evaluation procedure by plasma-arc heating for functionally gradient materials, in Proc. 3rd IUMRS Intern. Conference on Advanced Materials, Tokyo, Japan, Aug. 31-Sept. 4, 16B, 1291-1294.

86. Cherradi, N., Dollmeier, K., and Ilschner, B. (1993) PSZ-chrome nickel graded materials. Powder Technology-Thermal Properties, 229-236.

87. Kawasaki, A. and Watanabe, R. (1992) Thermal shock fracture mechanism of metal/ceramic functionally gradient materials, in Proc. Intern. Workshop on Thermal Shock and Thermal Fatigue Behavior of Advanced Ceramic, Schloss Ringberg, (eds. G. Petzow and G.A. Schneider), Kluwer Academic Publishers, Netherlands, (1993), 509-520.

88. Hashida, T. et at.. (1994) Burner heating method for determining thermal shock resistance of ceramic coatings for gas turbine rotor blades, in Proc. of 3rd IUMRS Intern. Conference on Advanced Materials, Tokyo, Japan, Aug. 31-Sept. 4, 16B, 1291-1294.

89. Yanagisawa, N., Sata, N., and Sanada, N. (1990) Fabrication of TiBz-Cu functionally gradient material by SHS process, ibid., 179-184.

90. Miyamoto, Y. et al. (1990) Gas-pressure combustion sintering of TiC-Ni FGM, ibid., 257-262.

91. Shimoda, N. et al. (1990) Production of functionally gradient materials by applying low pressure plasma spray, ibid., 151-156.

92. Uemura, S., Sohda, Y., and Kude, Y. (1990) SiC/C functionally gradient material prepared by chemical vapor deposition, ibid., 237-242.

160 Chapter 5

93. Kumakawa, A. et al. (1990) Experimental study on thermo-mechanical properties of FGMs at high heat fluxes, ibid., 291-295.

94. Fujii, K. et al. (1992) Functionally graded material of silicon carbide and carbon as advanced oxidation-resistance graphite, J Nucl. Mater., 187,204-208.

95. Yamamoto, O. et al. (1993) Preparation of carbon materials with SiC-concentration gradient by silicon impreg durations and its oxidation behavior, J Eur. Ceram. Soc., 12, 435-440.

96. Kude, Y. (1993) Carbon/carbon composites using high performance carbon fibers, in Proc. of 4th Symp. on High-Performance Materialsfor Severe Environments, 1-12.

97. Kawai, C., et al. (1990) Oxidation resistant coating with TiC-SiC gradient composition on carbon fiber reinforced composites by CVD, in Proc. of The First Int 'I. Symp. on FGM'90, (eds. M. Yamanouchi, M. Koizumi, T. Hirai, and I. Shiota), FGM Forum, Tokyo, Society for Non-traditional Technology, 77-82.

98. Sasaki, M., Hiratani, T., and Hirai, T. (1993) Corrosion resistance ofan SiClTiC FGMcoated stainless steel in a Br2-02-Ar atmosphere, in Ceramic Transactions, 34, Proc. Second Int'l. Symp. on FGM'92, (eds. J.B. Holt, M. Koizumi, T. Hirai, and Z.A. Munir) American Ceramic Society, Westerville, 369-376.

Chapter 6

PROCESSING AND FABRICATION

Keywords: FGM processing, constructive based processing, transport based processing, powder metallurgy, powder stacking, sedimentation, fiber stacking, slip casting, consolidation, sintering, hot pressing, infiltration, solid state sintering, differential sintering, percolation, pulsed electric current sintering, spark plasma sintering, plasma activated sintering, liquid phase sintering, hot isostatic pressing, microwave sintering, reaction sintering, self-propagating high temperature synthesis, combustion synthesis, underwater-shock explosion, melt infiltration, laser beam cladding, repair, spray deposition, spray forming, thermal spraying, electron beam physical vapor deposition, magnetron sputtering, chemical vapor deposition, reaction layers, transient liquid phase sintering, electrophoresis, electrodeposition, preform, solid state diffusion, thermal field, electrical field, PMA(THF) /PVC, SiC/C, PZT, settling, centrifugal casting, verneuil process, microgravity, solid state joining, transient liquid phase joining, solid freeform fabrication, CAD, laminated object manufacturing, stereolithography, selective laser sintering, 3-dimensional printing, fused deposition modeling, zirconia toughened alumina, drug delivery device, extrusion freeform fabrication, diffusion bonding, sol-gel infiltration, superplastic forming.

1. INTRODUCTION

Since the mid-1980s the processing of FGM materials and structures has become of increasing academic interest. This is reflected in the considerable number of papers that have been published on specific processing routes. During the first Japanese FGM program (1987 to 1991) processing methods were developed for FGM parts to be used as high temperature components of a hypersonic space plane [1, 2, 3]. These early methods included powder metallurgy, physical and chemical vapor deposition, plasma spraying, self-

162 Chapter 6

propagating high temperature synthesis (SHS), and galvanoforming (see Figure 6.1). Since 1991, many variations of the initially used methods as wel1 as a considerable number of new processing routes have been developed. Today, the spectrum of processing options ranges from methods already established before FGMs became a well-defined subject, such as processing similar to the case-hardening of steel, to more recently developed methods, such as solid freeform fabrication.

CAD System:

Inverse design model

Selection of composition &

microstructure

Optimization of gradation

Fuzzy function

( Stress Analysis by FEM )

Process Developments:

CVD, PVD, PM, Plasma Spray

SHS, Galvano Forming,

CVD/CVI, PM/CVD, SHS/HIP,

PS/GF

[]

Fractal & Percolation Theories:

Quantitative analysis of gradation

Micromechanical Modeling:

Correlation of graded

microstructures & properties

FGM Samples:

Disk: SiC/C, PSZlSUS, PSZlNi,

AIN/SiC,TiC/Ni, Cr3C2/Ni,

TiB2/Ni

Nose cone: SiC/CC

Rod: PSZJNi, TiB2/Cu

Figure 6.1. Major results of the 1987-1991 FGM research program on the "Fundamental Study on Relaxation of Thermal Stress for High Temperature Materials by the Tailoring of Graded Structures" [2].

Published comprehensive reviews on the processing of FGMs approach the systematic ordering of processing methods in different ways [4, 5, 6]. In one of these reviews the processing methods are classified into those based on constructive processing and those based on mass transport [4]. In constructive processing the FGM is constructed layer-by-Iayer starting with

163

an appropriate distribution of the FGM's constituents, often in a precursor of the component. These techniques are called constructive processes because gradients are literally constructed in space. Constructive processes are distinguished from a second class of FGM processes that depend on natural transport phenomena such as the flow of a fluid, the diffusion of atomic species, or the conduction of heat to create gradients within a component [4].

In this book the fabrication of FGMs is categorized into bulk, layer, preform, and melt processing, as shown in Figure 6.2. The major distinction made is the object into which the gradient is introduced. Processing to achieve this objective can include one or more constructive or mass transport mechanisms, or a combination of these, as well as solid, liquid, or gaseous aggregation states. Bulk processing utilizes those methods that initially create a bulk material that has graded porosity, composition, or phase configuration. This is accomplished largely by forming stacks of powder, fibers, or even sheets by means of normal gravity, centrifugal forces, or pressure induced flow. The stacks are then consolidated either by pore elimination which results in their shrinkage, or infiltration, essentially without any concomitant shrinkage.

c::>

c::>

Stacking - powder - fibers - sheets

Molecular deposition - vapor - electro

Diffusion - solid -liquid - vapor

Settling

Stack consolidation - pore elimination - infiltration

Mechanical deposition - lamination - spraying

Graded fields - thermal - electrical - others

Solidification

Figure 6.2. Processing methods for creating FGMs and their classification in this book.

164 Chapter 6

Layer processing can be achieved by mechanical deposition, which includes lamination and thermal spraying. It also can be achieved by the deposition of molecules or atoms as with physical and chemical vapor deposition, or with mechanisms based on electrotransport or chemical reaction. Most of the layer deposition methods also can be used to produce bulk FGMs if adequate time is allocated, or if other specific processing parameters are applied. In addition, layer processing involves interlayers that, for example, can be formed by the transient liquid phase bonding of bulk or sheet components. Preforms can be porous or dense, homogeneous or inhomogeneous, and mayor may not contain gradients introduced intentionally.

Preform processing is applied to initiate or to modify existing gradients in a preform. The conventional processing methods are solid state, liquid phase, or vapor phase diffusion. Graded fields can be used to introduce the gradients into the FGM.

Melt processing comprises elements both of constructive processing and of processing associated with mass transfer. Gradual phase separation under normal or enhanced gravity can be treated similarly to sedimentation in particulate processing. Specific solidification methods, such as the Verneuil (or flame-fusion)6 technique for growing graded single crystals, can be treated similarly to thermal spraying, and the formation of gradients during the solidification of melt pools can be regarded as a mechanism dominated by mass transport.

In Section 7, several advanced manufacturing techniques that include constructive and transport based processing steps are discussed in detail.

2. BULK PROCESSING

Bulk materials with graded properties, such as porosity, composition, mechanical response, or chemical reactivity, can be fabricated initially by bulk processes that include the stacking of powder, fibers, or even sheets under normal gravity or other enhancing forces. In addition to dry stacking, sedimentation in aqueous or organic fluids and spray and slurry deposition can be used. Slurry deposition can be treated as powder stacking if an arrangement of slurry layers is formed, or as stacking of individual sheets

6 In the Verneuil technique for growing crystals, also known as flame fusion, the powdered materials are passed through the high temperature flame of an oxyhydrogen torch, where they are melted, followed by their deposition on the tip of a rod, thereby promoting crystallization.

165

that are initially fabricated by a slurry method (or other processing route). Powder stacking by pouring, slurry, or spray deposition can be used to produce thin graded surface layers on graded or ungraded substrates. The spraying of suspensions is described in this section; thermal spraying is described in the section on layer processing; and slurry techniques are described in both of these sections.

Graded stacks are consolidated into components or partially finished material by methods that produce the desired functional properties. Starting with a particulate stack, processes such as sintering, hot pressing, or hot isostatic pressing are used to modify the contact areas between the individual particles or to modify the pores in bulk materials with residual porosity. These processes can reduce the porosity, which results in macroscopic shrinkage of the stacks. The infiltration of porous stacks decreases their porosity with little attendant shrinkage.

2.1 Powder Stacking

The bulk processing of FGMs by powder stacking, shown in Figure 6.3, involves the following sequential steps with a selected combination of metals and ceramics: determination of the optimum distribution of the composition for effective functioning of the FGM; stepwise or continuous stacking of premixed powder according to a predesigned spatial distribution of the composition (the composition profile); and consolidation of the powder stack [7].

FGM components with a compositional gradient can be readily produced from powder that consists of at least two powder species with different chemical compositions. All of the chemical compositions in-between the chemical compositions of the different unblended powders are considered to be equivalent to mixtures of the powders with defined volume ratios. To obtain compositional gradients, two different powders are distributed in the FGM component in volume ratios with equivalent gradients. The control of the minimum size of the spatial distribution of the composition depends on the particle size of the starting powders and on the method of powder stacking. Layer-by-Iayer stacking enables controlling its size to 0.2 mm [8]. While the spray deposition method enables controlling it to a minimum size of 0.01 mm [9].

To obtain FGM components with density gradients, a single powder can be stacked so that the packing density is gradually changed. Alternatively, mixtures of two or more powders that fill the space differently can be packed together.

166

o o o o

Sinter

Select powders

V Add sintering aids for ceramic

Mix intermediate compositions

___ ------L----r----~-

o o o o o o

Hot press

0 ..

<+ 0 .. o

Chapter 6

HIP

Figure 6.3. Flow chart of the processing steps involved in the fabrication of FGMs by particulate technology [7].

2.1.1 Powder stacking under normal gravity

Materials with unidirectional compositional gradients have been produced using a wide variety of experimental powder layering techniques. Scaling up any of these processes for commercialization depends on their potential for automation. One of the simplest methods, with respect to ease of reproducibility and minimizing the necessary process control, involves multilayer powder configurations with a discrete composition in each layer and stepwise changes in composition from one layer to the next. With advanced techniques such as thin sheet lamination, shown in Figure 6.4, layers can be formed from powder slurries 100 to 1500 Ilm thick [10]. A preform is obtained by removing the organic binders from the layers [11].

In one automated technique for powder stacking under normal gravity, a suspension is sprayed onto a heated substrate. During the spraying process the composition of the suspension is continuously changed, and variation in

167

the composition and the spray deposition site are continuously controlled by a simple computer program. Another computer-controlled method, shown in Figure 6.5, enables the formation of sheets with a wide variety of compositional gradients. A powder sieve is positioned above a moving conveyor belt that transports the substrate. Vibrating the sieve causes a continuous shower of powder to fall onto the passing conveyor belt. The thickness of the forming powder stack ranges from zero before the conveyer belt reaches the vibrating sieve to a maximum when it leaves the area under the sieve. The compositional gradient of the powder composition in the plane of the sieve results in gradients in the composition of the sheet perpendicular to the substrate [12]. This method has been investigated for fabricating metal strips with a composition at the upper and lower surfaces of Cu-15Ni-8Sn (in weight %) and a compositional gradient into the interior of the strip, which consists of pure Cu metal.

Ceramics powder

B'nder 1 Metal

'~:~~~:~~~ Mixing Defoaming Doctor blade

Figure 6.4. Stack formation by a multiple layer technique.

synchronized powder distributor

~A~Y .)\~A' vibrating ", • .,.v.,.._ Wl©lll. lID graded powder strip YL ~~ ----------powder rolling

+ C ~ sintering

x

Figure 6.5. Schematic of the automated powder stacking apparatus developed for strip production [II].

168 Chapter 6

In sedimentation processes, a powder or fiber stack is formed after removal of the liquid. The conventional sedimentation method is based simply on the force of gravity causing particles with different sizes, shapes, and densities to settle at different speeds. It is possible to control the sedimentation rates of different phases almost independently of their relative particle size or density. Therefore, if the sensitivity to the addition of agglomerating additives in the solvent is different for particles of different phases, it should be possible to control agglomeration by the specific addition of those additives. When the sedimentation process was tested with a nickel aluminidelalumina (NiAll Ah03) powder mixture in a hydrophilic methanol solution, there was a marked enhancement in the settling speed of the NiAI due to the presence of aqueous capillary bridges [13].

A variation of this process reduces drying stresses in FGMs [14]. To prepare a NiAll Ah03 FGM as a sequence of discrete layers with different compositions, premixed powders for each layer of the composition were mixed with dry hexane, then ultrasonicated, diluted by water, ultrasonicated again, and finally allowed to settle in a predetermined sequence using a perforated plate device mounted on the die of a hot press, shown in Figure 6.6. The die of the hot press was used to avoid any problems with handling the delicate, layered stack before consolidation by hot pressing. After all the layers had settled, solvent removal was slowly initiated to allow relaxation and complete drying. The powder stacks were then hot pressed [15].

2.1.2 Powder stacking under centrifugal forces

Powder stacking under centrifugal forces enables the formation of structures with continuous or stepwise gradients. It was initially developed for fabricating experimental samples [16] that were used for processing highly porous sintered parts with pore size gradients [17]. Either type of gradient can be obtained by stacking mixtures of powders of different sizes or shapes or of both. Figure 6.7 shows the setup for centrifugal powder forming with a unit for proportionally dispensing two different microsized powders, the filler, and the solid components of the binder. An additional reservoir contains the liquid component of the binder system, which is sprayed into a rotating centrifugal cylinder to fix the vertical arrangement of the stack by reacting with the solid components of the binder.

Stainless steel rings with graded porosity have been made from stainless steel powder by stacking controlled ratios of two different fractions: a coarse (40-63 f..l.m) spherical powder and a finer (15 f..l.m) irregular powder. The powder stacks were debinded and then consolidated by sintering in hydrogen [17]. The resulting rings had a gradient in their pore size due to the

169

difference in the average size of the powder fractions, and a gradient in their total porosity due to the difference in the shape of the powders.

Components

Proportioning steps

Mixing

Supply

Centrifugal forming

;-

[ 0 Teflon c olumn~

!tl ,»:

,»:

r-

000000[

')1 ,);

Stainle D~steeipi

ss ate

"Ill

~

~

~: Mo foil )(

~ ~ I

"iJ~ BN coating

Graphite foil

Figure 6.6. Settling of powders in the die of a hot press [13].

~ CJ:) 83 EB 0 = =

CD CD CD CD

binder liQuid

Centrifugal

Vibration supported mixing and powder transport

Cl Metal powder .. Solid binder

Figure 6.7. Centrifugal powder fonning equipment [15].

170 Chapter 6

Centrifugal powder stacking also can be used to produce graded tungsten-copper (W-Cu) components [18]. A mixture of Wand Cu powder with a continuously changing mixing ratio is deposited using a semiautomated computer controlled feeding system. Because of their relatively low density, the centrifugally stacked powder cylinders are consolidated by hot isostatic pressing (HIP). Graded W -Cu components also can be made by Cu-infiltration ofpresintered W powder stacks with graded porosity.

Centrifugal casting at high rotation speeds can greatly increase the sedimentation force. After centrifuging cylindrical containers filled with suspensions of powders of different sizes, shapes, and compositions for several hours, the supernatant is poured off, and the stacked cylindrical green bodies are dried and sintered. The settling behavior of the particulates is controlled by adjusting the pH and by the addition of various stabilizers and deflocculants. To make it easier to handle the green bodies (unsintered parts), binders such as polyvinyl acetate (PVA) are added, which also may prevent cracking during drying. Models have been developed for the stacking characteristics of complex suspensions [19].

Flat membranes of commercial alumina (Ah03) powders with a graded pore-size distribution from 40 to 250 nm across a membrane thickness of about 5 mm can be fabricated by centrifugal casting [20]. A narrower poresize distribution with pores no larger than 200 nm can be obtained using flocculated suspensions, and a graded pore structure is clearly seen in samples prepared from stabilized suspensions with a solids content of 10 volume %.

2.1.3 Powder stacking under pressure induced flow

Slip casting, a technique used for shaping ceramics, is a typical method for powder stacking under pressure induced flow. A cavity in a plaster of Paris mold is filled with a slurry containing particles of a ceramic material. The slurry liquid is absorbed by the mold through capillary forces and the ceramic particles, which are generally larger than the capillary channels, are left behind on the walls of the mold cavity. When a desired wall thickness has been reached, the excess slip is drained from the mold and the green body is removed for drying. Repeated slip casting of slurries with different compositions can be used to produce laminated FGMs.

A schematic of graded casting in which a slip flows continuously through the mold during the casting step is shown in Figure 6.8 [21]. Other configurations also have been proposed [22]. The common element is the presence of several reservoirs containing suspensions of the materials to be cast. The composition of the slip fed into the mold is controlled to correspond to the desired composition of the stacks being deposited. The slip

171

is continuously circulated through the mold, and suspensions from the reservoirs are added by flow pumps at computer-controlled flow rates. The mixing of the suspensions is enhanced by the use of in-line stationary, spiralshaped plastic mixers to create turbulence during the flow through the feed tubes.

Reservoirs Stirrer

A~ ~~~~LL~

Cast layer

Figure 6.8. Schematic of the process for producing graded materials by slip casting [21].

If appropriate models and data are used for predicting the casting kinetics for the slip being used, the process can be controlled to produce a specified compositional gradient [23]. Equations have been developed to predict the composition of the slip in the recirculating reservoir at any point in time. This can be done by knowing the initial slip composition in the reservoirs and how the various parameters such as the flow rate from the reservoirs, the volume absorption rate of the mold, and the total free volume of the mold, change as a function of time. These equations can be incorporated into the process control model.

When designing a mold to produce a part by graded casting, the path that the slip will follow when traveling through the mold must be taken into consideration to avoid any zones where the slip is stagnant. To prevent the formation of lateral compositional gradients, the slip can be injected at various points along the length of the mold. The slip casting approach can be used with any material that can be suspended in a suitable liquid. Slips typically consist of a material in powder form dispersed in a carrier such as water or an organic liquid. Slip casting has been used conventionally to manufacture ceramic parts. However, it has been modified recently to a slip

172 Chapter 6

casting-sedimentation process for fabricating graded metal-ceramic composites [24].

Examples of linear and parabolic changes of compositions produced in alumina zirconia (Ah03-Zr02) tubes are shown in Figure 6.9. When these tubes are cooled from the sintering temperature, the alumina-rich surface area is subjected to compressive stress because of the higher thermal expansion coefficient of the zirconia-rich inner area. Using the same design strategy, this process also can be used to produce other ceramic-ceramic composites such as mullite-zirconia (mullite is an alumina-silica mineral) [25].

0.9

'" 0 -<'< <{ c: 0

0.8 U co ~ Q) 0.7 E :::I (5 >

0.6

0.5 0 2 3

Distance from outer surface (mm)

Figure 6.9. Examples of linear and parabolic compositional changes produced in alumina zirconia composite tubes.

This approach can also be used to produce parts in which the degree of porosity is graded. For example, bubbles can be introduced by adding foaming agents to one of the reservoirs. Similarly to other stacking methods, microstructural gradients can be obtained by the time-controlled addition of particles or dopants as a second phase. This has been demonstrated with the addition of zirconia or mullite to alumina. Both retard grain growth in alumina [26, 27].

In the formation of graded ceramic gears by centrifugal ceramic casting, a ceramic slurry is poured into a rotating plaster of Paris mold with the negative shape of the gear [28]. By changing the composition of the slurry, a

173

gear-shaped green (presintered) ceramic body with compositionally graded layers can be formed. Tough zirconia gears have been fabricated that incorporate both compositional and morphological gradients from alumina platelets in the surface layer to fine alumina particles toward the interior.

2.2 Fiber and Sheet Stacking

The most important properties of carbon/carbon (CIC) composites after their mechanical properties, are their thermal conduction and thermal expansion. The potential relationships between tailored, graded CIC composite fiber configurations and their resulting thermophysical properties have been investigated [29]. Carbon fibers have highly anisotropic thermophysical properties. The room temperature thermal conductivity of carbon fibers made by chemical vapor deposition (CVD) can be as high as 2000 W/m·K parallel to the basal plane of the hexagonal graphitic structure (which is also parallel to the fiber axis), but only 10 W/m·K perpendicular to the basal plane. The fiber strength is high and the thermal expansion along the fiber axis is close to zero or even negative, whereas it is positive perpendicular to the fiber axis. Quasi-isotropic fiber configurations can be constructed with zero expansion between 20°C and 500°C by means of this anisotropic thermal expansion effect.

Unidirectional CIC composites made from pitch-based fibers in pitchderived carbon matrices maintain an anisotropic thermal conductivity of 700 W/m·K parallel to the fiber bundles. It has been shown that the considerable contribution of anisotropic thermal conductivity in CIC composites is due to the anisotropic behavior of the matrix. The matrix crystals tend to grow epitaxially to the carbon fibers, thereby enhancing the crystallographic anisotropy within and between fiber bundles. In a three-dimensional CIC composite fabricated with a total fiber fraction of 55 volume %, 75% of the fibers are oriented in one direction. This results in up to a fivefold difference in thermal conductivity between this direction and the other two directions.

FGMs can be produced by laminating or stacking thin sheets with different compositions, as shown in Figure 6.10 [30]. Mixtures with different compositions of the starting powders, Zr02 and Ni, are processed into aqueous slurries containing binder, deflocculant, and plasticizer additives. Air and excess water are removed in an evaporation step before film casting. Next the individual sheets are stacked by pressing them together, followed by drying the stack with slow heating, and then sintering it. The green sheets are flexible and can be easily formed into various geometries.

174 Chapter 6

<' »

~~W~~I~]~~ ? Laminating Oebinding Sintering

Figure 6.10. The production of compositionally layered bulk materials by sheet lamination.

2.3 Stack Consolidation by Pore Elimination

2.3.1 Solid state sintering

Besides the usual advantages and disadvantages of consolidating powder stacks by solid state sintering, there are several challenges to be met for the successful fabrication of FGMs. In general, with stacked powders that have different mixing ratios according to a predesigned composition, and with the desired sample size and shape, the compacting and sintering behavior normally varies from one layer to the other. If no special measures are taken, different sintering behavior will cause various localized sintering faults such as warping, necking, splitting, and crack formation [31]. The sintering behavior of the different mixing ratios or compositions is characterized by three parameters of the shrinkage curve: the onset temperature of shrinkage, the slope of the shrinkage curve as a function of the temperature, and the integral net shrinkage. In order to control shrinkage, these three parameters must be taken into consideration [32].

Figure 6.11 a shows an example of the different sintering behavior of various mixing ratios of stainless steel (SS304) and partially stabilized zirconia powder. The purpose of controlling sintering is to obtain similar shrinkage in all areas of the FGM part independent of the green density and composition of the compact. Among the methods used to control sintering are varying the particle sizes and introducing temperature gradients. Figure 6.11 b shows an example of sintering control by varying the powder particle size for the same material systems shown in Figure 6.11a [32]. A similar approach has been used for powder blends of Mo-AIN (molybdenumaluminum nitride) [33], for partially stabilized Zr02-Ni FGMs [34, 35] and

175

for ZrOrNi composites [36]. In the last study, shrinkage was equalized by using bimodal powders for the Zr02 phase.

PSZlSUS=0.18 ~m/0.09 ~m 25r-r---r---r---r-~~

20

Q)

g> 10 ~ c ''::: .t: (j) 5

6. 20 vol% PSZ .40 vol% PSZ .50 vol% PSZ o 60vol% PSZ

@

20

Q) g>10 ~

.§

.t: (j) 5

6. 20 vol% PSZ .40vol% PSZ o 60vol% PSZ .80 vol% PSZ

Temperature CC) (b)

Figure 6.11. a and b: Adjustment of sintering shrinkage by controlling the powder particle size in a mixture of stainless steel 202 and zirconia powder [31].

The addition of small amounts of sintering aids raises the densification rate during sintering of the ceramic phase in MgOlNi (magnesium oxide/nickel) and TiClNbAI (titanium carbide/nickel aluminide) FGMs. The addition of Fe203 (iron oxide) to MgO reduces its sintering temperature close to that of nickel [37], and the sintering rate of titanium carbide (TiC) approaches that of NbAI with the addition of chromium carbide (Cr3C2) [38]. Another way to equalize local sintering rates throughout an FGM is to densify the powder compact within an appropriate temperature gradient. This has been successfully accomplished using laser beam surface heating [39], microwave sintering [40], and pulsed current heating with a stepped or tapered die [41, 42, 43, 44].

In general, predicting the densification kinetics of an FGM structure requires an in-depth understanding of the densification behavior of all the compositions present. Various theories have been proposed to address the question of control by volume and size ratios. The distribution of the phases is determined by the volume fraction, the size of the phase, and the wetting behavior, for example the interface energy values of the boundaries between

176 Chapter 6

phase 1/phase 2, phase 1/phase 1, phase 2/phase 2, and between the solid phases and the gas phase.

When the interface energies between all the solid phases are approximately equal, low volume fractions of phase 2 in phase 1 lead to densification of the composite, which is controlled by the percolating matrix (phase 1) as long as the size of the microstructures of both phases is similar. At higher volume fractions, when significant percolation of phase 2 occurs, densification can change drastically. Above a certain volume fraction, if the presence of phase 2 dominates, the shrinkage of the composite is accelerated if phase 2 sinters faster than phase 1. However, if phase 2 sinters much more slowly than phase I, the percolating phase 2 strongly hinders the densification of the composite. This relation can be seen in practice when the size of the particles and the diffusion coefficients of phase 2 result in a zero sintering rate for this phase. A typical example is a metal-ceramic composite with a ceramic phase that does not sinter at the normal sintering temperature of the metal.

The exact volume fraction beyond which percolation of the inclusions essentially prevents the overall shrinkage of the composite powder compact, largely depends on the ratio between the size of the matrix powder and the size of the ceramic inclusions [32]. If the size of the matrix powder is fine, and the particle size of the ceramic inclusions is coarse, a high volume fraction of the inclusions is required for direct percolation between the inclusions. If the particle size of the inclusions is small compared with the particle size of the matrix powder, a volume fraction of the inclusions as low as 10% is sufficient to prevent effective sintering of the matrix.

Figure 6.12 shows the volume fraction of equal sized spherical inclusions of phase 2 above which there is sufficient percolation of this phase to prevent sintering of the matrix. These values for as-packed green powder compacts have been combined from both experimental data and calculated values for various FGMs [45]. These theoretical calculations assume that the lower limit that prevents densification of the composite is a percolation in which each inclusion has an average of three contacts with the other inclusions.

When there is good wetting of the inclusions by the matrix phase, the interface energy between phase I/phase 2 is much lower than the interface energies between phase 2/vapor phase and phase 2/phase 2. In this case, during sintering the matrix (phase 1) spreads in between all the phase 2 inclusions, which results in densification during sintering by rearrangement [49]. If the size of the inclusions is extremely small, even the presence of a small volume can suppress the sintering of the percolating phase. Fine inclusions at the grain boundaries of contacting matrix grains decrease the ability of the grain boundaries to act as vacancy sources or sinks.

0.6 r--,---..,--,---,

c:: o U w 0.5 co UJ ~CO -..c: wOo

~~ 0.4 OUJ >-c::CO 00 .~:: 0.3 _0 o c::

0.21....--..I...---...I...--....l.----1 o 1 2 3

Matrix particle radius/ inclusion radius

(a)

4

177

1 .0 r---,......---,----"T----,

c:: o :;::: w 0.8 OUJ ~co -..c: ~~0.6 ::J:= 15g ~Cii 0.4 00 '(i):: .20 o 0.2 c::

• computer simulation • experimental data •

O.OI....--.l....-_..L.-_..1.-_...J 0.01 0.1 10

Matrix particle radius! inclusion radius

(b)

100

Figure 6.12. (a) Volume fraction of an equisized spherical inclusion phase above which there is sufficient percolation of this phase to prevent the sintering of the matrix in an as packed green powder compact ([6.1 [9] after data from [45]). (b) Experimental data for the volume fraction of the second phase at the onset of percolation, measured in densified FGM composites produced by powder metallurgy, superimposed on the theoretical curve for (a) ([6.1 [9] after data from [45, 46, 47]).

Pulsed electric current sintering (also called spark plasma sintering or plasma activated sintering), shown schematically in Figure 6.13, is one of the more advanced sintering methods. For example, to form graded structures, cemented carbide powders with different cobalt contents are stacked into multiple layers on steel in a hot pressing die. This is followed by the application of pressure via two pistons plus extremely short simultaneous electric pulses via two electrodes on the upper side of the sample. Thick, dense cemented carbide layers are formed that are bonded to the steel without cracking or flaking off [50,51]. The FGM's residual stress properties can be controlled by supplementing the electric pulses with a temperature gradient produced by a graphite die with a specifically designed external shape. The supplementary temperature gradient for homogeneously sintering an FGM of Zr02ITiAI (zirconia/titanium aluminide) has been experimentally determined [50]. By appropriate modification of the temperature gradient, this FGM can be processed to contain residual stresses that are sufficiently compressive to suppress fracture in its zr02 rich regions [51].

178

pu

... OJ .0 E III ~ o C)

. 5 "8

P Upper +

nChele~e if///'

I"-

Upper f:7777J. v punch

" powt "- }Slntenng die o

~ ~

o<:S

E ::J ::J

~ >

pu

~ Lower punch

v /

Lower 31t--nch electrode

Chapter 6

Sintering press J

,.....'-... roo--0 iii ... ... ~ OJ (5 c: ... OJ 'E C) 0 OJ () (/)

""5 T Co

() 0 I I '-,..... Positioning

Operating environment (Vacuum& argon gas)

I Water cooling I I Thermometer I

Figure 6.13. The system configuration for pulsed electric current sintering [42].

2.3.2 Liquid phase sintering

During liquid phase sintering, a powder compact is heated to a temperature at which it is partially melted and a stationary or transient liquid phase is present. A molten volume fraction is selected that is low enough to retain the geometry of the compact. In compacts of fine powder « 1 0 /lm) the solid particles are rearranged by capillary forces so that the local density of their packing is higher. In addition, at optimal wetting conditions capillary forces exert a negative pressure on the liquid phase. If solution reprecipitation occurs it can cause the adjacent solid particles to conform to the compact's shape.

Generally, shape conformation causes shrinkage by increasing the volume fraction of the solid phase per unit volume [52, 53, 54]. Shape conformation at low volume fractions of the liquid phase refers to the development of polyhedral shaped grains with rounded corners or edges from spherical or irregularly shaped particles. It is associated with the dissolution of the solid phase in the melt, and its reprecipitation at sites with lower chemical potential. Since grain growth during liquid phase sintering is

179

based on the same principles, shape conformation and grain coarsening are closely related. In general, the composition of the reprecipitated solid phase is close to the equilibrium composition. Thus both grain growth and shape conformation lead to rapid chemical homogenization [55]. Furthermore, the relaxation of local stresses inhibits the buildup of long range stresses, which frequently hinders densification during solid state sintering. Liquid phase sintering is an established processing method for densifying and homogenizing materials such as the cemented carbide, tungsten carbidecobalt (WC-Co), a hard metal; tungsten-nickel-iron (W-Ni-Fe), a heavy metal; and SIALON ceramics (Si6-zAlzOzNs-z), at moderate temperatures and small volume fractions of the melt phase.

It is not always easy to retain the functional gradients of graded powder stacks during liquid phase sintering. In the porous areas, the melt tends to flow from the areas with larger pores toward those with smaller ones. These are usually areas with smaller particles or where the density of the compact is higher [56, 57]. At the same porosity and pore size, the melt is distributed evenly throughout the compact. This implies that the initial gradients of the melt forming phase become equalized during sintering. Nevertheless, the fabrication of graded cemented carbides, particularly WC-Co, by liquid phase sintering is an established process [58].

2.3.3 Transient liquid phase sintering

By using transient viscous flow sintering mechanisms in ceramic FGM systems, it is possible to realize the benefits of liquid phase sintering by avoiding the disadvantages of residual glassy phases. The transient liquid phase is added intentionally into the system as a chemically deposited thin coating on each particle of the different starting powders. The presence of the same amorphous coating surrounding the particles of various components in a discontinuously graded FGM leads to its simultaneous local densification at same rates and temperature similar to a non-graded material.

The presence of a viscous phase (i.e. Si02) on the powders prevents solid state sintering before densification due to the considerable reduction in densification temperature. Because of the absence of solid state sintering, the formation of extended rigid regions characterized by solid state bridges between the particles can be avoided. Further heating following the densification in the presence of a viscous phase causes extensive phase reactions between the core particles and the surrounding viscous matrix. By initially selecting stoichiometric compositions, all the components will fully crystallize and sinter to dense materials at lower temperatures « 1500°C) [59].

180 Chapter 6

For example, a mullite/SbN4 (silicon nitride) discontinuously graded composite was processed with a-Ah03 and a - S bN4 powders that were initially coated with a Si02 (silica) layer by a sol-gel process [60]. In this coating process, particles of a-Ah03 and a-SbN4 are dispersed in ethanol containing a dispersant and surfactants (triethanolamine and polyethylene glycol) and mixed with tetraethylorthosilane (TEOS) solution. The TEOS, which is hydrolyzed by addition of a basic aqueous solution (pH> 10), condenses on each particle as a very thin layer « 50 J,tm). After the composite is densified, the mullite and the X-phase (Si12Alls039Ns) are crystallized by a solution and reprecipitation process. The resulting FGM consists of graded layers of mullite and the X-phase, but not of Ah03 and Si02, or of Ah03, SbN4, and Si02 as in the starting composition.

2.3.4 Hot pressing and hot isostatic pressing

The undesirable effect of nonuniformity in sintering can be mitigated to a large extent by using hot pressing or hot isostatic pressing (HIP). External pressures up to 40 MPa during hot pressing and up to 200 MPa in hot isostatic pressing produce stress states in particle contact areas that are almost equivalent to the stresses created by sintering particles that are 50 or 10 nm in size. When the particles are larger, these stresses cause local plastic flow in the neck areas and, most importantly, lead to the activation of dislocation-based creep as the major densification mechanism [61, 62]. Densification by this power law creep is essentially independent of the particle size as long as the dislocation density is not influenced strongly by the size of the grains or the particles. One of the major benefits of both hot pressing and hot isostatic pressing is the partial destruction of the long range stresses, which can be a precondition for the achievement of the crucial complete densification of FGMs that have local variations in their green density.

Intermetallic nickel aluminide (NiAI) compounds containing a graded addition of Cr have been prepared by uniaxially hot pressing a large number of layers [63]. After blending in a planetary mill, NiAI powder « 44 J,tm in diameter) containing 2.5, 5, 7.5, 10, and 20 weight % Cr powder « 44 J,tm in diameter) was stacked in 1.5 mm thick layers, then uniaxially hot pressed. The final thickness of the consolidated pellets was 4 mm with a few percent residual porosity. Subsequent homogenization was partially achieved by heat treatment. In this as well as in other work, the pressure limitations « 60 MPa) in the preparation of hot pressed FGMs often result in residual porosity [63]. During cooling from the hot pressing temperature, fine crack patterns can develop in disc shaped samples that have gradients in their coefficients of thermal expansion. Densification during hot pressing often

181

can be enhanced by small amounts of sintering additives [55]. For example, adequate densification of the titanium carbide (TiC) rich areas of II-layer TiClNbAI FGM discs that are 30 mm in diameter and 6 mm thick can be achieved by the addition of 3 volume % chromium carbide (Cr3C2). After hot pressing the resulting discs are dense and flaw-free [64].

Hot isostatic pressing (HIP) is of additional value if near-net shaping can be obtained at the same time. It has been used to fabricate a turbine blade with a microstructural gradient in the transition region from the base to the foil by stacking titanium aluminide (TiAI) powder, containing varying amounts of alloying elements such as Cr, in a HIP can for a turbine blade. In the base of the can, the powder stack was Cr-free; in the transition region from the base to the airfoil the Cr content of the powder stack was increased sharply; and in the airfoil itself the Cr content was uniform. The gradient in the Cr content produced a predesigned microstructural and mechanical gradient in the turbine blade [65].

The first step in the fabrication of an FGM turbine blade by HIP is to form a near net shape steel can by die pressing followed by welding low carbon steel sheets. The HIP can is filled initially with TiAI powder that contains an increasing Cr content as described above, then degassed, sealed, and HIPed. To avoid reaction between the TiAI powder and the steel can, the HIP temperature is kept below 1 I 50°C. After consolidation, the steel can is removed by chemical pickling in sulfuric acid, and the near-net-shaped FGM TiAI blade is machined to final dimensions.

Hot isostatic pressing also has been used to produce tungsten (W) preforms with graded porosity for further infiltration with another metal such as molten copper (Cu) [66]. First, a layer of W powder is unidirectionally cold compacted at high pressure. This is followed by pouring a second layer of W powder on the compacted first layer, and then compacting the two layer stack at a lower pressure than the first layer alone. This procedure is repeated with decreasing pressure to obtain ultimately a compacted six-layer stack of W powder with graded density. During subsequent sintering the variations in density are partially retained.

Similar density gradients in sintered stacks of W powder have been obtained by using stacks of different W powder sizes. When the W layer with the highest density is sintered to a residual porosity below 8 volume % all the pores are isolated. This makes it possible to completely eliminate the pores by containerless HIP (also known as sinter-HIP). The open porosity that dominates the other layers is unchanged during the containerless HIP procedure. Using this method, Cu-infiltrated W components have been produced for the first wall of a fusion reactor in which a pore-free and Cufree W surface faces toward the plasma [66].

182 Chapter 6

2.3.5 Microwave sintering

Microwave heating with 2.45 GHz is a pressureless heating method that provides a "self regulating" heating pattern (thermal gradient) to a heated bulk material. The thermal gradient is governed by the material's compositional gradient [67]. Microwave radiation is dissipated as heat within the volume of a material according to several different mechanisms [68]. The basic electric and dielectric properties of the material govern both the penetration depth of the radiation and the conversion efficiency of the low frequency microwaves.

The heating behavior of a particular material is determined both by the amount of power dissipated and by the heat loss of the sample due to radiation or convection. In general, the energy dissipated by dielectric losses increases markedly with increasing temperature. When the surface of a material loses heat by convective or radiant heat transfer to the environment, self-enhancing temperature gradients can develop due to a reduction in the dissipation of microwave energy at the cooler surface areas. Selfenhancement is particularly pronounced for materials that have low thermal conductivity. This thermal gradient could be used intentionally, but then might lead to a catastrophic "thermal runaway" in the interior of the heated part.

Graded layered stacks of cubic magnesium-iron-doped nanocrystalline Zr02, Zr02-Ni, Ni-NiO, and steel can be produced by sequential slip casting onto porous steel substrates with different porosities. Porosity variations in the ceramic regions are obtained by the addition of different amounts of commercial Zr02 powder. After drying and calcining in a vacuum, the layered stacks are microwave sintered as shown in Figure 6.14, in a single mode resonant cavity or in a multi-mode cavity, using carbon as the susceptor material. As shown in Figure 6.15, after pressureless sintering, the graded layered stacks are crack-free. Efficient heating within the ceramicmetal mixture at the beginning of the microwave sintering process is particularly important. Coupling of the microwaves with the surface skins of the metal particles supplies sufficient local heating to facilitate dielectric dissipation in the ceramic phase. Compacts of Zr02 or Ah03 powder that do not contain metallic inclusions cannot be sintered without initial separate preheating.

In microwave sintering experiments that start with a homogeneous mixture of Ah03 powder plus 5 weight % Zr02 powder, selective transport and enrichment of the melt of the eutectic composition containing > 40% Zr02 results in a substantial enrichment of Zr02 in the interior of the sample. Microwave sintering of Ah03 powder compacts after preheating produces a measurable gradient in the grain size from the interior to the periphery [69].

/r----

,....----,1 L..I - ______ ...J

VCeramic = VMetal

Pyrometer -----.,

L-_~I .... 1 __ .....

FGM/SGM

/ VCeramic« VMetal

Thermal Insulation

Figure 6.14. Casketing of samples for microwave sintering experiments: (left) with an additional metallic heat sink to support a vertical thermal gradient; (right) with a susceptor composed of porous, laminated carbon-felt or SiC to improve the heating of a small amount of ceramic phase on top of a metal matrix [69).

c-ZrO:! Ni-Addition Ni+NiO Addition bulk steel

Figure 6.15. An FGM of cubic zirconia-nickel-nickel oxide-steel (c-Zr02-Ni-NiO-steel) microwave sintered.

2.3.6 Reactive powder processing (Combustion processing)

183

In combustion processes in which two or more phases of a powder stack or preform react exothermically, the reaction is sustained with the heat that is released. A combustion wave at high temperatures (up to 4000°C) and speed (up to 20 cm/s) expands the reaction area [70, 71]. There are three initial ways that an FGM can be produced by combustion processes [72]. Figure 6.l6 illustrates a general method to fabricate dense FGMs by synthesis from a graded powder stack. This method has been developed largely by Japanese researchers [73, 74]. A graded material also can be synthesized from a twolayered powder stack by controlled impregnation during reactive processing [75]. In addition, it can be produced from a compositionally homogeneous

184 Chapter 6

powder stack because the high-temperature, multi-component melt allows the phase separation to be controlled.

D I..---------.,;~

Ignition V /" .......

Dense FGM

Compositionally graded reactantpowder mixtures

Figure 6.16. A schematic of the combustion process to fabricate dense FGMs.

Combustion synthesis has been used particularly for producing hard materials such as TiC-Ni, TiB2-Al, and TiB2-eu; the intermetallics NiAl, TiNi, and TiAl; and also ceramic composites. Powder compacts can be heated homogeneously to temperatures at which the synthesis reaction is initiated. This results in uncontrolled and often multiple self-ignition at different sites in the compact. This process is referred to as self-ignition or thermal explosion reactive sintering.

During self propagating high temperature synthesis (also known as SHS) a powder compact is heated to a temperature below the self-ignition temperature, and ignition is initiated at one location on the compact. The self-sustaining reaction then spreads completely through all the unreacted regions. In most cases, the reaction results in the formation of a liquid, which densities the compact rapidly. However, in some cases, the combustion is controlled by reaction with a gas phase that is present, which can affect densitication. In general, the duration of the reaction is short compared with sintering. Therefore, the initial concentration gradient in the powder compact is largely retained. Sharp discontinuities in the concentration gradients in the initial powder stacks usually are leveled out during reactive sintering.

185

Materials produced by combustion synthesis often are highly porous due to the evolution of gases, the formation of phases with lower density, or incomplete densification during transient liquid phase sintering. The formation of porosity can be suppressed by combining combustion synthesis with hot pressing in closed dies, with hot isostatic pressing, or with other methods that exert uniaxial or hydrostatic pressure on the sample during the reaction.

Despite the success of combustion processing for producing composite powders, a number of problems are encountered when scaling up this process to produce FGMs. The control of the combustion temperature must kept within a range that ensures both achieving the desired densification and reaction state throughout the FGM part and preventing the unintentional vaporization of the material. The maximum temperature can be controlled by the addition of prereacted components that act as diluents for the reactants. This effect has been demonstrated by the addition of titanium diboride (TiB2) to the reactants Ti-Ni and Ti-Ni-B [76], and the addition of titanium carbide (TiC) to the reactant Ti-Ni-C [77]. These diluents also can act as "passive steric hindrance" for the densification of the compact. Optimization of the amount of diluents is therefore important to fabricate dense FGMs. The FGM material combinations that have been prepared by combustion processing are listed in Table 6.1.

Table 6.1. Various FGMs Produced by Combustion Processes.

FGMs Density-Increase Methods References TiClNi TiCINi3AI Cr3CzINi TiBzINi TiB2/CU

(MoSiz-SiC)/TiAI AIz03/Cu DiamondlTiB2-Si

Hydrostatic pressing Hot pressing Hydrostatic pressing Hydrostatic pressing Hydrostatic pressing Spring-pressurization Hydrostatic pressing Centrifugal pressing Dynamic pressing

2.3.7 Other stack consolidation processes

71,72 73 74 70 68,69,75 76 77 78 79,80

Underwater shock explosion has been used to produce stacks of silicon nitride (SbN4) [87], and also to consolidate a variety of FGMs such as CulSS304 (stainless steel 304 's composition is 0.08%C, 2%Mn, 1 %Si, 18-20%Cr, 8-10.5%Ni, 0.045%P, 0.03%S, the rest Fe), Ti/TiAI, and SbN4/SS304 [88]. The experimental apparatus consists of an explosive charge, a container for the water, and a container for the powder stacks. The energy of the shock wave increases because of its compression when

186 Chapter 6

traveling from the larger cross-section of the water containment area to the much smaller cross section of the explosion site and the powder stack container. Pressures as high as 11 GPa are created for several milliseconds.

3. LAYER PROCESSING

Coating a substrate is a conventional method for protecting it against oxidation and wear or to decrease the heat flux into the substrate. A coating layer can also be used to add a functional capability produced by the coating itself or by the combined effect of both the coating and the substrate. Additional functions can be obtained with coatings that have electric or magnetic properties different from those of the substrate. Coatings can also be designed to have multiple functions. For example, a wear resistant hard coating can both protect a cutting tool and act as a hard cutting edge. A metallic bond coating in a thermal barrier system, which produces a chemical bond between a turbine blade and a zirconia thermal barrier layer, also can act to protect the superalloy airfoil from oxidation.

Thin layers in the nanometer range are usually deposited by atomistic transport processes via diffusion in the solid state, the liquid phase, or the vapor phase; or by convective transport often supported by electric or magnetic fields. Chemical surface reaction methods such as conventional surface hardening by carbonization or nitridation result in surface modification by atomic solid state diffusion transport.

Much higher deposition rates and thus lower costs for producing thicker coatings are achieved by mechanical deposition processes that include microlamination, cladding, and spray deposition (e.g., spray forming or thermal spraying). Layer processing also includes the fabrication of interlayers for the transient liquid phase bonding of bulk and sheet components. Thin solid or liquid reaction layers also can form due to interdiffusion when two bulk materials with different compositions are brought into contact. The gradients can be broad or narrow depending on the system and the processing parameters. Short processing times produce transition layers. Prolonged processing times can result in the formation of graded bulk materials.

In the subsequent sections, the fabrication of thick coatings by cladding, spray forming, and thermal spraying is discussed. Deposition processes based on atomistic transport are categorized into those based on vapor transport, such as physical and chemical vapor deposition; on diffusion in the solid or liquid states; and on electromigration processes.

187

3.1 Cladding

In surface engineering, laser beams are used as a high intensity heat source for hardening, alloying, and cladding. The characteristic features of laser processes are high localized heating and cooling rates, short interaction times between the heated materials and their surroundings, and a very low integral heat input [89]. Laser cladding has been applied largely to steel substrates using high power CO2 lasers and conventional hardfacing materials, such as NiCrBSi; the Co-based superalloy, Stellite® (Co-28Cr-4W-3Ni-lC); and WC-Co. More recently, a wider range of materials has been clad successfully, for example, oxide ceramics on steel and aluminum substrates, and composite claddings containing diamonds and other hard particles that can be damaged by heat. The composites have been obtained by using solid state Nd:Y AG lasers that make it possible to improve the control of the cladding process at low intensities. In laser cladding processes the powder can be predeposited, or more commonly it is fed continuously into a melt pool on the surface of the substrate. The melt pool is generated and maintained through interaction with the laser beam. It forms a cladding track after solidification as the substrate is scanned relative to the beam, as shown in Figure 6.17.

Laser beam

stream

Relative movement ~

Figure 6.17. A schematic of the laser beam cladding process [89].

Solid freeform processes that use a laser require precision powder fed cladding. For example, Nd:Y AG lasers that allow the laser energy to be delivered by an optical fiber have a much simpler but more highly flexible

188 Chapter 6

laser beam guiding system. The use of this system produces good results both with graded coatings and with 3-dimensional contouring using a robot. With this process, thin, graded multilayer claddings are obtained using predeposited powder, thereby suppressing melt pool convection through buoyancy forces [90].

Figure 6.18 shows the variations of the microstructure, the concentration of the elements, and the hardness throughout the four layers (each about 200 J!m thick) of a NiCrBSi alloy cladding with increasing Cr3C2 content (0, 10, 30, and 50 volume %), deposited on a steel substrate using the powder feed process. The carbide distribution within each layer is relatively homogeneous. The final 0.8 mm thick coating, which contains only a small amount of residual porosity, has a surface roughness of RZ,ISO = 30 to 35 J!m and a maximum Vickers hardness of 800 HV [89].

3.2 Spray Deposition

3.2.1 Spray forming

Spray forming, shown schematically in Figure. 6.19, combines several techniques from other fabrication methods such as powder metallurgy and rapid quenching. Initially, a metallic melt is atomized into a spray of fine droplets, typically between 10 and 150 J!m. When these droplets solidify in free flight or remain individual particles after impinging on the spray chamber walls, the resulting powder is used as starting material for subsequent powder metallurgical processing. When the spray of fine droplets is directed toward a mandrel or a substrate, and if the droplets are still in a partially liquid state at the moment of impingement, the process is referred to as spray forming [91]. The gaseous atomizing medium, e.g., Ar or N2,

usually is unreactive with the metal being atomized. Thick plates or bars can be formed on the substrate by continued spray deposition [92].

Spray forming can be used to produce both monolithic and composite materials. For composites, fine reinforcement particles, such as SiC or A120 3, are added either to the initial metal or to the spray. By changing the deposition parameters, the amount of dendrites that form before the droplets impinge, the grain size, and other microstructural features can be tailored to produce an FGM. Similarly, continuously varying the quantity, and ultimately the quality, of the reinforcing particles added to the spray can produce well-designed FGM structures. Because spray forming is still a developmental method and experiments are costly to perform, little work in producing FGMs has been done to date. With respect to their relative rates of production, spray forming is considerably faster than either plasma or flame

189

spraying. Materials that can be spray formed successfully are limited to those that are relatively unreactive, can be melted, and cannot be dissociated.

(a)

~ 100~-----,------~----~----~r---------, :l

.!:!. c: 80 o

~ C 80

8 § 40 ()

C Q)

E Q)

ill O~--~~-===~~~~~--~~--~--~ (b)

900r------,------~----_r----,_----------,

~ 800 o > 700 J: ';;;'600 III Q) 500 c: 'E 400 IU J: 300 ~ Q) 200

""" 5 100

O~----~----~--~~--~----~----~ o 200 400 600 800 1000 1200 Distance from Surface [lJm]

(c)

Figure 6.18. a) Microstructural section of a four-layer cladding ofNiCrBSi graded by varying the Cr3C2 content. b) Concentration profile ofNi, Cr, and Fe. c) Gradient in the Vickers hardness [89].

190 Chapter 6

3.2.2 Thermal spraying

In thermal spraying, feedstock (in the form of powder, rods, or wire) is introduced into a combustion plasma or other heat sources, such as arcs or laser beams. Arc spray methods use electrically conductive wire as feedstock, while combustion methods use powder or wire. Powders are used for plasma or laser beam spraying. The particles melt in transit and impinge the substrate where they flatten, undergo rapid solidification, and form a deposit through successive impingement. Thermal spraying is used largely to produce metal, ceramic, or polymer protective coatings, but it is also used to form bulk ceramics [93]. The relatively high porosity of FGMs should be taken into consideration when fabricating them by thermal spraying [94].

1.METAL DELIVERY

2.ATOMIZATION

3.TRANSFER OF DROPLETS

4.CONSOLIDATION

5.PREFORM

.-Overpressure , Induction power

~ iii r. .. pr_ure ,..------State of the top surface

--Spray height

___ Substrate motion

Figure 6.19. A schematic of spray forming [92].

The majority of plasma spray torches are gas stabilized. The plasma originates within a gas that acts to form and sustain a flame. These torches generally operate for extended periods of time at 30 to 40 kW, and have a material throughput of 2 to 5 kg/h. Low pressure plasma spraying, shown in Figure 6.20, is usually conducted in a low pressure inert gas filled chamber

191

[95]. It has extended the capabilities of the deposition process to reactive metals and intermetallics. The higher particle velocities of low pressure plasma spraying compared with air pressure plasma spraying, deposit oxide free metallic coatings with low porosity for applications such as oxidation resistant protective coatings on turbine airfoils and other high pressure turbine parts [96].

t--~--r----r--r _. _ .. _ .. _. - _. _ .. _. _ .. _. _ .. _ ....

, ............. --- .. --.

, ' r - - - - - - - - - - - -:- - - - - I , I

: I Torch motion .-----+--, controller

Plasma control console

®®®® Ii Ii Ii Ii

t9 Torc,h

'~-1\ ~(\ :.-Arc/Powder gas

UUU .------i8

EB Normal power supply

tC;>. Reversing I(]:I transfer Arc

...--____ +-______ --; tfi:>. power '6' supply

Mandrel - C:=::::;:~=~==1

"-----------/

Envi ronmental chamber

I Deposit motion controller Rotation/Transiation

Heat exchanger

Filter

Figure 6.20. A schematic oflow pressure plasma spray forming [95].

Plasma spraying is a favorable processing method for functionally graded materials because highly refractory phases can be melted simultaneously with a metal, blending the two in ratios that can be predetermined by controlling the relative feed rates of the two powdered materials. Since the

192 Chapter 6

deposit is formed through the sequential buildup of layers, several approaches can be used to produce a graded deposit. These can be broadly classified as single torch multiple feeders for blended or composite powders; mUltiple torch independent feeding systems for each component; and combinations of wire/powder feed systems.

A basic requirement for fabricating FGMs is that dissimilar materials (e.g., a metal and a ceramic with large differences in their densities and melting points) are homogeneously mixed in the desired proportions in the planes non!lal to the thickness direction. In addition, the desired compositional pattern should be achieved with good reproducibility in the direction of the heat flow.

Considerable experimental work has been done on fabricating FGMs by plasma spraying. In the early stage of the development of graded spray coatings, FGMs of YSZlNiCrAIY (yttria partially stabilized zirconialnickelchromium-aluminum-yttrium) were produced on steel by air plasma spraying [97, 98]. An example of an FGM produced by low pressure plasma spraying is the fabrication of YSZlNiCr with a four-port plasma torch operating in a chamber in which ceramic and metal powders are simultaneously introduced into the plasma flame, shown in Figure 6.21 [99]. Identical types of powders are delivered through diametrically opposed ports so that the dissimilar particle mixture is uniformly distributed in the flame while maintaining the necessary axial symmetry with respect to the axis of the torch. High density coatings are produced at high deposition rates when the average particle size is 57 ~m. When this system is programmed for a PSZlNiCr powder feed ratio, and automatically sprayed according to the program, the FGM composition can be controlled with sufficient accuracy to meet the requirements for the coating. Figure 6.22 shows a cross sectional view of a graded PSZlNiCr coating on a stainless steel substrate [100].

The characteristic solidification patterns of these FGM coatings, with columnar microstructures in the single-phase regions normal to the substrate of both the NiCr and the PSZ, can be seen in transmission electron micrographs. This is a typical feature of plasma sprayed microstructures formed by the deposition of droplets with a high ratio of melt to solid [100]. The microstructure is transformed into more equiaxed grains if the NiCr droplets coalesce at a slower cooling rate. This can occur if the local heat source is shielded by the presence of a high volume fraction of the Zr02 phase, which has a low thermal conductivity of about 2 W/m·K.

Depending on processing parameters such as particle size, environmental pressure, and the distance between the torch and the substrate, the porosity of plasma sprayed layers can range from as little as a few percent to 20 percent. This makes it possible to introduce porosity gradients into plasma sprayed layers [101].

Current

Ceramic -to (through 2 ports)

,

Plasma gas

I I ~_M.ml j [] (through 2

ports)

~ Coating

Substrate

Figure 6.21. A low pressure plasma torch with multiple feeding ports for metal and ceramic powders [99].

NiCr O-O-~.

. .. 100 !1Il1 . ::. 4," . 0 '

. .... , .

193

Figure 6.22. Cross sectional transition electron micrograph of a low pressure plasma sprayed PSZI NiCr FGM coating [99].

194 Chapter 6

A laser beam is another heat source for thermal spraying. When powder mixtures of Al with 10, 30, and 50 weight % SiC are directed into a 2kW continuous wave CO2 laser beam focused on an Inconel 625 (Fe-Ni base superalloy) substrate, a graded multilayer structure is formed [90].

3.3 Physical Vapor Deposition

3.3.1 Electron beam - physical vapor deposition

Electron beam-physical vapor deposition (EB-PVD) is a promlsmg technology for the production of thermal barrier coatings. Among the advantageous features of EB-PVD are that the coatings produced have smooth surfaces without requiring additional polishing; they have good erosion resistance in service; and there is no closure of cooling holes. However, the most important advantage is their outstanding thermal shock resistance, thus considerably longer life, which is related to their columnar microstructure. The state-of-the-art-material for thermal barrier coatings is Zr02 stabilized with 6-8 weight % Y 203 (YSZ), which is composed of the non-transformable tetragonal t'-phase [102]. The equipment used for depositing thermal barrier coatings via EB-PVD, shown in Figure 6.23, consists of several containers for loading, preheating, and deposition plus a device for rotating and manipulating the samples. In order to ensure continuous deposition, ingots of the metallic or ceramic coating material are fed into crucibles below a deposition vessel. Evaporation of the coating materials is achieved by the application of high-energy electron beams (150 kW and higher). Vaporization during EB-PVD depends on the vapor pressure of each compound. This makes it difficult to evaporate materials simultaneously that have large differences in their vapor pressures (e.g., Zr02 and Ce02).

Using a single source coater only certain chemical gradients have been produced successfully, such as density graded YSZ and chemically graded bond coats [102, 103]. Another approach to obtain a graded structure is to use a mixture of aluminum, alumina, and zirconia as the starting composition in the form of pressed tablets on top of YSZ ingots [104]. Chemically controlled graded coatings can be fabricated readily by vaporizing from multiple sources using one or more electron guns. By adjusting the electron beam parameters, different deposition rates can be obtained for each source by independently adjusting the temperature of the melt pool. If the evaporation rates of two components, for example alumina and zirconia, are changed continuously, a defined gradient of the composition over the coating thickness is obtained

195

Because of the different deposition conditions that are required to maintain adequate evaporation of the alumina, the zirconia, and their various mixtures, the substrate temperature increases with increasing zirconia content in the vapor cloud. Initially, a thin, pure layer of alumina is deposited on top of a NiCoCrAlY bond coat, followed by compositionally graded aluminalYSZ layers, and topped with pure YSZ. The formation of a continuous chemical gradient can be seen in Figure 6.24 [105]. (For clarity, the graded layer is shown as a fractured cross section.) Although there is still a discontinuity in the gray scale of the scanning electron micrograph in Figure 6.24, the spatial distribution of the concentration shows only a slightly steeper change in the concentration in this area.

Alumina ~ vapor cloud

Electron gun

Jumping electron beam

- -

~ Zirconia vapor cloud

Crucible

PYSZ-ingot

Figure 6.23. A schematic of an electron beam - physical vapor deposition coater with two evaporation sources heated by a single jumping beam [105].

Experiments indicate that dual-source EB-PVD can produce a discrete graded layer even though only one electron-beam gun is used. The application of jumping beam technology for ceramics requires a fast beam deflection system capable of beam scanning frequencies up to 1 kHz. The key to reproducible evaporation behavior is to adjust the beam pattern and the beam focus for the different ceramics. This can be difficult to accomplish

196 Chapter 6

because zirconia requires two to three times more energy for evaporation than alumina, but the beam power cannot be changed quickly during the jumping procedure.

70 lr

60

~ 50 ~ c .2 40 ;

30 QI

!! 8 20

10

1 2 3 4 5 distance

Figure 6.24. Distribution of the elements (above) and fractograph (below) of a graded alumina-yttria stabilized zirconia (Ah03-YSZ) coating. [105].

3.3.2 Sputtering

Sputtering is commonly used for depositing thin films for applications in microelectronics, optics, energy generation, and mechanical and chemical engineering. The almost unlimited selection of substrates and coating materials includes plastics, metals, and ceramics for the substrates, and metals and ceramics for the coatings.

197

The principle of the sputtering process is illustrated in Figure 6.25. The deposition process is generally carried out in a vacuum chamber that is evacuated and refilled with an inert gas, usually argon, at from 10 to 10-1 Pa. For coating deposition, a high voltage is applied to both electrodes. (The target material is on the cathode.) The high voltage ionizes part of the inert gas producing a plasma. Because of the specific polarization, the inert gas ions are accelerated toward the target material and bombard its surface. When the energy of the impinging ions exceeds the binding energy of the target atoms, the target surface is sputtered. As shown on Figure 6.25, the ejected target atoms deposit on the substrate to form the coating. In many industrial applications, an additional magnet system is used to obtain higher deposition rates. In reactive sputtering, an element from the process gas is also incorporated into the coating. For example, titanium nitride (TiN) coatings for the wear protection of tools are produced by sputtering a pure Ti target in an Ar-N2 atmosphere.

Magnetsystem

....------<0 Power suppl y

~~~l1~arg~et~(c~ l ......

Q ~.kArgon ion

a 4-Target atom S ~Q

Q ~ ~. V Electron ~ ~ Q

o 0 0 De-coaling s-&Jbs trate

Figure 6.25. A schematic of magnetron sputtering [109].

The various sputtering processes for the deposition of single- and multilayered coating systems by using single andlor multi-source equipment is versatile state-of-the-art technology that is widely used in industrial applications. However, there are only a few applications for graded coatings. Surgical implants made of the titanium alloy Ti-6AI-4V are coated with compositionally graded TiN films by reactive DC (direct current) sputtering to improve the implant's biocompatibility and to relax the stresses concentrated at the interface between the alloy and the coating [106]. For this application, the nitrogen partial pressure of the Ar-N2 gas mixture is continuously increased during deposition in order to vary the composition

198 Chapter 6

from metallic titanium at the interface of the alloy and the coating to stoichiometric TiN at the coating's outer surface.

Another application of graded sputtered coatings is for protecting high temperature materials against oxidation. Molybdenum can be protected with a functionally graded coating of MoSh + SiC (molybdenum disilicide + silicon carbide), which is deposited by RF (radio frequency) magnetron sputtering from a composite target. The compositional gradient in the coating is achieved by varying two of the deposition parameters: the applied power and the argon pressure [107]. For obtaining large compositional changes in the coating, co-sputtering is more effective than varying the deposition parameters.

Intermetallic titanium aluminide (TiAI) coatings for the oxidation protection of titanium alloys can be produced by co-sputtering Ti and Al from different sputtering sources [108]. By modifying the applied power for each source, the composition of the coating theoretically can be varied from o to 100% of either Ti or Al or both. For this application, the composition is continuously varied from 25% Al at the substrate coating interface to 75% Al at the outermost surface, as shown in Figure 6.26. This compositionally graded coating, which was designed both to minimize interdiffusion and stresses between the substrate and the coating and to provide maximum oxidation protection, has excellent oxidation resistance under cyclic and isothermal conditions [109]. Graded TiAI coatings also have high mechanical performance under cyclic loading [110].

100 ~

<ll 0 '§ ::i. 80 (jj

.c c: ~

.g (f)

'w 60 0 a. E

40 0 u cu u 'E 20 <ll

.I::. ()

0

Distance [11m]

Figure 6.26. Graded Ti-AI coatings for the oxidation protection of titanium alloys: a) schematic and b) EDX profile of the graded coating [110].

25

199

3.4 Chemical Vapor Deposition

In chemical vapor deposition (CVD), shown schematically in Figure 6.27, a deposit is formed on a substrate by subjecting source gases (e.g., hydrides, bromides, or chlorides) that fill the reaction chamber, to various types of energy such as heat, light, and plasma [111]. FGMs can be synthesized at slow to moderate deposition rates by changing the ratio of the source gas mixture, or by controlling the deposition temperature, the gas pressure, or the gas flow rate. Because of its slow deposition rate, CVD is largely used for infiltrating stacks or preforms, or for the fabrication of thin layers.

Ribbon heater Pressure regulator

\ Water-<:ooled ---- -, - - -I I reaction

I I chamber II II II

. II Wort<. coil Constant reservor I I ~

I 0 temperature bath L - I 0

- I Mass flowmeter I I 0

~ I 0 I I Pressure I I gauge II

o

II

: ~ _______ j Pump

I.!; - - - - - - - - -I Pressure regulatcr

Figure 6.27. A schematic of the setup for the preparation ofa SiC/C FGM by chemical vapor deposition [111].

A typical example of an FGM application of CVD is the formation of a graded SiC coating on graphite in which there is a continuous transition from SiC to C. Compositionally controlled source gas mixtures of tetrachlorosilane-methane-hydrogen (SiCI4-CRt-H2) are used at deposition temperatures between 1400°C and 1500°C and pressures between 1.3 and 6.5 kPa. The computer controlled variation of the SiCI4/CH4 gas ratio controls the change in the SiC/C deposition ratio with time. Graphite substrates with FGM coatings of SiC/C between 200 and 800 /lm thick are shown in Figure 6.28 [112].

Similarly, graded coatings of zirconium carbide/carbon (ZrC/C) on a C/C composite can be deposited from gaseous mixtures of zirconium chloridemethane-hydrogen (ZrCI4-CH4-H2) at temperatures between 1400°C and

200 Chapter 6

1600°C [113]. Other FGM protective coatings that have been deposited by CVD on C/C composites are boron nitride/silicon nitride (BN/SbN4) and carbon/boron carbide/silicon carbide (C/B4c/SiC) [114]. In addition, FGM films of 1 mm thick titanium carbide/carbon (TiC/C) have been deposited to protect materials facing the plasma in fusion reactors.

Diamond is a useful material both for electrical components and machine tools because of its extreme hardness, thermal conductivity, and chemical and thermal stability. In a novel CVD process developed specifically to fabricate a diamond/WC FGM, plasma spraying is combined with the simultaneous synthesis of a diamond film by DC plasma jet CVD using a single torch apparatus, shown in Figure 6.29 [115]. Graded intermediate layers are formed by controlled feeding of powdered WC using hydrogen as the plasma gas. The intermediate layers improve the normally poor adhesion of diamond films due to their low thermal expansion coefficient, high deposition temperature, and high elastic modulus. These properties of diamond films cause high thermal stresses to develop at the interface between the diamond film and the substrate.

SiC/C graded zone

C

, 0.1 mm ,

Figure 6.28. Cross-sectional view of the microstructure of a SiC/C FGM prepared by chemical vapor deposition at 1500°C and 1.3 kPa [Ill].

H2 + CH4 He + Powi,er -0 .-----1

Substrate holder

Substrate III \\ /1 \

Figure 6.29. A schematic of the single torch apparatus specifically developed for the deposition of a diamond/metal FGM [113].

201

As shown in Figure 6.30, the intermediate layers, which are composed of a mixture of diamond and metal or ceramic, produce a composition gradient from the plasma sprayed layer to the diamond film. The composition gradient causes a continuous change in the thermal expansion coefficient of the intermediate layers thereby effectively relaxing thermal stresses and improving the adhesion strength tenfold.

Diamond is synthesized by feeding a mixture of hydrogen and methane between the two electrodes of a plasma gas initiated by a DC arc discharge. A composite with a diamond matrix and WC particles is produced by feeding WC powders to the plasma torch at the same time as the He carrier gas. The graded diamond film is formed by first depositing a plasma sprayed layer using only hydrogen as the plasma gas. Methane is then added to the plasma gas, and the feed rate of the WC powders is gradually reduced. In the last step, only diamond is deposited without feeding any powder.

3.5 Chemical Reaction

An FGM can be produced by a chemical reaction on the surface of a material. For example, a C/SiC FGM can be formed on the surface of a carbon material by heating it in Si powder at 1450°C for 3 hours [116]. Also joining similar or dissimilar materials with thin sheets of a different material

202 Chapter 6

inserted between them frequently results in the creation of new phases in the joining area [117, 118].

Figure 6. 30. Scanning electron micrograph and the compositional distribution of an FGM diamond film [114].

3.6 Constructive Deposition by Electrotransport

3.6.1 Electrodeposition

An electrodeposition method can be used to form a Cu-Ni FGM with a controlled concentration gradient in which up to 30 volume % of fine (lJ.lm) Ab03 powder is incorporated in the deposited Cu-Ni [119]. The volume fraction of AhO 3 particles incorporated in the Ni- Cu-Ah03 structure is controlled by the particle content in the liquid electrolyte [120].

Electrolyte jet streams directed onto the substrate have proven to be beneficial for enhancing the deposition of compositionally graded layers. They enable the deposition at high rates of metallic materials such as nickelphosphorus (Ni-P) alloys because of an increase in the limiting current density [121]. When SiC or Ah03 particles are added to the jet, their volume

203

fraction in the deposit changes with the jet velocity, while at the same time the P content is controlled by jet polishing.

As the velocity of the jets increases between 0.5 and 16 mis, the volume fraction of electrodeposited SiC particles decreases from 30 to 0 volume %. The mechanism of codeposition via jet electroplating is explained in the following way. Nickel hydration ions in the form of anion clusters are present on the surface of the SiC particles in the electrolyte. The SiC particles enveloped by the nickel hydration anions are transferred energetically to the surface of the cathode by the electrolyte jet. Following their dehydration and reduction, the nickel hydration ions adsorbed on the SiC particles become nickel atoms immediately adjacent to the cathode surface. The particles now behave differently in that they directly contact the cathode surface and are simultaneously covered with crystallizing nickel. Figure 6.31 shows the microhardness and X-ray intensity of an electrodeposited Ni FGM with a compositional gradient of SiC particles.

~~--~----~--~----~~950

900 u;-N 0 ci ?;:

::I:

~ ui 'en (fJ

850 Q) c c Q) "0 C ... ctj ..c 0 ... I:)

800 ~

~~~~--~----~---L~750

o 10 20 30 40

Distance from interface, d /).lm s

Figure 6.31. The microhardness and X-ray intensity of an electrodeposited Ni FGM with a compositional gradient of SiC-particles [121].

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3.6.2 Electrophoresis

Electrophoretic deposition is a colloidal process in which on the application of a DC electric field particles are deposited from a stable suspension onto a shaped electrode that has an opposite charge, shown schematically in Figure 6.32. Recently, electrophoretic deposition has been applied to fabricating composites and micro laminar ceramics [122 -124], nonplanar laminates [125], fiber composites [126], and FGMs [123, 127, 128]. The deposition rate is controlled by the deposition current density and the concentration of the particles in the suspension .

4 ~ •

• •

• • ---e· • • •

Figure 6.32. A schematic of an electrophoretic deposition cell showing the process [128).

The mechanism for electrophoretic deposition is still in question. One that has been proposed recently is the distortion of the spatial symmetry of the double layer (diffuse-double-layer or lyosphere) that forms around particles when submerged in a liquid. Oxide surfaces, for example, can be considered to undergo the following reactions to produce either H+ or OH-:

205

(at low pH)

and

(M-OHtsurface + OR === (M-Orsurface + H20 (at high pH)

Such a system could be an oxide particle with a first layer that is positively charged and a second, more extended, deformable layer of negatively charged ions. The overall double layer can still be positive. An electric field makes the particle with its double layer system move toward the negative electrode (cathode) where it is deposited after discharging. (Coagulation can have a marked effect on the process.) Due to the migration, the double layer becomes polarized. This implies that the thinning of the negatively charged second layer occurs in the migration direction of the system. All the oxide particles show this polarization, and the particles with bipolar double layers coagulate throughout the suspension. The coagUlation reverses when the electric field is removed. However, it can become irreversible if there is sufficient asymmetric thinning of the double layers for van der Waals interaction between the particles

FGMs with a continuous or stepwise transition from Ah03 to YSZ, AhOiMoSi2, Ab03INi, and YSZINi all have been prepared by electrophoretic deposition. The gradient is obtained by a stepwise or continuous change in the concentration of Ab03, YSZ, MoSb, or Ni particles in the suspension during deposition. Laminated and graded Ah03/Zr02 ceramics have been fabricated from ethanol suspensions at a pH of 3.5. Both suspensions are stable and positively charged at this highly acidic pH. After drying the deposits were sintered.

Figure 6.33 is a micrograph of an FGM about 4 mm thick composed of six layers in 20 volume % steps. The Ab03 (dark phase) and YSZ (light phase) phases are homogeneously distributed throughout the microstructure. The end composed of pure Ah03 has an average grain size of 6 11m that reduces to 2 11m in the 80 volume % layer. The sharp interface between the layers, which has an irregularity roughly corresponding to the scale of the grain sizes, is clearly visible in the microstructure.

The experimental set-up for the electrophoretic deposition of FGMs with continuously graded compositions is shown in Figure 6.34. The deposition starts when a Zr02 suspension and an Ab03 suspension are slowly injected using a syringe pump into the bottom of the electrophoretic bath where they become mixed (the mixing area is shielded from the deposit). The spatial distribution of the composition (composition profile) of the Ah03 can be precisely controlled via the deposition current density, the rate of pumping of the Ah03 suspension, and the concentration ofthe suspension.

206 Chapter 6

Figure 6.33. The scanning electron micrograph microstructures of a step-graded Al20 3IYSZ FGM. The dark phase is Al20 3 and the light phase is YSZ [\28].

207

Figure 6.34. The experimental set up for the synthesis of ceramic/ceramic and ceramic/metal FGMs with continuously graded compositions [128].

4. PREFORM PROCESSING

Stacks or other preforms that do not contain gradients initially can be graded using preform processing. In addition, preform processing can be used to enhance or reduce the gradients of graded preforms. The preforms can be porous or dense, or may even consist of an arrangement of bulk materials or layers that is itself homogeneous thus would not be considered anFGM.

Most of the methods of preform processing are based on the traditional transport mechanisms used to create gradients in materials. Heat and mass diffusion, for example, have been used for centuries to create functional, microstructural and/or compositional gradients in steel. A second group of methods is based on the application of external, largely stationary, fields. For example, thermal gradients can be applied to sinter porous preforms to different local densities, and electrical fields can produce a graded porosity when a porous preform is electrolytically dissolved.

208 Chapter 6

4.1 Solid State and Liquid Phase Diffusion

When diffusion couples with different compositions are heated, interdiffusion can create gradients. These gradients are obvious in solid solutions, but are also present to a widely varying degree in intermetallic compounds formed as reaction products during interdiffusion. Solid state diffusion methods have been used frequently to fabricate FGMs.

During annealing at about 2000°C, solid solution gradients are formed between a rhenium (Re) layer and a molybdenum (Mo) layer that have been deposited by chemical vapor deposition (CVD). Inserting a thin tungsten (W) layer between the Re and Mo layers reduces the interdiffusion between the Re and Mo [129]. When diffusion couples of Cu-lO weight % Sn (tin) and AI-0.5 weight % Sn are heated at 500°C, four intermetallic compounds develop based on Cu-AI.

In alumina (Ah03) uniformly doped with Ti, a dopant gradient is developed during sintering in a carefully controlled atmosphere [130]. When compacts of high-purity Ah03 powder that either are not doped, or are doped with 500 ppm Ti4+ and 2000 ppm Ti4+ , are sintered at 1500°C in a vacuum, the Ti4+ is reduced to Ti3+ in their peripheral regions. This is indicated by their pink color, which is characteristic of Te+-doped alumina. The centers of the compacts remain white, however, indicating that the Ti4+ oxidation state of the dopant is unchanged. The gradient in the oxidation state of the dopant has a pronounced effect on both the microstructural development and the rate of grain boundary migration. The peripheral regions contain equiaxed grains whereas the central regions show the onset of grain elongation and evidence of faceting.

When a polymethylacrylate (PMA) solution is poured on a polyviny1chloride (PVC) substrate, the PVC starts to dissolve in the PMA and diffuses in the solution to the side facing the air (see Figure 6.35) [131]. Diffusion continues as long as the solvent tetrahydrofuran (THF) is present in the PMAIPVC solution. When the solvent has completely evaporated, the local PMA/PVC composition ratio is frozen in. Different initial ratios of PMA and the solvent THF lead to different times required for complete evaporation of the THF, and thus to different concentration profiles resulting from the diffusion of the PVC into the PMA. Small volume ratios of THF/PMA produce steep gradients close to the initial interface between the solid PVC substrate and the PMA solution, with the PVC and PMA phases still present. Higher THF/PMA ratios produce extended zones between the PVC and PMA phase, with smooth compositional gradients. Still higher THF/PMA ratios provide sufficient time for the complete consumption of the initial PMA and PVC phases by interdiffusion. This results in the formation of a solid solution with a narrow composition gradient.

~ Evaporation

PMA solution

DiSsohtion and Diffusion

Figure 6.35. Schematic of the PMA(THF)IPVC system to fabricate FGMs [131].

4.2 Liquid and Gaseous Flow

209

When zirconium phosphide (ZrP) powder is uniaxially pressed at 75 MPa then isostatically pressed at 100 MPa, bars are formed with a homogeneously distributed porosity of about 38 %. After the powder compacts are evacuated and pressure infiltrated with epoxy resin for varying lengths of time, a concentration distribution of the epoxy resin is observed even following prolonged infiltration. The kinetics of the infiltration process are diffusion controlled, and the infiltration depth varies as the square root of the ratio of the pores to the viscosity [132]. An identical pore filling process has been demonstrated when pressed bars of the superconductors yttriumbarium-copper-oxide (YBCO) and barium-strontium-copper-oxide are infiltrated with epoxy resin [133, 134].

To produce FGMs of WC-Co cemented carbides, presintered compacts of WC-l weight % Co are rapidly immersed in a melt of WC-20 weight % Co, which only partially eliminates their high porosity. The resulting material has a gradient structure in which the residual porosity increases with increasing distance from the surface of the compact that was in contact with the melt. The hardness (Vickers) ranges from 1200 HV close to the contact areas, to between 1020 and 1080 HV furthest away, corresponding to Co concentrations of6 and 12 weight %, respectively [135].

A compositionally graded silicon carbide-silicon carbide/graphite (SiCSiC/C) material composed of a SiC surface coating, an intermediate SiC/C layer, and a graphite matrix can be made by reacting gaseous SiO (silicon monoxide) with graphite followed by CVD coating with a homogeneous SiC

210 Chapter 6

layer. The SiO is vaporized in a high purity helium carrier gas stream at 1300°C that transports the SiO through isotropic, fine-grained graphite at 1350°C. The reaction between the SiO and the graphite results in the formation of SiC within the graphite, with the development of a narrow gradient in SiC content between the surface and the interior of the graphite. The maximum Si content within the graphite can be varied between 0.17 and 5.1 weight % by controlling the reaction rates [136].

4.3 Processing in Thermal Fields

With the exception of the self-propagating high temperature synthesis (SHS) process, when producing a metal-ceramic FGM by powder metallurgical methods, the ceramic is usually heated without the application of a thermal gradient [137]. However, a density gradient can be created by applying temperature gradients during sintering. For example, a cylindrical lead zirconate titanate (PZT) powder compact was sintered by heating one side externally with an infrared lamp [138]. During sintering (1 hour), a stationary temperature gradient of 150°C was maintained between the heated surface and the inside of the cylinder, at a distance of 5 mm from the heated surface. After sintering, the density in the surface area was considerably higher than in the corresponding interior of the cylinder, and this difference was paralleled in the piezoelectric properties of the PZT material.

Microwave heating is a pressureless heating process that provides a selfregulating heating pattern governed by the compositional gradient. Radiation at a frequency of 2.45 GHz has been identified as a general heating method with a self-regulating potential for compositional gradients [139]. The creation of compositional and grain size gradients in homogeneous ceramic materials as a consequence of certain characteristic mechanisms of microwave heating also has been investigated [140].

4.4 Processing with Electrical Fields

In a novel method based on the infiltration of refractory porous preforms with a molten metal or polymer, a gradient in the electrochemical potential is set up inside a porous preform. This produces a gradient in the rate of electrochemical dissolution or deposition of the preform material, and consequently creates a graded porosity. A macroscopic electrokinetic model of the gradation process has been developed, and the influence of experimental parameters such as current density, electrode and electrolyte resistivity, and geometrical factors on the spatial distribution of the gradation has been compared with experimental observation. An FGM of

211

tungsten/copper (W/Cu) was produced, and some of its properties have been determined as a function of position [141].

4.5 Processing with Other Fields

Gradients in plastic strain can be used to produce FGMs with graded magnetic properties [142]. The paramagnetic phase in austenitic stainless steels, such as Fe-18Cr-8Ni, is transformed into the ferromagnetic alpha prime (a ') phase by plastic deformation at low temperatures. Since the amount of the deformation induced martensite increases with the local strain, the saturation magnetization of the deformed austenitic stainless steel should also increase with increasing strain. As indicated in Figure 6.36, this could make it possible to produce large FGM parts with gradients in their magnetic saturation.

0.5 0.5 • Specimen (B) ra

"iii

0.4 • Specimen (C) .2l 0.4 ...... c: 0

c: ~ 0.3 .~ N

~ iii c: Q) 0) ::J ra 0.2 ,=: E

c: 0

0.1 ~ 0.1 ::J iii en

10 20 30 40 50 60 0 0 10 20 30 40 50 60

Distance / mm Distance / mm

(a) (b)

Figure 6.36. a) Distribution of the plastic strain (a) and saturation magnetization (b) in two austenitic stainless steel specimens with different geometries (B and C) [142].

212 Chapter 6

5. MELT PROCESSING

The use of melt processing to form FGMs has been very limited. Two reasons may be the difficulty in fine-tuning and controlling the formation of extended compositional gradients in melts and the incompatibility of many phases in the molten state.

5.1 Settling under Gravity

Mixtures of phases with considerable differences in their densities may separate under normal gravity when kinetic and steric conditions allow the heavier phases to settle [143]. In some respects, settling under gravity resembles powder s·ettling and stacking from aqueous and non-aqueous particle dispersions. In aqueous dispersions, the liquid phase is drained off after settling, producing a powder stack that is then dried. However, settling under gravity during melt treatment ends with the solidification of the melt. In other words, in melt processing the liquid phase is an integral part of the final solidified product.

A number of model experiments have been performed with W-Fe-Ni heavy metal alloys where the almost pure tungsten grains (density approximately 1900 kg/m3) are dispersed in a W-Fe-Ni melt (density approximately 1000 kg/m\ The volume fraction of the W-particles decreases from the bottom to the top of a cast column due to density gradients resulting from initial settling and subsequent enhancement of the differences in the density by shape accommodation of the W-based grains to allow a more optimal packing of the solid material [144].

5.2 Settling under Centrifugal Forces

Centrifugal force enables the creation of a gradient compositional distribution in a ceramic powder mixed with a metal because of the difference between the densities of the molten metal and the ceramic material. In the centrifugal casting of thick-walled rings of SiC/AI FGMs, molten Al containing 10 volume % SiC is poured into a rotating 90 mm long thick-walled tube with an outer wall diameter of 90 mm and a wall thickness of 10 mm [145]. The mold's rotational speed produces forces on the melt equivalent to 23 to 143 times the gravitational force. The molten material solidifies during the rotation after a certain rotation time has elapsed. Figure 6.37 shows a schematic of an FGM fabrication set-up that uses centrifugal force during solidification [146].

In Al-Ni melts, when AI-Ni alloys containing less than 30 mass % Ni are used, the intermetallic compound AhNi (containing more than 5.7 mass %

213

Ni) is initially nucleated during cooling [147]. Because of the difference in density between AhNi (= 4000 kg/m3) and molten AI-20 mass % Ni alloy (3930 kg/m3), a graded radial distribution of the AbNi phase within the cross section is obtained (see Figure 6.38). These model materials have been examined via a series of experiments that included an evaluation of their graded Young's modulus and the measurement of their residual stresses after cooling to room temperature.

Figure 6.37. Schematic ofan FGM fabrication set-up that uses centrifugal force during solidification [145].

5.3 Solidification

The potential of the Verneuil process (also called the flame fusion process) to fabricate FGMs has been demonstrated with various aluminum oxide based materials [148]. The main component of a conventional Verneuil system is a vertical, inverted oxyhydrogen burner composed of several concentric tubes. Hydrogen is transported through the outer tube to feed the flame that burns in a ceramic muffle, and oxygen and the crystalline powder are fed through the inner tube. A seed crystal is positioned in such a manner that only a thin film of the liquid phase is present at the top surface of the crystal. The crystal grows at constant conditions if the rate that the molten droplets are fed to the melt film and the rate that the melt film solidifies on the crystal (i.e. the crystal growth rate) are in proper balance [149]. The standard material used for the Verneuil process is Ah03 (aluminum oxide). Because of the small volume of the liquid phase, this process is well-suited for growing graded chromium doped alumina (ruby-

214 Chapter 6

sapphire) crystals with sharp transitions between the Cr-rich (40 %) and the Cr-poor (15 %) regions, as well as for growing crystals with transition regions of several centimeters. The recent development of a low-temperature flame fusion technique makes it possible for the Vemeuil process to work in the temperature range between 800°C and 2500°C.

60 120

0

50 110

>- III a.. Z 40 100 {!J

M -... <i: (/l

:J

'0 30 90 "S ~

"C 0 0 E Q)

E 20 - : Estimated V, curve 80 (/l :J -0)

"0 X : Experimental V, for G=23 t: > :J

10 o : Experimental V, for G=147 70 0 >-

- - - : Young's modulus

0 60 0.0 0.2 0.4 0.6 0.8 1.0

Specific thickness

Figure 6.38. Distribution profile of the Al3Ni phase in the aluminum based phase and the corresponding graded distribution of the Young's modulus. (G represents the centrifugal acceleration of the mold) [147].

Under gravity conditions, buoyancy convection and gravitational segregation caused by density differences often produce compositional gradients in melts. When gravity conditions are rapidly changed into microgravity conditions, these compositional gradients can be stabilized for a brief time. Research using this method to produce FGMs has been carried out by synthesizing indium antimonide (lnSb) with various compositions [150].

6. JOINING

Joining dissimilar materials is complicated by unanticipated chemical reactions and by stresses produced during the joining process or in service.

215

From an end-use perspective, applications at elevated temperatures (e.g., for gas turbines or heat exchangers) require the use of interfacial layers with high temperature properties sufficient to exploit the potential of joints of ceramic or other high temperature materials. To keep production costs low and to limit thermal stresses and reactions, a rapid mass production method is desirable that could be applied at low joining temperatures.

6.1 Solid State Joining

Combining different nickel-base superalloys is desirable when a variety of high temperature properties are required for turbine engine components. The joining of various components via solid state bonding has been studied in the laboratory. A bladed disk can be manufactured by joining a finegrained powder metallurgy material such as Astroloy® (Cr 15.0 weight %, Co 17.0 weight %, Mo 5.0 weight %, Ti 3.5 weight %, Al 4.0 weight % and balance Ni) to blades made from directionally solidified or single crystal materials by either hot isostatic or hot uniaxial pressing. Coextrusion has been used to join two disc materials with different microstructures, one for the hub that has high strength and good low cycle fatigue, and one for the rim with good creep resistance. The gradation in the grain size and the secondary strengthening phase (gamma prime precipitation) in the microstructure occurs over distances that are controlled by diffusion, and that also can be manipulated by the insertion of nickel-base (Ni-20Cr) foils as intermediate layers between the fine-grained and the coarse-grained components. During the joining process, the microstructural development is characterized by the elimination of pores, the formation of submicrometer oxides or carbides at the joint, and by gamma prime phase precipitation and diffusion phenomena in the vicinity of the interface. The mechanical tests of uniaxial tension, creep, and fatigue show reproducible values exceeding those of the weakest material used as a component [151].

Graded silicon nitride/titanium nitride (ShN4ITiN) ceramic matrix composites (CMCs) and TiNlNi metal matrix composites (MMCs) have been prepared by hot isostatic pressing (HIP) to demonstrate a joining concept aimed at reducing the residual stress in joints between SbN4 and Incoloy 909 (a Fe-Ni base superalloy). Due to the abnormal grain growth in the superalloy at higher temperatures, the hot isostatic pressing temperature for simultaneously joining and densifying the TiNlNi MMC was limited to 980°C. To obtain layer regions with low Ni matrix content and high strength, Ni-coated coarse TiN powders with fine TiN and Ni additions were used [152].

A new welding technology for joining dissimilar materials has been tested with two different stainless steels. A short, bar-shaped transitional

216 Chapter 6

joint with a graded composition that varies along its length, made by blending the two parent powders then hot isostatic pressing, was placed between the two different steels before joining them. Because of thermodynamic interactions between the hot isostatically pressed powder particles of the parent materials, the transition zone contains phase constituents different from those in the parent materials. Using fundamental thermodynamic and kinetic principles, the microstructure of the transition zone was predicted correctly. Investigation of the creep properties show values above those of the weaker of the two parent materials with decreasing stress level and increasing test duration [153].

6.2 Transient Liquid Phase Joining

Transient liquid phase (TLP) joining was originally developed for joining Ni-based superalloys with appropriately designed interlayers at low temperatures and low bonding pressures. It produces joints that can be used subsequently at elevated temperatures [154, 155].

FGM structures need to be designed with gradients in the interlayers for joining that are temporary and serve largely to facilitate processing [156]. The interlayers generally consist of a thick core layer of a high melting point metal combined with a thin cladding layer of a low melting point material, as illustrated in Figure 6.39. The former affects the physical and thermal properties, while the latter makes it possible to reduce the processing temperature. The components are selected using phase diagrams, so that they interact to form a refractory alloy or compound. Isothermal solidification of the liquid layer occurs due to the diffusion of the low melting point component of the core layer [157]. To minimize the processing time and to maximize the diffusion rate of the low melting component in the core layer, or in the compound layer formed by the reaction, the thickness of the liquid film should be minimized. Good wetting is another essential requirement for superior bonds [158]. Whereas the wetting of metals by metal melts is adequate for many systems, the wetting of ceramics by liquid metals often requires facilitation by the addition of certain elements.

Inserting a graded CulNb/Cu foil between two Ab03 plates has been found to facilitate their bonding by means of a transient liquid phase [159]. During hot pressing in a vacuum at 1150°C, the molten Cu-rich regions of the foil react with the Nb to form a solid phase. The temporary presence of the Cu-rich melt is sufficient to produce good bonding between the two alumina plates, and the presence of the Nb core layer provides a close thermal expansion mismatch between the metal interlayer and the ceramic plates. Although the CulNb/Cu system does not become completely homogenized due to the low solubility and slow diffusion of Cu into Nb, the

217

joining of the plates is successful. Comparable wetting experiments indicate that the presence of Nb in the graded Cu foil reduces the contact angle of the Cu on the Ab03 plates.

t = 0

I Ceramic I Cladding

a) Cae

Cladding

I Ceramic I

t>O

b)

t = 00

I Ceramic I c) Interlayer

I Ceramic I Figure 6.39. An illustration of multilayer interiayer design and evolution. (a) Initially, a sharp discontinuity exists between the thicker core layer and the thinner cladding layers. (b) After some annealing and interdiffusion, the interiayer becomes homogenized, which increases the remelting temperature. (c) After prolonged annealing or use at elevated temperatures, a uniform interlayer is formed [155].

As shown in Figure 6.40, joining Ab03 ceramics with Cu/80Ni-20Cr/Cu interlayers [158] is relatively successful compared with CulNi/Cu interlayers

218 Chapter 6

[160] with respect to the average strength and strength distribution. The latter joining system shows high strength but the values are scattered. The scatter appears to be associated with two factors: incomplete wetting and the formation of a nickel spinel (NiAIz04) reaction product along the ceramic/metal interface. In the Cu/80Ni-20Cr/Cu joining system, wetting is improved by tailoring the chemistry of the transient liquid phase and no reaction product is formed. Among the graded systems for the transient liquid phase joining of SbN4 that have been investigated, Au/80Ni-20Cr/Au interlayers have been found to produce bonds with failure strengths approaching those of the SbN4 [161].

100 90 80 70 60 50

'# 40 ->-:!:: 30 :s

CIS Jl e 20 Q. G) ... .2 'ji LL

10

0

x

40 60 80 100 150 200 300 Failure strength (MPa)

Figure 6.40. The failure probability vs. the failure strength for unbonded reference Ah03 (X); for a CulNi/Cu interlayer bonded alumina (D); for a Cr-precoated CulNilCu interlayer bonded alumina ( ... ); and for a Cu/80-Ni20Cr/Cu interlayer bonded AhOl.) [155].

Partial transient liquid phase bonding has been developed for joining SiC ceramics via nonmetallic multilayer interlayers. It is characterized by a composition gradient in the low melting phase that limits the amount of melt formed. (By comparison, the low melting interlayers in transient liquid phase joining melt completely.) A Ge interlayer is used as a transient liquid phase former in conjunction with core layers of Si or reaction bonded SiC [162]. In

219

joining bulk SiC using Ge interlayers and liquid Si, functionally graded interlayers have been shown to prevent interfacial reactions and to level out the shear stresses from potentially weak sharp interfaces to thicker interfacial layers [163].

When used as an insert material, an alloy of Al with O.S weight % Sn to prevent the formation of an undesirable oxide film on the surface of the aluminum, relaxes the thermomechanical stress between ceramics that are diffusion bonded with metals. Good bonds are obtained with the AI-Sn alloy (ceramic/AI-Sn alloy/metal) for many pairs of ceramics with metals when the bonding is carried out at temperatures (SOO°C - 600°C) below the melting temperature of Al [164].

6.3 Liquid Phase Joining

Eutectic bonding utilizes the presence of a liquid phase throughout the joining procedure. Typically, a metal and an intermetallic compound are combined using a eutectic reaction between both materials to create an amount of melt sufficient for joining. For example, in the production of FGMs of TilTbSn and TiITi5Si3 by this method, solidification during cooling results in bonding of the intermetallic to the metal part, and also to the formation of a compositional gradient perpendicular to the initial eutectic melt layer. At controlled slow solidification rates of the eutectic, the ratio of metal to intermetallic compound changes almost linearly between the two adjacent plates. Rapid cooling produces a less defined compositional gradient. The difference between the thermal expansion coefficients of Ti and the intermetallic compounds is much smaller than between metals and ceramics. Therefore only moderate stresses are created in the solidified transition layer [16S].

7. ADVANCED MANUFACTURING TECHNIQUES

This section describes three state-of-the-art techniques to manufacture FGMs: solid freeform fabrication or layered manufacturing, a rapid prototyping technique; sol-gel processing and fiber stacking, a chemical technique; and diffusion bonding utilizing superplasticity.

7.1 Solid Freeform Fabrication

Solid freeform fabrication (SFF) is a developing technology of potential commercial importance because of its capability for rapid prototyping. Parts

220 Chapter 6

made of engineering plastics, ceramics, metals, and composites, with complex geometries, can be produced in days rather than the weeks or even months required using conventional processing technologies. The ability to do cost-effective and iterative design with these techniques enables ceramic components to be created from the start using ceramic design guidelines, rather than fitting ceramic parts into component geometries designed for metals [166]. Solid freeform fabrication is a computer controlled, layer-bylayer, additive process in which a standard CAD (computer aided design) file of a part is first converted to a rapid prototyping format (an STL file) and sliced into a series of horizontal planes. The method for transferring this layered CAD design information into a physical reconstruction of the actual part depends on the SFF technology used.

There are many ways to categorize the increasing number of SFF processes [167-170]. Some typical SFF systems are shown schematically in Figure 6.41. Laminated object manufacturing (LOM) builds threedimensional objects with complex geometries from flat sheets of material that are laid down one at a time and laminated to the previous layer (Figure 6.41a) [171]. The perimeter of each layer is cut by a CO2 laser directed over its surface. Information from a CAD file is used to determine the laser's path and speed. The laser also cuts the excess material into cubes, which act as a support during the process but can be removed easily manually to reveal the completed part.

Laminated object manufacturing was originally designed to produce paper models of parts from adhesive paper. These are then used to test form and fit in the same way as foam or cardboard prototypes. Ceramic or metal parts (e.g., alumina, aluminum nitride, silicon nitride, hydroxyapatite, and stainless steel) have been made in a similar way by replacing the paper with tape-cast, flat sheets consisting of fine ceramic or metal particles dispersed in a polymer matrix. The thin sheets or tapes are easily cut by the laser because the polymer decomposes at high temperatures. The resulting part is then fired to burn off the polymer and sinter the ceramic or metal powder to form a dense object [172]. Parts with complex geometries can be built. The LOM process can be adapted readily to create compositionally graded structures by using tape cast sheets composed of varying proportions of two different materials. However, compositional gradients in LOM-built components are limited to the region of the part along the build axis.

In the SFF process, stereolithography, lasers are used to selectively cure resins layer-by-Iayer to build complex shapes (Figure 6.41 b) [173]. Thin layers of liquid photopolymer are spread and a laser beam is tracked over the surface of the reservoir. The polymer's curing is initiated by the laser, and only the regions exposed to the beam undergo polymerization. Then a new layer of liquid polymer is spread above the cured component, which is

221

lowered incrementally. The speed of the laser scan and the intensity of the beam determine the depth of polymerization. When the laser scanning is completed, the uncured material is drained off revealing the finished part. The presence of excess uncured polymer facilitates removing the part. When building complex shapes with features such as overhangs or undercuts, temporary structures are added to support them.

x-v scanning x-v scanning mirror on Galvanometer drives

Laser beam z Elevator

UV curable liquid

Supply roll

x-v positioning system II

galvanometer I P d flIlfl: I drives ower t'of"' . .

be spread. Ink'Jet pnnt head ROllerl1 : . .

Formed object spreader :~ Binder Jet I x , : Formed object

....... -'T'"''---<-....---.;_ - • • o

Powder Powder

-Piston

t..J Figure 6.41. Schematics of various SFF techniques. (a) Laminated object manufacturing (LOM) builds components by laminating and cutting sheets of paper or thin tapes of plastics filled with ceramic or metal powders. (b) Stereo lithography uses a laser to initiate curing locally in a reservoir of the molten polymer. (c) Selective laser sintering (SLS) builds parts from fusible powder by locally heating the surface of a powder bed with a laser. (d) Three dimensional printing (3DP) builds parts by ink-jet printing binder onto a powder bed followed by conventional powder processing of the green part built. (e) A heated extrusion head is used to shape three-dimensional objects in the fused deposition modeling (FDM) process. (t) Extrusion freeform fabrication (EFF) with dual extrusion cylinders and a mixing head can be used to build FGMs.

222 Chapter 6

Many of the commercially available stereolithography machines are used to produce acrylic or epoxy components for prototyping applications. Some of these components are used as patterns for casting processes, and have proven to reduce the lead time for new products effectively. Although most of the work in stereo lithography has been with polymeric parts, ceramicfilled polymer parts have also been produced [174]. Composite structures made of ceramic particles in a polymer matrix are fired to decompose the polymer and densify the ceramic. Parts with varying compositions could be fabricated by filling the liquid polymers with two or more different materials, and structures with a one-dimensional compositional gradient could be built as with LOM.

A laser is also used for the SFF process, selective laser sintering (SLS), shown in Figure 6.41 (c). In this case, laser energy is directed toward the surface of the layer to initiate localized bonding [175, 176]. The bonding mechanism for SLS, unlike stereo lithography, involves partial melting and fusing of a thermoplastic powder. The powder is spread in thin layers and the laser beam is traced over its surface. This causes the thermoplastic powder to melt in the regions of the powder bed exposed to the laser beam. Subsequent resolidification results in local bonding within each layer as well as bonding to the layer immediately below. The powder spreading and laser scanning steps are repeated until the part is completed. The part is retrieved simply by brushing away the loose particles, since the area not exposed to the laser remains unsolidified. Unlike the stereolithography process, support structures are not required since the packed powder particles are rigid enough to prevent the solidified portions from moving. Compositionally graded structures can be created by SLS in the same way as by LOM and stereolithography. Green (unsintered) parts containing compositional gradients can be built by spreading powder mixtures that have varying compositions.

In three-dimensional printing (3DP), the SLS technology has been adapted to form objects by selectively binding the powder particles (Figure 6.41 d) [177, 178]. First, a thin layer of a powdered material is spread in a container. The printhead assembly then scans over the powder bed, depositing binder droplets in selected regions. 3DP employs ink-jet printing technology to generate and place the binder droplets. When a layer has been printed, the floor of the powder container lowers to allow another layer of the powder to be spread. Information about the new layer is relayed from a computer and the next layer is printed. These steps are repeated until the final layer is printed. The binder bonds the powder locally within the layer and also to the layer below. Loose powder is removed, and the 3-dimensional green part is now ready for strengthening and densifying processing.

223

The 3DP process has been used to produce complex components in a wide range of materials including ceramics [179], glass [180], metals [181], and engineering plastics [182]. The support provided by the powder bed makes it possible to exploit the versatility of the process for creating complex parts with overhangs, undercuts, and internal volumes without requiring temporary support structures. In addition, different materials can be dispensed through separate nozzles, which is a concept analogous to color ink jet printing. The materials can be deposited as particulates suspended in a liquid, or dissolved in a liquid carrier, or in the molten state. The proper placement of binder droplets can be used to control the local composition and to fabricate components with true three-dimensional compositional gradients. Compositional control is a unique feature of SFF processes that use the deposition of material to build parts.

In the SFF process fused deposition modeling (FDM), the thermoplastic starting material is in the form of filaments that are fed to a heated extruder (Figure 6.41e) [183]. Fine beads (0.026 - 0.125 cm in diameter) of the molten plastic are extruded from the extruder head, which moves in the x-Y direction, onto a fixtureless platform that moves in the Z-direction. Threedimensional objects are made by computer control of the movement of both the extruder head and the platform. Material is extruded and deposited layerby-layer only in those areas defined by the CAD program. Control of both the extrusion head's translational velocity and the extrusion rate are required to ensure accuracy in the component's dimensions. The rheology of the molten thermoplastic needs to be tightly controlled, since the final quality of the part is closely related to the absence of knit lines between adjacent beads and also the ability to make sharp cuts in the extruded beads. FDM has recently been adapted for fabricating ceramic parts by using filaments made of a thermoplastic filled with ceramic particles [184]. Parts with compositional gradients can be built by loading the thermoplastic filaments with different materials.

The fused deposition of ceramics and metals (FDC and FDMet) have been demonstrated successfully with silicon nitride, lead zirconate titanate, alumina, silica, and hydroxyapatite ceramics, as well as stainless steel and tool steel. Parts made by FDM have microstructures and properties identical to those made by conventional powder processes. Four different materials can be deposited at a time with a recently designed machine that has four extrusion heads [185].

In Extrusion Freeform Fabrication (EFF) which is similar to FDM, layers of self-supporting viscous suspensions of highly loaded (20 volume % to 60 volume %) thermoplastics are sequentially deposited using a computercontrolled extrusion head that builds a 3-D body (Figure 6.41f) [186]. For fabricating FGMs, the EFF system has been modified by using two extruders

224 Chapter 6

that dispense different materials into a small mixing head. The composition of the two-component extrudate is controlled by proportioning the feed rate of the raw materials from the extruders, thereby creating a continuous compositional gradient in the final component. Examples of ceramic to metal graded compositions successfully made with EFF include alumina (Ah03) to nickel aluminide (NiAI), zirconia (Zr02) to NiAI, Ab03 to 305 stainless steel, Ah03 to the nickel-base superalloy, Inconel 625, and tungsten carbide (WC) to NiAI [187].

In an investigation to fabricate FGMs by EFF, eight polymerizable slurries were loaded with ceramics or metals including Ah03, Zr02, NiAI, TiB2, TiC, WC, 304 stainless steel, and Inconel 625 [188]. The slurries were processed by ball milling the raw powders in two different acrylate monomer vehicles with solids loadings of 44 to 58 volume %. Their rheological properties showed thixotropic behavior (thinning when under shear stress but thickening on standing). This is advantageous for the EFF process since it enables low pressure extrusion and accurate deposition of the freeformed material with minimal spreading of the layers once the slurry is deposited. Initiators such as peroxycarbonate are added to the slurries just prior to extrusion to enable rapid curing or gelation after their extrusion.

In order to produce FGMs, an EFF machine has been configured with dual extrusion cylinders having separate slurry reservoirs for the individual ceramic and metal slurries (Figure 6.41 f). The flow of the individual slurries is passed through a Y -block into a small mixing head containing an in-line static mixer and out through a deposition needle. The extrusion head then sweeps out the designed path while depositing the liquid slurries to build up a 3-dimensional FGM body. Figure 6.42 shows the microstructure of an Ah03-304 stainless steel FGM made by EFF [189].

" . !. '-1' mm

!

----------- .Iu~:.'i

Figure 6.42. Scanning electron micrograph of a section of an Ah03·304 stainless steel (SS) FGM produced by extrusion freeform fabrication (EFF) [168].

225

After hot pressing, the FGM billets have a slightly bowed appearance -concave on the metal side and convex on the ceramic side. This would be expected given the differences in the thermal contraction coefficients of the two different materials. The density of the pure metal end is close to 100% of theoretical, while the pure ceramic end is relatively dense (84-94%) despite the fairly low hot pressing temperature (1175°C -1350°C).

Other systems for the localized deposition of material that use precision heads for laser surface cladding also are being used to make rapidly prototyped metal and ceramic parts [89]. The fabrication of threedimensional objects without the use of tooling has rapidly progressed from simple models to complex functional prototypes. Parts can be produced in a wide variety of materials including wax, thermoplastics, thermosets, photopolymers, paper, metals, ceramics, and glass fiber reinforced composites. It is now possible to create one, two, and three dimensional FGMs by several SFF techniques.

t-ZTA

Compositionally Graded Layer

~1~~+-+-~I~I~-4-+~1

o 1.5 3 mol% Y20 3

Figure 6.43. Schematic of a 3D-printed zirconia toughened alumina (ZrA) multilayer plate.

In the alumina-zirconia (Ah03-Zr02) ceramic system, zirconiatoughened alumina (ZTA), the continuous phase is alumina (70-95%) and the second phase is zirconia particulates (5-30%). The transformation of the monoclinic to the tetragonal phase (transformation toughening) and the stabilization of the tetragonal phase in the zirconia by addition of a suitable dopant results in the strengthening and toughening of this ceramic system [190]. Three-dimensional printing can be used to build a ZTA component

226 Chapter 6

with a compositional gradient. First a mixture of spray dried particles of alumina and undoped zirconia is spread, and next a polymeric binder is selectively deposited onto the powder layer to define the macroscopic shape of the component. A second nozzle then prints an aqueous solution of the yttrium nitrate (YN03k6H20 dopant onto selected areas of the part. The proportion of the tetragonal Zr02 in the final component is controlled locally by appropriate selection of the process parameters. Figure 6.43 shows a schematic of a 3D-printed ZTA plate with a compositional gradient. Once the 3DP build process is completed, post processing including isostatic pressing, binder burnout, and pressureless sintering is carried out to fully densify the component.

Figure 6.44 shows the relative volume fraction of the two different crystalline forms of Zr02 (monoclinic and tetragonal) as a function of position, based on x-ray diffraction analysis (XRD). The linear variation of the monoclinic Zr02 along the thickness illustrates successful fabrication of the compositionally graded multilayered ZTA by 3DP. Deviation from linearity near the monoclinic ZT A surface can be explained by the presence of very fine Zr02 particles that are not large enough to transform.

100 • •

C\I 0 80 • .... N

I

E '0 60 • c 0 • :;:= -0 40 ~ • -(]) E 20 • ::J

0 >

0% 0 250 500 750 1000

Position (/lm)

Figure 6.44. The calculated relative volume fraction of monoclinic zirconia content through the thickness of ZTA multilayer plates.

The evolution of residual stress and curvature within ZTA multilayers with a compositional gradient has been studied [191]. The mismatch in both the coefficients of thermal expansion and the zirconia transformation strain

227

were used to derive the analytical solutions for the residual stress and the curvature. The experimentally measured curvature from a 3D-printed ZTA multilayer was 6.4 m- I while the predicted values ranged from 3.9 to 5.7 m-\ shown in Figure 6.45 as the shaded area. Such analytical tools can be used to design compositional gradients within components to optimize the mechanical properties of 3D-printed ZT A parts.

0.5

E .s 0.3

c 0 0.1 fj

Measured Curvature (S.4m')

\. ~ Qj -0.1 0

\ Predicted curvature based on 3-5% Zr02 expansion (3.9 - 5.7m")

-0.3 -15 -10 -5 o 5 10 15

Distance from the center of the plate (mm)

Figure 6.45. The calculated and the measured curvature in 3D-printed ZTA multilayer plates.

Another example of an SFF-derived component with localized compositional variation is a 3D-printed drug delivery device (DDD). This is an example of a nonstructural application for a compositionally graded material. It is used for controlling the release rates of different drugs. The macroscopic configuration of a model device 3D-printed with a plastic matrix that undergoes hydrolytic degradation is shown in Figure 6.46. A number of compartments are built into this device and various drugs are inserted appropriately. When a DDD is inside a patient, either by implantation or ingested orally, resorption of the plastic matrix results in release of the drugs from the micro-reservoirs. The lag time required to release drugs from each of the compartments can be designed to meet the specific prescriptions of individual patients. The composition and microstructure of the matrix as well as the thickness of the compartment walls strongly affect both the rate and the mechanism of resorption. In a preliminary investigation, the controlled release of various dyes from 3Dprinted model devices was tested with promising results [189, 193].

7.2 Graded Ceramic Matrix Composites Made via a solgel technique

Graded ceramic matrix composites can be fabricated by a number of methods [193]. One innovative process, illustrated in Figure 6.47, fabricates

228 Chapter 6

functionally graded ceramic composites reinforced with woven fibers that can be oriented in different directions in the same composite. Sol-gel techniques are used to combine several different fiber and matrix materials that have various desired properties [194, 195]. First, the fibers in a twodimensional woven fabric are impregnated with a fiber-matrix interface precursor (a liquid alkoxide mixture) that reacts with the fibers to form an interface between the fibers and the matrix to minimize reaction between them. It also acts as a diffusion barrier to protect the fibers from chemical attack. Depending on both the matrix material and the fiber, several different interface precursors can be used either alone or in various combinations, as indicated in Table 6.2 [196, 197, 198]. The liquid alkoxide interface precursor mixture slowly hydrolyzes and polycondenses, forming a gel by reacting with atmospheric moisture.

(a)

(b)

P !=J Q Q

J Q 12. [\-- !=J Q

P. !=J Q (c)

Figure 6.46. (a) Schematic of a 3D-printed model drug delivery device (DOD). The top lid is open to show the internal shape. (b) A DOD with strategic placement of two different drugs. The differently shaded circles represent different drugs placed in the compartments. (c) A DOD with compositional variation to control the drug release rate. The thick lines represent the walls with a substantially lower rate of desorption than the walls represented by thin lines.

Tributyl Borate.In the next processing step, a fine, porous amorphous matrix precursor powder suspended in chi oro benzene is deposited on the

229

layers of woven fiber fabric. This powder is prepared by rapid hydrolysis of the appropriate mixture of alkoxides in propanol. The resulting gel is dried (750°C) to drive off most of the water in the polymeric oxide network. In the final processing step, the doubly impregnated layers of woven fabric are stacked in a graphite mold and hot pressed (1000°C - 1400°C) [199]. A liquid sintering aid (B20 3 or Ge03) is also used, which is eliminated either by incorporation into the matrix or by volatilizing. Therefore it does not contribute to the formation of second phases at the grain boundaries.

Special additives fj\ Woven fabrics preparation whiskers,powder I..:J ~

A ®~ Impregnation with the interface precursor

Matri~ precursor Infiltration + (ultraflne powder) Interface in situ gelation .. (~r~g~i~~~) ~hYdrOIYSiS-POIYCOndensatiOn)

DisperSion - - - ---, (+ SOlventS\ n cycles ® eventually

L~ J ~

--- .. Deposit of the

matrix precursor Stacking-sticking ~ Hot-pressing

@ _ ..... ~ ..... I Composite I 4 0 . . . . . . ~

Drying 0. . . . . . ®

Figure 6.47. Flow diagram of the sol-gel process for fabricating functionally graded ceramic matrix composites incorporating different fibers and matrices in the same composite [198].

Porosity in the ceramic matrix is minimized « 5%) by the use of a very reactive matrix precursor plus an interface precursor. This results in the formation of a liquid phase that acts both to densify the matrix (via liquid phase sintering) and to maximize the contact between the grains of the matrix precursor during the hot pressing step.

In order to improve the linear elastic limit of the composite, the matrix is further reinforced via submicrometer and nanometer size precipitates [196]. Since in the sol-gel process the homogeneity of the liquid alkoxide mixture that forms the gel is maintained up to the nucleation stage, which occurs at a relatively low temperature, the precipitates formed are in the submicrometer range. Because of the presence of these reinforcing submicrometer particulates, cracks are arrested immediately after their initiation. The net effect of this sol-gel process is to produce what could be regarded as a multi scale reinforced composite, which is reinforced at the millimeter scale

230 Chapter 6

by the woven fiber network, at the micrometer scale by the fiber-matrix interface, and at the nanometer scale by the precipitate/matrix interface.

Table 6.2. Examples of Fiber Reinforced Ceramic Matrix Composites with Functionally Tailored Layers. Composite (M/F)

Components

Mullite/SiC Mullite/SiC

MullitelMullite MullitelMullite NASICON/SiC

NASICONlMullite Celsiank/SiC CeisianJ/SiC

Zr02/SiC

3Ah03·2Si02·0.1B203. b Na29Zr2Si1.9P1.1012'

Matrix (M) Mullite' NASICONb

No

Mullite Mullite/Zr02

Mullite Mullite NASICON

Celsian Celsian

Zirconia (Zr02)

Interface Fiber (F)

No No

No Nicalon NLM 202c

NexteI 440d

ALOSi"+ TBBf Nicalon NLM 202c

ALOSI+ TBB/ZPg Nicalon NLM 202c

TEOGeh

ALOSI+TB NexteI 440 ZP Nextel440 ZP+ TBPi+ TBB Nicalon NLM 202c

+TEOSi Nextel440

ALOSI+TBB Nicalon NLM 202c

ZP+TBP+TBB Nicalon NLM 202c

+TEOS TEOS+TBB+ Nicalon NLM 202

CaSm/ZP+CaS

c Two-dimensional fabrics woven along the four directions within the plane. d Satin weave fabric. e Aluminum silicon ester f [(OBuh-Al-O-Si (OEthM g Zirconium propoxide. h Germanium ethoxide. i Tributyl phosphate.

i Tetraethylorthosilicate. k BaAhShOs. J Al20 doped CeIsian. m Calcium at 750·C in air

Figure 6.48 illustrates graphically the relative shrinkage of stacked layers of doubly impregnated woven fibers during the hot pressing cycle in the formation of a SiC reinforced mullite composite. At the beginning of the cycle, a low pressure of 2-5 MPa is applied to maintain optimum contact between the impregnated woven fiber layers. Above 500°C, when the polymerized interface precursor has been dehydrated, the pressure is increased to 25 MPa, which is just below the densification temperature (T d) of the matrix and the melting point of the sintering aid. By selecting the appropriate interface precursors, the dwell temperature required to achieve

231

maximum densification (T d = 200°C - 300°C) can be raised or lowered about lOO°C. This makes it possible to combine several different kinds of impregnated fiber/matrix layers in the same composite (for example using A and B materials as the matrix, the following associations are possible: AB, AABB, ABAB, CABBAC, etc.) thereby tailoring the physical and lor chemical properties for a particular application.

OJ Cl

Jl1 c: ~ Ul OJ > ~ Qj a:

,

, , , ,

dwell ., , -_.

2 3 h time

Figure 6.48. Shrinkage plots during the hot pressing cycle for two stacks of fabric layers (mullite matrix reinforced with SiC fibers) impregnated with different interface precursor mixtures. The arrows indicate the pressure increases. The dwell temperatures are l350DC and l300DC [198].

1 mm _ ::;;a::sr

Figure 6.49. Scanning electron micrographsofpolished sections of two different fiber reinforced ceramic composites. a) Four woven Nexte1440™ layers (a mullite-like aluminosilicate with low permittivity) with two layers ofNicalon NLM 202™ fibers (silicon carbide with semiconducting properties) and a NASICON matrix (Na29Zr2SiI9PU012, a superionic conductor). The Nextellayers lower the permittivity of the outer surface of the composite thereby favoring the transmission of electromagnetic energy, whereas the Nicalon layers create high dielectric losses and increase the composite's mechanical strength. b) In this composite, a mullite matrix (gray) plus a zirconia matrix (white) both reinforced with silicon carbide fibers combine mullite's good chemical stability with zirconia's hardness [198].

232 Chapter 6

Figure 6.49 shows examples of fabricated composites that combine various kinds of woven fibers (e.g., SiC, mullite) with various kinds of matrix materials (e.g., NASICON, celsian, zirconia, mullite). For example, radar absorbent composites can be made by combining materials that have different electromagnetic properties. While composites for durable aircraft components can be made by combining materials that have high hardness, such as zirconia, with those that have good corrosion resistance, such as mullite and celsian [200].

7.3 Superplastic Forming Combined with Diffusion Bonding

In superplastic forming certain fine-grained materials, ceramics as well as metals, can undergo exceptionally high tensile elongation at high temperatures, which makes it possible to produce near-net-shape large and complex-shaped components. Superplastic forming methods (e.g., blow molding, vacuum forming, thermoforming) are characterized by low strain rates, and can be particularly advantageous for materials that are difficult or costly to machine, such as metal matrix composites, intermetallic compounds, ceramics, and FGMs. Combining superplastic forming with simultaneous diffusion bonding further extends the fabrication range for large and complex-shaped components [201,202].

In conventional solid state diffusion bonding, the surfaces of components from similar or dissimilar materials are pressed together at elevated temperatures. During the initial bonding stage the original contact areas are increased by elastic and plastic deformation and grain and/or phase boundaries develop in the contact areas. The residual pores at the interfacial region are eliminated by grain boundary and volume diffusion. The rate of elimination depends on the applied pressure and the surface curvature of the pores as well as their size and shape. The size of the pores is reduced if the initially contacting surfaces of the components fit smoothly and if the flattening of the initial points of contact by plastic deformation occurs at low pressures. Thus materials with high yield strength and rough surfaces usually require high pressure and high temperatures (close to the sintering temperature) for complete diffusion bonding.

By using a superplastic metal or ceramic as the basic material for a component, or as an insert in the joining area, the temperature and pressure for plastic yield are substantially reduced. Therefore superplastic diffusion bonding makes it possible to achieve optimal bonding at moderate conditions without requiring mirror polishing of the bonding surface. At present, superplastic forming combined with diffusion bonding is used primarily for manufacturing aircraft parts from titanium alloys. The high cost

233

of this process can be justified by the increased flexibility of design, the reduced number of mechanical joints, and a significant reduction of the number of parts therefore reduced assembly costs.

Figure 6.50. The interfaces of a stack of six thin sheets each about 180 I-lm thick with Alz0 3

mole fractions ofO, 20, 40, 60, 80 and 100 % respectively, formed by diffusion bonding during superplastic deformation of the stack (the initial surface roughness of the thin sheets is 3 I-lm, the deformation rate is 0.01 mm/min , and the total strain is 17%) [201].

234 Chapter 6

It has been shown that a graded sheet 1 mm thick can be produced in the Zr02 (Y 203)-Ah03 composite system, by diffusion bonding a stack of six very thin sheets (each about 180 /lm thick) containing increasing mole fractions of Ah03 of 0, 20, 40, 60, 80 and 100 % respectively [203]. During diffusion bonding the total strain of the superplastic stack is 17 %. The internal interfaces of the graded sheet can be seen in Figure 6.50.

As innovative techniques for processing FGMs have been developed, these have led to many interesting new applications, as can be seen in the next chapter. However, the reverse is also true, creative ways to use FGMs have motivated the development of many of the innovative processes described in this chapter.

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2. Miyamoto, Y., Niino, M., and Koizumi, M. (1997) FGM research programs in Japan, from structural to functional uses, in Proc. of The Fourth Int 'I Symp. on FGM'96, (eds. I. Shiota and Y. Miyamoto), Elsevier Science B.V., Amsterdam, 1-8.

3. Ford, R.G. and Stangle, G.C. (1993) in High Temperature Ceramic Matrix Composites, Proc 6th EACM-HTCMC, 20-24 Sept. 1993, Bordeaux, (cds. R. Naslain, J. Lamon, and D. Doumeingts), Woodhead Pub!. Ltd., 795-811.

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11. Scherer, G. W. (1987) Sintering with rigid inclusions, JAm. Ceram. Soc., 70, 719-25. 12. Ilschner, B. (1996) Special contribution to this book. 13. Miller, D.P., Lannutti, 1.1., and Yanecy, R.N. (1992) Functionally gradient NiAlIAl20 3

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18. Joensson, M., Birth, U., and Kieback, M. (1997) Gradient components with a high Melting point difference, in Proc. of The Fourth Int'l Symp. on FGM'96, (eds. I. Shiota and Y. Miyamoto), Elsevier Science B.V., Amsterdam, 167-72.

19. Hong, C-W., Muller, F., and Greil, P. (1997) Fabrication of pore-gradient membranes via centrifugal casting, ibid., 173-78.

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26. Lange, F.F. and Hirlinger, M.M. (1984) Hindrance of grain growth in alumina by zirconia Inclusions, JAm. Ceram. Soc., 67(3), 164-68.

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33. Kawasaki, A, Li, J.F., and Matsubara, T. (1994) Fabrication of thermal barrier type of SiC-AlNlMo FGM by hot isostatic pressing and their thermomechanical performance, in Proc. of the Third Int 'l. Symp. on Structural and Functional Gradient Materials, (eds. B.

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36. Takemura, M. et al. (1990) Mechanical and thermal properties of FGM fabricated by thin sheet lamination method, in Proc. of The First International Symposium on FGM'90, Sendai (eds. M. Yamanouchi, M. Koizumi, T. Hirai, and I. Shiota), Functionally Gradient Materials Forum, 97-100.

37. Yuan, R.Z. et al. (1993) Design and fabrication of a MgOlNi functionally gradient material, J. Materials Synthesis & Processing, I, 171-179.

38. Zhang, L.M. et al. (1994) Preparation ofTiCINi3AI FGMs, sintering and structure, in Proc. of the Third Int 'I Symp. on Structural and Functional Gradient Materials, (eds. B. Ilschner and N. Cherradi), Presses Poly techniques et Universitaires Romandes, Lausanne, 59-64.

39. Yuki, M. et al. (1990) Temperature gradient sintering ofPSZlMo functionally gradient material by laser beam heating, in Proc. of The First Int 'I. Symp. on FGM'90, Sendai (eds. M. Yamanouchi, M. Koizumi, T. Hirai, and I. Shiota), Functionally Gradient Materials Forum, 203-208.

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172. Griffin, C., Daufenbach, J., and McMillin, S. (1994) Desktop manufacturing, LOM vs pressing, Am. Cer. Soc. Bull., 73[173], 109-103.

173. Jacobs, P.F. (1992) Rapid Prototyping & Manufacturing, Fundamentals of Stereolithography, First Edition, Society of Manufacturing Engineers, Dearborn MI.

174. Griffith, M.L. (1995) Stereolithography of ceramics, Ph.D. dissertation, Department Materials Science and Engineering, The University of Michigan, Ann Arbor, MI.

175. Deckard, C. and Beaman, 1 (1987) Recent advances in selective laser sintering, in Proceedings of the Fourteenth Conference on Production Research and Technology, 447-452, University of Michigan, MI.

176. Vail, N.K., Barlow, lW., and Marcus, H.L. (1993) SiC preforms for metal infiltration by selective laser sintering of polymer encapsulated powders, in Proceedings of the SFF Symposium, 204, University of Texas, Austin, TX.

177. Sachs, E.M. et al. (1992) Three-dimensional printing, rapid tooling and prototypes directly from a CAD model, 1. Eng. Ind., 114,481-488.

178. Sachs, E. et al. (1992) CAD-casting, The direct fabrication of ceramic shells and cores by three dimensional printing, Man. Rev., 5[167], 117-126.

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179. Yoo, J. et al. (1993) Structural ceramic components by 3D printing, in Proceedings 0/ the SFF Symposium, 40-50, Univ. of Texas, TX.

180. Cima, M.J. et al. (1995) Structural ceramic components by 3D printing, in Proceedings o/the SFF Symposium, 479-488, Univ. of Texas, TX.

181. Michaels, S., Sachs, E.M., and Cima, M.J. (1992) Metal parts generation by three dimensional printing, in Proceedings o/the SFF Symposium, 244-250, Univ. of Texas, TX.

182. Cima, M.J. et al. (1994) Computer-derived microstructures by 3D printing bio and structural materials, in Proceedings o/the SFF Symposium, 181-190, University of Texas, TX.

183. Wales, R. and Walters, B. (1991) Fast, precise, safe prototypes with FDM, in Proceedings o/the SFF Symposium, 115, University of Texas, Austin, TX.

184. Agarwala, M.K. et al. (1995) Structural ceramics by fused deposition of ceramics, in Proceedings o/the SFF Symposium.l-8, University of Texas, Austin, TX.

185. Danforth, S.C. (1998) Rutgers University, Piscataway, New Jersey, private communication.

186. Hilmas, G.E. et al. (1997) Advances in the fabrication offunctionally graded materials using extrusion free form fabrication, in Proc. o/The Fourth Int 'I. Symp. on FGM' 96, (eds. I. Shiota and Y. Miyamoto), Elsevier Science B.V., Amsterdam, 319-324.

187. Hilmas, G. (1996) Innovative technique for rapidly prototyping parts of polymers, metals, ceramics, composites, and functionally graded materials, Materials Technology, 11(6),226-228.

188. Hilmas, G.E. et al. (1998) Recent developments in extrusion freeform fabrication(EFF) utilizing non-aqueous gel casting formulations, in Proc. Solid Free/orm Fabrication Symposium 1996, (eds. H.L. Marcus, 1.L. Beaman, J.W. Barlow, D.L. Bourell, and R.H. Crawford), University of Texas, Austin.

189. Borland, S.W. et al. (1995) Solid freeform fabrication of reticulated structures from biomedical polymers, Therapeutic Medial Devices by 3D Printing Research Summary, June, 1994.

190, Cannon, W.R. (1989) Transformation toughened ceramics for structural applications, in Structural Ceramics, Treatise on Materials Science and Technology, (ed. J.B. Wachtman), 195-228, Academic Press, San Diego, CA.

191. Yoo, J. et al. (1996) Transformation-toughened ceramic multilayers with compositional gradients, JAm. Cer. Soc., 81, 21-32.

192. Wu, B.M. et al. (1995) Solid freeform fabrication of drug delivery devices, J 0/ Controlled Release, 40, 77-87.

193. Ford, R.G. and Stangle, G.c. (1993) Compositionally gradient materials-unconventional composites, in High Temperature Ceramic Matrix Composites, Proc.6th EACM-HTCMC, Bordeaux, (eds. R. Naslain, 1. Lamon, and D. Doumeingts), Woodhead Pub!. Ltd., 795-811.

194. Colomban, P. (1996) special contribution to this book. 195. Colomban, P. and Vendange, V. (1992) Sintering of alumina and mullite prepared by

slow hydrolysis of alkoxides, the role of the protonic species and of pore technology, J Non-Crystalline Solids, 1471148,245-250.

196. Colomban, P. et al. (1992) French Patent n02672283, European Patent n092400235.5, US Patent n0071830.904.

197. Colomban, P. (1995) Process for fabricating a ceramic matrix composite incorporating woven fibers and materials with different compositions and properties in the same composite, Materials Technology, 10(5/6),93-96.

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198. Mouchon, E. and Colomban, Ph. (1996) Microwave absorbent-preparation, mechanical properties and r.f.-microwave conductivity of SiC (and/or mullite) fiber reinforced Nasicon matrix composites, J Materials Science, 31, 323-34.

199. Colomban, P. (1989) Gel technology in ceramics, glass ceramics and ceramic-ceramic composites, Ceramics International, 15,23-50.

200. Colomban, P. and Mazerolles, L. (1991) Nanocomposites in mullite - zrOz and mulliteTiOz systems synthesized through alkoxide hydrolysis gel routes, microstructure and fractography, J Materials Science, 26, 3503-3510.

201. Nagano, T. (1996) special contribution to this book. 202. Nagano, T., Kato, H., and Wakai, F. (1990) Diffusion bonding ofzirconialalumina

composites,} Am. Ceram. Soc., 73(11), 3476-3480. 203. Nagano, T. and Wakai, F. (1993) Fabrication of ZrOz-Alz0 3 functionally gradient

material by superplastic diffusion bonding, J Materials Science, 28(21), 5793-5799.

Chapter 7

APPLICATIONS

Keywords: Apollo, space plane, SiC/C FGM, HOPE, CVI, rocket combustor, TBC, Nicalon® SiC fibers, Nextel® mullite fibers, Nasicon, EB-PVD, diesel engine, titanium aluminides, TiAI, thermal barrier coatings, superalloy, turbine blade, creep, ductility, solar receiver, thermionic converter, C/C cavity, thermoelectric converter, figure of merit, magnetic fusion reactor, W/Cu, fuel cell, SOFC, cutting tool, cemented carbide, WC-Co, throwaway chip, diamond/SiC, razor blade, Si3N4-Cu FGM, graded porosity, gas-reinforced materials, GASARs, graded bandgap, graded semiconductor, heterojunction, heterostructure, quasi-electric field, bipolar transistor, graded-index, GRIN, refractive index, step index, optical glass fiber, multimedia, optical polymer fiber, CVD, V AD, hip joint, Ti-6AI-4V, bone material interface, implant, porous metal, bone cement, hydroxyapatite, orthopedics, bioactive ceramics, biocompatibility

1. INTRODUCTION

The FGM concept is applicable to almost all material fields. Examples of a variety of real and potential applications in transport systems, energy conversion systems, cutting tools, machine parts, semiconductors, optics, and biosystems are described in this chapter. Potential applications include those structural and engineering uses that require combinations of incompatible functions such as refractoriness or hardness with toughness, or chemical inertness with toughness. In aerospace and nuclear energy applications, reliability rather than cost is the key issue. But in applications such as cutting tools, high temperature rollers, and engine components, which require wear, heat, mechanical shock, and corrosion resistance; the key issues are the cost/performance ratio and reliability.

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Two applications now commercialized are high performance cutting tools of tungsten carbide/cobalt (WC/Co) based FGMs and razor blades of an iron aluminide/stainless steel FGM (FeAlISS). A more reliable hip prosthesis in which a biocompatible graded interface forms between the bone and the implant has been demonstrated successfully clinically. The successful FGM applications for cutting tools and biomaterials have in common that their graded structures are formed in situ, and their relatively small size makes it feasible to process them cost effectively.

To protect the environment and conserve nonrenewable energy resources, energy conversion systems that do not produce atmospheric pollutants such as thermionic and thermoelectric converters, fuel cells, and solar batteries are under active development. The incorporation of the concept into their total FGM design can improve their conversion efficiency by optimizing their electronic potential, thermal stress resistance, and chemical durability.

Although the practical applications of FGMs have not yet been fully realized and exploited, it is the belief of the contributors to this book that the impact of the FGM concept itself, which can potentially impact almost all materials research and development, will begin to emerge in the 21 st century.

2. TRANSPORT SYSTEMS

2.1 Space Vehicle Components

2.1.1 Vehicle protection during reentry

Space vehicles flying at hypersonic speeds experience extremely high temperatures from aerodynamic heating due to friction between the vehicle surface and the atmosphere. Two types exist or are on the drawing board: vehicles like the U.S. space shuttle and the capsules formerly used for the Apollo missions that are launched vertically into space by a rocket propulsion system; and fully reusable spacecraft planned during the late 1980s such as the U.S. National Aerospace Plane (NASP), the Japanese Single Stage to Orbit (SSTO), and the German Sanger program. These latter are all based on a horizontal takeoff either from a ground-based runway or from a horizontally flying carrier.

In the first type, after sufficient acceleration the rocket system completely separates from the space vehicle. During reentry at velocities greater than 11 km/s, rapid heating of the leading edge, where the heat protection shield is

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located, occurs at altitudes between 120 and 50 km, and maximum temperatures (the radiant equilibrium temperature) above 2500·C develop. Because the relatively flat heat protection shield is exposed to the extreme heat of reentry for just a few minutes and is used only one time, it can be fabricated from ablative materials.

The reentry velocity of the U.S. space shuttle at an altitude of 120 km is below 8 km/s, and the maximum temperature experienced is about 1500°C for a few minutes. Structural components that experience the maximum exposure to heat such as the nose cone, the leading edges, the rudder, and the flapperons need to be made of non-metallic carbon/carbon composites (C/C) with adequate oxidation protection coatings. Nickel and titanium alloys can be used in other areas of the spacecraft that are exposed to less extreme heat. However, this may result in weight penalties. For temperatures up to 1200°C, ceramic tiles can be used, and for temperatures from 300°C to 550°C, alternative thermal barrier protection systems based on multi wall Ti sheets can be used.

Horizontally launched space planes that are accelerated by air-breathing engines (e.g., jet engines) fly in the atmosphere at hypersonic speeds for a longer time than vehicles launched vertically by rockets. Therefore, the space plane experiences its maximum exposure to heat during its launch into space. Initially, one of the main objectives of investigating FGMs deposited by chemical vapor deposition (CVD-FGMs) was to develop thermal barrier coatings for a space plane. In a comparison test, models of the components of a nose cone (hemispherical C/C composites 50 mm in diameter) were coated with an ungraded 100 ).lm thick protective layer of SiC (silicon carbide). Similar C/C composite models were coated first with a graded SiC/C FGM by penetrative CVD followed by deposition of the 100 ).lm thick SiC protective layer. All the coated nose cone models were subjected for 1 minute at 1900°C to a supersonic gas flow (at Mach 3) containing an amount of oxygen approximately equal to a standard atmosphere. The nose cones with the SiC/C FGM intermediate layer showed no discernible change in structure even after 10 cycles. In contrast, those without the intermediate SiC/C layer between the C/C substrate and the ungraded SiC coating deteriorated after the first cycle [1]. Sheets of SiC/C FGMs produced by CVD provide excellent thermal stability and thermal insulation at 1227°C, as well as excellent thermal fatigue properties and resistance to thermal shock [2].

2.1.2 Rocket and scramjet engines

A C/C combustion chamber with a SiC/C FGM protective layer has been developed for the reaction control system engine for HOPE, a Japanese

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space shuttle under development. The simplified design shown, schematically in Figure 7.1, has high heat resistance at thermal conditions simulated by a hot gas flow test [3]. The 30 Jlm thick graded layer of SiCIC was made by chemical vapor infiltration (CVI) and subsequently coated with a 100 Jlm thick SiC layer by CVD. Repeated hot gas flow tests indicated that the FGM coated material was very resistant to delamination and cracking [4]. However, after stationary or pulsed combustion for 500 seconds, some partial delamination and corrosion occurred at the SiC layer. This was attributed to the cyclic thermal stress along the inner wall of the thrust chamber [5]. Figure 7.2 shows the SiCIC FGM coated engine during a test [3].

SiC layer (100l-lm) CVD/CVI

FGM layer (40l-lm) ~

Inj~-:.I.. ~ NTO c;> ~=--'

MMHC;>r:-I~ ~~L;~

CIC combu stion chamber

~--------248~--------~

Figure 7.1. Schematic of the carbon/carbon (C/C) composite combustion chamber for the engine of the reaction control system of the Japanese space shuttle, HOPE, with an FGM protective layer of silicon carbide/carbon (SiC/C) [3]. The propellants are NTO (nitrogen tetroxide: N20 4) and MMH (monomethylhydrazine: N2H3CH3).

Other CVD-SiCIC FGMs produced for rocket combustors have undergone critical tests with nitrogen tetroxide and monomethyl hydrazine propellants at firing cycles of 55 seconds with subsequent quenching by liquid nitrogen. The maximum outer wall temperature of these model combustors was 1376°C to 1527°C, while the inner wall temperature reached 1677°C to 2027°C. No damage to the combustors was observed after two test cycles [I].

Most rocket engines use TBC (thermal barrier coating) materials that have been previously developed for turbine engine applications. The heat flux in the path of the hot gases is much greater in rocket engines than in turbine engines. Here the TBCs are exposed to a hostile environment, that is higher temperatures and more severe thermal transients, but for shorter mission cycles. Hence, the TBCs are mainly deposited as thin structures «

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0.2 mm thick) to reduce the probability of coating failure. In large combustion chambers, the heat flux is so high that high conductivity copper alloys are used to diffuse the heat away from the inner surface. In this application, TBes are not ordinarily used because the heat cannot be dissipated fast enough to avoid local hot spots and coating failures [6].

Figure 7.2. The engine shown in Figure 7.1 during a test carried out in both stationaJ)' and pulsed modes using the mixed propellants NTO and MMH). (Photo courtesy of National Aerospace Laboratory, Japan.) [3].

In large liquid propellant rocket engines, TBes are mainly used in the high pressure hydrogen and oxidizer turbopumps shown in Figure 7.3 [6]. In the production of both hydrogen and oxidizer turbopumps, TBes have been used as liners for the spark igniters and pre burners, for turbo housing liners, for turbine blade shanks (located between the blade platform and root), and for vane shrouds. Experimental coatings have been used on the turbine blade platforms and vane airfoils. In addition, TBes have potential applications in the upper part of the main combustion chamber as coatings on the interpropellant plate, spark igniter, and injector baffle tips.

Graded TBes have been considered also for other rocket engines such as small regeneratively cooled thrust chambers in orbital maneuvering systems [7]. These zirconia/nickel (Zr02INi) FGM chambers are prepared by a combination of galvanoforming and plasma spraying. The graded layer is first deposited (up to 25% Zr02 on a Ni metal chamber) by galvanoforming and subsequently coated to 100% Zr02 by plasma spraying. No delamination

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of zr02 was observed after 550 seconds of combustion. In order to assure the reliability of the Zr02/Ni FGM, it was necessary to engineer the microstructure to form strong layers as well as to further optimize the graded structures, and also to control the reaction with a propellant [8]. As noted above, graded TBCs are potentially applicable for engine and airframe structures in reusable hypersonic vehicles [9].

HIGH PRESSURE HYDROGEN TURBOPUMP

MAIN COMBUSTION CHAMBER

HIGH PRESSURE OXIDIZER TURBOPOMP

Figure 7.3. Cross sectional schematic of a rocket engine showing the potential location of thermal barrier coatings (TBCs) in the high pressure hydrogen turbopurnp (left), main combustion chamber (center), and high pressure oxidizer turbopurnp (right) [6].

2.1.3 Stealth missiles

Stealthiness is now a required specification for modern weapons. Parts made of specific materials can be used to absorb the emitted electromagnetic energy to minimize waves reflected in the direction of the enemy radar receiver. In some applications, e.g. high velocity missiles, the materials can be subjected to high thermomechanical stress. For these applications, the most promising new materials are ceramic matrix composites reinforced

253

with ceramic woven fabrics. The use of long, continuous ceramic fibers embedded in a refractory ceramic matrix creates a composite material with much greater toughness than monolithic ceramics, which have an intrinsic inability to tolerate mechanical damage without brittle rupture.

The conducting properties of these ceramic composites depend on the fibers, the matrix, the interfaces, and other parameters such as the topology of the arrangement of the various phases. Nicalon® SiC fibers, which have semiconducting properties, and Nextel® mullite (3Ah03_ 2Si02_ 0.1 B20 3) fibers, which are completely dielectric, are used in the preparation of oxide matrix ceramic composites. Nasicon matrix composites reinforced with long semiconducting and/or dielectric fibers can have mechanical and electrical properties, ranging from dc to microwave frequencies [10]. The Nasicon solid solution, structural formula Na\+xZr2SixP3-x012 (0 ::s; x ::s; 3), which has an electrical conductivity that varies by four orders of magnitude as a function of x, is a useful system for investigating the preparation and properties of ceramic matrix composites with tailored microwave properties.

2.2 Aeroengines

Thermal barrier coatings are used for military and commercial aeroengines as well as for gas turbine engines for automobiles, helicopters, marine vehicles, and electric power generators [6, 11]. The TBCs are mainly used where hot-gas pathways are located in order to increase the temperature of the turbine inlet. Compared with coatings for diesel engines, the coatings for turbine engines operate under higher heat fluxes, higher temperatures, and greater thermal transients [12]. In addition, turbine engine coatings are subject to hot corrosion and particulate erosion. Coatings for the hot-gas pathways are usually thin « 0.4 mm) to reduce spalling. For other applications, e.g. seals that are not in the path of the hot gas, the coatings can be thicker.

Figure 7.4 shows a schematic cross section of a military turbine aircraft engine with a wide variety of TBC applications. Although the commercial turbine engines have similar applications for TBCs, they do not have augmentor (afterburner) or nozzle sections. TBCs in commercial aircraft are deposited on the inside liners (i.e., panels and walls) of combustors where the fuel ignites with air [13], and on the platforms of turbine vanes and blades where the hot gases expand into the turbine section [14]. Coatings on the airfoils of blades and vanes where the temperatures, thermal fatigue, and corrosion are critical are largely produced by electron beam-physical vapor deposition (EB-PVD ). For the other applications, plasma spraying is generally used. Thick (2.5 mm) plasma sprayed TBCs are used for abradable blade outer air seals where the rotating blades cut a gas-path seal

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in the porous coatings [15]. In military aircraft turbines, TBCs also are used on augmentor components, e.g. tail cones, flame holders, heat shields, and duct liners, and in the nozzle section they are being used experimentally on the verging/diverging flaps and on seals where the hot gases exit the engine [6].

Combustor Liners Turbine V

Blade Outer Air Seals

Figure 7.4. Cross sectional schematic of the turbine engine ofa military aircraft showing the general location ofTBCs on components (redrawn by [6] from a chart supplied courtesy of United Technologies).

Turbine blades are one of the most highly stressed rotating parts in gas turbines. In order to increase the efficiency and performance of turboengines, gas inlet temperatures in the high pressure turbines must be increased, and component cooling must be decreased. Here, ceramic TBCs with a low thermal conductivity applied on turbine components playa key role. EB-PVD is a promising technology for the production of TBCs with some advantageous properties. TBCs deposited via EB-PVD have smooth surfaces without requiring additional polishing, good erosion resistance in service, and no closure of the cooling holes. But the most important advantage is their outstanding thermal shock resistance, which is related to their columnar microstructure. This results in their having considerably extended lifetimes. The state-of-the-art-material for TBCs is zirconia (zr02)

stabilized with 6-8 weight % yttria (Y 203) that consists of a tetragonal t/phase (nontransformable to the monoclinic phase) [16]. Ceramic TBCs are connected to components by thin metallic bond coats, which also protect the components from hot corrosion and oxidation. Conventional bond coats are single layers of MCr AIY (where M = Ni or NiCo) or Pt-AI based materials.

To increase the lifetime of the MCrAIY layers the dissolution of the yphase in the superalloy by interdiffusion of Al and Cr must be minimized or

255

if possible, prevented. Therefore, interdiffusion between the superalloy and the bond coat must be low. This is achieved by increasing diffusion barrier elements such as platinum or palladium and reducing the Al and Cr contents at this interface. In contrast, at the interface between the bond coat and the ceramic TBC, the concentration of oxide forming elements, such as Al and Cr, in the bond coat should be as high as possible to build up a dense, stable, and thus protective, alumina (Ab03) scale. The optimal concentration distribution could be met by grading the composition across the coating thickness [17, 18].

The interface between the bond coat and the ceramic top coat is the most critical region with respect to the lifetime of EB-PVD processed TBCs. One of the major failure mechanisms is the formation of thermally grown oxides (TGO), particularly alumina, that are generated during processing and grow by diffusion of oxygen through the protective zirconia layer, and subsequent reaction with Al at the interface of the zirconia and the bond coat. The formation of these oxides produces stresses that ultimately cause spallation of the TBC.

Graded TBC systems are potentially advantageous compared with TBC systems that have ungraded layers. Gradients can be introduced that combine the thermal insulation of zirconia with the low oxygen diffusivity of alumina. Laterally graded alumina/zirconia coatings have been produced by EB-PVD in order to investigate the morphology, the phases, and the chemical compositions of the different Ab03-Zr02-mixtures that correspond to the different regions of a graded TBC [19]. The use of FGMs to join high temperature materials is being actively investigated.

2.3 Thermal Protection in Diesel Engines

TBCs are utilized in diesel engines for trucks, buses, locomotive, marine vehicles, tanks, military transport engines, and farm vehicles [20]. Their advantages in this application are increased power density, reduced heat loss, and reduced fuel consumption [21, 22]. In addition, TBCs have been shown to reduce exhaust emissions [23]. Figure 7.5 shows the commercial application of TBCs at various locations on a diesel engine. Thick (2.5 mm) TBCs are used on piston crowns, and thinner (0.5 mm) ones are used on valve faces and cylinder heads. Experimental TBCs have been tested on cylinder liners, exhaust valve systems and valve seats [12]. It has been shown that a 5% reduction in fuel consumption is obtained by insulating the combustion chamber with 2 mm thick functionally graded TBCs [24]. This performance gain could be increased to an overall 54% thermal efficiency for certain advanced diesel engine concepts. It has been shown that graded TBCs have a much longer lifetime [24-26].

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Valve Face;:;.. ~~II'"

Figure 7. 5. Cross sectional schematic of a diesel engine showing the location of rBCs on various components [6].

3. ENERGY CONVERSION SYSTEMS

3.1 Components for Conventional Fuel Burning Systems

The majority of today's power stations still burn conventional fuels. By optimizing combustion techniques and combining stationary gas turbines with steam turbines, efficiencies close to 60 % have been achieved. The incorporation of advanced material concepts such as FGMs could further improve the efficiency of these systems [27].

Of particular interest is the replacement of the heavy superalloys by lighter materials particularly for large fast rotating components. Basic requirements for widespread application of a replacement material are higher specific creep strength than superalloys and ductility levels of at least 1 % -2 % [28]. The low specific weight of 3.9 kg/m3 makes y-titanium aluminides (TiAI) candidate materials for turbomachinery applications at intermediate temperatures (600°C to 800°C) with potential to replace cast superalloys [29]. Unfortunately, y-titanium aluminides are either strong or adequately ductile depending on the microstructure, but do not fulfill both requirements

257

at the same time. Heat treatment in the a.-phase field results in fully lamellar microstructures with excellent creep strength but poor ductility. Heat treatment in the a. + ~ two-phase field results in duplex microstructures with acceptable creep strength and ductility that while low is also acceptable.

In gas turbine blades, the ductility has to be maximized in the root block where temperatures are low due to cooling via the turbine disc. Creep performance is of minor concern in the root area. In contrast, the airfoils have to exhibit maximum high temperature strength and creep resistivity. Ductility is of secondary importance at the blade area. A gradient from a fully lamellar microstructure at the foil to a duplex microstructure at the root would provide the desired mechanical properties at both sites.

The underlying principle to obtain the desired gradient of properties is to create a gradient in the equilibrium volume ratio of the a.+~ phase during isothermal annealing. This is achieved by introducing a gradient of ternary alloying elements such as Cr thereby shifting the volume ratio. At a homogeneous annealing temperature of 1375°C a duplex microstructure forms in Ti-48AI, and a near to fully lamellar microstructure develops if sufficient Cr is present. Table 7.1 quantifies the continuous gradient with respect to the volume fraction of lamellar grains obtained with a concentration gradient of the ternary alloying elements. Turbine blades of titanium aluminide with gradients in Cr content have been produced by hot isostatic pressing. Measurement of the mechanical properties of machined pieces cut from tested Ti4sAhCr2Nbrri46AbCrsNb2 Ta turbine blades that were evaluated after heat treatment at 1350°C for 2 hours, confirms the presence of the expected microstructural and mechanical gradients [29].

Table 7.1. The Microstructure of TiAl Alloys as a Function of the Heat Treatment Temperature and the Alloying Additions [29].

Ti48Al duplex duplex duplex Ti47AhCr duplex near lamellar fully lamellar Ti48AhCr2Nb2Ta duplex near lamellar fully lamellar Ti47AhCr3N~ Ta fully lamellar fully lamellar fully lamellar

In the hot turbine sections of stationary gas turbines combustion gases at about 1400°C enter the turbine section. However, an internal air cooling system with a highly advanced design reduces the surface temperature of the turbine blades to 1050°C or lower. Unfortunately, both the superalloy material used for the airfoils and the cooling technology are developed to a performance limit that is unlikely to allow a sizable increase in the inlet temperature in the near future, unless it becomes possible to develop ceramic TBCs with a guaranteed service life of at least 25,000 hours and several thousand restarts.

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Premixing the fuel and air before combustion is a method that has a high potential of achieving an optimized fuel efficiency at low emission levels of soot, hydrocarbon, and nitrogen oxide (NOx) gases. The critical step for this combustion concept is the reliable decoupling of the evaporation of the liquid fuel from the burn zone. This is accomplished by spraying the liquid fuel on the porous outer surface of an evaporator tube. The vaporized fuel is transported by the flowing air to the inner side of the tube where combustion takes place. Therefore, the inner side of the tube must be hermetically sealed, and there needs to be a porosity gradient from the outer side to the inner side of the tube.

Porous silicon carbide ceramics are proving to be the most promising materials for liquid fuel evaporator tubes in gas turbine combustors with premix burners. During the operation of these tubes, temperature gradients that are 1500·C at the inner and 550·C at the outer tube wall, are produced perpendicular to the tube walls. A comparison of finite element modeling calculations for various systems with both stepped and continuous graded functions indicates that the use of a specifically designed porosity can reduce the probability of failure significantly [30, 31].

3.2 Components for Integrated ThermioniclThermoelectric Systems

The goal of the second Japanese FGM program [32] was to develop highefficiency (-40%) hybrid energy conversion system (HYDECS) using different types of converters adopted for different temperature ranges [33]. Specifically these were a thermionic element at 2000 K, thermoelectric elements at a lower temperature of 11 OOK, and a heat radiator at an about 300 K [34]. Figure 7.6 shows a schematic of a hybrid direct energy conversion system proposed in the second Japanese FGM program [35]. In order to develop an efficient and durable device, an optimized system with low heat loss and minimal degradation had to be developed. A number of interface problems needed to be solved with respect to heat and carrier transportation, materials joining, thermal stress, electrical contact, and insulation under severe thermal conditions. In the proposed design, the solar heat receiver is an FGM cavity made of a carbon/carbon (C/C) composite [34].

259

Graded CIC heat reservoi"

TIC emitte

Graded SiGe ---~

Graded PbTe

Graded AI NNoI radiator

Figure 7.6. Schematic of a hybrid direct energy conversion system consisting of thermionic and thermoelectric converters.

3.2.1 Solar receiver system

The heat flow through C/C composites via conduction can be controlled by the design of the fiber architecture. Figure 7.7 a and b shows a C/C composite cavity for a sunshine heat receiver produced with a graded arrangement of the fibers [36]. Optimal thermal management in the receiver requires a high heat flux from the top surface, where the concentrated sun beams are absorbed, to the bottom surface that transmits the heat to the transformers. The loss of heat due to heat flux perpendicular to the path from the top surface to the bottom surface must be minimized. The C/C cavity is shielded by multiply folded molybdenum cylinders.

A properly designed cylindrical C/C receiver should provide high thermal conductivity from the flat top surface to the bottom surface, and low thermal conductivity to the outer cylinder walls. The graded C/C composite cavity has a gradient both in fiber volume fraction, which increases toward the central axis of the cavity, and fiber orientation, which in the central areas is essentially parallel to the axis of the cylindrical cavity. Toward the outer

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regions of the cavity more and more fibers are directed circumferentially to the cylinder [36]. The temperature at the transmitting bottom planar surface of the receiver is increased by 100 to 1S0°C due to the FGM design of the cavity as indicated in Figure 7.8.

Solar Rays

,~t ~~

~'--""-'--.""

,u""" Figure 7.7. A C/C composite cavity for a solar heat receiver fabricated using an FGM alignment of the fibers. (a) Schematic showing the fiber alignment. (b) A C/C composite cavity indicating typical temperatures at the inner and outer walls [29].

2000

1900

g 1800 ~ ::l 1700 iii .... Q) c. 1600 E Q) f- 1500

1400 0 10 20 30

Distance from center of back wall (mm)

Figure 7.B. Comparison of the temperatures at the bottom transmitting plane of an FGM and a non-FGM C/C solar heat receiver [29].

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3.2.2 Thermionic converter system

Thermionic conversion is based on the principle that electrons discharged from a hot emitter will move to a low temperature collector located on the opposite side [34]. By applying the FGM concept, the performance of the thermionic converter can be optimized by decreasing the energy loss between the emitter and the converter (the barrier index). For this objective the work function of the emitter needs to be optimized, while the work function ofthe collector needs to be decreased as much as possible.

The radiated heat from the bottom of the C/C cavity is received by a titanium carbide (TiC) plate. Titanium carbide has a high melting point and a high emissivity ("" 0.9), which allows an efficient absorption of heat from a wide range of the solar spectrum at high temperatures [37, 38]. The heat is conducted to a rhenium (Re) plate that operates as a thermionic converter emitter. The transition layers between the heat receiving TiC plate and the Re emitter should have excellent heat conductivity, should relax the thermal stresses, and should act as a diffusion barrier between the Re and TiC. This transition was achieved by a combination of graded layers of TiC/Mo -MoW - WRe [35]. The collector electrode was made of sputtered niobium oxide with a very low work function (1.38 eV) on a molybdenum (Mo) electrode [39]. Using this arrangement, a thermionic conversion system was constructed with a maximum output of 80 K W 1m2 at emitter-collector temperatures of 1600°C -760°C and a cesium reservoir temperature of 330°C [38] .

The conventional emitter electrode consists of a tungsten (W) plate. The dual work function emitter, which is composed of high work function metalceramic composite layers on a low work function W plate, has good electron emission characteristics. The work function of the collector is decreased by coating conventional W or Mo electrodes with their metallic oxides. These two measures combined are expected to result in a major improvement in the power output. Fabrication of this composite emitter electrode is difficult because of the high thermal stresses and the interfaces between the ceramic layers and the metal plates. It is anticipated that by applying the FGM concept the thermal stresses in these components will be reduced.

3.2.3 Thermoelectric converter system

The efficiency of a thermoelectric power converter is proportional to the temperature difference between its hot and cold sides. In the case of a homogeneous material, the thermoelectric figure of merit, Z, shows a distinct temperature dependence with a peak value that depends on the nature of the carriers and their concentration. At each temperature a composition exists

262 Chapter 7

that has the highest possible Z value. The pronounced temperature difference between the hot and cold sides of the converter may require different material systems to attain the highest figures of merit as indicated in Figure 7.9. Thus from 1300 K to 900 K, the silicon germanium compound, Si.8Ge.2, could be used, while lead telluride (PbTe) and bismuth telluride (Bb Te3) would be advantageous from 900 K to 500 K and 500 K to 300 K, respectively (see Figure 7.8) [40, 41, 42].

3.5

SZ ""- 3.0 ..-'-"

C')

0 2.5 ..->< N ..... 2.0 . ;:: Q)

E 1.5 -0 Q) , .... :::J 1.0 '" Cl ,\ u: PbTe

0.5

0 200 400 600 800 1000 1200

Temperature T (K)

Figure 7.9. Estimation of the figure of merit for an FGM composed of three compounds (straight line) with partly adapted compositions ofBi2 Te3, PbTe and SiGe in limited temperature ranges (dashed line).

Figure 7.10 compares the maximum output power of graded PbTe having different carrier concentrations (a-layer: 3.51 x 1025 /m3, b-layer: 2.6 x 1025/m3, c-layer: 2.26 x 1025 1m3) with the maximum output power when each layer has homogeneous carrier concentrations. The PbTe containing a graded concentration of the dopant lead iodide (PbI2) was prepared by laminating together powders consisting of three different concentrations of the dopant, followed by hot pressing. The output power was measured at the temperature of the cold side (500 K). It was shown that PbTe with a graded carrier concentration has a maximum power of 253 W 1m for a temperature

263

differential of 486 K, which is 11 % higher than the highest power for homogeneous PbTe with the a-layer carrier concentration composition.

300 -E • FGM ~ ~a-Iayer (jj ~b-Iayer 3: 200 0 -0-- c-Iayer a. E :::J E 'x ItS 100 E Q) > U Q) :t: W

0 0 100 200 300 400 500 600

Temperature difference (6 K)

Figure 7.10. Variation of the figure of merit in an n-type lead telluride (PhTe) conversion unit with the carrier concentration and temperature, plus an estimation of the figure of merit in the case of an optimized gradient in the carrier concentration.

The presence of both hot and cold sides requires attaching durable electrodes to the thermoelectric components that are both compatible with the thermal expansion mismatch and are sufficiently electrically conductive. As noted above, SiGe is one of the materials under consideration for use in thermoelectric conversion at high temperatures. Dense graded SiGe units with electrodes have been fabricated by a one-step sintering process using hot isostatic pressing (HIP) with glass encapsulation, shown in Figure 7.11 [43]. Materials with low electrical resistivity, tungsten, molybdenum disilicide, and titanium diboride (W, MoSi2, and TiB2) were selected for the electrodes. They were blended with silicon nitride (ShN4 ) in order to reduce the thermal expansion mismatch in the joints between the electrodes and the thermoelectric conversion unit.

Two types of units have been designed: WlMoSh/SiGe for phosphorus doped n-type SiGe and W/TiB2/SiGe for boron doped p-type SiGe. The graded design of the electrodes provides a smooth profile for the electrical resistivity that decreases continuously from the SiGe to the electrodes, and ensures the stability of the thermoelectric properties at least up to 1100 K. A conversion unit made of an n-type SiGe FGM with gradation in the

264 Chapter 7

concentration of the phosphorus dopant shows a marked improvement in output power characteristics.

W+40 vol.% Si,Ns

MoSi2+55 vol.% Si3N4

SiGe (0.5 at.% P)

111111\111111111\111111111\1111 \ I 1 2 3 4

Figure 7.11. A dense, graded n-type silicon germanide (SiGe) conversion unit produced by single-step hot isostatic pressing with sintering. (ShN4: silicon nitride, MoSh: molybdenum disilicide).

A system with an optimized FGM configuration is predicted to provide a conversion efficiency of 20.6% compared with approximately 18.7% for a conventional ungraded system. Optimization includes determination of the most advantageous length for each stage of the configuration and also of the most advantageous number of devices that are connected in series and parallel. In addition, radiation heat loss is taken into consideration because of the elevated operation temperature of the SiGe stage. A 2- dimensional finite difference method was developed requiring the upper and lower end temperatures, the temperature dependent values of thermal (K) and electrical (0") conductivity, and the Thomson coefficienC 't ('t = T dex./dT, ex. = the Seebeck coefficient8) [44]. Other computational design procedures use electron and phonon transport coefficients calculated by band theory combined with an elastic thermal stress analysis by 2-dimensional finite element modeling [45]. Transport properties also have been calculated that

7 The Thomson coefficient is defined as the ratio of the voltage between two points on a homogeneous conductor to the difference in temperature of those points when an electric current flows between them.

8 The Seebeck coefficient is defined as the ratio of the open-circuit voltage to the temperature difference between the hot and cold junctions of dissimilar conductors in the same circuit.

265

include in part the influence of microstructural features such as grain boundaries [46, 47, 48, 49].

3.3 Components for the Fusion and Nuclear Reactor Field

The fabrication of the components that are directly exposed to the plasma in steady-state magnetic fusion reactors, such as the diverter plate and the first wall, is the key technology for the successful development of these reactors in the near future. These components are subjected to extremely high heat fluxes and incident particle fluxes that cause thermal shock, thermal fatigue, and erosion. Materials for components exposed to plasma have been investigated worldwide with particular emphasis on materials for armor, on bonding techniques, and on heat flux tests [50, 51]. FGMs have been proposed as possible solutions for reducing the deleterious effects of thermal stresses [52].

A diverter plate of a graded tungsten/copper material has been produced by sintering a tungsten skeleton with graded porosity to a mechanical stability that allows its subsequent melt infiltration with copper [53]. The tungsten side of the FGM can withstand the highest temperature hot spots in the plasma, while the opposite copper-rich side has sufficient thermal conductivity for adequate cooling with water. This graded composite has proven to be effective up to heat fluxes of 15 MW/m2 (see Figure 7.12), which is the required qualification for diverter plates exposed to the plasma in the International Thermonuclear Experimental Reactor (ITER).

The graded tungsten/copper target of the beam attained a steady state after 10 seconds of irradiation. The maximum surface temperature of the tungsten did not reach 830°C, and no surface fissures, cracks, or spalling were observed after up to 40 seconds of irradiation [54]. Using the same electron beam irradiation system and a particle beam engineering facility, chemical vapor deposited FGM coatings of titanium carbide/carbon 1 mm thick were evaluated at a surface heat flux of up to 70 MW/m2 for several seconds. The FGM film sustained temperature differences as high as 1500°C without cracking or melting [55].

3.4 Components for Fuel Cells

Fuel cells are electrochemical devices that convert chemical energy directly into electrical energy. Electricity and heat are generated by the electrochemical combination of a fuel with an oxidant. They offer several advantages including a relatively high conversion efficiency, low emission

266 Chapter 7

of pollutants, a potential for cogeneration of electricity and heat, fuel and product flexibility, and minimal siting restrictions compared with other power plants.

1500 4 • ....t.rIail

"" CU

12125 :~~ ml !II, I i

~1~~·_~ __ ~ __ ~ __ ~_S_wH-.~.-__ ~ __ ~~T-41 ~ n ~ n 1ii .. ~ 5OOj--t~~~~~~-r--~~ ~

initial t8l'l1Ml'atll'e of coollnl water (298K)

o 5 10 15 Electron beam power density (MW IniI-)

i !

.5 .. i ~ ...

~I

~ I'" o. ~ 0.. ...

I

;'

f- -

(a) TI

n

In.. ...,.

~\. TC

" 0

(b)

Figure 7.12. The results of electron beam irradiation tests. (a) The temperature distribution during heating to a stationary heat flux of 15 MW/m2. (b) The effect of the electron beam power density on measured temperatures.

Fuel cell systems are categorized based on their electrolyte materials. One type is the solid oxide fuel cell (SOFC) system. This consists of a cathode on the air side and an anode on the fuel side, with both sides

267

separated by an oxide electrolyte with a high ionic conductivity. Oxygen from the air is reduced at the cathode side to 0 2- by gaining two electrons, and diffuses through the electrolyte to the anode. Fuels such as H2, C~ or natural gas, are oxidized at the anode side to H20 or CO thereby releasing electrons to the external circuit at the cathode. Solid oxide fuel cells require a number of functional electrochemical components and contact materials as shown in Figure 7.13 [56]. The electrodes are connected to the electrolyte by electrode/electrolyte interfaces with low interfacial polarization and chemical compatibility. This transition region can have a large triple-phase boundary length (electrode/electrolyte/gas) or can be a mixed conductor.

Cu nent collector

Interface electrode/interconnect good mechanicaVelectrical contact chemical compatibility

Electrode main strucbJre

Electrode: 1I333333333333333333333:IB=-r-_ low in· plain resistivity, porosity for gas diffusion ... Interface electrode/electrolyte

Electrolyte bw nterfacial polarization more electrode/electrolyte/gas il terfaces or mixed conductor chemical compatibility

Figure 7.13. Electrochemical components and materials required for the electrode/electrolyte layers of solid oxide fuel cells [56].

The main electrode requires a low in-plane resistivity and sufficient porosity for gas diffusion. The current collector is connected to the electrode by an interface region that must maintain good mechanical and electrical contact between the electrode and the current collector and also adequate chemical compatibility. The operating conditions require lifetimes of several ten thousands of hours at temperatures between 800°C and 1000°C in an oxidizing atmosphere. Most of the components are made of ceramic materials or metals with high-temperature oxidation resistance. For example, yttria stabilized zirconia or YSZ (Y 203-Zr02) has been used for the electrolyte, lanthanum strontium manganite (Lal_xSrxMn03) for the cathode (air electrode), nickel-zirconia (Ni-Zr02) for the anode (fuel electrode), and lanthanum strontium chromite (Lal_xSrxCr03), or high temperature alloys for the interconnects (current collectors).

For solid oxide fuel cells that can operate at lower temperatures to be technically feasible, the electrolyte resistance needs to be lowered. This can be achieved by reducing the thickness of the YSZ-electrolyte and by minimizing the electrolyte/electrode interfacial losses by using more efficient electrodes. In addition, the fabrication conditions and the operation

268 Chapter 7

of all of the components must be compatible. Gradient configurations should improve the operation of this fuel cell [56].

For current SOFC systems a large part of the voltage losses arise on the cathode side (air side). Therefore, it would be desirable to optimize the cathode layers, the cathode/electrolyte, and the cathode/interconnect interfaces. From a material selection standpoint the cathode material should fulfill the following requirements: high electronic conductivity, high ionic conductivity, high catalytic activity for oxygen reduction, chemical compatibility with the electrolyte and interconnects, compatibility of the thermal expansion coefficient with the other components in the fuel cell, stability in air at high temperatures, absence of destructive phase transformations, and the ability to be formed into films. The use of a gradient material in this application is most likely to satisfy the many seemingly contradictory requirements [56].

One approach is to use a graded multilayer configuration of strontium doped lanthanum chromite (La\.xSrxCr03) because the thermal expansion coefficient of this material and the electrolyte are similar, and also because it is chemically compatible with zirconia (zr02), the fuel cell's main material. In order to increase the electrical conductivity of the homogeneous La\. xSrxCr03, the Sr content is increased toward the electrode side in two graded steps. Replacing the fuel cell's original interconnect made of the homogeneous material with the graded three-layer interconnect increases its power output by 40% [56].

Strontium doped lanthanum manganite (LSM) is a material of choice for the cathode of solid oxide fuel cells because of its chemical stability, good electrical conductivity, and relatively low overpotential for oxygen reduction. It has been shown experimentally that addition of the electrolyte, yttria stabilized zirconia (YSZ), improves the cathode's performance. During actual operation the LSM and YSZ content as well as the porosity and pore size need to be continuously varied in order to optimize the triple phase boundary length and the current collection. When vacuum plasma sprayed cathodes with different gradations in their LSM-YSZ content were tested in planar substrate-type solid oxide fuel cells, both the cathode performance and the cell's lifetime were affected by the different concentration profiles [57].

The use of the more active lanthanum cobaltite (LaCo03) in cathodes creates compatibility problems with the YSZ electrolyte. Ceria-based materials (Ce02), however, show no interaction in contact with LaCo03. Therefore the incorporation of a cerium-gadolinium-oxide (CGO) protective layer between the electrolyte and the cathode was considered a possible solution. A low temperature densified ceria material and interdiffusion phenomena between YSZ and CGO were investigated to determine the

269

feasibility of cofiring the double layered ceria-zirconia electrolyte. The best results were produced by a graded composition of the solid solution phase of YSZ-CGO [58].

By using materials with eutectic microstructures (regular arrays of lamellae or rods) for the anodes of solid oxide fuel cells (SOFCs), both their electrochemical activity and their thermal stability can be increased. To produce a eutectic of NiO-YSZ, a microwave melting process was developed that allows crystallization to occur within a wide temperature gradient. The microstructure resulting from melting the electrodes directly onto the top of the YSZ electrolyte substrate is composed of partly aligned lamellae perpendicular to its surface. The thickness of the alternating NiO-YSZ lamellae is graded over the thickness of the electrode; it increases with increasing distance from the substrate. In terms of functionality, this means that there is an inverse gradient between the electrochemically active triple phase boundaries (gas - Ni - YSZ) after the NiO is reduced to Ni. During the reduction process, the microstructural gradient of the thickness of the lamellae (or the inverse gradient of their density) is converted into a functional gradient of electrochemically active sites. This is because the graded density of YSZ lamellae coated with Ni particles represents an increasing number of triple phase boundaries per unit area. This model elucidates how this type of eutectic microstructure could also lower the overpotential of SOFCs [59].

Natural gas, which consists mainly of methane, is the most promising fuel for stationary applications of SOFCs. The endothermic steam reforming of this fuel within the anode chamber of the SOFC stack (internal reforming) is applied advantageously to reduce the cost and increase efficiency. The reforming reaction, which is catalyzed by the NilYSZ anode, is rapid, and depends to a great extent on the temperature distribution within the SOFC stack. However, within the stack's fuel inlet area, a considerable degree of cooling can occur resulting in large temperature gradients. To prevent this local cooling, the catalytic activity for reforming of the NilYSZ anode must be decreased at the top layer of the anode material, which is in contact with the initial fuel gas mixture, without decreasing the activity in the interior of the anode. Therefore, for optimal performance, the anode should be designed as an FGM. To reduce the NilYSZ anode's catalytic activity, a sol-gel method has been used to coat and infiltrate it with YSZ while retaining its gas permeability and electrical conductivity [60].

In an SOFC with a planar design, a metallic bipolar plate is used to separate the ceramic single cells. These ceramic single cells consist of a YSZ electrolyte, cathodes of strontium doped lanthanum manganite, and anodes of a Ni-cermet. During their operation (at 950°C) an electrical contact between the metallic plate and the screen printed electrodes has to be

270 Chapter 7

maintained. This problem can be solved using intermediate layers between the metallic plate and the single cells. However, these layers must be ductile, at least before the initial heating of the stack. Porous layers fulfill this requirement. In addition, however, because the used metal forms a Cr03 scale that volatilizes as CrO at 950°C in an oxygen atmosphere, a protective layer is needed to prevent Cr03 evaporation from the metallic bipolar plate. This is because the Cr03 condenses in the cathode and leads to a decrease in the long term stability of the SOFC. In contrast to the contact layer this protective layer must be dense.

In the initial attempts to develop a gradient material as a solution for this problem, different compositions of the system (La, Sr) (Mn, Co) 0 3 were investigated for the contact layers, and both doped LaCr03 and doped YCr03 were investigated for the protective layers [61].

4. CUTTING TOOLS

4.1 Wear Resistant Bulk Materials

The Japanese sword exemplifies a classic application of the FGM concept. It is composed of steel with a gradient in carbon content from the surface (-0.6% carbon) to the interior (-0.25%) [62]. The graded carbon composition results in a tough sword with a sharp edge that remains sharp during prolonged use due to its increased wear resistance. This sophisticated technology, which was established about 1000 years ago, gave these classic swords and knives superior hardness and toughness.

More recently, in order to extend machining efficiency, the cutting speeds and feed rates have been substantially increased. Therefore, modern cutting tools need to withstand considerable thermomechanical stresses and require substantial thermochemical resistance against reaction with the parts being machined. Various techniques to fabricate compositionally graded tools and graded multilayer coatings have been developed to cope with the increasingly harsh conditions at the cutting tool tip and in the bulk of the parts being machined [63].

Construction parts and tools generally are subjected to a range of loads that cannot be satisfied by a single material or composition. For example, cemented carbide (tungsten carbide-cobalt; WC-Co) tools often require extreme hardness at the surface regions of the cutting edges but a stronger and tougher base material. If the size and configuration of the carbide particles remain the same, the hardness of the WC-Co system depends almost linearly on the binder (Co) content, with Vickers hardness of HVI 0 =

271

2100 at 5 volume % binder and 1100 at 30 volume % binder. At the same binder content, the hardness increases with decreasing grain size. Unfortunately, the transverse rupture strength and the fracture toughness (K1c) decrease with increasing hardness almost independent of compositional and microstructural details. These relationships between microstructure and properties are the basis for FGM cutting tools.

Cylindrical or bar-shaped compacts (formed at 150 to 300 MPa pressure) with a three-layered configuration have been fabricated [64]. The composition and/or grain size of the two peripheral layers are different from that of the central layer. The cobalt (Co) content varies between 5.5 and 11 weight %, the tungsten carbide (WC) grain size in the sintered alloys is between 0.4 J.lm and 4 J.lm. During liquid phase sintering (1400°C- 1450°C) the melt flow is controlled by the phase distribution and the grain size of the carbide phase. As expected, if the layers contain carbides of equal size, the cobalt content tends to equalize throughout the compacts during sintering almost independent of the initial Co distribution. If the size of the WC particles in the layers differs, melt enrichment in those areas with smaller sized carbide particles is observed. Figure 7.14 (a) and (b) shows the Co distribution in a three-layered bar before (a) and after (b) liquid phase sintering.

Functionally-graded cutting tools for high speed cutting or for machining at high feed rates have been developed by Sumitomo Electric Industries Ltd. One is a graded WC/Co throwaway chip [65, 66]. It is designed with a decreasing Co concentration from the surface to the interior, which causes the hardness at the cutting tool's surface to be higher than its interior. Graded and ungraded cutting tools are compared in Figure 7.15. This gradient in hardness results in both considerably higher damage resistance and higher wear resistance than a cutting tool with a homogeneous composition. The graded composition in WC/Co is produced during sintering by controlling the atmosphere and the rates of heating and cooling.

In addition, the WC/Co FGM is coated with a layer of titanium nitride (TiN), a layer of alumina (AI20 3), and a layer of titanium carbonitride (TiCN) by chemical vapor deposition. These graded and multiply coated WC/Co FGM cutting tool chips are very resistant to flank wear. Furthermore, they have the advantage of high machining speed combined with a high feed rate. Their graded composition can also control the internal stress arising from the thermal expansion mismatch. A simple, asymmetric gradient in composition such as in a ceramic/metal FGM can relax thermal stress, while a symmetric or radial gradient can induce a sizable compressive stress at the outer ceramic layer, resulting in stress reinforcement similar to tempered glass or prestressed concrete [67, 68].

272 Chapter 7

20 -0-- before sintering 18 wC-Co wC-Co

• after sintering ;g 16 0.

"0 14 ~ 12 C Q) 10 C 0 8 () .... 6 Q) "0 C

CD 4

2 0

0 2 3 4 5 6

Distance (mm)

(a)

2200

wC-Co WC-CO 2000

S 1800 > wc-TiC-TaC-Co ~ en 1600 en Q) c 1400 "0 .... -0-- before sintering CIS I

1200 • after sintering

1000 0,0 1,0 2,0 3,0 4,0 5,0 6,0

Distance (mm)

(b)

Figure 7.14. Flow in a 3-layer bar caused by compositional differences in the carbide phase before and after sintering: (a) binder (Co) composition, (b) microhardness. (After [64]).

Tough FGM cutting tools based on this model were commercialized in 1996. To fabricate them, green (unsintered) compacts of a mixture of powdered TiCN, 40 weight % WC, 10 weight % Co, and 5 weight % Ni are sintered at 1400°C in a vacuum for 1 hour under controlled nitrogen pressure. The WC in the outer layer dissolves in the Co-Ni melt and

273

reprecipitates as a solid solution of (TiW)(CN) at the surface. The molten metal continues to flow onto the surface when the atmosphere and other key parameters such as the cooling rate are controlled. This process results in a gradation in cobalt content as seen in Figure 7.16, thus hardness from high at the surface to low in the interior after the surface layer of metal is removed because the hardness decreases with increasing Co metal content [69]. The thin surface layer of the cutting tool is composed almost completely of ceramic without any metal binder. This results in a high hardness of 22 GPa, and the compositional gradient in the Co metal phase produces a high surface compressive stress of 0.8 GPa, as shown in Figure 7.17. The high surface hardness and compressive stress plus the toughness of the interior almost doubles the wear resistance, and increases the tool life as much as fivefold compared with conventional cermet tools.

«i' 18r----------------. & IAC151 -rn rn Q)

1 6

c 14 "0 ..... ctl .!:

rn ..... Q)

..lo:: ()

:> 6 ..... .~ ::

1 2

1 0

8

Conventional coated chip

o 50 100 150 200 250

Distance from the surface (J.I. m)

Figure 7.15. Comparison between the hardness at the surface of a cemented carbide (wc/co) FGM (ACl5) cutting tool and a conventional tool [65).

The in-situ formation of a graded structure from a homogeneous green body by controlling the atmosphere and processing parameters such as the cooling rate, is a promising approach for a cost effective process. Other process controls such as the heating rate and the nonuniformity of heating or cooling could also be useful for the in situ formation of graded structures.

Graded cutting tools have also been made for interrupted cutting from cermets of TiC-NiMo in which the percentage of TiC in the graded layer ranged from 95 weight % at the top surface to 86 weight % at the transition

274 Chapter 7

site to plain steel [70]. A gradient structure was obtained by brazing together segments having different compositions. Successful cutting results demonstrate that enhanced performance can be obtained using the gradient concept to prevent damaging tensile loads. The fabrication method for these graded cutting tools was based on previous work to produce gradient armor [71 ].

- 50 (J) c... ~ 40 0 () - 30 0

>-:=: C/l 20 c: Q) -c:

>- 10 en ... , x 0

0 20 40 60 80 100 120

Distance from the surface ( f.!m)

Figure 7.16. The cobalt concentration (X-ray intensity) as a function of the distance from the surface ofa TiCN/WC/Co-Ni FGM measured by electron probe microanalysis [69].

275

-«S 24 1.0 Q. (ij" ~ Q.

(/) 22 0.8 ~

(/) (/) CD 20 (/) c CD '0 0.6 ... ... -«S 18 (/)

.r::. .~ ~ 0.4

~ 16 (/) (/)

(,) 0.2 CD ... :> 14 0.

I E 0 ... 0 0 (,) () ~ 20 40 60 80 100 120

Distance from the surface (~m)

Figure 7. J 7. The hardness and the compressive residual stress as a function of the distance from the surface of a TiCNIWC/Co-Ni FGM, indicating the high hardness and compressive strength at the surface [69].

Diamond cutting tools are increasingly used for high precision machining of soft components, such as plastic contact lenses, polygonal mirrors for laser printers, and hard disk substrates made of aluminum alloys. Conventional diamond cutting tools are manufactured by joining a diamond crystal onto a metallic alloy shank with a silver solder containing active metals. However, the machining accuracy is relatively poor due to the silver solder's lack of stiffness, which causes vibrations during machining. This problem was solved in 1992 with the development of extremely stiff FGM diamond tools with a graded layer of diamond/SiC between the diamond chip and the SiC shank [72].

These FGM diamond tools are produced by a reaction sintering process shown schematically in Figure 7.1S. The shank is formed from a green compact made of several graded layers of SiC powder mixed with from 0 -SO volume % diamond powder and a polymer binder. The compact is heated to carbonize the binders and subsequently infiltrated with molten silicon. The silicon and carbon react to form new SiC grains that bond the existing SiC grains with the diamond particles. As shown in Figure 7.19, a thin layer of the diamond chip at the interface with the graded diamond/SiC layer reacts with silicon and converts to SiC, resulting in the formation of a strongly integrated body without metal interfaces. Finite element analysis indicates that the formation of the graded layer can reduce the thermal stress at the joint of the diamond chip and the SiC shank from 400 MPa to 150 MPa. An almost sixfold improvement in machining precision has been reported for these tools, with as much as a 30% extension in tool life.

276 Chapter 7

However, despite their performance advantages, FGM cutting tools are still too expensive to manufacture to be commercially cost competitive.

Diamond particle ~ .... ~ ... , ... SiC particle Polymer

1":_ I ........ I ........ ...·t·,.· , ~OMiXing

~...-~ ~ Pre-compaction t=:J of non-FGM ~ Shaping of the tool's shank

with a FGM joining layer

Carbonization of polymer Diamond chip

~R . .. h _-----_ eactlon slntenng toget er

with a diamond chip

Diamond tool FGM layer

Figure 7.18. Schematic of the process for fabricating diamond/SiC FGM cutting tools [72].

Nevertheless, it is easier to apply the FGM concept to cutting tools than to larger and more complex FGM components, since their effectiveness can be assessed quite quickly with relatively simple tests. For this reason, it is anticipated that the practical applications of functionally-graded cutting tools will continue to grow, and the transfer of the resulting advances in the technology to other applications for FGMs will soon follow.

4.2 Wear Resistant Coatings

Several types of graded coatings for the cutting edges of tools have been developed recently. These are coatings in which the carbide/nitride content is

277

varied from the substrate interface to the surface, and also coatings in which nitrides of different metals are alloyed so that their final compositions are graded. For specialized applications such as drilling and milling, bifunctional coatings are used that are built with anti galling hard coatings like titanium nitride and self-lubricating moderately hard coatings like WC/C, in which a diamond like carbon matrix is strengthened by incorporation of WC crystals [73].

Graded cemented carbide coatiQgs on steel substrates relieve the thermal stress associated with homogeneous WC/Co coatings. These graded coatings are made by stacking on the steel substrate, multiple layers of cemented carbide powders in which the Co content is compositionally graded. This is followed by sintering via pulsed electric current sintering. Because of the low sintering temperature and the short sintering time the desired graded composition of the cemented carbide is maintained, and a crack-free WC/Co coating is obtained on the steel substrate [74].

In 1995 Matsushita Electric Industries introduced an advanced electric shaver with thin FGM blades of stainless steel that have a hard intermetallic compound of an iron aluminide (Fe-AI) precipitated on their surface. The graded composition results in a hard but flexible surface aimed at producing a smooth and comfortable shave [75].

Figure 7.19. Schematic of the bonding between a diamond chip and a diamond/SiC FGM [72).(CD - diamond chip, ~ -bonding phase, @ - diamond particles, @ - reaction sintered SiC).

5. MACHINE PARTS

There is little published information about the applications of graded materials as engineering components other than for coatings and wear resistant materials [76]. Therefore, it is difficult to classify the application of an FGM that exhibits several functions concurrently, for example, a silicon carbide reinforced carbon-carbon composite (SiC/C-C) that acts as a

278 Chapter 7

structural part of the heat collector for an energy conversion system and also provides thermal stress relaxation, heat conduction, and oxidation protection.

The major application of FGMs for machine parts is for joints, largely metal-ceramic joints for gas and steam turbines [77, 78, 79]. The advantage of using an FGM joint is chiefly for thermal stress relaxation and improving the strength and toughness of the joints. Figure 7.20 shows a schematic of an alumina-nickel superalloy specimen joint for mechanical testing made by plasma spraying and diffusion bonding [77, 78, 79]. Because the rupture strength of graded joints is 3-8 times higher than for directly bonded joints, they are expected to provide longer service life at elevated temperatures.

100

-~ 80 '0 ~

60 CD III 1\1 .r. 40 Q.

~ 20 ~

0 234 5

Thickness of the graded layer (mm)

Figure 7.20. Schematic of an alumina - nickel (AIz03-Ni) alloy graded joint for hightemperature applications. The graded layer is composed of AIzOiNiCrAIY plasma sprayed layers with different contents of the metal phase [78].

279

Another FGM application that involves thermal stress relaxation and a low coefficient of friction is in the welding apparatus used, for example, for the automated electric arc welding of the large aluminum sheets used in building huge ships such as liquid natural gas (LNG) tankers. This welding device uses sliding water-cooled copper "shoes" that move along the seam and support the molten aluminum bath. Figure 7.21 indicates the location of the FGM component. The electric arc between the melt and the electrode wire oscillates between the shoes on the sheets being welded together. The inserts, -which prevent both reaction between the molten aluminum and the copper and the shoes from sticking to the hot aluminum surface, are usually made of silicon nitride to obtain a low friction coefficient and a good seam. However, their life is limited under the thermal cycling conditions during normal operation. This problem is alleviated with a SbN4-Cu FGM fabricated by a process, shown in Figure 7.21, that combines powder metallurgy and infiltration. The FGM design improves the stability of the inserts, reduces their thermal stress, and retains their low coefficient of friction [80].

Engine components are also being designed using ceramic-based FGMs to improve their thermal fatigue life by making them more heat-insulating. For example, in a hybrid design for the pistons of a diesel engine for passenger cars, the ceramic content is gradually reduced with increasing distance from the piston head [76, 81].

Materials with graded porosity produced by conventional powder metallurgical methods are used for filters and membranes [76, 82, 83]. A new technique, gas-metal eutectic transformation in metal-hydrogen systems, discovered in Russia, can be used to produce graded near net-shape components of strong, permeable, porous materials called GASARs (the Russian word for gas-reinforced), shown in Figure 7.22 (a) [84, 85]. The transformation involves the decomposition of a molten metal that does not form a stable hydride (e.g., copper, aluminum, iron, or brass) but reacts with hydrogen at high pressures to produce both a solid and a gas phase via a eutectic. The solidification processing involves the charging or saturation of the liquid metal melt with gaseous hydrogen until the eutectic composition is reached, followed by directional melt solidification. The hydrogen level in the melt, the gas pressure over the melt during solidification, the direction and rate of heat removal, and the chemical composition of the alloy, all influence the amount of porosity, and the shape, size, and orientation of the pores. Pore sizes can range from 5 J.!m to 10 mm and pore volume from 5% to 75%.

280

Welded sheets

Chapter 7

Oscillation of the guide ~

Electrode wie

Sliding oopper shoe (water-cooled)

Movement direction

Figure 7.21 . The process for arc welding two large aluminum sheets, and the microstructure of the Si)N4-Cu FGM insert plate [77].

An FGM rectangular bronze GASAR with grading in the diameters of the pores, is shown in Figure 7.22 (b) [78]. FGM GASARs can be made both by controlling the gas-eutectic in the phase diagram and by appropriately designing the cooling system [86]. GASARs are claimed to be the strongest

281

permeable porous materials, thus usable for structural load-bearing components. Other suggested applications include as filters, catalysts, mufflers, heat exchangers, self-lubricating bearings, silencers, vibration dampers, and shock absorbers [84]. The potential for grading their porosity plus varying the shape of the pores and their distribution in radial and axial directions can give these materials additional useful properties such as anisotropic heat conductivity, liquid/gas permeability through the thickness, and nonlinear thermal elastic-plastic behavior [85] .

(b)

. / --' " /. /,;,~

(~\I,/' 6,' \~S.~: ~~lf/t1/

.... _ .. '" ~9 •• '

/'--/'9--~

/' /'/' /'? . . 'l I ••....• ........ '"

• ••• •• 6 I ••• • ••• ••••• ~'1 •••• • •••• . d~' ~

• ~ •••. ! ••.•• ; It" I e ..•• ~ .•... ~ •• ' /' .. ...•. .•... /' .. '., •......... :

Porosity gradient •

Figure 7.22. Various geometrical structural types ofGASARs (gas-reinforced materials), produced under different processing conditions of gas-eutectic reactions; (a) ungraded GASARs, and (b) a graded GASAR[80].

Structural components have also been made of nickel- and iron-based superalloy FGMs containing a Ti(C,N) graded metal phase [87]. In coppergraphite FGMs made by controlled mechanical alloying, that is by arresting the process at predetermined intervals, then arranging the various fractions sequentially and hot pressing them, both the graphite concentration and the particle size distribution are graded. Casting molds with high thermal conductivity and wear resistance, and a relatively low coefficient of friction

282 Chapter 7

have been made by joining this FGM to a graphite part by vacuum brazing with a copper-based solder [78].

6. GRADED BANDGAP SEMICONDUCTORS

The concept of graded bandgap structures in semiconductor research was introduced in 1957 with the theoretical investigation of their potential application to transistors [88]. This section describes the unique and desirable functions and applications of graded band-gap structures in semiconductors.

6.1 Semiconductor Heterojunctions

Semiconductor heterojunctions can be regarded as an extension of multiple heterostructures. In considering the band diagrams of abrupt heterojunctions (where the composition changes from that of one to that of the other within one atomic layer) without interface defects, two semiconductors are assumed with different energy bandgaps, Egl and E g2 ,

electron affinities, Xl and X2, and Fermi energies, EFI and EF2, as shown in Figure 7.23 (a). Electron affinity is defined as the energy to transfer an electron from the edge of the conduction band to the vacuum level. Unlike the work function, which is defined as the energy to take an electron from the Fermi energy level to the vacuum level, electron affinity is a characteristic quantity for each semiconductor because it depends on the concentrations of impurities.

vaCUUJn level

CB

X2 ¢lz XI ¢II

!~E(' CB

- - - - - - - EFl Egz Egi

~ - --- - - -VB ~·v -----

c

semiconduclor I semiconductor 2

(a) (b)

Figure 7.23. (a) The essential band parameters for semiconductors, and (b) the energy band diagram of the heterostructures composed of the semiconductors shown in (a).

283

When these two semiconductors are brought into contact, the energy differences of the conduction band, Ec=Ec\-Ec2' and the valence band, Ev'=Evt-EV2, are conserved, because it can be assumed, as a first order approximation, that the bulk properties of the semiconductors are conserved even in the vicinity of the interface. At the same time, electrons close to the junction in semiconductor 1 move toward semiconductor 2, resulting in band bending as shown in Figure 7.23 (b).

The significant features of heterojunctions are the presence of energy steps at the junction owing to band discontinuities. Therefore, staircaseshaped energy band profiles can be prepared by stacking different semiconductors one after another. A smooth energy band profile, rather than a staircase-shaped profile, should then be achieved readily by reducing the distance between the interfaces and gradually changing their compositions.

The precise values of energy band discontinuities for various semiconductor pairs have been studied both theoretically and experimentally for a long time, since they are among the most important parameters for designing heterostructures. However, experimentally determined band discontinuities are somewhat different from estimated values that are based on the simple assumption discussed above. This is because crystalline periodicity is not present at surfaces and interfaces.

Figure 7.24 indicates the relative energy levels of the bandgaps for typical semiconductors when heterostructures are prepared that are based on a theoretical model [89]. Band lineups of heterojunctions composed of any two semiconductors shown in Figure 7.24 are possible. However, given the variety of possible combinations, heterojunctions of practical importance are quite limited, since lattice mismatch causes dislocations and dangling bonds that generally act as effective scattering and recombination centers for carriers. As an approximation, the lattice mismatch of heterojunctions should be reduced to less than 0.3 %.

6.2 The Functions of Graded Bandgap Structures

The functions of graded structures in semiconductors can be classified as: crystallographic, electronic, and optical. The crystallographic function is the simplest. Heterostructures are usually prepared using epitaxial growth techniques on suitable substrate crystals. However, epitaxial substrates are limited to elemental or binary semiconductors such as silicon (Si), gaIIium arsenide (GaAs), and indium phosphide (InP), because it is difficult to produce crystals of alloys with uniform and accurately designed lattice constants. When substrates with matching lattices cannot be obtained, a thick buffer layer, in which the lattice constant is changed from that of the substrate to that of the material to be grown, is deposited on the substrate

284 Chapter 7

prior to the growth of the desired layer. Misfit dislocations are introduced gradually during the growth of a buffer layer so that substantial reduction of dislocation density in the required layers can be obtained. Although this function is very primitive, such a buffer layer has been utilized even in commercially available devices, such as orange-colored, light-emitting diodes.

r'--''''

I I , I 1 ! I i ! I ~ ! ~ i i I ~ i

-Un SI G.

1:" .... "--, , I

I i !'".''''~' !'··_·-t

II I I I r'''-1

, ·8- '-"f1-D-~Gg ~ GaA. InA. AISb

AlAs GaP InP

AlP

CdT.

ZnS.

Figure 7.24. The band discontinuities ofheterojunctions are shown as relative positions of bandgaps based on a theoretical model [88]. The dashed line indicates the midgap energy, which is the energy to be aligned when the heterojunction is formed.

Electronic functions are of particular importance and specific to graded bandgap structures. In graded semiconductor structures, the conduction band edge energy, Ec (r), and the valence band edge energy, Ev (r ), which are functions of the coordinates, Ec (r ) and Ev (r ), are the lowest energies for electrons and holes, respectively. In addition, the energies of carriers with higher energies can be represented as the sum of the kinetic energy, K, and the band edge energies. Therefore, Ec (r ) and Ev (r ) can be regarded as potential energies for electrons and holes, respectively.

When the potential energy of a body, U (r ) has a gradient, the body receives a force of F = - grad U (r ). The forces on electrons and holes exerted by external electric fields are equal in magnitude but opposite in direction, as shown in Figure 7.25 (a). Similarly, electrons and holes in graded structures receive forces due to their position that depend on the band edge energies as if they were in an electric field. However, gradients in the conduction band edge energies and the valence band edge energies should no longer be equal in graded structures, as shown in Figures 7.25 (b) and (c).

285

Forces on electrons and holes can be controlled by designing graded structures. This type of quasi-field effect for graded structures has been previously proposed [88] . Therefore, based on this effect, it should be possible to use the potential profile to control the behavior of carriers.

~ """~ ..

(a) (b) (c)

Figure 7.25. The forces on the carriers (electrons and holes) in a uniform semiconductor in (a) an external electric field, and (b) and (c) in graded bandgap structures.

E E

~: rI---~,'

[, /

p ---+--.p

-----II 1,---(a) (b)

Figure 7.26. Examples of energy levels and densities of state in micro-heterostructures. (a) A square well and (b) a harmonic potential well.

Energy spikes formed at abrupt heterojunctions act as potential barriers for carriers and substantially reduce the current flow normal to the heterojunction. However, such energy barriers can be removed by gradually changing the composition of the alloy. Moreover, even the scale of the

286 Chapter 7

potential profile can be reduced to about 10 nm where quantum size effects dominate and the concept of the potential profile is still operative. A variety of wave functions and densities of state can be obtained by designing graded band gap structures. Examples of quantized levels and the density of state p, with graded potentials are shown in Figure 7.26. Several practical devices based on quantum effects, such as high electron mobility transistors (HEMTs) and quantum well lasers, have already been developed.

In general, the dielectric constants of semiconductors are inversely proportional to their bandgap energies. Photonic devices such as semiconductor laser diodes include waveguide structures that consist of semiconductor heterojunctions.

6.3 Applications of Graded Bandgap Structures

Bipolar transistors were the first application of graded bandgap structures investigated [88]. Figure 7.27 indicates the energy band profile of a bipolar transistor. The power gain of a bipolar transistor depends on the injection efficiency of the emitter current, which is defined as the ratio of the current injected from the emitter to the base region and the total emitter current. Since the injection efficiency of a homojunction bipolar transistor is proportional to the ratio of the concentration of the impurity dopant between the base and the emitter regions, the concentration of the base dopant must be reduced in order to obtain high injection efficiency. However, this results in high base resistance and inferior frequency response.

electron inject~~

Emitter

Base

h~e injection

Collector

Figure 7.27. Schematic of the energy band profile ofa bipolar transistor under operating conditions.

287

The injection efficiency of heterojunction bipolar transistors (HBTs) that have wide bandgap emitters can be improved without reducing the concentration of the base impurity due to the presence of the energy steps in a heterojunction. A band profile of this type of HBT with abrupt heterojunctions is shown in Figure 7.28 (a). However, as discussed above, the energy spikes at a heterojunction reduce the injection current. This has led to the development of the graded emitter HBT structures shown in Figure 7.28 (b). The graded base HBTs, shown in Figure 7.28 (c) are another type of structure. Their frequency response is improved by the acceleration of the injection of minority carriers into the base layers by a built-in field. However, the base layers in the most recent HBT structures are so thin that the acceleration effect is no longer significant. The current structure of standard HBTs is intermediate between those shown in Figures 7.28 (a) and (b).

Graded bandgap structures are also being considered for heterojunction solar cell structures in which wide bandgap materials are used to reduce the ineffective photo-absorption at the exterior of p-n junctions. A graded bandgap structure rather than a abrupt heterojunction is expected also to further improve solar cells, since the built-in field can lower the recombination rate of photogenerated carriers and improve their collection efficiency at the electrodes [90].

~ ~ ! Collector

Emitter

(a) (c)

Figure 7.28. Ihe three major heterojullctioll bipolar transistor (RBI) structures are: (a) an abruptjullctioll RBI, (b) a graded bandgap emitter RBI, and (c) a graded bandgap base RBI.

A semiconductor laser contains a waveguide structure, a so-called double heterostructure (DH), in which a narrow bandgap layer is sandwiched between wide bandgap materials. The threshold current for the operation of this laser is sharply reduced by the inclusion of DH structures, due to the simultaneous confinement of photons and carriers within a narrow bandgap layer. Separate confinement heterostructures (SCH), in which carriers and

288 Chapter 7

photons are confined by different DH structures, are of particular importance for quantum well lasers. Graded index (GRIN) SCH lasers, shown in Figure 7.29 (b), have superior characteristics compared with conventional SCH lasers, shown in Figure 7.29 (a), because photoabsorption in the cladding layers can be reduced, and the carrier capture into a well layer is enhanced by the built-in field in the cladding layers [91].

(a) (b)

Figure 7.29. The energy band profiles of single quantum well lasers. (a) A conventional separate confinement heterostructure (SCH) laser, and (b) a graded index (GRIN) SCH laser.

7. GRADED INDEX MATERIALS

The terms gradient index and graded index, abbreviated GRIN, are used to indicate an inhomogeneous medium in which the refractive index varies continuously. Such GRIN materials possess unique and useful optical properties that cannot be achieved with conventional optical materials, which have a constant refractive index. The most common application is for optical glass or polymer fibers in which the refractive index is changed stepwise or gradually along the radial direction. A light wave can propagate reflecting or curving with the change of refractive index through a fiber. Many applications for GRIN materials have been reported for optical fiber communications and microoptics. In this section, graded index optics and some applications are described [92].

7.1 Graded Index (GRIN) Optics

In a conventional optical system, the refractive index within each optical component is considered to be homogeneous. Therefore, in the design of such systems, focusing and imaging properties are determined by varying the curvature of each lens component. However, if the refractive index varies

289

continuously within a lens material, a light beam is gradually curved according to the index distribution inside the lens. This results in changing focusing properties.

The ray trajectory through such GRIN media, where the refractive index varies as a function of position, satisfies the ray equation [93]:

d [ dr] - n(r)- = Vn(r) ds ds

(7.7.1)

where r = xi + yj + zk is the position vector of a point on the ray, n is the refractive index distribution, and ds is a differential element of each path length along the ray. There are three types of gradient index lenses, as shown in Figure 7.30; axial, radial, and spherical. When the refractive index linearly decreases along the z axis in the axial GRIN lens, all parallel light rays focus into one point without any spherical aberration. However, a conventional homogeneous lens with the same dimension causes a large spherical aberration.

A radial or cylindrical GRIN rod consists of a cylinder with the refractive index distribution indicated in Equation 7.7.2 :

(7.7.2)

where no and nCr) are the refractive indices at the central axis and at a distance r from the central axis respectively, and A is a positive distribution constant.

Axial-GRIN Radial-GRIN Spherical-GRIN

z ---{}-----t-_m)--z t) Figure 7.30. The three types of graded index (GRIN) lenses.

Since the refractive index is symmetrical on its axis, in a paraxial condition Equation 7.7.1 is reduced to:

1 dn(r) (7.7.3)

nCr) dr

290 Chapter 7

Substituting Equation 7.7.2 into Equation 7.7.3 and assuming n(r) = no gives:

Solving Equation 7.7.4 gives:

r = ro cos( -fAz) + ~ (dr) sin( -fAz) -v A dz z=o

(7.7.4)

(7.7.5)

In such a GRIN rod, rays injected at r = ro follow approximately sinusoidal paths according to Equation 7.7.5, as shown in Figure 7.31. The symbol L denotes the period of the sinusoidal ray paths and is related to the distribution constant A.

L= 2n -fA (7.7.6)

Since all rays in Figure 7.31 pass through one point at Ll2 and L, for an object at one end of a GRIN rod, the rod will form an inverted image with unit magnification at a distance Ll2 and an upright image after a distance L. Such GRIN rod lenses have been used as connectors and couplers for optical fibers, and as imaging lens arrays in photocopiers.

L-----~~----~------~------~------~z o

Figure 7.31. The paths of light rays through a radial GRIN rod.

291

7.2 GRIN Glass Fibers

There are three types of optical glass fibers as illustrated in Figure 7.32. Step-index type fibers (a and c) are composed of a core glass with a high refractive index and a cladding glass with a low refractive index. The light passes through the fiber and is reflected at the interface of the core and the cladding glass. A graded-index type fiber (b) is fabricated so that the refractive index continuously changes along the radial direction. The light is curved in this fiber according to the graded index and is propagated without leaking. Fibers (a) and (b) have several paths for the light wave to travel, while fiber (c) is limited to a single path. Only single-mode, step-index fibers, indicated by (c), are used for optical communication because of their superior data-carrying performance.

Amp 1\ R~frdeacUve •• ~ '~~~~~~~~j~ Clad

In x ._. ' • .... +--I40-100llm ..... Core

•• : .::::~:::] Clad

a) Multimode step-index fiber

A1\mp R~fr~t~e •• ~ ~~~~~~::::j~Clad Index

40-100llm - -Core

--: ~:~~:~~:~:J C~d t

~ t

b) Multimode graded-index fiber

,-,A<-mp_~Re_:;tl:~~;~:.~E: ~ t t

c) Singlemode step-index fiber

Figure 7.32. Schematic showing how the pulse spreads through step-index type and gradedindex type optical fibers.

In the multimode step-index fiber (a), light spreads an impulse over a time interval that is equal to the difference of the arrival times of the slowest and fastest modes. In the multimode graded-index fiber (b), if the index profile is optimum, all modes propagate at the same velocity without

292 Chapter 7

spreading an impulse. This results in a significant increase in the bandwidth of the carrier wave, or its data-carrying capacity, compared with the multimode step-index fiber (a).

Glass fibers can be made of silica glass, borosilicate glass, soda-lime glass, or other multicomponent glasses. A silica glass fiber is produced by chemical vapor deposition (CVD) [94] or vapor axial deposition (VAD) [95]. In the CVD process, gaseous silicon tetrachloride (SiCI4) plus gaseous germanium tetrachloride (GeCI4) or phosphorus oxychloride (POCI3) as dopants, flowing through a silica (Si02) tube with oxygen as the carrier gas, are heated to above 1300'C. The resulting combustion products, fine particles of silicon dioxide (Si02) and germanium Ge02 or phosphorus pentoxide (P20 5), are deposited on the walls of the silica tube. The presence of the Ge02 or P20 5 increases the refractive index of the silica glass. The glass preform is fabricated by heat treatment above 1700'C followed by hot drawing to a fiber about 100 /lm in diameter.

In the V AD process, a porous rod is produced by the deposition of fine particles of Ge02 or P20 5 using an oxygen-hydrogen flame. In order to fabricate fibers with a graded refractive index, the amount of Ge02 or P20 5

deposited is controlled to increase at the center of the preform and decrease toward its periphery. The V AD process produces high performance optical fibers in which there is a low attenuation of light below 0.5 dB/km.l Eliminating impurities that absorb light, such as hydroxyl ions (OK), transition metals, and rare earth ions as well as minimizing the presence of structural imperfections that can scatter the light are key factors in decreasing attenuation of the propagated light.

Figure 7.33 is a schematic of a process for producing graded index fibers of borosilicate glass using a double crucible [96]. The inner crucible is charged with a thallium monoxide-boric oxide-silicon dioxide (ThO-B20 3-

Si02) glass, which has the higher index, and the outer crucible is charged with a sodium monoxide-boric acid-silicon dioxide (Na20-B20 3-Si02) glass, which has the lower index. During hot drawing of a glass fiber from the bottom of the molten glasses, the two glasses come into contact and TI+ ions diffuse out to the cladding glass, while Na+ ions diffuse into the core glass, resulting in the formation of a graded refractive index.

The diameter of the core is between 40 - 100 /lm for multimode wave fibers and between 5 - 15 /lm for single mode wave fibers. Because the attenuation of silica fibers is at a minimum between 1.3 - 1.55 /lm, this wavelength band has been adopted for optical communication system by using Inl_xGaxAsyPl_y semiconductor lasers.

293

11 glass (Core glass)

Double cnx:ible

Figure 7.33. Schematic for fabricating graded-index glass fibers by an ion exchange method.

7.3 GRIN Polymers

The advantage of GRIN polymers is the ease of obtaining GRIN materials with a short reaction time and a large GRIN region compared with GRIN glasses. In general, GRIN polymers are produced by the copolymerization of two or more kinds of monomers (M!, M2, •.. ), which results in a material with two or more different refractive indices. If the copolymer composition is controlled in the axial, radial, or spherical direction, the corresponding GRIN material is produced.

In the copolymerization of monomers M\ and M2, the refractive index n of the copolymer is related to the copolymer composition by the LorentzLorenz equation:

nZ -1 nr -1 n~ -1 --=--v +--v nZ + 2 nr + 2 \ n~ + 2 Z

(7.7.7)

where n\ and nz are the refractive indices of the monomers M\ and M2,

and VI and V2 are the volume fractions of each monomer unit, respectively. A number of fabrication techniques have been proposed for GRIN polymers.

294 Chapter 7

The basic mechanism for forming the graded index can be classified into two types: a diffusion process and a process that utilizes the difference in the reactivities of the monomers [92].

7.3.1 The diffusiou process

Figure 7.34 shows the distribution of the refractive index obtained by the vapor phase diffusion of the monomer, 2,2,2-trifluoroethyl methacrylate, into a diethylene glycol bis (allyl carbonate) gel substrate at 70°C, followed by heat treatment to complete polymerization [92]. Since the refractive index of the diethylene glycol gel (nD= 1.50) is higher than the 2,2,2-trifluoroethyl methacrylate (nD = 1.42), the refractive index, n, of the polymer gradually increases with the distance x. The label of the ordinate, np , is the refractive index of the substrate's surface. The curves with broken lines are theoretical. They were derived from the diffusion equation by converting the concentration at a distance x to the refractive index using equation 7.7.7.

c-o c. .s 0.00

x Cll 'C .£: Cll .~ t5 Cll ....

'O:i .... 15 Cll (J c: Cll .... Cll ;: 15

a 3 6 9

Distance x (mm)

Figure 7.34. The distribution of the refractive index obtained by the diffusion of 2,2,2-trifluoroethyl methacrylate vapor into a diethylene glycol bis(allyl carbonate) gel substrate for lhour (squares) and 2 hours (circles). The broken lines were calculated from diffusion theory.

7.3.2 The monomer reactivity process

In the copolymerization reaction between monomers Mi and Mj , the monomer reactivity ratio rij is defined as:

295

[i = 1,2, ..... ,nJ

kii • 'ij = - ] = 1,2, .... ,n

k. '1 ••

l:t]

(7.7.8)

where kiiand kij are the propagation rate constants in the following copolymerization reactions:

kij >M.M .. I ,

kij >M.M.· I 1

(7.7.9)

In the photocopolymerization process to produce a radial GRIN rod in which the refractive index monotonically decreases from the central axis to the periphery of the rod, a glass tube including the monomer mixture is rotated on its axis and irradiated by UV light from the side, as illustrated in Figure 7.35 [97]. The copolymer gradually forms from the inner wall of the tube, and solidifies upward through the tube's central axis. In the case of a binary monomer system (M\ and Mz), the monomer pairs should satisfy the condition:

where n\ and nz are the refractive indices of the homopolymers M\ and Mz respectively. In the case where monomer MJ is more reactive than monomer Mz, the polymer formed at the initial stage of polymerization at the periphery has a lower refractive index. But the polymer formed during the final stage of polymerization, in the center region of the tube, has a higher refractive index. Consequently, the refractive index of the resulting radial GRIN rod is at a maximum at the central axis and gradually decreases toward the periphery.

7.4 Low loss, High Bandwidth GRIN Polymer Optical Fibers

Recently, there has been considerable interest in the development of a polymer optical fiber (POF) for short-distance communication applications such as local area networks (LANs), data links, and multimodal bus networks, because their ease of processing and large diameters enable high efficiencies of fiber coupling and beam insertion [98]. A GRIN POF (GI POF ) has a much higher bandwidth (> 500 MHz·km) than a multimodal step-index (SI) POF (2-5 MHz·km). For short-distance communications, which require many junctions and connections of optical fibers, a flexible

296 Chapter 7

POF with a large diameter would be most desirable. However, the SI POFs will not be able to cover the entire bandwidth (about several hundred megahertz) that will be required in the near future for LANs and for fast data link applications. Therefore, GI POFs have been explored to provide light wave media with large data-carrying capacity.

A

8

C

· · ~ · · , ,

Monomer Mixture

........................... "' .................... , , ,

······· .. · ...... r ...... ·· ............ Liquid Phase

Gel ,0.;, •• , .. ............... _-<10$'& ,

, ......................... i ......................... , , ,

, , , ,

Glass Tube

; Shading

, "'//1'//.'

... ... .. UV ... .. '//////,

Figure 7.35. A schematic of the photocopolymerization process for fabricating radial GRIN polymer rods.

8. FUNCTIONALLY GRADED BIOMATERIALS FOR ARTIFICIAL JOINTS

Several of the organs of animals such as skin, blood vessels, and bones are composed of multilayers that have different properties. These layers bind together creating, in effect, a functionally graded material. Therefore, incompatibility and separation at the interface never occur under normal physiological conditions. Consequently, the ideal technique would be to

297

mimic such natural bonding in order to obtain a fixation of a prosthesis to bone that will be stable for many years. In this section, examples of orthopedic and dental implants are described as well as the development of hip joint materials of nitrided Ti and the alloy, Ti-6AI-4V, with graded structures and functions.

8.1 A Graded Interface for Bone in Orthopedic Implants

Conventional methods of fixing an artificial bone and joint prosthesis to bone include: (l) total close contact of the prosthesis to the bone as shown in Figure 7.36, (2) direct mechanical fixation with screws or spikes, and (3) filling the space between the prosthesis and bone with polymethylmethacrylate (PMMA) bone cement (see sections 8.1.2 and 8.1.3). In addition, prostheses coated with a porous metal have been used recently for fixation. In this case, the prosthesis is expected to become mechanically fixed to the bone due to bone ingrowth into the pores of the porous metal coating. However, there are still many problems to overcome with these methods because of the mechanical nature of the bonding.

Recently, several calcium phosphate ceramics that can bond to a bone physicochemically, such as the bioactive ceramic, hydroxyapatite (HAp), have been studied extensively, and their clinical application has been rapidly adopted [99-108]. Furthermore, although binding bone to bioactive ceramics is an excellent method in itself, since these bioactive ceramics lack sufficient strength, many investigators have developed methods of coating bioactive ceramics on the surface of metals or on the ceramic alumina (aluminum oxide: Ah03) to create a composite material.

8.1.1 Cementless fixation

8.1.1.1 A smooth surface and a HAp coating on a smooth surface In this method, the prosthesis is fixed to the bone without bone cement

(see Figure 7.36). During surgery, a prosthetic component with a smooth surface on the area that contacts the bone is tightly inserted into a bone cavity. To maximize the contact area, the bone cavity is reamed into the same shape as the component. In this case, a connective tissue membrane (living soft tissue) is intermediate between the component and the bone. This may cause pain during weight bearing while walking due to micromotion of the prosthesis in the bone, and subsequently the prosthesis may even loosen in the bone. To prevent these potential problems, the surface of the component is coated with a 50-100 /lm layer of hydroxyapatite (HAp) by a plasma spraying method commonly used for coating HAp. The HAp-coated

298 Chapter 7

component bonds chemically to the bone by means of the HAp. However, this also may result in failure if the component is moved before the onset of chemical bonding, normally 2-3 weeks after surgery when a connective tissue membrane has formed.

Conneclive Tissue

Artificial Joinl ___ -' Componenl

I Cementless Fixation I I Smooth Surface I (a) (b)

Bone

HAp Coaling

I HAp Coating I (c)

Figure 7.36. Diagram of the interface between the prosthesis and the bone in a cementless fixation. Figures (b) and (c) are based on microscopic observation. Figure (b) indicates that when a prosthesis with a smooth surface is implanted into the bone, connective soft tissues will interpose following the surgeI)'. Figure (c) indicates that when a prosthesis with a smooth surface on which HAp is coated is implanted into the bone, the HAp coating will form a bond with the bone.

There is a further potential problem, several years after total hip replacement the HAp coating may separate from the base metal of the component, or the HAp coating may break off in the body after continued weight bearing during walking, because of the repetitive strong shear stress on the stem in the femur. The breaking off of HAp from the component might be prevented if the coating is graded. However, because the HAp coating on the outermost layer (closest to the bone) is pure, the possibility of its breaking at the interface between the bone and the HAp still exists.

In animal experiments using the tibia of beagles, (see Figure 7.37) the adhesive strength (shear strength) to bone of test samples coated with HAp were 3-5 MPa, 6-12 MPa, 8-11 MPa, and 12-14 MPa at 2, 4, 6, and 12 weeks respectively after implantation. During push-out tests, breaking

299

frequently occurred at the area coated with HAp. In clinical cases as well, breaking at the HAp coating and also HAp peeling-off from the base metal has been reported frequently.

Figure 7.37. Histological section of an animal bone at 6 wecks after implantation of the prosthesis coated with HAp. (B: bone, H: hydroxyapatite coating, Ti: titanium).

8.1.1.2 Porous metal and HAp coating on porous metal A more effective method for adhering a prosthesis to bone is to coat it

with a porous metal because new bone ingrowth into the pores occurs after implantation (shown in Figure 7.38). The rate of adhesion is fastest and the strength is highest when the pore sizes are 300 to 600 ~m. A porous surface can be created by spraying with metal, or coating with beads or a fiber mesh. Plasma spraying a HAp coating on the porous metal further increases the adhesion strength and rate of binding to the bone. To create the porous metal surface, beads, 750 ~m in diameter, of a titanium alloy (Ti-6AI-4V) are coated in two layers. Changes after the implantation of specimens in the bones of mature rabbits and goats under unloaded conditions are measured at regular intervals. In the HAp-coated specimens, after 7-10 days the ingrowth of bone tissue into every specimen is comparable to that seen in non-HApcoated specimens after about 3 weeks. Ultimately, almost all of the HAp coated on the beads become bound to the bone by growing into the pores. Consequently, there is a large amount of bone ingrowth into the pores between the titanium alloy beads.

After push-out tests, the adhesion of the HAp-coated specimens to the bone at 2, 4 , and 6 weeks after implantation is 4 times (2 MPa on average), 3 times (5 MPa on average), and 2 times (15 MPa on average) respectively, greater than the adhesion of non-HAp-coated specimens. It becomes almost equivalent after 12 weeks (26 MPa on average). During the push-out tests, breakage occurs at the area of the bone around the implants. From these

300 Chapter 7

results, the interface between the bone and the coated specimen appears to be adequately fixed. When a cross section of this interface is examined, three graded zones are seen: a zone of titanium alloy, a zone of titanium alloy with bone ingrowth, and a zone of bone, as shown in Figure 7.39.

(a)

Bone

Rough Surface

Metal Base

Bone Ingrowth Bone Ingrowth

Late Postoperative Period (Over 12 weeks)

(b)

Early Postoperative Period (1 • 2 weeks later)

(e) (d)

Figure 7.38. Diagram of the interface between porous metal and bone. (b) From microscopic examination it is seen that immediately after surgery there is no bone ingrowth. (c) At 1-2 weeks bone ingrowth begins in some areas. (d) At over 12 weeks after surgery bone ingrowth into the pores is accomplished.

When beads of the titanium alloy are coated with HAp, a larger amount of bone ingrowth occurs earlier and bonds to the beads physicochemically, thereby providing stronger adhesion. In animal experiments using beagles, when an uncoated prosthesis is implanted, bonding to the bone frequently fails during weight bearing. However, the probability of successful fixation via bone ingrowth increases greatly when a prosthesis coated with HAp is implanted. Therefore, porous metal with a HAp coating appears to remedy the biggest drawbacks of cementless prostheses. It promises to prevent pain while walking caused by micromotion or loosening of a prosthesis fixed without bone cement, and also allows weight bearing earlier after implantation.

301

1000J,lm

Figure 7.39. Histological examination at 6 months after the implantation of HAp coated granules on a prosthesis under loaded condition (back scattered electron image). (B: A layer of bone, P: A layer of porous beads with bone ingrowth, M: A layer of metal.)

Therefore it can be concluded that when a smooth surface is coated with HAp the coating may separate from the metal. Even when a HAp layer is applied as an FGM, the outermost layer can be destroyed. When porous metal is not coated with HAp, even if there is bone ingrowth into the porous beads, pain may occur due to micromotion because of spaces between the bone tissue and the beads. But when the beads are coated with HAp, no micromotion can occur because a large amount of bone grows into the pores earlier and bonds to the HAp physicochemically. In addition, the layer between the material and the bone is also graded.

8.1.2 Polymethylmethacrylate bone cement fixation

In this conventional technique, shown in Figure 7.40, polymethylmethacrylate (PMMA) bone cement is used to fix the prosthesis to the bone. The bone cement is prepared during surgery by mixing and kneading the polymeric powder with the liquid monomer. The mixture, in the form of a slurry, is poured into the bone cavity, and the prosthesis is then inserted into the bone cavity and fixed in place. The cement hardens (polymerizes) in about 10-15 minutes after kneading. This is a conventional

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bone cement fixation technique. A prosthesis can be completely fixed in the bone immediately after surgery. However, a connective tissue membrane (living soft tissue membrane) can become interposed between the bone and the bone cement several weeks after implantation (see Figure 7.40 (b». Occasionally, this may cause the prosthesis to loosen at the interface between the bone cement and the bone. Figure 7.40 (c) illustrates the use of interface bioactive bone cement fixation to overcome this difficulty.

Fixation with Bone Cement (PMMA)

(a)

Connective Tissue Bone Cement

Artificial Joint Component

Bone Cement

Interface Bioactive Bone I Bone Cement Fixation I Cement Fixation

(c)

Figure 7.40. Diagrams of the interface between polymethylmethacrylate (PMMA) bone cement and bone, and also of the interface in IBBC (interface bioactive bone cement) fixation. Figures (b) and (c) are based on microscopic observation. (b) When the prosthesis is fixed to the bone with PMMA bone cement, connective soft tissues will often subsequently become interposed between the bone cement and the bone. (c) When 1-3 layers of HAp granules, 100-300 mm in diameter, are interposed between the bone cement and the bone at the time of cementing, the bone cement, HAp, and bone will become bonded in a functionally graded condition after several weeks.

8.1.3 Interface bioactive bone cement fixation

Interface bioactive bone cementation (IBBC) is a technique developed to overcome the problems that can occur at the interface between PMMA bone cement and the bone. Specifically, one to three layers of fine HAp granules (100-300 /lm in diameter) are placed between the bone and bone cement.

303

Therefore only the interface becomes a bioactive bone cement. The bone cement mechanically adheres to the HAp granules and the HAp granules chemically bond to the bone (see Figure 7.41). In an animal experiment in which interface bioactive bone cementation was performed in the femoral condyles of mature rabbits, ingrowth was observed in almost all spaces between the HAp granules after 2 weeks, and fixation was completely accomplished after 6 weeks. After a push-out test, the bonding strength to the bone at 2, 6, 12, and 24 weeks was on average 3.8, 5.0, 5.5, and 9.0 MPa, respectively.

Interface Bioactive Bone Cementation

Bone Cement HAp Granules Bone (PMMA) .....--------.

Immediately After Surgery

(1.1)

Bone Ingrowth

Late Postoperative Appearance

(b)

Figure 7.41. The fonnation of a porous surface on the surface of the bone by coating it with granules of HAp.

Histologically, at 6 weeks there were four different graded zones at the interface between the bone, the HAp granules, and the bone cement: a zone of bone, a zone of HAp granules with bone ingrowth, a zone of HAp granules with bone cement, and a zone of bone cement (see Figure 7.42). After push-out tests at 6 weeks, breakage occurred at the zone of the bone or

304 Chapter 7

the zone of the HAp granules with bone ingrowth. This can be explained by the presence of an FGM layer at the interface.

(e) 100,um

Figure 7.42. Back scattered electron image of the histological examination of the interface between the bone and the bone cement at 7 months after surgery using the IBBC fixation technique. (B: bone, H.B.: a layer of HAp granules with bone ingrowth, C.H.: a layer of bone cement with HAp granules, C: a layer of bone cement, B.C.: bone cement)

In the case of the HAp coating on a smooth surface, after the push-out test breakage occurred at the outermost layer of the HAp coating. From these results, it can be concluded that the IBBC technique combines the advantageous properties of cement fixation with those of a cementless fixation of a HAp coating on a smooth surfaced prosthesis. Specifically, a good bond can be obtained immediately after surgery, and adequate fixation can be obtained indefinitely.

A HAp coating on porous metal is disadvantageous compared with the IBBC technique because the prosthesis is covered with the HAp coated porous metal only at the proximal part of the femor so that the entire surface of the prosthesis cannot be covered with bone. Therefore, the proximal part of the prosthesis will bind to the bone too strongly. This will cause the bone surrounding the prosthesis to atrophy and weaken because the weight of the body is not transmitted to the bone but to the prosthesis. In the IBBC technique the entire surface of the prosthesis can be covered completely with bone. Therefore, when the prosthesis binds to the bone with sufficient

305

strength, the load of the body's weight is largely transmitted to the bone, and thus the bone will not become atrophied or weakened.

The IBBC technique has been used clinically in Japan since 1985 at Osaka-Minami National Hospital for more than 2500 cases involving total hip and total knee replacement. The absence of radiolucent lines at the bone interface in X-rays post surgery indicates that no spaces are present between the bone cement, the HAp granules, and the bone. Because the morphology of the living body resembles that of an FGM, it is advantageous for a prosthesis, such as an artificial bone and joint, to possess a similarly graded form in order to achieve an organic linkage between the prosthesis and the bone.

8.2 Dental Implants

Functionally graded dental implants composed of titanium plus HAp have good biocompatibility and mechanical toughness [109]. A schematic of the structure of such an FGM dental implant is shown in Figure 7.43. The TilHAp FGM is produced by sintering (at 1300·C in argon, using radio frequency) a graded green body consisting of various compositions of Ti, 5 weight % Pd, and 20 to 30 weight % HAp. The bending strength (150 MPa) of the resulting material is similar to human bone. Animal implant tests show that newly formed bone is in contact with most of the TilHAp FGMs, but only partially in contact with an implant of Ti metal. After 8 weeks, no inflammation was observed in either the Ti or Ti/HAp animal implants [110].

8.3 A Nitrided Titanium Alloy Stem

Titanium and its alloy, Ti-6AI-4V, have high biocompatibility relative to other metallic alloys that are currently used in orthopedics. However, since serious cell toxicity caused by vanadium has been reported, there is concern about the potential for local and systemic effects due to vanadium ions released from the Ti-6AI-4 V alloy. Despite its desirable properties of stiffness and fatigue strength, this alloy has a higher coefficient of friction and lower wear resistance than ceramics such as alumina (Ah03) and zirconia (Zr02). Therefore, it is unsuitable for the frictional component of an artificial joint.

The combination of Ti-6AI-4V and ultrahigh molecular weight polyethylene (UHMWPE) for prosthetic joints generates an unacceptable quantity of Ti-6AI-4V and UHMWPE wear particles due to the relatively poor frictional properties of this alloy. These wear particles have been implicated as a contributing factor in the development of aseptic loosening

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of the prosthesis [111, 112]. Moreover, once the protective oxide film on the alloy's surface is worn off by an electrolytic solution such as the body fluid, the alloy becomes corroded rapidly [113]. The titanium, aluminum, and vanadium ions released have been reported to inhibit the formation of hydroxyapatite [114, 115]. This could affect normal osteoid mineralization and remodeling, which could result in loosening of the implant. Vanadium metal is also very toxic to cells, although a trace amount of vanadium is an essential element nutritionally [116, 117].

.s::. ..... C') c: Q) .... ..... (j)

A ""'''----• ---....;>~B o

,Q ..... ctI c.. E o C,) o III

Figure 7.43. Schematic of the structure of an FGM dental implant.

A number of surface modification methods have been developed to prevent the release of harmful ions or to harden the surface of metals. In the case of Ti and Ti-6AI-4V, titanium nitride or oxide is synthesized on the surface. Because Ti has an affinity for nitrogen as well as for oxygen, the nitridation is successfully accomplished by heating the Ti or Ti-6AI-4V to about 800'C in a gaseous nitrogen atmosphere. Compared with other techniques including ion implantation and sputter coating, nitridation has the advantage of low operating costs and minimal limitation on the sample's shape. The biocompatibility of the nitrided Ti-6AI-4V is promising both in vitro and in vivo [118].

Figure 7.44 shows the Vickers hardness of surface-nitrided Ti and Ti-6AI-4V at various depths from the surface, measured by a 25 g indenter. It is maximum at the surface: 1168 for the surface-nitrided Ti and 925 for the surface-nitrided Ti-6AI-4V. The hardness gradually decreases toward the interior to 116 at a depth of 60 !lm for the surface-nitrided Ti, and to 210 at a depth of 50 !lm for the surface-nitrided Ti-6AI-4V. Because the Vickers

307

hardness of 116 and 210 correspond to that of Ti and Ti-6AI-4 V respectively, the thickness of the nitrided layer is estimated to be 50-60 11m for both materials. Nitrogen diffusion forms a compositionally graded layer that results in a tough, nitrided surface. The frictional properties of surface nitrided Ti-6AI-4V are superior to alumina for the frictional component of an artificial joint plus it has high corrosion resistance.

1200

1000 en en Q) 800 c '0 .... <11 L: 600 en .... Q) ~ 400 ()

:> 200

~ • Surface-nitrided Ti-6AI-4 V

~ -0 Surface-nitrided Ti

~\ 1\ ~\ ~

0 o 20 40 60 80 100

Depth from the surface (11m)

Figure 7.44. Vickers hardness of surface-nitrided Ti and Ti-6AI-4V at various depths from the surface.

Within only ten years the thinking about ways to use FGMs has grown from the initial limited one of relieving stress when combining two dissimilar materials like a metal and a ceramic to the many and varied applications described in this chapter. It is anticipated that these represent only the beginning for potential applications of this useful concept.

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79.Nuutinen, S. and Heikinheimo, L. (1997) The coating-braze combinations in ceramicmetal joints (FG-joints), The State Research Center of Finland (VTT), Report VALB215, Espoo, Finland.

80.Gasik, M. (1995) Principles of functional gradient materials and their processing by powder metallurgy, Acta Polytechnica Scand., Ch. 226.

81.Henning, W., Melzer, C., and Mielke, S. (1992) Keramische Gradientenwerkstoffe fiIr Komponenten in Vemrennungsmotoren (Ceramic gradient materials for components in passenger cars), Metall., 46 (5) 436-9.

82.Joensson, M. and Kieback, M. (1995) Highly porous sintered parts with a pore size gradient made by centrifugal powder metallurgy, in Proc. o/The Third Int '/. Symp. on Structural and Functional Gradient Materials, (eds. B. IIschner and N. Cherradi), Presses Polytechniques et Universitaires Romandes, Lausanne, 33-9.

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83.Hong, c.-W., MUller, F., and Greil, P. (1997) Fabrication of pore-gradient membranes via centrifugal casting in Proc. of The Fourth Int'!. Symp. on FGM'96, Tsukuba, Japan, (Eds. I. Shiota and Y. Miyamoto), Elsevier Science Pub!., 173-8.

84.Shapovalov, V.1. (1994) Porous metals, MRS Bull. 19(4) 24-8. 85.Piekarczyk, J. and Jeremenko, M.D. (1992) Properties of ani sotropica I metal materials

GASARS, lnzyn, Mater., Poland, 12 (2-3) 64-8. 86.Shapovalov, V.I., Eryomenko N.D. (1989) The structure and properties of composite

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88.Kroemer, H. (1957) Quasi-electric and quasi-magnetic fields in nonuniform semiconductors, RCA Review, 18 (3) 332-342.

89.Tersoff, J. (1984) Theory of semiconductor heterojunctions, The role of quantum dipoles, Phys. Rev. Let., 30 (8) 4874-4877.

90.Hutchby, J.A. and Fudurich, R.L. (1976) Theoretical analysis of AlxGa'_xAs-GaAs graded band-gap solar cell, J App. Phys., 47 (7) 3140-3151.

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98.Emsile, C. (1988) Review polymer optical fibers, J Mater. Sci., 23, 2281-2293. 99.Bobyn, J.D. et al. (1980) The optimum pore size for the fixation of porous-surfaced metal

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105.00nishi, H. et al. (1989) The effect of hydroxyapatite coating on bone growth into porous titanium alloy implants, J Bone Joint Surg., 718,213-216.

313

106.0onishi, H. (1990) Mechanical and chemical bonding of artificial joints, Clin. Mater., 5, 217-233.

107.Pilliar, R.M. (1983) Powder metal made orthopaedic implants with porous surface for fixation by tissue ingrowth, Clin. Orthop., 176, 42-51.

108.0onishi, H. (1991) Orthopedic applications of hydroxyapatite, Biomaterials, 12, March, 171-178.

109.Watari, F. et af. (1995) Functionally graded dental implant composed of titanium and hydroxyapatite, in Proc. of The Third Int 'I. Symp. on Structural and Functional Gradient Materials, (eds. B. Ilschner and N. Cherradi), Presses Poly techniques et Universitaires Romandes, Lausanne, 703-708.

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Chapter 8

SUMMARY AND OUTLOOK

Since the first international symposium on FGMs, held in Sendai, Japan in 1990, research and development incorporating this concept has expanded to many different fields worldwide. From the 70 papers at the Sendai Symposium the number has grown to 240 at the 5th symposium held in Dresden, Germany in 1998. The topics covered at this meeting included computer-aided design and modeling; measurement; characterization; fabrication by bulk, layer, melt, preform and other processes; joining; thermal barrier coatings; various applications to cutting tools; heat resistant materials; energy conversion; functional components with graded ferroelectric, magnetic, and optical properties; and biomedical systems. In addition, a number of papers have been published in scientific journals and presented at numerous meetings with FGM sessions.

This book was written to provide a comprehensive description of the design, processing, and applications of FGMs developed during the past ten years. In the second chapter entitled, Lessons from Nature, the architecture of graded structures in living bodies is shown to embody the essence and image of the FGM concept. Indeed, it is surprising to see how living organisms form and optimize graded structures in order to become accommodated to complex environments and sustain life. The graded distribution of fibrous tissue and porosity or graded texture can increase the specific strength and protect against breakdown. Biomedical implants with graded structures in the bone/apatite/metal system or in a ceramic are gaining acceptance as both safe and effective prostheses. Also it has been observed that the combination of graded structures with piezoelectric stress sensing can control the growth or healing of living tissue. The incorporation of sensing functions in graded structures will enable the production of a

316 Chapter 8

variety of materials that are sensitive and responsive to external and internal environmental stimuli including damage, fatigue, and degradation.

The investigation of the mechanical and thermomechanical behavior of FGMs has created a new field of study in materials engineering. For example, analyzing crack propagation or delamination in a nonhomogeneous material that has a gradually changing composition and structure presents a challenging problem. Local structures and properties and the external load and temperature are correlated with an FGM's geometry. The profile of stress or strain is determined by this type of correlation, which strongly affects an FGM's mechanical and thermomechanical stability. The role of stress is essential for both the structural and the functional applications of FGMs.

The metamorphosis of FGMs due to atomic diffusion and/or chemical reaction in their graded composition and structure not only can cause degradation but also can provide active functions such as for the drug delivery device described in Chapter 6. Dynamical modeling of the mechanical, thermal, and chemical behavior of FGMs is needed to develop the design methodology for fabricating active FGMs that are stable.

A wide variety of processing techniques to fabricate FGMs have been proposed. However, only those that are cost effective probably will become feasible for commercial application. The development of relatively simple and productive fabrication processes by means of powder stacking, laminating, or coating graded layers, as well as the fabrication of small size FGMs such as machine tools, machine parts, small engine components, and hip joints are all expected to encourage commercialization in the near-term. In the future, FGMs with complex properties and shapes including threedimensional gradients will be produced by computer-aided design and manufacturing techniques. Various combinations of freeform fabrication and compositional gradation techniques are under development. These advanced processes should soon make it possible to eliminate the present relatively laborious and time-consuming methods for processing and optimizing graded structures and functions.

Various examples of the design and fabrication of FGMs to use for transportation, energy conversion, mechanical systems, semiconducting applications, and optical and biological systems are introduced in this book. Other applications to electromagnetic, medical, and agricultural systems are being actively investigated [1].

A completely or partially graded structure can be formed in any material and component. Therefore, the application of the FGM concept is virtually unlimited. Materials science as developed in the 20th century is based on homogeneous material systems. It is likely that the 21 st century will see the development of a new materials science and technology for non-

317

homogeneous materials and systems that have optimized structures like FGMs.

REFERENCES

I. Hirai, T. and Chen, L. (1999) Recent and prospective development of functionally graded materials in Japan, in Functionally Graded Materials 1998, ed.W.A.Kaysser, Materials Science Forum, vols. 308-311, Trans Tech Publications Ltd., ZUrich, 717-722.

Index

Acoustic emission, 89, 140, 144, 145 Acoustic microscope, 89, 112, 113,

129, 130, 156 Actuator, 96, 97, 98 Alumina zirconia, 172 Arc spray methods, 190 AuJ80Ni-2OCr/Au, 218 Bal_.Sr.Ti03, 96, 97 Bamboo, 1,7,8, 10, 11, 12, 13, 14,

15,17, 18,22,26 BaTi03, 96, 97 Bending test, 112, 114, 127 Betti number, 60, 89, 92, 94 BizTe3,262 Biaxial compression, 141 Bioactive ceramics, 247,297 Biocompatibility, 197,247,305,306,

313 Biocompatible graded interface, 248 Biological structure, 7 Bipolar transistor, 247, 286, 287 Bismuth telluride, 262 Blow molding, 232 BN/Si3N4, 200 Bond coat, 144, 195,255

Bone, 1,5,7,8,13,14,20,21,22, 26,248,297,298,299,300,301, 302,303,304,305,312,315 bone cement, 247, 297, 300, 301, 302,303,304,305 bone material interface, 247

Boron nitride/silicon nitride, 200 Boundary area density, 35 Boundary conditions, 50, 56, 70, 84, 102, 103, 130 Boundary surface, 153 Brittle-to-ductile transition

temperature, 142 Bulk modulus, 63, 69, 70, 73 Bulk processing, 163 Buoyancy convection, 214 Burner heating test, 139, 143, 145 ClB4C/SiC, 200 C/C, 152, 173, 199,240,247,249,

250,258,259,260,261 C/C cavity, 247, 259, 261 C/C composites, 173, 200, 240, 249,

259 C/SiC FGM, 201 CaC03,22 CAD, 161,220,223,243

320

Capacitance, 89, 96, 97 Capacitor, 96 Capillary forces, 170, 178 Carbon, 31, 87, 109, 110, 146, 151,

152,160,173,181,182,183,201, 235,240,250,258,265,270,275, 277,307,309

Carbon fibers, 160, 173 Carbonlboron carbide/silicon carbide,

200 Carbon/carbon composites, 173,249,

307 Cellular structure, 7 Celsian, 232 Cemented carbide, 177, 179,237,

270,273,277,311 Centrifugal casting, 161, 170, 212,

235,312 Centrifugal forces, 168 Centrifugal powder stacking, 170 Ceramic actuator, 89, 155 Ceramic-ceramic composites, 172,

245 Characteristic dimensions, 29, 37,45,

48 Characteristic length scales, 44 Chemical reaction, 164,201,316 Chemical surface reaction methods,

186 Chemical vapor deposition, 5,31,87,

146, 151, 152, 159, 160, 161, 164, 173,186,199,200,208,209,240, 247,249,250,271,292

Chemical vapor infiltration, 146,250 Cladding, 186, 187, 188, 189,216,

217,225,239,244,288,291,292 Coefficient of thermal expansion, 70,

71, 77, 127 Coextrusion, 215 Cold compaction, 115, 116 Combustion plasma, 190 Combustion processing, 183

Index

Combustion synthesis, 161, 184, 185, 238,241

Composites, 7, 68,74, 123,253,297 Composition profile, 50, 51, 56, 57,

165,205 Compositionally graded layer, 307 Compositionally graded

semiconductors, 94 Compositionally graded silicon

carbide-silicon carbide/graphite, 209

Compositionally graded TiN films, 197

Compositionally graded tools, 270 Conductivity profile, 90, 94, 96 Connectivity, 29, 38, 54, 60, 61, 67,

83,92 Consolidation, 161, 165, 168, 181,

185,236,239,312 Constitutive relations, 67, 84 Constructive processes, 163 Constructive processing, 161, 162,

164 Contact angle, 39, 217 Containerless HIP, 181 Contiguity, 32, 38, 39, 53, 89, 90 Continuous multi phased composites,

73 Continuum mechanics, 66, 69 Continuum models, 63, 66 Continuum solid mechanics, 117 Controlled spatial distributions, 41 Convective transport, 186 Coordination number, 39, 54 Copper-graphite FGMs, 281 Corrosion, 4, 41,130,151,152,232,

247,250,253,254,307,313 Corrosion resistance, 152 Co-sputtering, 198 Cr3CjNilCr3C2 , 89, 125 Crack driving force, 154 Crack opening displacement, 118

Index

Crack propagation, 90, 126, 127, 128, 129,158,316

Creep, 63, 84, 89, 112, 126, 127, 128, 155,158,180,215,216,256,257

Critical power density, 144 Critical surface temperature, 138 Critical volume fraction, 38, 60 Crystal structure, 32, 34 Cu/80Ni-2OCr/Cu,217 CulNb/Cu, 216 CulNi/Cu, 89, 117,218 Cu/SS304, 185 Cu-Ni FGM, 202 Curie point, 96 Cutting tools, 4, 186,247,248,270,

271,272,273,275,276,311,315 Cyclic heat loads, 130 Cyclic thermal loading, 130 Defects, 32, 282 Deformation, 22, 46, 61, 63, 83, 114,

123,130,132,142,211,232,233 Densification, 175, 176, 179, 180,

181,184,185,230 Dental implants, 297 Deposition current density, 204 Design, 2, 3, 5, 19,24,42,45,47,61,

63,64,84,85,86,87,130,131, 152,157,172,217,220,227,233, 235,240,248,250,257,258,259, 260,263,264,269,279,288,308, 310,315,316

Diamond, 200,201, 202, 238,240, 275,276,277,311

Diamond/SiC, 247, 275, 277 DiamondlWC FGM, 200 Dielectric constant, 89,96,97,98 Dielectric permittivity, 89,96,98 Diesel engine, 64, 65, 247, 255, 256,

279,308 Diffusion, 40, 44, 48, 50, 56, 57, 69,

90,109,110,111,158,163,176, 186,207,208,209,215,216,219,

321

228,232,233,234,241,242,243, 245,255,261,267,278,294,307, 311,316

Diffusion bonding, 161,219,232, 234

Diffusion equations, 50, 57 Digital image correlation, 99, 155 Dihedral angle, 39 Dilatometer,99 Discontinuous multi phased composites, 73 Discontinuously graded FGM, 179 Dispersed phase microstructures, 54,

60 Dispersion strengthened alloys, 53 Divergence theorem, 153 Drug delivery device, 161, 227, 228,

316 Drug delivery system, 2 Effective field, 96 Elastic constant, 113, 114 Elastic limit, 115, 229 Elastic modulus, 54, 63, 69, 73 127,200 Elasticity theory, 69, 112 Electric shaver, 277 Electrical conductivity, 61, 89, 90,

92,93,253,268,269 Electrical resistivity, 93, 263 Electrodeposition, 161,202 Electromigration, 186 Electron beam-physical vapor

deposition, 161, 194, 195,239, 240,247,253,254,255,308

Electrophoresis, 161 Electrophoretic deposition, 204, 241 Electrotransport, 164 Energy conversion systems, 247, 248,

256 Enthalpy of solution, 110 Epitaxial growth, 283 Error function, 56, 57

322

Eshelby's equivalent inclusion method,46

Eutectic bonding, 219 Extrusion freeform fabrication, 161,

224,244 Fabrication, 60, 163, 166, 174,316 Failure criteria, 84 Fatigue, 41,89,90, 112, 126, 127,

128, 130, 145, 148, 150, 155, 158, 215,240,249,253,265,279,305, 316

Fatigue crack propagation, 127, 158 FGM processing, 161,234 Fiber bundles, 11, 13, 173 Fiber composites, 204 Fiber reinforcement, 7 Fiber stacking, 161, 173,219 Fick's second law, 56 Field parameter, 29, 45 Figure of merit, 247, 261, 262, 263,

310 Film casting, 173 Finite element method (FEM), 46,

63, 119, 131,309 Flame fusion, 164,213,242 Foaming agents, 172 Forced-resonance technique, 112 Fourier number, 89, 107, 108 Fractal dimension, 60, 93, 95 Fracture, 54, 61, 83, 84, 89, 90, 112,

115,117,119, 121, 123, 124, 125, 126, 130, 138, 140, 142, 146, 154, 155,159,177,241,271 fracture initiation, 117, 154 fracture mechanics, 89, 112, 117, 118, 119, 121, 157 fracture mechanism map, 146 fracture toughness, 54, 61, 89, 112, 123,124,125,126,154,271

Fuelcell,247,266,268, 310 Functionally graded biomaterials, 96

nitrided titanium alloy stem, 305

Index

Functionally graded dental implants, 305

Fused deposition modeling, 161 Fusion reactor, 181 Fuzzy logic, 63, 75 Gallium arsenide, 283 GASARs, 247, 279, 280, 281 Geometrical models, 39, 40, 52 Geometrical parameters, 32, 33 Glass fibers, 292 Graded Alp/Zr02 ceramics, 205 Graded band engineering, 94 Graded band semiconductor, 89 Graded bandgap, 95, 247, 282, 284,

285,286,287 Graded bulk materials, 186 Graded C/C composite, 173 Graded C/C composite cavity, 259 Graded casting, 170, 171 Graded cemented carbides, 179, 237 Graded ceramic gears, 172 Graded ceramic matrix composites,

227 Graded chromium doped alumina,

213 Graded CulNb/Cu, 216 Graded fields, 164 Gradedindex,98,247,288,289,291,

292,293,312,294 Graded index polymer optical fiber,

295 Graded magnetic properties, 211 Graded metal-ceramic composites,

61,172 Graded multilayer coatings, 270 Graded porosity, 279 Graded properties, 164 Graded PSZlNiCr coating, 192 Graded radial distribution, 213 Graded semiconductor, 247, 284 Graded sheet, 234 Graded SiC coating, 199

Index

Graded silicon nitride/titanium nitride, 215

Graded single crystals, 164 Graded structures, 2 Graded thermal barrier coatings

(TBCs),251 Graded tungsten/copper, 265 Graded tungsten-copper, 170 Graded Young's modulus, 213 Gradients

arrangement of phases gradient, 40 asymmetric gradient, 271 composition gradient, 40 compositional gradient, 41, 132, 137,142,143,165,167,171,182, 198,203,210,219,222,224,226, 240,273 compositional gradients, 49, 128, 137,165,166,167,171,207,208, 210,212,214,220,222,223,227, 244 continuous gradients, 168 density gradients, 165, 181,212 grain size gradients, 210 magnetic saturation gradients, 211 mechanical gradient, 181 microstructural gradient, 181 one-dimensional gradient, 42 self-enhancing temperature gradients, 182 shape gradient, 40 size gradient, 40 stepwise gradients, 168 temperature gradient, 90, 100, 109, 110,130, 131, 145, 175, 177,210, 236,241,269 thermal gradients, 65, 207 three-dimensional gradient, 42 two-dimensional gradient, 42 volume fraction gradient, 40

Gravitational segregation, 214

323

GRIN, 247, 288, 289, 290, 291, 293, 295,296

GRIN polymers, 293 HAp, 5, 297, 298, 299, 300, 301, 302,

303,304,305 Hardwood, 15 Hashin and Shtrikman, 69, 72, 77, 78,

79,80,81,82 Heat capacity, 80,99, 106, 109 Heat flow, 90,100,102,192,259 Heat flux, 99, 103, 104, 105, 106,

147,150,186,236,250,259,265, 266,308,310

Heat of transport, 110 Heat resistance, 99, 100,250 Heterogeneous materials, 100 Heterojunction, 247, 284, 285, 287 Heterostructure, 247, 287, 288 HIP, 124, 125, 146, 158, 170, 180,

181,215,263,311. See hot isostatic pressing

Hip implants, 5 Hip joint, 247, 297 Hip prosthesis, 248 Hot isostatic pressing (HIP), 124,

146,161,165,170,180,185,215, 216,235,257,263,264

Hot pressing, 159, 161, 165, 168, 177,180,185,216,225,229,230, 231,262,281

Hutchinson-Rice-Rosengreen theory, 123

Hybrid energy conversion system (HYDECS), 258

Hydroxyapatite (HAp), 5, 20, 220, 223,247,297,299,306,312,313

Hypersonic space plane, 3 Implant, 5,197,247,248,306,313 Inclusions, 72, 176, 182,234 Indium antimonide, 214 Indium phosphide, 283 Inelastic deformation, 149

324

Infiltration, 161, 163, 165, 170, 181, 209,210,235,241,243,265,279

Intelligent adaptive system, 17 Intelligent material system, 7, 14 Interconnected microstructures, 60 Interconnected second phase, 54 Interdiffusion, 50, 56, 186, 198,208,

217,254,268,310 Interdiffusion rates, 52 Interface bioactive bone cementation,

302 Interface energy, 175, 176 Interfaces, 32, 39,44,55,90, 114,

219,233,234,253,261,267,268, 275,283

Interlayers, 64, 66,103,131,164, 186,216,217,218,240,242,243

Intermetallic compounds 208, 219, 232 AIIAI3Ni, 89, 112 iron aluminide/stainless steel FGM,248 NiAI,89, 114, 168, 180, 184,224, 234,235,237 nickel aluminide, 112, 168, 175, 180,224 titanium aluminide, 158, 181, 198, 257,309

Isothermal solidification, 216 Japanese sword, 270 Jet electroplating, 203 Joining, 30,134,201,214,215,216,

217,218,219,232,240,242,243, 258,275,282,315

Joining dissimilar materials, 214 Kerner's model, 70, 72, 75, 76, 77,

78,79,80,81,82,86 Lal.,Sr,Cr03, 268 Laminate theory, 115 Laminated object manufacturing, 161 Lamination, 164, 174,235,236 Lanthanum cobaltite, 268

Index

Laplace transformation, 106 Laser beam cladding, 161, 187 Laser beam surface heating, 175 Laser cladding, 187, 225 Laser flash method, 100, 105, 109 Laser interferometry, 99 Laser processes, 187 Layer processing, 163, 186 Layer-by-Iayer stacking, 165 Layered manufacturing, 219 Lead telluride, 262, 263 Lead zirconate titanate, 210, 223 Linear elasticity, 69 Liquid phase diffusion, 164 Liquid phase joining, 219 Magnetic fusion reactor, 247 Magnetron sputtering, 161, 197, 198 Material ingredient, 1, 3 Mean intercept length, 36, 44 Mean linear spacing, 37 Mean-field, 46, 71 Mechanical design, 7 Mechanosensor,7, 14 Melt infiltration, 161 Melt processing, 163,212 Metal matrix composites, 232 Metal-ceramic composite, 176,261 Metal-ceramic joints, 278 MgOlNi,175 Microcrack, 90 Microgravity, 161,214,242 Microhardness, 22, 90, 203, 272 Microlaminar ceramics, 204 Microlamination, 186 Micromechanical approach, 63, 70,

73,80,81,83 Microscopic length scale, 67 Microstructure, 7

microstructural analysis, 29, 30, 31,45,47 microstructure characterization, 29

Index

microstructural features, 30, 32, 37,39,40,41,42,43,44,47,55, 84,188,265 microstructural transition, 38, 53, 62, 155, 159 microstructural variation, 40, 47

Modified rule of mixtures, 74 Modulus of elasticity. See Young's

modulus Moire interferometry, 99 Mollusk shell, 7, 8, 27 Mullite, 232 MullitelSi3N4, 180 Mullite-zirconia, 172 Multifunctionality, 7, 8 Multilayered material, 106 Multilayered structure, 55 Multimedia, 247 Multiple graded features, 41 Nap-Bp3-Si02,292 NASICON, 232, 245, 247, 253, 308 Near-net shape, 181 Nextel, 230, 231, 247, 253 Ni/MgO,60 NiAI, 89, 114, 168, 180, 184,224,

234,235,237 NiAl/AI,89 Ni-AIP3' 76,87,234 Nicalon, 230, 231, 247, 253 Nickel aluminide, 112, 168, 175, 180,

224 Nickel/magnesium oxide, 60 Nickel-base superalloys, 215 Nickel-phosphorus, 202 Ni-NiO, 182 NiO-YSZ, 269 Nitrided titanium alloy stem, 305 Nondestructive evaluation, 128, 130 Non-linear deformation, 142 Nonplanar laminates, 204 Numerical solution techniques, 66,

73

325

Optical communication system, 292 Optical fibers, 98, 187,247,288,

290,291,292,295,312 Optical microscope, 130 Optical polymer fiber, 247 Orthopedic implants, 297 Osteoblasts, 13 Osteoclasts, 13 Ostwald ripening, 50, 90 Oxidation resistance, 2, 151, 152,

198,267 Oxidation resistant protective

coatings, 191 Palm tree, 7,17,18,19,20,24 Paris' law, 127 Partially stabilized zirconia, 60, 63,

83,85, 109, 114, 131, 132, 146, 174,192

PbTe,262 Percolation, 38,54,60,89,93, 161,

176,177,236 percolation clusters, 60 percolation models, 54 percolation threshold, 38

Perturbation method, 106 Phase distribution, 59, 69, 132, 139,

142,148,271 Phase equilibria, 48, 50 Phase shape, 36, 37 Phase stability, 52, 308 Piezoelectric, 13,21,26,96,97,98,

155,210,241,315 Piezoelectric effect, 13,21,26 Piezoelectricity, 26, 89 Pixels, 33 Plasma spraying, 109, 145, 146, 159,

161,190,191,192,200,239,251, 253,278,297,310 air plasma spraying, 192 low pressure plasma spraying, 192

Plaster of Paris mold, 170 Plastic deformation, 116, 132, 232

326

Plasticity, 73, 83, 84, 123 Poisson's ratio, 63, 71, 89, 112, 114,

120,121, 154 Polyhedral shaped grains, 178 Polymer optical fiber, 295, 296 polymethylmethacrylate (PMMA)

bone cement, 301 Porous metal, 247, 297, 299, 300,

301,304,312 Powder stacking, 161, 164, 165, 166,

167,170,316 Power-law, 63, 84 Preform, 161, 164, 166, 183,207,

210,292,315 Preform processing, 163, 164,207 Pressure induced flow, 170 PSZ, 60,83,87, 109, 114, 130, 131,

132, 133, 134, 136, 137, 138, 139, 140,141,142,144,145,146,149, 150,156,159,192,193,236,239

PSZlNi,89 PSZlNiCrAlY, 89 PSZJSS,60,89, 114, 130, 131, 133,

134, 136, 137, 138, 141 Pyrometer, 104, 145 PZT, 161,210,241 Quantum well, 286, 288 Quasi-electric field, 89,95,96,247 Radar absorbent composites, 232 Rapid prototyping, 219

extrusion freeform fabrication, 161,224,244 fused deposition modeling, 161 laminated object manufacturing, 161 stereolithography, 161,220,222 three-dimensional printing, 161

Rapid quenching, 188 Ravichandran model, 72, 80, 81, 82,

83,87 Rayleigh wave, 113 Reaction layers, 161, 186

Index

Reactive powder processing, 183 Reactive sputtering, 197 Refractive index, 98, 247, 288, 289,

291,292,293,294,295 Refractory porous preforms, 210 Representative volume element, 63 Reuss estimate, 68 Rocket combustor, 247 Rule of mixtures, 16,46,63,68,69,

73,74,75,76,77,78,79,80,81, 82,84,115,123

Saturation magnetization, 211 Sedimentation, 161, 164, 168, 170,

172 Seebeck coefficient, 264 Self-adaptive modeling, 7 Self-healing effect, 152 Self-optimization, 7, 21 Self-propagating high temperature

synthesis (SHS), 124, 125, 146, 158,159,161,162,184,210,238, 311

Semiconductor heterojunctions, 282 Separate confinement

heterostructures, 287 Settling, 161, 168, 170,212

settling under centrifugal forces, 212 settling under gravity, 212

Shape conformation, 178 Shape parameters, 37 Shear modulus, 18,63,69,70, 71,

73,76,80,119,154 Sheet stacking, 173 Shrinkage curve, 174 Si.sGe.2, 262 Si)NjfiN, 215 Si)N4-Cu FGM, 247, 279, 280

Si().zAl,OzNs-z' 179 SIALON,179 SiC/C, 151, 159, 161, 199,200,240,

241,247,249,250,277,307

Index

SiCffiC/CC, 89, 152 SiC-SiC/C, 209 Silicon, 283 Silicon germanium, 262 Sintering, 52

differential sintering, 161 liquid phase sintering, 161, 179, 271 liquid sintering aid, 229 microwave sintering, 161, 175, 182 plasma activated sintering, 161, 177 pulsed electric current sintering, 115,161,177,178,277 reaction sintering, 161 selective laser sintering, 161 sintering faults, 174 sinter-HIP, 181 solid state sjntering, 161, 174, 179 thermal explosion reactive sintering, 184 transient liquid phase sintering, 161, 179 transient viscous flow sintering, 179

Slip casting, 161, 170, 171, 182,235 Slip casting-sedimentation process,

172 Slurry deposition, 164 Softwood, 14, 15 Solar receiver, 247 Sol-gel infiltration, 161 Sol-gel process, 180,219,229 Sol-gel techniques, 228 Solid freeform fabrication, 161, 162,

219 Solid freeform processes, 187 Solid oxide fuel cell, 247, 266, 268,

269,270,310 Solid state diffusion, 161, 164, 186 Solid state diffusion bonding, 232

Solid state joining, 161 Solution reprecipitation, 178 Space plane, 159, 161,247,249 Spalling, 146, 148, 150, 152, 253,

265 Spalling resistance, 146 Spark plasma sintering, 161, 177 Spatial gradation, 2

327

Spatial variation, 29, 30, 40, 41, 44, 47,48,55,56,67

Specific boundary area, 32, 35, 36 Specific heat, 71, 100, 101, 153 Specific surface area, 32, 39 Spray deposition, 65, 161, 165, 167,

186,188,239 Spray forming, 161, 186, 188, 190,

234 Sputtering, 196 Step index, 247 Stereolithography, 161,220,222 STL file, 220 Strain

inelastic strain, 142 isostrain, 63, 72, 74 plane strain, 118, 120 strain energy release rate, 155 strain hardening, 63, 74, 89, 115, 116, 117, 124 strain to failure, 115 stress-strain, 74, 83, 116 stress-strain behavior, 74, 84, 115 stress-strain curve, 112, 116

Strength,2 bending strength, 114, 115 tensile strength, 15, 18,20,21,22

Stress 2 axial stress, 133, 134, 137, 138, 141 asymptotic stresses, 154 circumferential stress, 134, 137 compressive stress, 127, 140, 142, 152,271,273 flow stress, 63, 74

328

isostress, 63, 72, 74 maximum tensile stress, 133 principal stress, 7, 13 residual stress, 89, 112, 115, 127, 142,152,226,227 residual stresses, 30, 43, 85, 115, 116,117,127,157,177,213 residual thermal stress, 63 shear stress, 7, 134, 137,224,298 stress corrosion cracking, 155 stress distribution, 7, 10, 15,43, 127, 133, 140 stress intensity factor, 89, 120, 121, 122, 127, 154, 155 stress singularity, 118 stress state, 115, 118, 142 stress-strain, 74, 83, 116 stress-strain behavior, 74, 84, 115 stress-strain curve, 112, 116 tensile stress, 151 thermal residual stresses, 65, 127 thermal stress, 43,89, 130, 131, 132, 133, 138, 139, 148, 149,271, 279 thermal stresses, 30, 65, 76, 90, 130,138,141,149,157,200,201, 215,261,265 thermal stress relaxation, 278 transient thermal stress, 142 yield stress, 123, 124

Strontium doped lanthanum chromite, 268

Strontium doped lanthanum manganite, 268

Subcritical crack propagation, 155 Superalloy, 87,186,187,194,215,

224,247,254,257,278,281,311 Superiattice, 94 Superplastic diffusion bonding, 232 Superplastic forming, 161,232 Superplasticity, 219 Surface crack formation, 140

Surface curvature, 39, 232 Surgical implants, 197 Tensile test, 114 Thermal runaway, 182 Thermal analysis, 65, 105

Index

Thermal barrier coatings (TBCs), 44, 64,86,143,158,194,239,247, 249,250,252,308,309,315

Thermal conductivity, 38, 54,69,70, 71,77,78,79,89,98,99,100, 101, 103, 104, 105, 109, 131, 146, 147,148,149,150,151,173,182, 192,200,254,259,265,281

Thermal cycling, 64, 99, 146, 148, 149,150,279

Thermal diffusion, 107, 110 Thermal diffusivity, 89, 98, 99, 100,

106, 107, 109, 156 Thermal expansion, 65, 75, 76, 77,

80,89,90,99,112,127,128,130, 131,152,156,172,173,180,200, 201,216,219,226,263,268,271

Thermal expansion coefficient (TCE), 65, 75, 76, 77,80,99, 112, 127,128,131,156,172,200,201, 219,268

Thermal fatigue, 89, 144, 146, 150 Thermal shock, 90, 128, 130, 138,

144,145,158,159,194,249,254, 265

Thermal spraying, 161, 164, 165, 186, 190, 194

Thermal stability, 90,109,200,241, 249,269,309

Thermionic conversion, 261 Thermionic converter, 247, 261, 309 Thermocouple, 104 Thermoelectric converter, 247, 261 Thermoforming, 232 Thermomechanical behavior, 67, 316 Thermomechanical response, 64

Index

Thermophysical properties, 67, 68, 69,98,99,100,105,106,173

Thin sheet lamination, 166 Thomson coefficient, 264 Three-dimensional FGMs, 223 Three-dimensional printing, 161 Ti(C,N) graded metal, 281 TilHAp FGMs, 305 Tiffi3Sn FGM, 219 TiffisSi3,219 Ti-6AI-4V, 197,240,247,297,299,

305,306,307,313 TiAI, 181, 184, 185, 198,237,238,

247,256,257,309 TiBffi,89 TiBjCu, 89, 146, 148, 185 TiB2-AI, 184 TiB2-Cu, 184 TiC/C,2oo TiCINi, 89, 149 TiCINi3AI, 175, 181, 185,236,237 TiC-Ni,89, 124, 184 TiNINi,215 TiNi,184 Titanium aluminide, 158, 181, 198,

257,309 Titanium carbide/carbon, 200

TlP-BP3-Si02,292 Torsional rigidity, 18, 19 Toughening ratio, 89, 123, 124 Transient heating, 98, 105, 108 Transient liquid phase bonding, 164,

186,218,242,243 Transient liquid phase joining, 161,

218 Transient temperature response, 105,

106 Transition zone, 216 Transport based processing, 161, 164 Transport properties, 38, 54 Transport-based methods, 38, 57 Tungsten carbidelcobalt, 248

Tungsten carbide-cobalt, 179,270 Tungsten/copper, 211 Turbine airfoils, 191

329

Turbine blade, 181, 186,247,251 Two-phase material, 35, 38, 39, 53 Two-phase mixtures, 38, 52, 54 Ultrasonic motor, 97 Underwater shock explosion, 161,

185 VN2C, 89, 109, 110, 111 Vacuum forming, 232 VAD, 247, 292 Van der Waals interaction, 205 Vanadium/vanadium carbide, 109,

111 Vapor phase diffusion, 164 Variational approach, 69 Variational principles, 69 Verneuil process, 161, 213 Viscous phase, 179 Voight estimate, 68 W/Cu, 81,211,247 Wakashima model, 61, 71, 72, 76,

77,78,79,80,81,82,86,156 Water-immersion ultrasonic method,

128 Waveguide, 286,287, 312 WC/Co,4, 103, 104, 105,248,271,

273,274,275,277 WC-Co, 179, 187,209,236,241,

247,270 Wear resistance, 270, 271, 273, 281,

305 Wear resistant coatings, 276 Weight-cost, 7, 22, 24 Welding technology, 215 W-Ni-Fe, 179 Wolfs law, 13 Young's modulus, 18,24,63,72,73,

80,81,82,89,112,114,154,156 YSZJNiCrAIY, 192 YSZlSS304, 83

330

Yttria-stabilized zirconia, 63, 78, 85 Zirconia, 232 Zirconia toughened alumina, 161,

225 Zirconia/titanium aluminide, 177 Zirconium carbide/carbon, 199 Zirconium phosphide, 209 ZrC/C, 199 Zr02 (yp3)-AIP3,234 ZrO/fiAI, 177 Zr02-Ni, 174, 182 ZTA, 225, 226, 227

Index

functionally graded materials: design, processing and applications

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