1. M.S. Blanter I.S. Golovin H. Neuhauser H.-R. Sinning Internal Friction in Metallic Materials A Handbook With 65 Figures and 53 Tables 123

2. Professor Dr. Mikhail S. Blanter Moscow State University of Instrumental Engineering and Information Science Stromynka 20, 107846, Moscow, Russia E-mail: mike@blanter.msk.ru Professor Dr. Hartmut Neuhauser Institut fur Physik der Kondensierten Materie Technische Universitat Braunschweig Mendelssohnstr. 3 38106 Braunschweig, Germany E-mail: h.neuhaeuser@tu-bs.de Professor Dr. Igor S. Golovin Physics of Metals and Materials Science Department Tula State University E-mail: golovin@relax.tsu.tula.ru Lenin ave. 92, 300600 Tula, Russia Professor Dr. Hans-Rainer Sinning Institut furWerkstoffe Technische Universitat Braunschweig Langer Kamp 8 38106 Braunschweig, Germany Series Editors: Professor Robert Hull University of Virginia Dept. of Materials Science and Engineering Thornton Hall Charlottesville, VA 22903-2442, USA Professor R.M. Osgood, Jr. Microelectronics Science Laboratory Department of Electrical Engineering Columbia University SeeleyW. Mudd Building New York, NY 10027, USA E-mail: hr.sinning@tu-bs.de Professor Jrgen Parisi Universitat Oldenburg, Fachbereich Physik Abt. Energie- und Halbleiterforschung Carl-von-Ossietzky-Strasse 911 26129 Oldenburg, Germany Professor HansWarlimont Institut fur Festkorper-undWerkstofforschung, Helmholtzstrasse 20 01069 Dresden, Germany ISSN 0933-033X ISBN-10 3-540-68757-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-68757-3 Springer Berlin Heidelberg New York Library of Congress Control Number: 2006938675 All rights reserved. No part of this book may be reproduced in any form, by photostat, microfilm, retrieval system, or any other means, without the written permission of Kodansha Ltd. (except in the case of brief quotation for criticism or review.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media. springeronline.com Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data prepared by SPi using a Springer LATEX macro package Cover: eStudio Calamar Steinen Printed on acid-free paper SPIN: 11014805 57/3100/SPI 5 4 3 2 1 0

3. To our families

4. Preface Internal friction and anelastic relaxation form the core of the mechanical spec-troscopy method, widely used in solid-state physics, physical metallurgy and materials science to study structural defects and their mobility, transport phe-nomena and phase transformations in solids. From the view-point of Mechan-ical Engineering, internal friction is responsible for the damping properties of materials, including applications of high damping (vibration and noise reduc-tion) as well as of low damping (vibration sensors, high-precision instruments). In many cases, the highly sensitive and selective spectra of internal friction (as a function of temperature, frequency, and amplitude of vibration) contain unique microscopic information that cannot be obtained by other methods. On the other hand, owing to the large variety of phenomena, materials, and related microscopic models, a correct interpretation of measured internal fric-tion spectra is often difficult. An efficient use of mechanical spectroscopy may then require both: (a) a systematic treatment of the different mechanisms of internal friction and anelastic relaxation, and (b) a comprehensive compila-tion of experimental data in order to facilitate the assignment of mechanisms to the observed phenomena. Whereas the first of these two approaches was developed since more than half a century in several textbooks and monographs (e.g., Zener 1948, Krishtal et al. 1964, Nowick and Berry 1972, De Batist 1972, Schaller et al. 2001), the second requirement was met only by one Russian reference book (Blanter and Piguzov 1991), with no real equivalent in the international literature. The present book, partly based on the Russian example, is intended to fill this gap by providing readers with comprehensive information about published experimental results on internal friction in metallic materials. According to this objective, this handbook mainly consists of tables where detailed internal friction data are combined with specifications of relax-ation mechanisms. The key to understand this very condensed information is provided, besides appropriate lists of symbols and abbreviations, by the introductory Chaps. 13: after the Introduction to Internal Friction in Chap. 1, defining and delimiting the subject and clarifying the terminology, the relevant

5. VIII Preface internal friction mechanisms are briefly reviewed in Chaps. 2 (Anelastic Relaxation) and 3 (Other Mechanisms). Although somewhat more space is obviously devoted to the former than to the latter, this part should not be understood as a systematic analysis of the physical sources of anelasticity and damping; in that respect, the reader is referred e.g., to the above-mentioned textbooks. The data collection itself, as the main subject of the book, can be found in Chaps. 4 and 5. The tables, generally in order of chemical composition, include the main properties of all known relaxation peaks (like frequency, peak height and temperature, activation parameters), the relaxation mechanisms as sug-gested by the original authors, and additional information about experimental conditions. Other (e.g., hysteretic) damping phenomena, however, could not be considered within the limited scope of this book, with very few exceptions. Chapter 4, which represents the main body of data on crystalline metals and alloys, is divided into subsections according to the group of the main metal-lic element in the periodic table, with alphabetic order within each subsec-tion. Chapter 5 contains several new types of metallic materials with specific structures, which do not fit well into the general scheme of Chap. 4. A short summary or specific explanations are included at the beginning of each table. Although the authors made all efforts to be consistent in style throughout the book, some difficulties in evaluating individual relaxation spectra led to slight deviations, concerning details of data presentation, between the different chapters and subsections. Since some of the data were evaluated from figures, the accuracy should generally be regarded with care; in cases of doubt, the original papers should be consulted. Over 2000 references published until mid 2006 were included, among which many earlier ones are still important be-cause certain alloys and effects are not covered by the more recent literature. Latest information, if missing in this book, might be found in three confer-ence proceedings published in the second half of 2006 (Mizubayashi et al. 2006b, Igata and Takeuchi 2006, Darinskii and Magalas 2006), as well as in forthcoming continuations of these conference series. This book is intended for students, researchers and engineers working in solid-state physics, materials science or mechanical engineering. From one side, due to the relatively short summary of the basics of internal friction in Chaps. 13, it may be helpful for nonspecialists and for beginners in the field. From the other side, its probably most comprehensive data collection ever published on this topic should also be attractive for top specialists and experienced researchers in mechanical spectroscopy and anelasticity of solids. The authors acknowledge gratefully the help of Ms. Tatiana Sazonova with the list of references, of Ms. Brigitte Brust with figures, and of Ms. Svetlana Golovina with tables. We are also grateful to the Springer team, in particular Dr. Claus Ascheron, Ms. Adelheid Duhm and Ms. Nandini Loganathan, for good cooperation. Moscow, Tula, Braunschweig Mikhail S. Blanter, Igor S. Golovin January 2007 Hartmut Neuhauser, Hans-Rainer Sinning

6. Contents 1 Introduction to Internal Friction: Terms and Definitions . . . 1 1.1 General Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Types of Mechanical Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Anelastic Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Thermal Activation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Other Types of Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Measurement of Internal Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Anelastic Relaxation Mechanisms of Internal Friction . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Point Defect Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 The Snoek Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2 Relaxation due to Foreign Interstitial Atoms (C, N, O) in fcc and Hexagonal Metals . . . . . . . . . . . . . . . 28 2.2.3 The Zener Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.4 Anelastic Relaxation due to Hydrogen . . . . . . . . . . . . . . . 36 2.2.5 Other Kinds of Point-Defect Relaxation . . . . . . . . . . . . . . 48 2.3 Dislocation Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.1 Intrinsic Dislocation Relaxation Mechanisms: Bordoni and NiblettWilks Peaks . . . . . . . . . . . . . . . . . . . 51 2.3.2 Coupling of Dislocations and Point Defects: Hasiguti and SnoekKoster Peaks and Dislocation- Enhanced Snoek Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.3.3 Other Kinds of Dislocation Relaxation . . . . . . . . . . . . . . . 73 2.4 Interface Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.4.1 Grain Boundary Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.4.2 Twin Boundary Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.4.3 Nanocrystalline Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.5 Thermoelastic Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7. X Contents 2.5.2 Properties and Applications of Thermoelastic Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.6 Relaxation in Non-Crystalline and Complex Structures . . . . . . . 95 2.6.1 Amorphous Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.6.2 Quasicrystals and Approximants . . . . . . . . . . . . . . . . . . . . 113 3 Other Mechanisms of Internal Friction . . . . . . . . . . . . . . . . . . . . . 121 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.2 Internal Friction at Phase Transformations . . . . . . . . . . . . . . . . . 121 3.2.1 Martensitic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.2.2 Polymorphic and Other Phase Transformations . . . . . . . 129 3.2.3 Precipitation and Dissolution of a Second Phase . . . . . . . 133 3.3 Dislocation-Related Amplitude-Dependent Internal Friction (ADIF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.4 Magneto-Mechanical Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 3.5 Mechanisms of Damping in High-Damping Materials . . . . . . . . . 148 4 Internal Friction Data of Crystalline Metals and Alloys (Tables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.1 Copper and Noble Metals and their Alloys . . . . . . . . . . . . . . . . . . 158 4.2 Alkaline and Alkaline Earth Metals and their Alloys . . . . . . . . . 189 4.3 Metals of the IIAVIIA Groups and their Alloys . . . . . . . . . . . . 196 4.4 Metals of the IIIB Group, Rare Earth Metals and Actinides . . . 223 4.4.1 Rare Earth and Group IIIB Metals . . . . . . . . . . . . . . . . . . 223 4.4.2 Actinides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 4.5 Metals of the IVB Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.5.1 Titanium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.5.2 Zirconium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.5.3 Hafnium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.6 Metals of the VB Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 4.6.1 Vanadium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 4.6.2 Niobium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 4.6.3 Tantalum and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 4.7 Metals of the VIB Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4.7.1 Chromium and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4.7.2 Molybdenum and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . 338 4.7.3 Tungsten and its Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 4.8 Metals of the VIIB group: Mn and Re . . . . . . . . . . . . . . . . . . . . . . 352 4.9 Iron and Iron-Based Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 4.9.1 Fe (pure) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 4.9.2 FeInterstitial Atoms (C, H, N), Other Elements (As, B, Ce, La, P, S, Y) ES) which explains the observed rate of the GB relaxation.

97. 82 2 Anelastic Relaxation Mechanisms of Internal Friction Table 2.17. Parameters of grain-boundary maxima in some pure metals (f = 1 Hz) (after Ashmarin 1991) Me peak Tm (K) Tm/Tmelt H (kJ mol 1) Al A 480610 0.50.65 117160 C 795 0.85 294 Cu A 473638 0.350.47 113168 B 703820 0.520.60 189210 C 9251025 0.680.76 202263 Ni A 670783 0.390.45 185294 B 843893 0.490.52 217246 C 9531075 0.550.62 260328 A: low temperature peak; B: medium temperature peak; C: high temperature peak. In pure metals three GB-induced damping maxima can exist (Ashmarin 1991): (a) a low-temperature peak with Tm (0.350.65)Tmelt (sometimes called Ke peak), (b) a medium-temperature peak with Tm (0.50.6)Tmelt associated with special grain boundaries but not observed in all metals, and (c) a high-temperature peak with Tm (0.550.85)Tmelt observed in coarse-grained samples. Examples of these peaks are displayed in Table 2.17. In low-concentrated substitutional solid solutions we may distinguish the low-temperature peak (at the same temperature as in a pure metal) from an additional peak at higher temperature (so-called impurity grain-boundary peak), which might be connected with the aforementioned change in the rate-controlling mechanism. The role of grain-boundary relaxation may become dominant in materials with extremely fine grains, where the GB regions constitute a substantial part of the total sample volume. These nanocrystalline materials (produced e.g., by extreme plastic deformation), with GB structures mostly far from equilibrium and particular mechanical properties, may require special model descriptions of deformation and anelastic relaxation beyond those mentioned earlier, and will be considered separately in Sect. 2.4.3. 2.4.2 Twin Boundary Relaxation A twin boundary is a very special type of grain boundary, separating two twin crystallites that are, with respect to their lattice, mirror images of each other (which is possible only at a well-defined misorientation angle). If the twin boundary is identical with the mirror plane, usually a low-indexed, close-packed crystallographic plane, it is called a coherent twin boundary. Since the twin crystals can be transformed into each other by a shear transfor-mation parallel to the mirror plane, the formation of twins (twinning), which may occur under sufficiently high stress or during recrystallisation (in metals and alloys of low stacking-fault energy), represents an additional deformation mechanism. Also the perpendicular shift of a twin boundary (growth of one twin at the expense of the other) means a shear deformation.

98. 2.4 Interface Relaxation 83 For anelastic relaxation and internal friction peaks to occur by stress-induced movement of twin boundaries, these boundaries must be sufficiently mobile. As crystallographic coherency exists across the twin interfaces, the relaxation mechanism cannot involve interfacial sliding (Nowick and Berry 1972). However, certain types of twin boundaries can be shifted as the result of movement of partial dislocations (Hirth and Lothe 1968); then, the corre-sponding dislocation mechanisms will be involved in twin boundary relaxation. Examples for the few existing experiments are those by Siefert and Worrell (1951) on Mn12at%Cu, De Morton (1969) and Postnikov et al. (1968b, 1969, 1970) on InTl alloys. Twinning is most frequently accompanying diffusionless phase transfor-mations (e.g., from cubic to tetragonal structure), which themselves involve a shear that can be accommodated by twinning in order to retain the external shape and to avoid high residual stresses in the sample. The high density of twin boundaries often produced in this case may give rise to large effects of anelastic relaxation and internal friction. The relation to martensitic trans-formations will be treated in Sect. 3.2.1. 2.4.3 Nanocrystalline Metals Subject of this section are polycrystals with ultra-fine grain sizes in the nanometer range. Such nanocrystalline materials form a special group of nanostructured materials (or nanomaterials) which also include other types of nanosized structures in one, two or three dimensions. Owing to the extremely rapid development of the field, a generally accepted terminology of nanomaterials has not yet been fully established. From the viewpoint of materials science, nanostructured materials may be classified into different groups according to the shape (dimensionality) and chemical composition of their constituent structural elements (Gleiter 1995, 2000); however, a less precise, synonymous use of terms like nanocrystalline, nanostructured, or nanophase is also found in the literature. There is also no commonly agreed grain size limit to define nanocrystalline materials. In the physical concept of highly disordered solids, it is the fraction of atoms situated in the cores of defects (grain boundaries, interfaces) which should be as high as possible. Under this viewpoint the grain size should be below 10nm (Gleiter 1989, 2000), but also limiting values of 15, 20 or 30nm have been mentioned. Engineers developing new materials, on the other hand, are sometimes using the prefix nano for length scales almost up to 1 m, which generally lacks a physical justification. An application-oriented delimitation of nanocrystalline grain sizes should rather be linked to spe-cific properties, which are expected to be different from those of conventional materials if dominated by the high density of grain boundaries. In some recent reviews, an upper limit of about 100nm is introduced (Tjong and Chen 2004, Suryanarayana 2005), which seems to be a reasonable compromise.

99. 84 2 Anelastic Relaxation Mechanisms of Internal Friction Many different methods and techniques have been employed to produce nanocrystalline metallic materials (n-Me), like inert gas condensation, mechanical alloying, severe plastic deformation, devitrification of amorphous precursors and many others. They all have their specific advantages or draw-backs concerning the compositions, properties or shapes (e.g., porous or fully dense, bulk or thin film) of materials produced; for example, the amorphous route is well established to produce nanocrystalline soft magnetic alloys since the successful development of FINEMET (Yoshizawa et al. 1988). With respect to improved mechanical properties, severe plastic deforma-tion (SPD) is one of the most important and widely used routes, as bulk and fully dense materials with ultra-fine grain (UFG) structure can be obtained in this way (Valiev et al. 2000, Mulyukov and Pshenichnyuk 2003). The surpris-ingly high temperature stability of such UFG structures has been attributed to a high GB diffusivity and low driving force of recrystallisation (Valiev 2002). In many cases it is not clear, however, whether there is a significant difference in GB diffusivity between the nanocrystalline and annealed states (Kolobov et al. 2001) or not (Wurschum et al. 2002). The small grain sizes of genuine n-Me lead to distinct changes in the mechanical properties including increased yield strength and hardness. A particular feature is the breakdown of the HallPetch relation at grain sizes around 20nm or below. In this range, a decreasing grain size leads to anom-alous softening, referred to as inverse HallPetch behaviour, which is associ-ated with the operation of diffusion-controlled mechanisms combined with GB sliding (e.g., Schitz et al. 1998, 1999, Yamakov et al. 2002a,b, Van Swygen-hoven 2002, Van Swygenhoven et al. 2003). The cross-over from normal to inverse HallPetch behaviour has been treated in a two-phase model (Kim et al. 2000, Kim and Estrin 2005), in which the grain boundaries deform by a diffusion mechanism, and the grain interiors by a combination of dislocation glide and diffusion-controlled mechanisms. Anelastic grain-boundary relaxation (Ke 1999) is considered, in a recent theory of non-equilibrium GBs (Chuvildeev 2004), to be hardly detectable in UFG metals below a certain temperature (0.35Tmelt), unless the dislocation density at the GBs is decreased. Alternatively, disclination concepts have also been discussed in connection with relaxation processes in n-Me (Romanov 2002, 2003). The conditions for GB relaxation are even less clear in multi-component n-Me produced by the amorphous route, where the same factors which favour glass formation may also lead to stabilised and more densely packed GB structures, being less susceptible to relaxation. This latter type of n-Me (for which, to our knowledge, no systematic studies of GB relaxation exist) is not included in the following considerations, but will be mentioned further below in connection with the respective amorphous alloys (Sect. 2.6.1). Experimental studies of anelastic properties of n-Me were undertaken by several research groups using different mechanical spectroscopy techniques. The main results of some systematic and therefore reliable studies, including the materials studied and the main effects observed, are summarised briefly in

100. 2.4 Interface Relaxation 85 Table 2.18. In most of these cases, the total temperature-dependent internal friction Q1(T) can be written as 1(T) = Q Q 1 b (T) + Q 1 r (T), (2.46) 1 b (T) and Q1 where the terms Q r (T) represent a background of internal friction and a superposition of different anelastic relaxation peaks, respectively. The first term, closely related to composition and microstructure of the respective alloy phases as well as to the dislocation structure, can depend on the annealing time, as well on the amplitude and frequency of the imposed oscillatory strain. This contribution was reported to be substantial enough to consider nanostructured Cu (Mulyukov and Pshenichnyuk 2003) and Mg (reinforced by different microparticles, Trojanova et al. 2004) as high-damping materials (see Sect. 3.5) even for low vibration amplitudes. The second term, which may contain contributions not only from GB relaxation but also from almost all relaxation mechanisms related to dislo-cations or point defects, is time-independent but frequency-dependent and can often be described by the Debye equation. Because of the lack of a com-bined study of nanostructured metals by different mechanical spectroscopy techniques, varying as many experimental parameters (frequency, tempera-ture, amplitude, annealing conditions) as possible, it is not easy to distinguish between pure anelastic relaxation mechanisms (most important: GB relax-ation) and irreversible mechanisms of structural relaxation, which are in most cases due to changes in the density and distribution of dislocations. Summarising the pertinent results published in the literature (partly pre-sented in Table 2.18), one can draw the following conclusions: Almost all UFG and nanostructured metals (except those produced from the amorphous state, see above) exhibit an IF peak (very roughly with an activation energy of about 1 eV), which is not often found in well annealed (coarse-grained) metals. The nature of this peak is still not entirely clear: some authors report a thermally activated, reversible anelastic GB relaxation consistent with the Ke approach (Ke 1999), while others attribute the effect to irre-versible structural changes like recovery and recrystallisation, connected with short-range GB diffusion in non-equilibrium GB. Internal friction can generally be correlated with superplastic proper-ties and thus can be used for determining the optimum temperature for superplastic deformation. Structural changes in severely deformed metals occur already around ambient temperature, as indicated by a group of low-temperature IF peaks observed after high-pressure torsion in Ti, Mg and several Fe-based alloys, which is extremely sensitive to heating. In some SPD-processed metals like Cu, a high damping capacity was reported in a broad range of strain amplitude and temperature, however, has not been reproduced all published works (see Sect. 3.5).

101. 86 2 Anelastic Relaxation Mechanisms of Internal Friction Table 2.18. Mechanical spectroscopy studies of UFG metallic materials materials mechanical spectroscopy short summary references Pd 1 TDIF, 15 Hz. Several IF peaks. IF peak H = 62.7kJmol 1 (reordering phenomena) Weller et al. (1991) Cu: 99.997% 99.98% 2 TDIFat 10 Hz and 100 kHz; TD- and ADIF 5MHz. IF peak 420 K: reversible dynamic GB rearrangement Akhmadeev et al. (1993) Au 99.99% 3 TDIF, 300500 Hz: Bordoni peak 120 K, dislocation peak 460 K, GB peak 750 K. Okuda et al. (1994) Al, Ni 4 TDIF, 130 Hz , 0.01200 Hz, 0.110 kHz. Relaxation IF peak 159 kJ mol 1 (Al: 475K 1 Hz, GB relaxation) Bonetti and Pasquini (1999) Cu (99.98%) Ni (99.98%) 2 TDIF, 17 Hz and 1kHz (time-dependent IF). Irreversible changes in the structure Gryaznov et al. (1999) Cu, Fe18Cr9Ni 5 TDIF, 2.5Hz, ADIF, 35 Hz. High damping; IF peaks at 54 and 475K Mulyukov and Pshenichnyuk (2003) Mg 6 TDIF, 0.5, 5, 50 Hz. Relaxation 1 due to peaks at 70K (116 kJ mol dislocations) and at 620K due to GB Trojanova et al. (2004) Mg alloys: Mg6Zn Mg9Al 2 TDIF, 10 Hz. Irreversible IF peak 530570 K, 87 kJ mol 1 enhanced GB diffusion Chuvildeev et al. (2004a) Fe25Ni 7 TDIF, 0.55 Hz. IF peaks due to martensitic transformation Wang et al. (2004a) Fe0.8C 5 TDIF, 12 kHz. Irreversible IF peak 550 K: recovery Ivanisenko et al. 2004 Fe25Al 5 TDIF, 0.52 kHz. IF peaks (150300 K) due to dislocations and self-interstitial atoms; unstable with respect to heating. Golovin et al. (2006a,c) Ti grad2 5 TDIF, 2 kHz. IF (Hasiguti) peak 210 K: dislocations and self-interstitial atoms; possibly hydrogen-related effect at 410 K. Golovin et al. (2006a) Fabrication methods: (1) evaporation, condensation, compaction; (2) equal channel angular pressing (ECAP); (3) gas deposition; (4) mall milling; (5) high-pressure torsion; (6) ball milling, compaction, hot extrusion; (7) consolidation.

102. 2.5 Thermoelastic Relaxation 87 2.5 Thermoelastic Relaxation 2.5.1 Theory Physical Principle In every solid, there exists a fundamental thermoelastic coupling between the thermal and mechanical states (e.g., between stress and temperature fields), with the thermal expansion coefficient as the coupling constant. The best known phenomenon of thermoelastic coupling is thermal expansion, as the response of the mechanical state to an applied change in temperature. Con-versely, fast adiabatic (i.e., isentropic) changes of the dilatational stress com-ponent result in (small) temperature changes, known as the thermoelastic effect. If such stress variations are spatially inhomogeneous either exter-nally according to the mode of loading (e.g., bending) or internally in a material with heterogeneous mechanical properties temperature gradients are produced which can then relax by irreversible heat flow (thermoelastic relaxation), causing entropy production and dissipation of mechanical energy. The resulting thermoelastic damping1 not to be confused with damping due to thermoelastic martensite2 (Sect. 3.2.1) represents the most fundamen-tal among all mechanical damping mechanisms, since it does not require any defects but exists in all solids with non-zero thermal expansion, even in the most perfect crystals. Assuming that the mean free path of the phonons is small compared to the length scale of the stress inhomogeneities, which is generally the case except for very low temperatures and high frequencies, the heat flow during thermoelastic relaxation can be described as a classical diffusion process. Biot (1956) pointed out that it is the entropy which satisfies the diffusion equation. Zeners Theory Thermoelastic damping is known since the late 1930s, when Zener was the first to give both a detailed theory (Zener 1937, 1938b) and a collection of related experimental results (Zener et al. 1938, Randall et al. 1939). The theory was developed in scalar (one-dimensional) form mainly for the transversal vibra-tion of homogeneous reeds and wires, but some other cases like spherical cav-ities or polycrystals with randomly oriented crystallites were also considered 1 As a fundamental thermodynamic phenomenon, thermoelastic damping is some-times also referred to as thermodynamic damping (e.g., Panteliou and Dimarogonas 1997, 2000). Other authors have called it elastothermodynamic (Bishop and Kinra 1995, 1997; Kinra and Bishop 1996) because the cause is elastic and the effect is both thermal and dynamic (i.e., time-dependent). 2 A martensitic transformation is called thermoelastic if its thermal hystere-sis and transformation energy is relatively small, comparable in magnitude with usual elastic strain energies. This alternative use of the term thermoelastic has nothing to do with thermoelastic coupling considered here.

103. 88 2 Anelastic Relaxation Mechanisms of Internal Friction by Zener. The simplest and best described case is certainly that of alternating transverse thermal currents (Nowick and Berry 1972) between the compressed and dilated sides of a homogeneous and isotropic, rectangular beam, vibrating in flexure with the frequency f. The thermoelastic damping of such a beam is in good approximation given by 1(f, T) = T Q f f0 f2 + f2 0 (2.47) with the relaxation strength T = 2EUT/Cp (2.48) and the peak frequency f0 = /2h2Cp, (2.49) where is the linear thermal expansion coefficient, EU the unrelaxed Youngs modulus, the density, Cp the specific heat capacity at constant pressure (or stress)3, the thermal conductivity and h the thickness of the beam (i.e., the distance over which heat flow occurs). Equation (2.47) has the same functional form as (1.8) and represents a Debye peak as a function of frequency, with a single relaxation time T = 1/2f0 = h2/2Dth (2.50) where Dth = /Cp is also called the thermal diffusivity. The analogy between (2.50) for the thermoelastic and (2.17) for the Gorsky relaxation, respectively, reflects the more general analogy between thermal and atomic diffusion already pointed out by Zener (1948). In the same way, the intercrystalline Gorsky effect introduced in (2.18) and (2.19) is analogous to the case of intercrystalline thermal currents (Zener 1948, Nowick and Berry 1972) with IT = R(3)2KUT/Cp (2.51) IT = d2/32Dth, (2.52) where R is an elastic anisotropy factor (see Zener 1938b for an estimate for cubic metals with randomly oriented crystallites), 3 denotes the volumetric expansion coefficient, KU the unrelaxed bulk modulus and d the dominating grain size in the polycrystal. Despite this analogy between atomic and thermal diffusion, it should be noted that the Arrhenius relation of thermal activation, (1.9), only holds for 3 Here we understand Cp per unit mass as found in most data collections; if con-sidered per unit volume as in Zeners original equations, the density does not appear in these equations. Instead of Cp, the symbol C (for constant stress) has also been used in the literature. If, on the other hand, Cp or C is replaced by C or C (at constant volume or strain), a small error in T (of the order of T 2) is introduced (Lifshitz and Roukes 2000).

104. 2.5 Thermoelastic Relaxation 89 the former but not for the latter having a comparatively weak temperature dependence. Thus, unlike the Gorsky relaxation, the thermoelastic Debye peak is found only as a function of frequency but not of temperature. Instead, both T /T in (2.48) and f0 in (2.49) are only weakly temperature-dependent (at least above the Debye temperature), so that thermoelastic damping is nearly proportional to the temperature. Another possibility of thermoelastic damping is related to longitudinal thermal currents between the hills and valleys of longitudinal elastic waves. In this case, treated in some detail by Lucke (1956), the relaxation time is itself frequency-dependent as 2 because the thermal diffusion dis-tance is given by half the wavelength, which means that in contrast to the normal case adiabatic conditions are expected here in the low-frequency (!) limit. With expected peak frequencies in the GHz range or even higher, longi-tudinal thermoelastic damping is usually negligibly small (Nowick and Berry 1972). Advanced Theories More extended and fundamental, three-dimensional and mathematically more rigorous treatments can be found in many later theoretical papers (e.g., Biot 1956, Alblas 1961, 1981, Chadwick 1962a,b, Lord and Shulman 1967b, Kinra and Milligan 1994, Lifshitz and Roukes 2000, Norris and Photiadis 2005). However, although the general thermoelastic equations and also some specific solutions (most often for the transversely vibrating EulerBernoulli beam) are now well known, it is up to the present date still difficult to calculate the thermoelastic damping explicitly for more complex cases beyond those already treated by Zener. An exact solution for the thin EulerBernoulli beam was given by Lifshitz and Roukes (2000), who also showed that Zeners approximation is valid within 2% in most of the relevant frequency range, except for the high-frequency side of the peak far above f0 where the deviations grow up to a 20% underestimation in the limit f . Therefore, the still widely spread use of Zeners (2.47)(2.49) is sufficiently accurate for many practical purposes, at least in the classical case of transverse thermal currents during flexural vibration of homogeneous samples. The analysis of Kinra and Milligan (1994) formed the basis for further model calculations of thermoelastic damping also in heterogeneous structures like fibre- or particle-reinforced composites (Milligan and Kinra 1995, Bishop and Kinra 1995), hollow spherical inclusions (Kinra and Bishop 1997), lami-nated composite beams (Bishop and Kinra 1993, 1997; Srikar 2005b) or some specific cases of pores and cracks (Kinra and Bishop 1996, Panteliou and Dimarogonas 1997, 2000; Panteliou et al. 2001). In the special case of flexural resonators made of polycrystals (e.g., of silicon) with particularly low thermal conductivity across the grain boundaries compared to that in the crystals,

105. 90 2 Anelastic Relaxation Mechanisms of Internal Friction a preliminary fast equilibration of the transverse thermal currents is possible inside the grains, which has been called intracrystalline thermoelastic damp-ing (Srikar and Senturia 2002). Another branch of theories is devoted to resonators with more complex external shape, usually in form of planar structures made of thin, flat plates vibrating predominantly either in flexure or in torsion. Although the ther-moelastic loss should be zero in case of pure shear, it is important to note that even the nominally torsional vibration modes almost always contain some flexural component which can produce significant thermoelastic damping. To solve this problem, a flexural modal participation factor (MPF) has been defined as the fraction of potential elastic energy stored in flexure (Photiadis et al. 2002, Houston et al. 2002, 2004). Assuming classical transverse ther-mal currents for this flexural component, the thermoelastic damping of any particular vibration mode is then obtained by multiplying the MPF with the classical result for the flexural beam e.g., from Zeners theory. The MPF itself can be calculated by integrating the curvature tensor of the vibration mode over the volume of the sample, provided the displacement field of the mode is known (Norris and Photiadis 2005). The problem then mainly consists of determining the mode shape, e.g., with the help of finite element modelling and/or advanced experimental techniques like laser-Doppler vibrometry (Liu et al. 2001). 2.5.2 Properties and Applications of Thermoelastic Damping To judge the practical importance of thermoelastic damping in a given mate-rial, we have to consider primarily the magnitude of the transverse relaxation strength T and the related peak frequency f0 according to (2.48) and (2.49). A detailed compilation of room-temperature relaxation strengths, including results of four data collections from the literature as well as re-calculated data using (2.48), is given in Table 2.19 for many pure metals and also a limited number of non-metallic materials. It is typical that in Table 2.19 the T values taken from different sources never match exactly. This scatter may come from unspecified microstructural influences (defects, textures) causing some variation mainly in the possibly anisotropic quantities and E, among which deviations in have a partic-ularly strong effect due to the quadratic dependence in (2.48). For our own re-calculations of T , the underlying basic data were checked for reliability by comparing different sources wherever possible. Ideally, the data in Table 2.19 refer to random polycrystals at least in case of metals. Exceptions are Si and Ge where single crystal values are given, according to the [100]-oriented wafers from which most of the respective resonators are fabricated. The second practically important quantity is the peak frequency f0 or, vice versa, the sample thickness h(f0) which belongs to a pre-selected peak frequency according to (2.49). With thermal diffusivities Dth usually in the

106. 2.5 Thermoelastic Relaxation 91 calc calculated from Table 2.19. Thermoelastic relaxation strengths T of pure metals and some other selected materials at 300 K: T Lit taken directly from the literature the intrinsic properties , E, and Cp,andT reference for T Lit 103 T 103 1) 1 K 3) 1) 6 K calc material (10 E (GPa) (kgm Cp (J kg T Lit Ag 18.9 83 10490 235 3.6 2.43.5 Zener (1948), Riehemann (1996), Srikar (2005b) Al 23.1 70 2700 904 4.7 4.65.1 Zener (1948), Kinra and Milligan (1994), Riehemann (1996), Srikar (2005b) Al2O3 2.73.7 Kinra and Milligan (1994), Srikar (2005b) Au 14.2 78 19300 129 1.9 1.72.2 Zener (1948), Kinra and Milligan (1994), Riehemann (1996), Srikar (2005b) Be 11.3 287 1850 1820 3.3 4.6 Zener (1948) Bi 13.4 32 9780 122 1.44 1.4 Zener (1948) Cd 30.8 50 8650 231 7.1 10 Zener (1948) Co 13.0 209 8900 421 2.8 3.4 Riehemann (1996) Cu 16.5 130 8920 384 3.1 3.03.7 Zener (1948), Riehemann (1996), Srikar 2005b Fe 11.8 211 7874 449 2.5 2.22.6 Zener (1948), Riehemann (1996), Srikar 2005b Ge 6.0 100 (Srikar 2005b) 5323 321 0.63 0.45 Srikar 2005b In 32.1 11 7310 233 2.0 3.1 Riehemann (1996) Ir 6.4 528 22650 131 2.2 2.35 Riehemann (1996) Mg 25 (Weast 1973) 45 1738 1025 4.7 4.85.4 Zener (1948), Kinra and Milligan (1994), Riehemann (1996) Mo 4.8 329 10280 251 0.88 0.86 Riehemann (1996)

107. 92 2 Anelastic Relaxation Mechanisms of Internal Friction Table 2.19. Continued reference for T Lit 103 T 103 1) 1 K 3) 1) 6 K calc material (10 E (GPa) (kgm Cp (J kg T Lit Nb 7.3 105 8570 265 0.74 0.71 Riehemann (1996) Ni 13.4 200 8908 445 2.7 2.62.9 Zener (1948), Riehemann (1996), Srikar (2005b) Pb 28.9 16 11 340 127 2.8 2.52.8 Zener (1948), Riehemann (1996) Pd 11.8 121 12 023 244 1.72 2.02.5 Zener (1948), Riehemann (1996) Pt 8.8 168 21 090 133 1.39 1.5 Zener (1948) Rh 8.2 275 12 450 243 1.8 0.7 Zener (1948) Sb 11.0 55 6697 207 1.44 1.51.8 Zener (1948), Riehemann (1996) Si 2.6 160 (Srikar 2005b) 2330 712 0.2 0.19 Srikar (2005b) SiC 0.350.6 Kinra and Milligan (1994), Srikar (2005b) Si3N4 0.22 Srikar (2005b) SiO2 0.003 Srikar (2005b) Sn 22.0 50 7310 227 4.4 4.04.8 Zener (1948), Riehemann (1996) Ta 6.3 186 16 650 140 0.95 0.3 Zener (1948) Ti 8.6 116 4507 522 1.1 0.81.2 Kinra and Milligan (1994), Riehemann (1996),Srikar (2005b) TiC 1.4 Kinra and Milligan (1994) W 4.5 411 19 250 132 0.98 0.81.3 Zener (1948), Riehemann (1996), Srikar (2005b) Zn 30.2 108 7140 388 10.7 5.818(!) Zener (1948), Riehemann (1996), Srikar (2005b) Zr 5.7 68 6510 278 0.37 0.68 Riehemann (1996) References: unless noted otherwise, the intrinsic properties , E, and Cp were taken from WebElements [http://www. webelements.com/].

108. 2.5 Thermoelastic Relaxation 93 range of 106 to 104 m2 s1, samples have to be prepared mostly with thick-nesses between 0.05 and 0.5mm in order to have maximum thermoelastic damping at 1 kHz. Metallic Materials In Table 2.19 the strongest effect is predicted for Zn with a relaxation strength as high as 0.01 (according to T calc) and a maximum thermoelastic damping Qm 1 = T /2 0.005, followed by Cd, Al, Mg, Sn; but also for Ag, Be, Co, Fe, Ni and Pb the thermoelastic loss factor at room temperature can exceed 103. Although such values are easily observable and practically significant, the interest in thermoelastic damping of metals has as yet been rather limited from both the fundamental and applied sides, and systematic experimental studies are very scarce. On the fundamental side, mechanical spectroscopy is usually concerned with thermally activated relaxation peaks, measured as a function of tempera-ture to study defects and transformations in solids. Thermoelastic damping is then noticed mainly as a linear background to be subtracted, but very rarely studied for its own sake. This has also experimental reasons: to trace out the full peak after (2.47), flexural frequency and sample thickness have to be mutually adjusted and varied accordingly, e.g., over at least two orders of magnitude in frequency, which requires more effort than just varying the temperature on a single sample. In addition, to observe the pure thermoelas-tic losses, other kinds of damping have to be suppressed effectively e.g., by suitable alloying. Only in the early days before many other mechanisms were known thermoelastic damping in metals was a subject of intense study as a main source of energy dissipation. The probably still most careful measurements of the thermoelastic relaxation peak come from that time, like the study of Bennewitz and Rotger (1938) on German silver, and in particular that of Berry (1955) on -brass which gave an impressively exact confirmation of Zeners theory of transverse thermal currents without any adjustable para-meters (see also Nowick and Berry 1972). Based on this fundamental work, the height and position of the thermoelastic peak were occasionally used later to determine coefficients of thermal expansion and conductivity, respectively, e.g., for some metallic glasses (Berry 1978, Sinning et al. 1988) or commercial Al and Mg alloys (Goken and Riehemann 2002). As an example, Fig. 2.33 shows the annealing-induced shift of the thermoelastic Debye peak, according to an increase in thermal conductivity from 7 over 11 to 17WmK1, due to structural relaxation and subsequent crystallisation of an amorphous Ni alloy (Sinning et al. 1988). In this case, the measurement temperature had been lowered to 170K to reduce the amount of other damping contributions, partly still visible in Fig. 2.33 on the low-frequency side of the peak for the as-quenched state; therefore, the thermoelastic peaks in Fig. 2.33 are almost a factor of two smaller than they would be at room temperature.

109. 94 2 Anelastic Relaxation Mechanisms of Internal Friction Fig. 2.33. Frequency-dependent internal friction of a rapidly quenched, meltspun Ni78Si8B14 ribbon (thickness h = 0.05mm) at T = 170K after different annealing treatments (the solid lines are fits to (2.47)): (a) as-quenched amorphous state, f0 = 1200 Hz; (b) after 2 h at 618K (structurally relaxed amorphous state), f0 = 1730 Hz; (c) crystallised, f0 = 2750 Hz (Sinning et al. 1988) Also worth mentioning in this context is the early work of Randall et al. (1939) on -brass with systematically varied grain sizes, which seems still to represent the only known example of a reliable observation of intercrystalline thermal currents. On the side of application, the main problem is that damping due to trans-verse thermal currents is available only in a relatively narrow frequency range around f0, depending on the geometry of the respective structural compo-nent. On the other hand, it might be possible in certain cases to adjust the geometrical dimensions or the thermal conductivity (by alloying) according to the technical requirements of damping properties. Much more interesting from the applied viewpoint are those thermoelas-tic damping contributions that occur in heterogeneous metallic materials like composites or porous metals. Three types of effects may be expected from such heterogeneities: 1. The introduction of new internal length scales, in addition to the sample dimensions, will distribute the dissipation processes over a much wider frequency range. This effect has been discussed qualitatively for metallic foams (Golovin and Sinning 2003b, 2004). 2. Thermoelastic damping will no longer be confined to flexural vibrations but will occur also in other deformation modes. 3. Additional heterogeneities cause additional temperature gradients and thus additional dissipation processes, i.e., more damping will be produced. This is the most promising but also least understood aspect: in fact, model calculations for specific arrays of pores (Panteliou and Dimarogonas 1997, 2000) have predicted a strong increase of thermoelastic damping with porosity, up to much higher values than in the case of classical trans-verse thermal currents; but the consequences for real materials are not yet clear. There is a strong need for theoretical as well as experimental

110. 2.6 Relaxation in Non-Crystalline and Complex Structures 95 research in this field, which then might open new perspectives towards the development of heterogeneous metallic materials with tailored properties of thermoelastic damping. Applications in Microsystems The recently renewed interest in thermoelastic damping is, in its main part, not related to the aforementioned perspectives of metallic materials but has a completely different reason: the rapid development of micro- and nano-electromechanical systems (MEMS and NEMS) which include silicon-based micromechanical resonators as central elements. Irrespective of the specific application (e.g., force sensors, accelerometers, bolometers, magnetometers, high-frequency mechanical filters or ultrafast actuators), the performance of the micromechanical system (e.g., sensor sensitivity) critically depends on the quality factor Q of the resonator which should be as high (i.e., the damping Q1 as low) as possible. That is, contrary to the metallic case discussed ear-lier, the aim is here not to produce damping but to avoid it. If in the most perfect silicon resonators all defect-induced sources of dissipation are removed, the thermoelastic damping remains and can be influenced only by a proper geometrical design and fabrication of the resonator. Especially with more complex-shaped resonators like single- or double-paddle oscillators (Kleiman et al. 1985, Liu et al. 2001, Houston et al. 2004) attempting quality factors as high as 108, or in case of layered structures including metallic or ceramic coatings (Srikar 2005b), this is not a trivial task. Since most of the recent theoretical progress on thermoelastic damping since about 2000 (see above) was without doubt strongly motivated by the needs of MEMS and NEMS, we have briefly sketched these important new developments here although their basic material, silicon, is as a semicon-ductor not included in the main data collections of this book. Finally, it should be mentioned as well that thermoelastic damping is also an important factor limiting the ultimate sensitivity of interferometric gravi-tational wave detectors (Black et al. 2004). 2.6 Relaxation in Non-Crystalline and Complex Structures With the important exception of the universal thermoelastic damping treated in the preceding section, most mechanisms of anelastic relaxation comprise the motion of defects interacting with an applied stress. According to the classical understanding of defects as structural imperfections in (periodic) crystals, such relaxation mechanisms are traditionally defined for crystalline solids (Nowick and Berry 1972). This classical line was also followed in the Sects. 2.22.4 on point defects, dislocations and interfaces, where the respec-tive microscopic processes of relaxation were introduced for the crystalline

111. 96 2 Anelastic Relaxation Mechanisms of Internal Friction case. An extension of such defect-related mechanisms to non-crystalline struc-tures is not obvious, except for some special cases like interstitial diffusion jumps of hydrogen atoms (if not coupled with the motion of matrix defects, see Sect. 2.2.4). In this context, the term non-crystalline is traditionally understood as opposed to periodic crystals, which then includes both amorphous solids and quasicrystals. To some extent this is still common practice (and practically useful), although it deviates from crystallographically correct terminology. In proper crystallographic terms, quasicrystals are in fact crystals in the wider sense of quasiperiodic crystals, which include both periodic and aperiodic, long-range ordered structures (Lifshitz 2003). From the viewpoint of anelastic relaxation of metals, on the other hand, quasicrystals and amorphous structures have many things in common, at least in case of icosahedral short-range order (cf. Sect. 2.2.4). There is a borderline, however, between common periodic crystals (in most practical cases with rel-atively simple crystal structures) on the one side, and other metallic structure types amorphous alloys, quasicrystals and to some extent even structurally complex periodic crystals with giant unit cells (Urban and Feuerbacher 2004) on the other side: in the former case, most defect-related mechanisms are quantitatively well understood and classified within the systematic and well-founded concepts of anelastic relaxation in crystalline solids (Nowick and Berry 1972; at that time crystals were always understood as periodic crys-tals), whereas in the latter case many details of the theoretical concepts have still to be developed. In principle we may distinguish roughly, in relation to the classical relax-ation processes in crystalline solids, between three types of relaxation mech-anisms in non-crystalline structures (in the above traditional meaning including quasicrystals): (a) Mechanisms which are independent of the structure type and exist in the same way in crystalline as well as in non-crystalline structures, with only numerical differences. Examples are thermoelastic damping and the Gorsky effect (at least in the basic form of transverse thermal or atomic diffusion currents), where relaxation strength and time may vary according to the values of the respective parameters, but all essential characteristics of the relaxation remain unchanged. (b) Mechanisms which are modified by the structure type, i.e., which are based on the same principle but with some conceptual differences calling for a modified or extended theoretical treatment. Examples are the Snoek-type relaxation in the generalised form as introduced for hydrogen in Sect. 2.2.4, or a hypothetical dislocation relaxation in an amorphous structure which can only be treated using a more general dislocation concept (independent of a crystal lattice). (c) Mechanisms which are specifically found in non-crystalline but not in (simple) crystalline structures. Examples are cooperative processes of

112. 2.6 Relaxation in Non-Crystalline and Complex Structures 97 directional structural relaxation or viscous flow (e.g., near the glass tran-sition) in metallic glasses, or some types of relaxation related to phasons in quasicrystals. While there is no reason to mention again type (a), we will focus in the following on mechanisms of types (b) and in particular (c) which can not always be differentiated clearly from each other. The aim is to give an intro-duction into those aspects of anelastic/viscoelastic relaxation in amorphous (Sect. 2.6.1) and quasicrystalline (Sect. 2.6.2) structures that have not yet been considered in the previous parts of this chapter. 2.6.1 Amorphous Alloys The most important aspect to be considered in amorphous alloys, also called metallic glasses, is the relation between structural and mechanical relaxation which are closely connected. To discuss this relation, it is first necessary to know the a-priori different definitions and characteristics of both kinds of relaxation. Since mechanical (anelastic or viscoelastic) relaxation has already been introduced in Chap. 1, a brief introduction into structural relaxation will be given here. Structural Relaxation In the literal sense, any time-dependent equilibration of the atomic structure of condensed matter, after any kind of external perturbation, may be called structural relaxation (SR). This may in principle include production, anni-hilation and rearrangement of defects in crystals (like equilibration of thermal vacancies after changes in temperature, or recovery and recrystallisation after plastic deformation or irradiation), and even certain cases of phase transfor-mations. However, it is more common to use the name structural relaxation more specifically for continuous changes of amorphous structures in partic-ular in glass-forming systems which are not so easily expressed in terms of defect concentrations but rather appear as integral modifications of the whole structure. For instance, temperature changes generally give rise to SR due to the temperature dependence of amorphous structures in (stable or metastable) equilibrium. The Glass Physics Approach Understanding SR in glass-forming systems is the key to understand glass per se, i.e., the formation and nature of glasses and the glass transition below which SR is largely frozen. According to many renowned experts, this is still the most challenging unsolved problem in condensed matter physics. The dif-ficult task of summarising the state of knowledge in this complex field was

113. 98 2 Anelastic Relaxation Mechanisms of Internal Friction tackled by Angell et al. (2000), by posing detailed key questions and review-ing the best answers available as given by experts and specialists in about 500 references. The subject was divided into four parts, i.e., three tempera-ture domains AC with respect to the glass transition temperature Tg, and a fourth part D dealing with short time dynamics which can be skipped here. The main emphasis in the review by Angell et al. (2000) is put on the high-temperature domain A of the (supercooled) viscous liquid at T > Tg where the system is ergodic (i.e., its properties have no history dependence). Impor-tant items to be understood are the temperature dependences of transport properties and relaxation times, e.g., in form of the VogelFulcherTammann (VFT) equation and deviations from it, as well as non-exponential relaxation functions of the form exp[(t/ )] with 0 < < 1 (KohlrauschWilliams Watts (KWW) or stretched exponential function, which was given a physical meaning e.g., by Ngais coupling model of cooperative many-body molecular dynamics (Ngai et al. 1991, Ngai 2000)). The VFT equation, e.g., for the viscosity , can be written as = 0 exp[D T0/(T T0)], (2.53) with the so-called fragility parameter D and VFT temperature T0, which are coupled with respect to the glass transition according to Tg/T0 = 1+D / ln(g/0) 1 + D /39, (2.54) where g and 0 represent the viscosities at T = Tg and T , respec-tively (Angell 1995). The fragility parameter D is used to distinguish between strong liquids or glasses with large D and almost Arrhenius-like behaviour (which would be exact for D = implying T0 = 0), and fragile ones with small D, a pronounced curvature in a Tg-scaled Arrhenius plot, and a very rapid breakdown of shear resistance on heating directly above Tg. A similar temperature dependence is also found for the relaxation time , which in this range A is so short that the structure can generally be considered to be in a relaxed state of internal equilibrium. The low-temperature domain C of the truly glassy state (T Tg), on the opposite side, can be defined as the range where the cooperative SR of the viscous liquid (also called main, primary or relaxation) is completely frozen. Here the properties change essentially reversibly with temperature (as they do in range A) but now depend strongly on history, i.e., on the initial time-temperature path on which the system was frozen. Relaxation in this glassy range is possible only by decoupled, localised motion of easily mobile species (also called secondary relaxations4). 4 These secondary relaxations are sometimes classified further as , , , . . . relaxations, which is more appropriate for polymers where the stepwise freez-ing of various local degrees of freedom may be associated with specific molecular groups, than for anorganic or metallic systems.

114. 2.6 Relaxation in Non-Crystalline and Complex Structures 99 In the intermediate temperature domain B near and not too far below the glass transition (T Tg), primary SR must be considered explicitly as it occurs continuously on all experimental time scales, but without reaching equilibrium except for long annealing times. This is the most difficult range in which structure and properties depend on both history and actual time during the measurement. A first approach relies on the principle of thermorheologi-cal and structural simplicity (Angell et al. 2000) which relates the molecular or atomic mobility to the structural departure from equilibrium, as described by a single parameter like the so-called fictive temperature Tf . As depicted in Fig. 2.34, the fictive temperature can be found by projecting the actual value of a certain property p (like volume, enthalpy, entropy, etc.) on the equilibrium curve for the liquid extrapolated from range A, using the slope p/T from the frozen range C. Structural relaxation in range B can then be described as a relaxation of Tf , in the simplest case according to T f = (T Tf )/ (2.55) with limiting conditions Tf = T in range A and Tf = const. in range C, respectively. The relaxation time now depends on both T and Tf, as expressed first by Tool (1946) (T,Tf) = 0 exp[xA/kT + (1 x)A/kTf )], (2.56) where x is a dimensionless non-linearity parameter (0 < x < 1, typically x 0.5), and A is an activation energy (Jackle 1986, Angell et al. 2000). Fig. 2.34. Definition of the fictive temperature Tf in different relaxing or frozen glassy states: (1) during and (2) after rapid cooling, (3) during slow cooling, (4) during heating after slow cooling. Indicated are also the temperature ranges AC (Angell et al. 2000; see text). For frozen states like in case (2), Tf may be considered identical with Tg for a given heating or cooling rate

115. 100 2 Anelastic Relaxation Mechanisms of Internal Friction In this simple form the fictive temperature concept has been useful for mod-elling relaxation in the difficult temperature range B; however, some ambiguity remains as regards which property p is chosen, and also the non-exponentiality (KWW function), found here as well, is not accounted for. The latter point is addressed by more advanced concepts like that of hierarchically constrained dynamics, considering elementary atomic relaxation events to occur not in parallel but in series (Palmer et al. 1984). The link in relaxation dynamics between ranges A and B is also underlined by correlations between the para-meters , D, A and x (Angell et al. 2000). Up to this point, the synopsis of SR under the viewpoint of glass physics applies to all kinds of glasses (polymers, metals, oxides), necessarily neglecting more specific aspects in these different classes of materials. In particular, for certain characteristics of SR in metallic glasses, some different viewpoints exist independently in the traditions of solid-state physics and materials science rather than of glass physics. SR in Metallic Glasses An obvious difference, as compared to non-metallic glasses, is that in metallic glasses SR has long been noticed mainly as a strong irreversible (irrecoverable) effect deep in the solid range (T Tg) existing even at room temperature, rather than as a phenomenon originating in the reversible properties of the undercooled melt above Tg as introduced earlier. This is a consequence of the high cooling rates used during production, especially in case of rapidly quenched conventional metallic glasses being in a highly unstable state far from equilibrium (high Tf ). The undercooled melt, on the other side, is more difficult to study and has been totally inaccessible before the development of bulk metallic glasses which, although first prepared by Chen (1974), became popular not before the 1990s (see Wang et al. 2004b for a review). On this historical background, some conceptually restricted usage of the term structural relaxation has partly developed for metallic glasses, regard-ing SR as being absent in the state of metastable equilibrium above Tg (e.g., Fursova and Khonik 2000) as observed macroscopically. This would however unnecessarily exclude from the term those fast dynamic processes in the vis-cous liquid which are needed to maintain equilibrium (e.g., during tempera-ture changes), and which in glass physics just form the core of SR, being only slowed down below Tg. To avoid this obvious inconsistency, in this chapter we use structural relaxation in its general physical meaning and only speak of different types or components of SR if necessary. It was shown long ago that the irreversible type of SR in metallic glasses, e.g., during annealing of a rapidly quenched PdSi glass, can increase viscos-ity by five orders of magnitude (Taub and Spaepen 1979, 1980), indicating enhanced atomic mobility in the initial unrelaxed state. In other words: this irreversible SR, affecting virtually all physical and mechanical properties p

116. 2.6 Relaxation in Non-Crystalline and Complex Structures 101 (Cahn 1983), cannot be a secondary relaxation in the frozen temperature range C but should be considered as a primary one in range B, kinetically extended to lower temperatures. At this point it seems surprising that at temperatures so far below Tg, there is also a reversible (recoverable) component of SR being even faster than the irreversible one, as observed e.g., for Youngs modulus (Kursumovic et al. 1980, Scott and Kursumovic 1982) or enthalpy (Scott 1981, Sommer et al. 1985, Gorlitz and Ruppersberg 1985), but hardly for density or vol-ume (Cahn et al. 1984, Sinning et al. 1985). This (selective) low-temperature reversible SR component, to be distinguished from reversible behaviour at the glass transition, is difficult to understand in terms of fictive temperature or primary/secondary relaxations, but at least roughly consistent with an earlier hypothesis by Egami (1978) relating reversible and irreversible SR, respec-tively, to changes in chemical and topological (or geometrical, Egami 1983) short-range order. The (also non-exponential) kinetics of such solid-state SR phenomena in metallic glasses, extensively studied in both conventional and bulk metallic glasses during the past three decades, have been widely analysed in terms of an activation energy spectrum (AES) model, introduced by Gibbs et al. (1983) on the basis of earlier work by Primak (1955), and subjected to some later extensions and modifications. This model is based on a wide non-equilibrium distribution of Debye-type relaxation events, which during annealing is gradually cut down from the low-energy side. While mathemati-cally equivalent to the use of a KWW function, the physics behind this model seems to be more consistent with the idea of independent relaxation centres (see later), instead of the picture of true cooperative motion associated with a KWW function. For a microscopic understanding of SR in metallic glasses, the oldest and maybe still most widely spread concept is that of free volume, which was introduced by Cohen and Turnbull (1959) and worked out later by van den Beukel and coworkers, incorporating also Egamis distinction between topolog-ical and chemical short-range order (e.g., van den Beukel 1993 and references therein). Alternative concepts were added more recently, for example based on interstitialcy theory (describing an amorphous solid as a crystal contain-ing a few per cent of self-interstitials; e.g., Granato 1992, 1994, 2002; Granato and Khonik 2004), or on the theory of local topological fluctuations (of atomic bonds and atomic-level stresses; Egami 2006). As SR is closely related to diffu-sion, much can be learned from the recent progress in understanding diffusion mechanisms in metallic glasses (Faupel et al. 2003), which generally revealed highly collective atomic processes (contrary to crystalline metals): according to molecular dynamics simulation supported by critical experiments, atomic migration mainly occurs in thermally activated displacement chains or rings. Being rather local at low temperature, these chains grow in size and concen-tration with increasing temperature until they finally merge into flow.

117. 102 2 Anelastic Relaxation Mechanisms of Internal Friction Relation Between Structural and Mechanical Relaxation Any structural relaxation whatever the exact microscopic mechanism is must involve atomic movements directed to lower the Gibbs free energy under the acting external perturbation, generally including anisotropic atomic-level distortions oriented in different directions (like the above displacement chains). If the external perturbation is isotropic, e.g., in case of a purely ther-mal deviation from equilibrium, such local anisotropies may be averaged out so that only a macroscopically isotropic volume change is observed. In the presence of a mechanical stress, however, the distribution of the local events may become asymmetric producing a net distortion in the direction of energet-ically favoured orientations, i.e., a mechanical relaxation due to a directional structural relaxation (DSR). In this generality, and using the widest meaning of SR which in principle applies to crystalline structures as well (see above), every mechanical relax-ation mechanism based on the motion of defects, including all cases considered in Sect. 2.22.4, might be called a DSR: under this viewpoint, DSR forms a very general principle of mechanical relaxation which of course also applies to amorphous structures. Thus, the connection between structural and mechan-ical relaxation is generally a rather close and direct one. More specifically, the different types and temperature ranges of SR in glass-forming systems must be considered. In the range of the primary relaxation around the glass transition, the same cooperative atomic motions cause both viscous flow and SR (i.e., SR occurs by viscous flow), so that relaxation time and viscosity can directly be converted into each other (for which, in spite of non-exponential relaxation, often a simple Maxwell model with = /EU is used, cf. Chap. 1). Therefore, in the range where a mechanical (e.g., internal friction) measurement is dominated by viscous flow, the result directly reflects the structural relaxation. There is a superabundant number of (mechanical and other) studies of the relaxation over wide frequency and temperature ranges in more stable non-metallic glass formers, whereas in metallic systems the relaxation is accessible only under favourable conditions using the best bulk metallic glass formers and low frequencies (see later). The situation is less clear in metallic glasses at temperatures further below Tg down to about 400K where the above-mentioned, specific types of irreversible and reversible SR are found, mechanical relaxation is at least partly anelastic (recoverable) in nature (Berry 1978), but plastic de-formation still occurs mainly by homogeneous flow. By assuming spatially separated structural relaxation centres represented by two-well systems, Kosilov, Khonik and coworkers developed a specific DSR model which applies in this range not only to mechanical relaxation but to mechanical prop-erties in general (e.g., Kosilov and Khonik 1993; Khonik 2000, 2003 and references therein). The relaxation centres (two-well systems) were divided into irreversible (highly asymmetric) and reversible (rather symmetric) ones, the former being responsible for mainly viscoplastic low-frequency internal

118. 2.6 Relaxation in Non-Crystalline and Complex Structures 103 friction, plastic flow and even for reversible strain recovery (Csach et al. 2001), whereas the latter cause anelastic processes seen at higher frequencies (Khonik 1996, Eggers et al. 2006). At still lower temperatures where plastic deformation of metallic glasses is known to change to a highly localised shear band mode, the primary SR is eventually frozen (range C in Fig. 2.34, in many cases below about 400 K). If speaking of DSR in this range at all, this can only mean secondary relaxations of special, easily mobile species, like those of interstitially dissolved hydrogen which have already been treated in Sect. 2.2.4. However, since such anelastic processes in metallic glasses classified as type (b) in the intro-duction to non-crystalline structures at the beginning of this section have more in common with crystalline structures than primary DSR, a true solid-state picture with a clear distinction of the relaxing defect might be more appropriate in this low-temperature range than the more general viewpoint of DSR. Internal Friction Phenomena in Metallic Glasses General Aspects Amorphous alloys have to be produced with the help of some non-equilibrium procedure (like rapid cooling from the melt, mechanical alloying, various kinds of deposition, etc.), during which the formation of the thermodynami-cally stable crystalline state is kinetically hindered. Therefore, all amorphous alloys crystallise when heated into a temperature range with sufficient atomic mobility, which is always connected with a maximum of internal friction at a temperature close to the onset of crystallisation (Fig. 2.35). In fact this crystallisation peak, with a position usually depending on heating rate but not on frequency (e.g., Zhang et al. 2002), is not a true relaxation peak but a transitory effect. It basically reflects the irreversible transition from the high and monotonically increasing IF in the glassy amorphous phase to a much lower damping level in the crystalline state, but can be a quite complex-shaped superposition of many different effects in the frequent case of a multiple-step crystallisation process. Once passed during heating, the crystallisation peak completely disappears during subsequent cooling or during a second heat-ing run. It has been used in some cases to study details of the crystallisation process including kinetics and activation energies (Sinning and Haessner 1985, Klosek et al. 1989, Nicolaus et al. 1992). In contrast to the high damping level at the onset of crystallisation, the internal friction in metallic glasses is generally low at room temperature and below, and at acoustic (vibrating-reed) frequencies often reduced to the ther-moelastic background (see Sect. 2.5) if no special low-temperature effects are there (see later). The temperature dependence of IF is rather weak up to about 400500 K, where a stronger, often exponential increase sets in which

119. 104 2 Anelastic Relaxation Mechanisms of Internal Friction Fig. 2.35. Comparison of the low- and high-frequency IF behaviour, at a heat-ing rate of 0.3Kmin 1, for two Ni-based glasses with (Ni60Pd20P20) and without (Ni78Si8B14) a glass transition before crystallisation. (1) Ni60Pd20P20, 0.08Hz; (2) Ni60Pd20P20, 450 Hz; (3) Ni78Si8B14, 0.095 Hz; (4) Ni78Si8B14, 400 Hz. The maxima of all curves (at 600K for Ni60Pd20P20 and 700K for Ni78Si8B14) correspond to the onset of partial (primary) crystallisation followed by further transformations (Sinning and Haessner 1988a) continues up to crystallisation. At all temperatures damping is higher at 0.1 Hz than at acoustic frequencies, indicating a broad spectrum of additional low-frequency processes. In this context, two main groups of metallic glasses have to be distin-guished: those which crystallise from the solid state before reaching the glass transition, and those which first show a glass transition and then crys-tallise from the undercooled melt (which largely corresponds to the distinction between conventional and bulk metallic glasses, except for a few inter-mediate cases like CuTi showing a Tg in a torsion pendulum at 0.30.5 Hz without being a bulk glass former (Moorthy et al. 1994)). Glass Transition and Relaxation As shown in Fig. 2.35 for a still moderate example, the occurrence of a glass transition has a dramatic effect on the height of the crystallisation peak at low frequencies which easily exceeds tan = 1, while the high-frequency IF peak remains unaffected and shows about the same height (tan < 0.1) as without a glass transition. The reason for this dramatic low-frequency IF increase, seen in Fig. 2.35 as the strong upward bend of curve 1 at Tg which is missing for the conventional metallic glass (curve 3), is the onset of dominating viscous damping Qv 1 due to the relaxation (described as Qv 1(T) = EU/(T) using a Maxwell model). It has been shown that this viscous onset,

120. 2.6 Relaxation in Non-Crystalline and Complex Structures 105 shifting to higher temperature with increasing frequency, is located just at the dynamic glass transition (assuming g = 1012 Nsm2) if the frequency is around 0.1 Hz; under certain conditions, it could be used for determining Tg at heating rates much lower than possible with the common DSC technique (Sinning and Haessner 1986, 1987, 1988b; Sinning 1991a, 1993a). It is important to note, however, that such maxima in the loss factor tan (or Q1) remain always transitory crystallisation peaks as mentioned earlier, even in presence of a glass transition: there is no glass transition peak or relaxation peak in tan in metallic (or more generally in low molecular weight) glasses, contrary to occasional misinterpretations in the lit-erature. The glass transition alone, without the intervention of crystallisation, produces an relaxation peak only in the loss modulus E (or G in case of shear) but not in tan = E/E which would in this case grow infinitely as E goes to zero in the supercooled liquid. The typical situation, producing a peak in tan , is depicted in Fig. 2.36 for Zr65Al7.5Cu27.5 (a moderate bulk glass former not very different from Ni60Pd20P20 in Fig. 2.35): whereas the loss modulus E shows two separate peaks, being identified with the relaxation and with losses during crystallisa-tion, respectively (Rambousky et al. 1995), the single maximum in tan does not reflect these two peaks. It is rather dominated by the behaviour of the storage modulus E in the denominator, which falls down in the supercooled liquid above Tg by more than one order of magnitude, to a sharp minimum that is solely determined by the onset of crystallisation (note the different, logarithmic and linear scales for the moduli and tan , respectively). There-fore, only the rising part of the damping peak may be associated with the relaxation. Fig. 2.36. Storage modulus E , loss modulus E and damping tan of as-quenched amorphous Zr65Al7.5Cu27.5, measured at 1 Hz during heating with 10Kmin 1 using a dynamic mechanical analyser (Rambousky et al. 1995). Tg denotes the onset of the calorimetric glass transition

121. 106 2 Anelastic Relaxation Mechanisms of Internal Friction For studying the relaxation by mechanical spectroscopy, it is therefore more appropriate to look at E and E (or G and G) separately, rather than just considering internal friction. To trace out the full relaxation peak in the loss modulus as a function of either temperature or frequency, it is important to have a wide supercooled liquid range, i.e., to use the best bulk metallic glasses available. Meanwhile such studies have been performed on several more advanced Zr- and Pd-based bulk glasses (e.g., Schroter et al. 1998, Pelletier and Van de Moort`ele 2002a, Pelletier et al. 2002b, Lee et al. 2003a, Wen et al. 2004); an example is shown in Fig. 2.37. The results follow the time-temperature superposition principle, well known from non-metallic glass formers: all curves fall on a master curve when shifted by a temperature-dependent relaxation time which usually obeys the VFT equation. Occasional low-temperature shoulders of the peak in E are sometimes interpreted as a relaxation (Pelletier and Van de Moort`ele 2002a). Contrary to the loss modulus, the loss compliance J does not show an relaxation peak either, but monotonically falls (like tan ) with increasing frequency or decreasing temperature. An analysis of its frequency dependence, with an exponent typically changing from 1 at low to 1/3 at high fre-quencies, may be used to separate Newtonian viscous flow from relaxation components and to discuss related models (Schroter et al. 1998). In addition, the interest in more specific questions of glassy dynamics in this range (which are beyond the scope of this chapter), calling for advanced or extended experimental conditions, has triggered some remarkable new experimen-tal developments in mechanical spectroscopy: for instance, a non-resonant Fig. 2.37. The relaxation of the Zr46.75Ti8.25Cu7.5Ni10Be27.5 bulk metallic glass studied by dynamic mechanical analysis. (a) Temperature dependence during heat-ing with 1Kmin 1 at different frequencies; (b) isothermal frequency dependence and fit to the KWW equation with an exponent = 0.5 (solid lines) at different temperatures (Wen et al. 2004)

122. 2.6 Relaxation in Non-Crystalline and Complex Structures 107 vibrating-reed technique with an extremely wide frequency range (Lippok 2000), or a special double-paddle oscillator for studying thin films (Liu and Pohl 1998) applied to glassy alloys at high temperatures and high frequencies (Rosner et al. 2003, 2004). The results of such fundamental studies on bulk metallic glasses generally confirm the main characteristics of the relaxation in the high-temperature domain A as briefly outlined above, known from non-metallic glass formers: in this respect, the underlying physics appears to be the same for quite different classes of glass-forming systems. Intermediate Temperature Range This is the classical range of materials science in which the study of inter-nal friction in metallic glasses began (Chen et al. 1971), and where most of our knowledge is still based on results obtained on rapidly quenched samples (although in this range there seems to be no big difference to bulk alloys; Berlev et al. 2003, Eggers et al. 2006). The main feature is here the exponen-tial increase of IF with temperature mentioned earlier, e.g., in form of the (on the logarithmic scale) linear rise of curves 24 in Fig. 2.35 towards the max-imum. The following main characteristics have been reported for this rising part of the IF spectrum: 1. It is reduced in its lower part (or shifted to higher temperature) by irreversible structural relaxation. For instance, if an as-quenched sam-ple is heated with a constant rate to successively increasing temperatures (Fig. 2.38), in each heating run the IF is reduced compared to the previous one, in close correlation to an irreversible increase of Youngs modulus or resonance frequency (seen more clearly in isothermal experiments; Morito and Egami 1984a, Neuhauser et al. 1990). Such an annealing behaviour is often analysed in terms of the AES model mentioned earlier, resulting in a broad spectrum for irreversible structural (not mechanical) relaxation ranging from about 100 to 200 kJ mol1 in case of Fig. 2.38. 2. Using appropriate isothermal anneals, Morito and Egami 1984b and Bothe (1985) have been able to cycle the IF spectrum reversibly (Fig. 2.39), proving an effect of reversible SR as well. 3. At frequencies about 1 Hz and below, the IF of as-quenched samples depends on the heating rate (Bobrov et al. 1996, Yoshinari et al. 1996b). 4. After a stabilising anneal and subtraction of the thermoelastic background Q 1 , the IF at constant frequency often shows a straight line in an B Arrhenius plot (Fig. 2.40), i.e., it is of the empirical form 1 Q Q 1 B eA/kT . (2.57) The slope parameter A increases with annealing (Berry 1978). 5. The IF increase shifts to higher temperatures at higher frequencies, i.e., it is a thermally activated relaxation effect (Fig. 2.40). The apparent activation enthalpies H, taken from cuts at constant damping according to

123. 108 2 Anelastic Relaxation Mechanisms of Internal Friction Fig. 2.38. Variation of (a) frequency and (b) damping of a vibrating amor-phous Ni78Si8B14 reed during heating-cooling cycles with 1Kmin 1. The vertical arrows indicate the onsets of structural relaxation and crystallisation, respectively, (Neuhauser et al. 1990) ln(f2/f1) = (H/k)(T 1 1 T 1 2 ), (2.58) vary between 115 and 250 kJ mol1 for different Pd- and Fe-based glasses and annealing treatments (with sometimes unphysically high attempt frequencies 1 0 ), and are in most cases much higher than the related slope parameters A (22125 kJ mol1) (Soshiroda et al. 1976, Berry 1978, Neuhauser et al. 1990). The problem of this discrepancy could be solved by Kruger et al. (1993) by showing theoretically that the r