diffusion and related phenomena in bulk ...the grain boundary diffusion in nanostructured materials...

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Corresponding author: A. P. Zhilyaev, e-mail: [email protected]

Rev.Adv.Mater.Sci. 2 (2001) 1-43

© 2001 Advanced Study Center Co. Ltd.

DIFFUSION AND RELATED PHENOMENA IN BULKNANOSTRUCTURED MATERIALS

M. D. Baró1, Yu. R. Kolobov2, I. A. Ovid´ko3, H.-E. Schaefer4, B.B. Straumal5,

R. Z. Valiev6, I.V. Alexandrov6, M. Ivanov2, K. Reimann4, A. B. Reizis3, S. Suriñach1

and A.P. Zhilyaev1, 6

1 Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain2 Institute of Strength Physics and Materials Science, Tomsk, Russia

3 Institute of Problems of Mechanical Engineering, St.-Petersburg, Russia4 Universitaet Stuttgart, Institut fuer Theoretische und Angewandte Physik ,70550 Stuttgart, Germany

5 Institute of Solid State Physics, 142432 Chernogolovka, Moscow District, Russia6 Institute for Physics of Advanced Materials, Ufa State Aviation Technical University, 450000 Ufa, Russia

Received: June 9, 2001; received in revised from: August 27, 2001

Abstract. The paper reviews results of experimental and theoretic studies of diffusion and relatedphenomena (grain growth, creeps, superplasticity) in bulk nanostructured materials. The particularattention is paid to the recent development in fabrication of bulk nanostructured materials andtheir microstructural characterization, evolution of bulk nanostructured materials during heating.Experimental study and theoretical modeling of grain boundary diffusion in bulk nanostructuredmaterials are described. Related phenomena (grain growth, creep, superplasticity) in suchmaterials and also ordering-reordering kinetics are presented.

CONTENTS

1. Introduction2. Recent developments in fabrication of bulk

nanostructured materials using severe plastic de-formation

3. Thermal stability and microstructure evolution insevere plastic deformed (SPD) materials duringannealing

4. The grain boundary diffusion in nanostructuredmaterials

5. Theoretical modeling of grain boundary diffusionin bulk nanostructured materials

6. Related phenomena (grain boundary phase phe-nomena, superplasticity) in nanostructured ma-terials

7. Ordering-reordering kinetics in Ni3Al base alloys

processed by ball milling8. Concluding remarks

1. INTRODUCTION

Since the introduction of nanocrystalline materialsand the progress in preparation techniques, manyinvestigators have hoped to discover extraordinary

properties (physical and mechanical) in pure met-als and alloys with nanocrystalline structure. Thishope is broadly based on the fact that grain bound-ary (GB) related phenomena should effectivelymanifest themselves as grain size decreases tonanorange scale. Therefore, nanocrystals, materi-als with mean grain size of 100 nm and less, haverecently become the subject of noteworthy inter-est. They demonstrated novel and often extraordi-narily properties such as a decrease in the elasticmoduli, decrease of Curie and Debye temperatures,enhanced diffusivity and improved magnetic proper-ties [1]. With respect to mechanical properties e.g., high strength, wear resistance, ductility, highstrain-rate superplasticity were observed in manyexperiments [2]. Recently Nature published thepaper on finding of low temperature and/or high strainrate superplasticity in nanocrystalline metals andalloys [3]. These results have remarkable scientificvalue for materials science and allow developing ofnew technology for superplastic forming of complexshape components. It is supposed that such su-perplastic properties of nanocrystalline materials isdue to abnormally high diffusivity and has been indi-

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rectly confirmed in some publications on diffusionmeasurements in nanocrystalline materials [4, 5].Moreover, both high diffusivity and high strain ratesuperplasticity may be induced by the grain bound-ary phase transitions (see Chapter 6 of this review).Therefore, the application of the recently proposedGB phase diagrams will allow to explain the unusualproperties of nanocrystalline materials and to de-velop the advanced nanocrystalline alloys with novelproperties.

A comprehensive study of physical and mechani-cal properties has become possible as innovativemethods such as severe plastic deformation (SPD)and ball-milled powders consolidation, were ad-vanced. By these methods bulk nanostructuredsamples from pure metals, alloys and usuallythought brittle intermetallics can be processed [6,7]. However, nanostructured materials processed bydifferent methods often differ drastically in proper-ties in spite of the fact that they have comparablemean grain size. There is no consensus on what isa reason for such discrepancy.

As mentioned above the interfacial structuresand their evolution in mechanically synthesizednanostructured solids are essentially different fromthose in conventional coarse-grained polycrystalsand nanocrystalline solids synthesized by non-me-chanical methods due to high density of severe-de-formation-induced non-homogeneities in the inter-facial structures as well the presence of high-den-sity ensembles of lattice defects in grain interiors,that strongly interact with the interfaces. In manycases the complicately arranged interfacial struc-tures and their effect on diffusion phenomena innanostructured metals and alloys can not beunambiguously identified by contemporary experi-mental techniques [8,9]. This can be studied bytheoretical modeling of both the interfacial structuresand diffusion processes in materials under studyneeded to understand their nature and evaluate theirpotential for practical use.

This review highlights a scientific progress maderecently by different groups of scientists united inframework of the project INTAS 99-1216.

Part 2 is devoted to recent developments in fab-rication of bulk nanostructured materials by meansof severe plastic deformation. Using finite elementmodeling (FEM) of equal channel angular pressing(ECAP) a distribution of plastic deformation inten-sity and contact pressure during different frictionconditions were obtained. The modeling has showna high sensibility of plastic deformation uniformityto friction conditions between ingot and die. Theresults were employed to optimize ECAP process

and to fabricate bulk ingots with uniformnanostructured states in hard-to-deform tungstenand titanium. New results in microstructuralcharacterization of nanostructured materials ob-tained by high pressure torsion (HPT) is presented.An influence of HPT parameters such as appliedpressure, total strain, temperature on themicrohardness and microstructural evolution duringHPT in samples of pure nickel are discussed.

Part 3 represents some new results on micro-structural evolution of ECAP nickel during annealing.Thermal stability is most important issue for ultra-fine grained (UFG) and nanostructured materials.Any successful utilization of nanostructured mate-rials depends upon the subsequent thermal stabil-ity of the materials. In practice the microstructuresproduced by severe plastic deformation are in ametastable condition so that relaxation may occurat significantly lower temperatures than in coarse-grained materials.

Part 4 presents experimental study of GB diffu-sion in nanocrystalline and nanostructured materi-als. Higher diffusivity observed in nanostructurednickel processed by SPD methods has been ex-plained by non-equilibrium state of grain boundariesin the materials under study. The consistent modelof ordering and diffusion by means of vacancy mi-gration for nanocrystalline and coarse grained ma-terials has been presented. This is a well definedstarting point to separate the atomic migration inthe crystallites and in the grain boundaries ofnanocrystalline materials

Part 5 is devoted to theoretical modeling of grainboundary diffusion in bulk nanostructured materialswith special attention paid to the role of non-equilib-rium grain boundaries in the diffusion enhancement.

Part 6 has a deal with the influence of GB phasetransitions like wetting, prewetting, premelting onthe properties of materials. Application of the re-cently proposed GB phase diagrams is illustrated,particularly for the description of superplasticity inalloys and semi-solid processing of materials. Forthe first time it was suggested to explain high strainrate superplasticity by the existence of the equilib-rium GB liquid-like layer close to the bulk solidus.

Part 7 is related to ordering-reordering kineticsin nanostructured intermetallics processed by me-chanical alloying. Short introduction to ball millingis given. The reordering of mechanically disorderedNi

3Al-based alloys was experimentally studied by

means of differential scanning calorimetry and mag-netization measurements. It is shown that the ap-parent activation energies of the second stage arevery close to the reported values of the activation

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energies for ordering and vacancy migration in thesealloys. Reordering occurs through the formation andcoarsening of ordered domains.

In Conclusion all results are summarized fromthe viewpoint of advantage of nanostructured mate-rials in comparison to their coarse-grained counter-parts. Some examples of application of NS metalsand alloys are presented.

2. RECENT DEVELOPMENTS INFABRICATION OF BULKNANOSTRUCTURED MATERIALSUSING SEVERE PLASTICDEFORMATION

There has been considerable interest over the lastdecade in processing materials using various tech-niques of severe plastic deformation in which intenseplastic straining is achieved under a high confiningpressure [6,10]. These techniques have the poten-tial of refining the microstructure of metals and al-loys to the submicrometer or even the nanometerrange. The two most important SPD techniques forproducing bulk non-porous samples are equal-chan-nel angular pressing and high-pressure torsion. Fig.2.1 shows a schematic view of ECA (a) and HPT(b)processing.

Let us review briefly methods of severe plasticdeformation suitable for formation of nanostructures.These methods have to meet a number ofrequirements taken into account while developingthem for formation of nanostructures in bulk samples

and billets. These requirements are as follows. First,it is important to obtain ultra fine-grained structureswith prevailing high-angle grain boundaries since onlyin this case a qualitative change in properties ofmaterials may occur. Second, the formation ofnanostructures uniform within the whole volume ofa sample is necessary for providing stable proper-ties of the processed materials. Third, thoughsamples are exposed to large plastic deformationsthey should not have any mechanical damage orcracks. Traditional methods such as rolling, draw-ing or extrusion have not met these requirements.Formation of nanostructures in bulk samples is im-possible without application of special mechanicalschemes of deformation providing large deformationsat relatively low temperatures as well as withoutdetermination of optimal regimes of material pro-cessing. At present the majority of the obtainedresults are connected with application of two SPDmethods: high pressure torsion [11-24] and equalchannel angular pressing [11-15, 24-31]. There areknown some investigations on formation of nano-and submicrocrystalline structures in various met-als and alloys by means of multiple forging [32-39].

2.1. ECA Pressing

The method of ECA pressing realizing deformationof massive billets via pure shear was developed bySegal and co-workers in the beginning of the eight-ies [24,27]. Its goal was to introduce intense plas-tic strain into materials without changing the cross

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section area of billets. Due to that, their repeat de-formation is possible. In the early 90s the methodwas further developed and applied as an SPD methodfor processing of structures with submicron andnanometric grain sizes [11,13,31]. In these experi-ments the initial billets with a round or square crosssection were cut from rods, from 70 to 100 mm inlength. The cross section diameter or its diagonaldid not exceed 20 mm, as a rule.

During ECA pressing a billet is multiple pressedthrough a special die using an ECA facility in whichthe angle of intersection of two channels is usually90°. If necessary, in the case of a hard-to-deformmaterial, ECA pressing is conducted at elevatedtemperatures.

A general relationship to calculate the strainvalue of the billet during ECA pressing for N passeshas the following form [25]:

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φ φΨ Ψ Ψ (2.1)

Here ψ and φ are outer and inner angles, respectively.From this relationship it follows that for the angles,φ =90°, Ψ=20°, each pass corresponds to an addi-tional strain value approximately equal to 1.

During ECA pressing the direction and numberof billet passes through the channels are veryimportant for microstructure refinement. In papers[25,28-30] the following routes of billets wereconsidered (Fig. 2.2): orientation of a billet is notchanged at each pass (route A); after each pass abillet is rotated around its longitudinal axis throughthe angle 90° (route B); after each pass a billet isrotated around its longitudinal axis through the angle180° (route C). The application of all three routesresults in an increase in values of yield stress andstrength of a processed material which after severalpasses achieve saturation [26]. In [31] it was shownalso that the first three passes at ECA pressing ofCu and Ni samples lead to a growth of a strain load.Further, there is a stable stage of strengthening andthe load does not almost change.

Recently, a series of papers described the mi-crostructural evolution occurring during ECAP hasbeen published and it was shown that the meangrain size and microstructure homogeneity weredependent upon the precise pressing procedure suchas routines and temperature described in[28,31,40,41] and friction and strained condition ofsamples during ECA pressing [42]. Below we dis-cuss recent results on optimization of ECA process-

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ing in bulk nanostructured metals and alloys includ-ing hard-to-deform ones.

Experimental and Finite Element Modeling ofECA processing. The investigations of the strainedcondition were conducted on Cu (99.9%) cylindri-cal ingot with diameter 20 mm and length 100 mm,subjected to one pass pressing at room tempera-ture in ECA die-set with two channels intersectingunder 90° (Fig. 2.1). The graphite lubricant was usedto decrease friction between ingot and die. At thisthe coefficient of ingot friction on die walls matchesabout 0.2. The Al plugs with 2 mm in diameter wereembeded into an ingot. The plugs were placed per-pendicular to the longitudinal axis in different places

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(Fig. 2.3a). As far as SPD by ECA pressing is ful-filled just by simple shear the strained condition wereestimated according to the intensity amount of de-formation by shear, calculated in respect to the plugsinclination angle ϕ from their initial state. At this theintensity of deformation by shear was calculated bythe formula Γ = tan ϕ.The picture of the strainedcondition in the vertical longitudinal cross sectionof the ingot coinciding with channels plane indicatesthe non-uniformity of plastic flow (Fig. 2.3b). Thedomain of non-uniform deformation about to the lowerpart of the ingot in cross section and amount toabout 10-12% of cross section square. One maysuppose that this area can essentially influence theformation of homogenous structure in the ingot dur-ing a multiple repeat of ECA pressing. The domainof non-uniform deformation of the upper part ofsample cross section is very small and its role maynot be considerable during structure development.It was also determined that a strain rate during ECApressing did not effect essentially the character ofnon-uniformity in distribution of shear strain. Fric-tion between ingot and die is one of the mostimportant parameters affected the ECA pressing andstructure forming character in the wrought ingot.However, it is very difficult to investigate experimen-tally the influence of this parameter for frictioncoefficient value k

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Calculation of the contact pressures betweeningot and die aimed to choose the material provid-ing workability and safety of forming tool was ful-filled as well. The proportion of applied pressure andcontact pressure in the walls of the die-set indicatesthe necessity to use the die produced of high-strength steels.Application of Modeling Results. The results ofthe experimental simulation and FEM of plastic flowand strained condition in SPD ingots enable to chosethe most optimal conditions for application of ECApressing to obtain bulk uniform nanostructured in-gots due to fitting of optimal friction conditions. Bypresent time, techniques of obtaining of bulk ingotswith uniform ultrafine-grained nanostructures in Cu,Ni and others ductile and relatively easy wroughtpure metals and alloys were worked through. Tech-niques and problems of obtaining suchnanostructured states in bulk ingots of hard-to-de-form metals including Ti and W could be work outnow by modification of the ECA die-set and pro-cessing regimes.

As an example, we managed to develop anuniform ultrafine-grained structure with a mean grainsize less than 1 µm (Fig. 2.5) by ECA pressing oflow ductile and hard-to-deform tungsten in order toenhance strength and plasticity in comparison withthe initial state.

By FEM of ECA pressing a distribution of plas-tic deformation intensity and contact pressure dur-ing different friction conditions were obtained. Thereis a high sensibility of plastic deformation uniformityto friction conditions between ingot and die. Theapproaches of enhancement of ECA pressinguniformity due to optimization of friction conditionsbased on the obtained results of experimental and

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FEM have allowed to fabricate bulk ingots with uni-form nanostructured states in hard-to-deform tung-sten and titanium.

2.2. High Pressure Torsion

Devices, where high pressure torsion (HPT) wasconducted, were first used in [44,45]. Their designis a further development of the Bridgeman anvil typedevice [46]. In the first work these devices were usedfor investigation of phase transformations duringheavy deformation [44] as well as evolution of struc-ture and changes in temperature of recrystallizationafter large plastic deformations [17,18]. Successfulformation of homogeneous nanostructures with high-angle grain boundaries via severe torsion straining[6,12,40] was a very important step allowing one toconsider this procedure as a new method for pro-cessing of nanostructured materials.

The samples fabricated by severe torsion strain-ing are usually of a disk shape, from 10 to 20 mm indiameter and 0.2-0.5 mm in thickness. A signifi-cant change in the microstructure is observed alreadyafter deformation by 1/2 rotation [19], but for forma-tion of the homogeneous nanostructure severalrotations are required, as a rule.

Recent investigations also showed that severetorsion straining can be used successfully not onlyfor the refinement of a microstructure but also forthe consolidation of powders [21-23,47,48]. It wasrevealed that during torsion straining at room tem-perature high pressures equal to several GPa canprovide a rather high density close to 100% in theprocessed disk type nanostructured samples. Forfabrication of such samples via severe torsion strain-ing consolidation not only conventional powders but

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also powders subjected to ball milling can be used.The HPT consolidation of nanostructured Ni powderprepared by ball milling [47] can be given as an ex-ample. The conducted investigations showed thatthe density of the fabricated powders is close to95% of the theoretical density of the bulk coarse-grained Ni. TEM examinations showed the absenceof porosity. The mean grain size is very small, it isequal to 17 nm, and, consequently, a large volumefraction of grain boundaries is present. Since grainboundaries have a reduced atomic density the au-thors assume that the given samples demonstratea decrease in the theoretical density in the materi-als with a very small grain size and strong distor-tions of a crystal lattice. It is also very interestingthat a value of microhardness of the Ni samplesfabricated by HPT consolidation is about 8.6 GPa.The given value is the highest value of microhardnessmentioned in literature for the nanocrystalline Ni.

Speaking about HPT processing we can notethat only recently detailed investigations have beenconducted to evaluate the parameters influencingHPT (e.g., applied pressure, total strain, tempera-ture). In earlier reports, it was found for several met-als and alloys that a pressure of ~5 GPa and morethan five rotations of the sample in torsion were gen-erally sufficient to produce a reasonably homoge-neous microstructure with a mean grain size closeto ~100 nm throughout the sample [49-52]. Belownew results [51] on the microhardness and micro-structural evolution during HPT in samples of purenickel are discussed in details.Microstructure Evolution in Nickel during High-pressure Torsion. High purity nickel (99.99%) wasselected as the material for investigation. This sectionreports the results obtained on Ni samples subjectedto annealing prior to high-pressure torsion, where allsamples were annealed for 6 hours at 700 °C to givean initial grain size of ~100 µm and a microhardnessof ~1.4 GPa. During the early stages of this study itwas found that the final microstructure produced byHPT is independent of the initial grain size withinthe range of ~1 – 100 µm. Electro-dischargedsamples in disk shape, with diameters of ~10 mmand thickness of ~0.3 mm, were subjected to HPTstraining under different applied pressures and withdifferent numbers of whole revolutions at room tem-perature. Fig. 2.1b schematically depicts the HPTmethod. A disk is placed between anvils and thencompressed and deformed under an applied pres-sure (P) of several GPa at room temperature. Thelower holder rotates and surface friction forces de-form the disk by shear so that deformation proceedsunder a quasihydrostatic pressure. Further details

of HPT processing are given elsewhere [10, 40]. Toevaluate the influence of the applied pressure,samples were produced with 5 whole turns at sepa-rate pressures of 1, 3, 6 and 9 GPa. The influenceof the accumulated strain was evaluated by using apressure of 6 GPa and totals of ½, 1, 3, 5 and 7turns. Precise microhardness measurements weretaken on both sides of the samples along two per-pendicular diameters and with incremental steps of~1.25 mm between each measurement. At everypoint, the average microhardness was determinedfrom four separate measurements. The total num-ber of measurements was not less one hundred foreach sample. For the samples deformed at pres-sures of 1 and 6 GPa, foils were prepared from boththe centers and the perimeters of the disks for ex-amination by TEM.Experimental results. The processing of nickel byHPT leads to an increase in the microhardness, Hv,due to significant microstructural refinement that isstrongly influenced by the processing parameters.Fig. 2.6 shows (a) the influence of the applied pres-sure on the microhardness profile across the diskand (b) the average values of Hv measured in thecentral region and close to the edge. In Fig. 2.6a,these microhardness values are higher than for theunprocessed nickel (~1.4 GPa) but at the lowerapplied pressure of 1 GPa there is a non-uniformdistribution of microhardness across the sample di-ameter with significantly lower values in the center.An increase of pressure to 9 GPa leads to an in-crease in the values of Hv, especially in the centerof the disk. Thus, there is an increased uniformityin Hv across the disk with increasing applied pres-sure such that the variation in Hv is not more than~7% for the sample subjected to a pressure of 9GPa. Figure 2.6b reveals the dependence on ap-plied pressure of the average microhardness takenfrom the center and edge of the disks, with themicrohardness at the edge appearing to exhibit aplateau. By contrast, the central region has arelatively low level of microhardness up to a pres-sure of 6 GPa but thereafter the microhardness in-creases and is close to the value measured at theedge. The apparent slight decrease in Hv for ap-plied pressures from 1 to 3 GPa is within experi-mental error and is probably not significant. A TEMexamination was performed for two different appliedpressures of 1 and 6 GPa and with foils cut from thecenter and the edge of the samples. Figures 2.7aand b show two bright field images of the micro-structures at the center and edge regions of thesample after HPT at 1 GPa: the corresponding se-lected area electron diffraction (SAED) patterns are

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shown as inserts. It can be seen that the micro-structures have important differences. Microstruc-tural refinement is higher near the edge of the diskand the mean grain size in this region was measuredas ~0.17 µm. The SAED pattern in Fig. 2.7 b con-tains many spots situated around circles indicatingthe presence of boundaries having high

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misorientation angles. By contrast, the SAED pat-tern in the center of the disk in Fig. 2.7 a consistsof separate spots showing the presence of low anglesubboundaries. The measured average size of thesubgrains in Fig. 2.7 a is almost two times largerthan the grain size in Fig. 2.7 b. In both cases, theazimuthal spreading of spots in the SAED patterns

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indicates the presence of high internal stresses. Thesituation was different for the sample subjected toHPT at 6 GPa, as shown at the edge of the samplein Fig. 2.7 c. The microstructures for this conditionwere reasonably similar at both the center and theedge, and the SAED patterns consisted of ringswith many diffracted beams so that there were manysmall grains with multiple orientations within theselected fields of view. The average grain size wasmeasured as ~ 0.17 µm for both areas. Detailed high-resolution electron microscopy (HREM) and TEMwere reported recently for this condition [53] and itwas shown that the grain boundaries in HPT-pro-cessed Ni generally have high misorientation anglesand they exhibit zigzag configurations and irregularfacets and steps that are indicative of high-energynon-equilibrium structures. Figure 2.8 shows theinfluence of the number of rotations on Hv in samplesprocessed at an applied pressure of 6 GPa. As inthe profile shown in Fig. 2.8 a, there is a differencein the microhardness values measured at the cen-ter and the edge of the disk. However, this differ-ence becomes relatively less after larger numbersof turns because of the overall increase in themicrohardness level. An increase in the number ofrotations leads to an increase in Hv at both the cen-ter and the edge of the sample, as shown in Fig.2.8 b.

It is known for torsion deformation that the shearstrain, γ, at a radius R is given by

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where N is the number of turns and L is the thick-ness of the sample. According to this relationship,the true logarithmic strain is larger than 6 for fiverotations of a sample having a thickness of 0.3 mmand at the radius of 5 mm. Such very large colddeformation produces a very significant refinementof the microstructure and the formation of an arrayof ultra-fine grains leading to increasedmicrohardness [40]. At the same time, it followsfrom Eqn. (2.2) that significant grain refinement can-not occur in the center of the samples because thestraining is then very much reduced. In practice,however, this is true only for small numbers of turnsand relatively low applied pressures and the presentexperiments demonstrate that an increase in thestraining leads to an increase in the homogeneityof the microstructure (Fig. 2.8). It is concluded thatsamples processed with more than ~5 turns arecapable of producing reasonably homogeneousultrafine-grained microstructures. This must be as-sociated with the accumulation of dislocations withinthe disks but the mechanism of microstructuralhomogeneity requires further investigation.

An additional important processing parameter isthe applied pressure. The role of pressure is notapparent from Eqn. (2.2) but the experimental re-sults show that higher pressures facilitate the for-mation of ultrafine-grained microstructures. A simi-

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lar conclusion was reached recently from HPT stud-ies of iron [20] and titanium [22]. It is well knownthat the applied pressure can reduce the rate of dif-fusion and consequently delay the recovery kinet-ics [54] and this may explain the smaller grain sizesgenerally achieved by HPT at higher pressures.

3. THERMAL STABILITY ANDMICROSTRUCTURE EVOLUTIONIN SPD MATERIALS DURINGANNEALING

Any successful utilization of nanostructured mate-rials will depend upon the subsequent thermal sta-bility of the ultrafine-grained structures and in prac-tice the microstructures produced by severe plasticdeformation are in a metastable condition so thatrelaxation may occur at significantly lower tempera-tures than in coarse-grained materials. It’s knownthat the onset of grain growth begins innanostructured materials at relatively low tempera-ture (0.4 T

M). An investigation of low thermo-

stability’s origin is very important for understandingprocesses occurring in NS materials duringannealing. The microstructure evolution and graingrowth were a main object in recent publications[49,55-63]. TEM investigations and x-ray analysiswere carried out in pure metals: nickel [51,58,61],copper [49,50,62], cobalt [61], iron [55] where thetemperatures of grain growth onset have been iden-tified and intensive relaxation processes have beennoticed. However, the characteristics of grain growthhave many particularities and need to be studiedmore extensively.

There are earlier reports of microstructural evo-lution and thermostability in copper [49,50] and nickel[51] processed by HPT but there are no detailedstudies for materials processed using ECAP. Thismethod is especially attractive because it has thepotential of producing relatively large billets that mayhave use in industrial applications. This part of reviewpresents recent findings [64] of the changes inmicrohardness and microstructure occurring in purenickel during annealing after processing by ECAP.

Experimental Material and Procedure. Cylindri-cal billets, with diameters of 16 mm and lengths of130 mm, were machined from high purity (99.99%)nickel. The initial grain size was ~30 µm. The billetswere subjected to ECA pressing at room tempera-ture using the equipment shown schematically inFig. 3.1. All samples were pressed repetitivelythrough a total of 8 passes under an imposed pres-sure of 800 MPa. Each sample was reversed fromend to end and rotated by 90° about the longitudinal

axis between each pass. The total logarithmic strainaccumulated in the samples was approximatelye = 8. Further details on processing by ECAP aregiven elsewhere [28,30,40]. For annealing andmicrohardness testing, small disks were electro-discharged from the central part of the as-pressedcylinders. The annealing treatment was conductedin air for 3.6 ks at temperatures from 373 to 773Kand also at 523K for periods of 3.6, 21.6 and 43.2ks. After annealing, the surfaces of all samples wereground to remove any oxide film and they were thenmechanically and electrochemically polished formicrohardness testing. The microhardness, H

v, was

measured using a Vickers diamond pyramidal in-denter under a load of 100 g. Samples were exam-ined by transmission electron microscopy (TEM)using a JEM–100B electron microscope. For thesamples annealed at the highest temperatures of573 – 773K, the microstructures were examinedusing both TEM and optical microscopy.

Experimental Results and Discussion. After pro-cessing by ECAP, the microhardness of Ni wasmeasured as H

v = 2.6 GPa and the mean grain size

was ~0.35 µm. By comparison, the initialmicrohardness of an unpressed sample annealedfor 3600 s at 773K was Hv0 ≈ 1.0 GPa. Samples wereannealed for 3600 seconds at different temperaturesand the measured values of the microhardness andthe mean grain size, d, are summarised in Table3.1: grain sizes were measured by TEM up to an-nealing temperatures of 523K and by optical mi-croscopy at the higher temperatures. For conve-nience, the experimental data are plotted in Fig.3.1. The microhardness tests reveal a slight de-crease in H

v for annealing temperatures up to 473K,

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a sharp drop at temperatures from 473K to 573Kand then a slow decrease to a value of ~1.0 GPaappropriate to the well-annealed condition.

All annealed samples were examined by TEMto evaluate the microstructural characteristics. Typi-

Table 3.1. Microhardness and mean grain size as a function of annealing temperature.

Temperature, K 293 373 473 523 573 673 773Microhardness, GPa 2.6 2.4 2.3 2.0 1.4 1.1 1.0Mean grain size, µm 0.35 0.40 0.45 0.80 4.5 6.0 53

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as-pressed microstructure is typical of pure metalsafter severe plastic deformation: there is an array ofessentially equiaxed grains, the boundaries haveirregular configurations and the azimuthal spread-ing of the spots in the selected area electron dif-fraction pattern suggests the presence of high in-ternal stresses. Annealing at temperatures of 373and 473K has a minor effect on the grain size butthere is noticeable grain growth at the relatively lowtemperature of 573K in Fig. 3.2 e corresponding to~ 0.33T

M, where T

M is the absolute melting tem-

perature of the material. There is also evidence atthis temperature for some twining in the grain interi-ors. At the higher annealing temperature of 673K inFig. 3.2 f, there are few intragranular dislocations,the boundaries have equilibrium configurations andthe microstructure is typical of well-annealed coarse-grained materials.

Typical grain size distributions are shown in Fig.3.3 for the samples annealed at the three highesttemperatures. It is apparent that the grain size dis-tributions in Figs. 3.3 a and b are close to log-nor-mal corresponding to normal grain growth but thegrain size distribution in Fig. 3.3 (c) is bimodal witha small secondary peak appearing at large grainsizes of ~130–140 µm. This suggests the onset ofabnormal grain growth at 773K. Fig. 3.1 demon-strates that the grain growth kinetics and themicrohardness measurements are consistent: thereis a gradual increase in the grain size in the tem-perature range of 373–473K and a sharp increasein d, by a factor of 10 times, from ~ 0.45 µm at473K to ~ 4.5 µm at 573K. Fig. 3.3 shows the influ-ence of the annealing time on microhardness andgrain growth kinetics at a temperature of 523K. Thetime dependence of the microhardness and the

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mean grain size therefore demonstrate similarbehaviour. It is well established that grain growthkinetics can be described by the relationship [65]:

d d kt n20

2− = , (3.1)

where d0 is the initial grain size, k is a constant

depending on the driving force for grain growth andthe mobility of the grain boundaries and n is aconstant typically lying in the range from 0.5 to 1.

Grain growth kinetics with n = 1 correspond tothe parabolic law and normal grain growth. By fit-ting the experimental points to Eq. (3.1) by meansof a least squares method, the value of the constantk was determined as 0.6165 µm2.h-1 or 170 nm2.s-1.The solid line in Fig. 3.4 is plotted using Eq. (3.1)with n = 1 for the parabolic law. Assuming parabolickinetics are fulfilled for all annealing temperatures,it is possible to calculate the activation energy, Q,and the pre-exponential factor, k

0, for the grain growth

process using the expression k = k0 exp(-Q/RT),

where Q is the activation energy for grain growth, Ris the gas constant and T is the absolute tempera-ture. From a best fit to the experimental data, itwas found that k

0 = 3.9.10-6 m2.s-1 and Q = 103 kJ.

mol-1. This value for Q is very close to the values ofthe activation energies reported for grain boundarydiffusion in nickel (115 kJ.mol-1 [66], 131 kJ.mol-1

[67] and 108.8 kJ.mol-1 [68]). In addition, the valueof the kinetic constant may be evaluated directly bymaking use of the relationship [65]:

kD

kTb

0 2

04= ⋅γ

δ

δΩ, (3.2)

where γ is the grain boundary surface energy (~ 0.7J·m-2), Ω is the atomic volume (1.1.10-29.m3), δ is

��

the grain boundary width (~0.7.10-9 m), δ·D0b

is thepre-exponential factor for grain boundary diffusionmultiplied by the grain boundary width (~3.5.10-15

m3·s-1), k is Boltzmann’s constant and T is the ab-solute temperature. Taking a value of T = 500K, itfollows from Eq. (3.2) that the value calculated fork

0 is given by (k

0)

cal ≈ 4.6.10-5 m2.s-1. This value is

consistent with the experimental value to withinapproximately one order of magnitude, therebyconfirming the validity of the present approach.

Thus, the calculation demonstrates that graingrowth kinetics in ultrafine-grained nickel processedby ECAP is controlled by grain boundary diffusionand possibly by other short-circuit diffusion pathsthat may exist in nanostructured materials. Thisresult is consistent also with the recent conclusionthat grain boundary diffusion dominates grain growthin nanostructured nickel prepared by electrodepo-sition [66]. This investigation shows that therelaxation processes in ultrafine-grained nickel, pro-cessed by equal channel angular pressing, undergo3 distinct stages. The first stage corresponds tothe temperature range of 373-473K and representsa relaxation of the internal stresses arising from theECAP process. The second stage takes place withinthe temperature range of 473-573K where there is asharp drop in the microhardness and an abrupt in-crease in the grain size due probably to subgraincoalescence and dislocation annihilation. The thirdstage relates to normal grain growth at tempera-tures above ~573K (~0.33T

M) where the rate-con-

trolling process is grain boundary diffusion. This lat-ter temperature is relatively low for normal graingrowth but it is consistent with the decrease in ther-mal stability observed in ultrafine-grained metals

processed by severe plastic deformation. The re-sults demonstrate a good correlation between themicrohardness measurements and microstructuralobservations following ECAP and annealing.

4. GRAIN BOUNDARY DIFFUSION INNANOSTRUCTURED MATERIALS

4.1. The characteristics of coppergrain boundary diffusion innanocrystalline andnanostructured materials

Diffusion in nanocrystalline materials is very impor-tant for their properties. Therefore, it is studied veryactively [69-78]. The keen interest in this theme isstipulated, among other reasons by the fact thatdiffusion exerts primary control over the kinetics ofrecovery and recrystallization processes occurringin nanocrystalline materials at significantly lowertemperatures relative to the coarse-grained materi-als. It has also been proven experimentally thatnanocrys-talline systems have “anomalously” highdiffusivity, which is an additional reason for this highinterest [69,70,73]. It has been found that the highdiffusivity results not only from the high volume frac-tion of the “GB phase” in such materials but alsomostly from residual porosity, impurity segregationand other factors related to a processing routine ofnanocrystalline materials [76-78].

In particular, porous nanocrystalline metals pro-duced by the methods of gas condensation and elec-trolytic deposition are found to reveal low-tempera-ture anomalies of GB diffusion. Their diffusivities areseveral orders of magnitude higher and the activa-tion energies of self-diffusion and of substitutionimpurity diffusion are about two orders lower incomparison to the coarse-grained materials[69,72,73]. In case of interstitial impurities, the dif-fusion processes depend essentially on the kind ofimpurity [74,78]. The “anomalies” of GB diffusionare not observed in materials after recovery andrecrystallization [75].

Non-porous nanocrystalline materials (in particu-lar those obtained by explosive pressing) are foundto reveal no “anomalous” diffusion behavior [75-78],which suggests that intrinsic free surfaces play asignificant role in the evolution of diffusion processesin nanocrystalline materials [72].

The existing physical models take into accountthe peculiar properties of GBs in nanocrystallinematerials. Thus the model of glue-like grain bound-ary state, which was developed on the base ofcomputer simulation results, takes into account the

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correlation effects, which are exhibited by ultrafine-grained materials and which are associated, in par-ticular, with the absence of relaxation effects due tothe generation of dislocations induced in such ma-terials [79]. One of the basic assumptions of thecluster model based on HREM examinationconsists of the important role played by the bound-aries of ensembles of adjacent grains – “clusters”,on which grain-boundary pores occur and the mainportion of excess (free) volume of nanocrystals isconcentrated [73].

Investigators to date focus their attention on therole of triple junctions. The density of these lineardefects in nanocrystalline materials is considerablyhigher relative to their coarse-grained counterparts.The occurrence in the bulk of a continuous networkof triple junctions and concentration of most porevolume in the latter assists a creation of short diffu-sion paths in nanocrystalline materials [74].

The available data on diffusion in nanocrystallinematerials and the results of other physicalinvestigations of these materials are summarizedin [69,72,75,78].

R. Würschum, H.-E. Schaefer and others[69,72,75,78-81] were the first to investigate diffu-sion in nanostructured materials produced by se-vere plastic deformation, in particular 59Fe diffusionin Pd. While the authors of the given work performeddiffusion experiments at the temperature above410K they failed to reveal any significant increasein the grain boundary diffusivities of the investigatedmaterials. However, the experimentally measureddiffusivities observed for nanostructured materialsat lower temperatures (371K and 471K) were byabout two orders of magnitude higher relative to therespective coarse-grain samples [75]. As in the caseof nanocrystalline metals, the activation energy ofgrain-boundary diffusion in nano-Pd turned out tobe significantly lower (by 1.5 times) as comparedto coarse-grained Pd. It should be noted that in theexperimental conditions of diffusion annealing [75]GB migration was found to occur, which has to betaken into account in choosing a particular modelfor evaluating materials diffusivity. Moreover, duringgrain boundary migration a change in the grainboundary state may occur and it can influence thediffusivity value. Thus it appears that material’s dif-fusion parameters may be significantly affected bythe recovery of the grain boundary structure innanostructured-Pd that occurs during diffusion an-nealing at elevated temperatures [75]. Recently theinvestigation of self-diffusion in nanostructured-Fewas performed; the results obtained indicate thatno “anomalous” increase in the grain boundary

diffusivity takes place in the material investigated[76,77].

A comparative study of the diffusion parametersof nanostructured and coarse-grain metals wasconducted for copper diffusion in nickel [82-84]. Theconcentration of Cu in Ni was measured at variousdepths by the method of secondary ion-mass spec-troscopy using the technique described elsewhere[83]. The activation energy of the grain boundarydiffusion of Cu in coarse-grained Ni, Q

b, was deter-

mined in the temperature interval of 773 to 873K;the value obtained was 124.7 kJ/mol (Fig. 4.1). As-suming that in the entire temperature intervalconsidered, the Arrhenius dependence parametersfor the coefficients of grain boundary diffusion areconstant, the respective values were calculated forcoarse-grained Ni at lower temperature [84].

According to TEM data reported in [83], no GBmigration in nanostructured Ni is likely to occur inthe temperature interval of 392–448K and the vol-ume diffusion may be virtually suspended. There-fore, the values of the GB diffusivity were calculatedbasing on the assumption that the experimentalconditions of diffusion annealing are close to regimeC. In the C regime the pure GB grain boundary diffu-sion proper takes place without leakage of thediffusant into the bulk [85]. It has been found thatthe GB diffusion coefficients, Dn, obtained for Cudiffusion in nanostructured Ni are by 4 to 5 orders ofmagnitude higher relative to coarse-grain Ni (Fig.4.1) [84].

From the temperature dependence of Dn of Cu in

nanostructured Ni we obtained the activation energyof grain boundary diffusion Qb

n = 43.5 kJ/mol [84].The latter value is by a factor of two lower than therespective value for coarse-grained nickel samples(Fig. 4.1). Unlike the diffusion in nanocrystallinemetals containing pores at GBs [69-78], the activa-tion energy of grain boundary diffusion innanostructured pore-free nickel is close to that ofsurface diffusion [84]. Moreover, the activation energyvalue of the grain boundary diffusion of Cu innanostructured Ni is close to the activation energyof the GB self-diffusion(46 kJ/mol, Fig. 4.1) [73].

As noted above, the GBs in nanostructuredmetals are characterized by a highly nonequilibrium(high-energy) state, while the GB state innanocrystalline and coarse-grain metals is close toan equilibrium one. Therefore, the values Dn obtainedfor nanostructured nickel are significantly highercomparing to nanocrystalline Ni [73,84] and coarse-grain counterparts. It suggests that the GB diffu-sion of impurity atoms in the investigated materialsis mainly determined by GB state and less by other

��

1.2 1.6 2.0 2.4 2.8 3.2 3.610-22

10-19

10-16

10-13

10-10

Ni(63Ni) nanocrystalline:Claster-compactedPressing (4.4 GPa, 773 K),d=0.07 µm (Bokstein et al.)

Ni coarse-grainedself-diffusion

Ni(Cu) nanostructured:severe plastic deformation,pre-annealing at 523 K,d=0.3 µm

Ni(Cu) nanocrystalline:electrodeposited,d=0.04 µm

Ni(Cu) nanostructured:severe plastic deformation,d=0.3 µm

Ni(Cu) coarse-grained,d=20 µm

1/T [1/1000 K-1]

Db, m2/s

900 700 500 300T, K

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4.2. Grain boundary diffusion – order-ing coupling in nanocrystallinematerials

The transport of atoms by diffusion determines, e.g.,the thermal stability, sintering properties, creep orion conductivity in oxides. The high number of inter-faces in nanocrystalline materials can change themacroscopic diffusion behavior considerably withrespect to coarse grained materials. For example,the self-diffusivity in grain boundaries of metals isnormally strongly enhanced due to the relative loosepacking of atoms in these interfaces. Therefore, theactivation enthalphy for jumps of atoms is lower thanin the perfectly ordered crystal and vacancies likevoids, which act as diffusion vehicles, can be formedeasily.

However, for a full understanding of the diffusionalproperties, one has to take into account not onlythe diffusion in the interfaces, but also in thecrystallites. The volume diffusion into the crystallitesbecomes more pronounced at longer diffusion timesor higher temperatures, which are often applicationrelevant. E.g., the oxygen diffusion in nanocrystallineZrO

2 [78, 87], which could be used as a solid state

electrolyte for fuel cells, is determined by both diffu-sion processes.

Fig. 4.2 gives an overview on the experimentalwork done or in progress to obtain an atomistic un-derstanding of the diffusion processes in nano-crystalline materials. Structural studies (X-ray dif-

diffusion in nanocrystalline materials

diffusion in interfaces(grain boundaries)

• local atomic structure(wetting, free volumes, ...)

• orientation of crystallites(grain boundarycharacter)

• chemical composition(segregation)

diffusion in crystallites

• athermal vacancycontent

• state of order

TEM

X-ray

common mechanism: jumps of atoms byvacancy formation and migration

e+

separation by analysis of tracer diffusion profiles

fraction and transmission electron microscopy(TEM)) provide the necessary input for conclusionson the correlation between diffusion and structure.Positron annihilation spectroscopy is a techniquespecific and sensitive on vacancies or vacancyclusters and was successfully employed toinvestigate voids at grain boundaries [88]. Furtherinsight comes from theoretical studies to modelequilibrium (see, for example, Chapter 6) and non-equilibrium (Chapter 5) grain boundaries.

High-resolution TEM studies on nanocrystallinematerials prepared by cluster condensation andcompaction concur with a randomly distributedcrystallite orientations and mostly high-angle grainboundaries (see Fig. 4.3 [78]). In many nano-crystalline pure face-centered cubic (fcc) metalsthese boundaries exhibit diffusivities similar to high-angle grain boundaries in coarse grained materials[89]. In nanocrystalline Fe [76] and Ni (Chapter 4.1)diffusivities were found which are higher than for high-angle grain boundaries in the corresponding coarsegrained materials. The structure of the grain bound-aries of these nanocrystalline materials relax withtime towards the equilibrium state of coarse grainedmaterials as indicated by the decreasing diffusivities.

Diffusion studies on composite materials or al-loys become more and more important because ofa more pronounced stability against grain growth.Nevertheless, studies on pure systems remainimportant as reference for the diffusion in compositematerials or alloys. Fig. 4.4 shows as an example

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the diffusion profiles of pure, unstabilizednanocrystalline ZrO2 [87]. A profile can be sepa-rated between volume diffusion (diffusion in thecrystallites) and grain boundary diffusion. Since ZrO

2

is used as an ion conductor of oxygen, it is ofimportance that oxygen diffuses much faster in the

grain boundaries than in the crystallites. An inter-esting experiment, which could be used to answerthe question whether the fast diffusion is due tostructural or thermal vacancy-like defects in the grainboundary, would be to measure the oxygen diffu-sion in nanocrystalline Y-stabilized ZrO

2. Y causes

the formation of structural oxygen-vacancies in thecrystallites of coarse grained ZrO

2.

Now, we report on the migration of atoms withinthe nanocrystallites in dependence on the state oforder. This is also important for the material trans-port within the grain boundaries, because orderingcould substantially change the volume diffusionconstant and thereby the type of grain-boundarydiffusion kinetics from type B to C. The preparationtechniques for nanocrystalline materials, especiallystrong plastic deformations, offer the unique possi-bility to disorder bulk intermetallic alloys like FeAlwhich hardly can be disordered as coarse grainedmaterials. Additionally, metastable, one-phasecompositions, e.g. Al-rich FeAl, can be prepared.

Diffusion and ordering are closely correlated toeach other. Both are goverened by the migration ofvacancies. The only difference is the distance overwhich the atoms migrate. For ordering a few jumpsper atom should be enough to arrive at the correctlattice site so that the diffusion length after the orderingis about one lattice parameter (roughly 1 nm).

The ordering due to annealing of nanocrystalline,disordered FeAl, NiAl and Fe

3Si were measured by

x-ray diffraction [90-93]. The mean temperature ofordering is ascribed to the temperature of 50%ordering. Fig. 4.5 shows the diffusion lengths fordiffusion by thermal vacancies. It is evident that ther-mal vacancies account for the ordering in Fe

3Si [94],

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but the thermal vacancy concentration at the order-ing temperature is much to small for ordering of FeAland NiAl. Therefore we conclude that the orderingin FeAl and NiAl occurs by vacancies which wereintroduced during the preparation of thenanocrystalline alloys. Furthermore, these vacan-cies migrate faster in the disordered than in theordered state [90], and they anneal out only after theordering is complete. The reason for the annealing ofvacancies in Fe

3Si before the ordering is complete

originated from the low vacancy migration enthalphyon the Fe-sublattice [95].

In summary, we arrived at a consistent model ofordering and diffusion by means of vacancy migra-tion for nanocrystalline [94,96] and coarse grainedmaterials. This is a well defined starting point to sepa-rate the atomic migration in the crystallites and inthe grain boundaries of nanocrystalline materials.

5. THEORETICAL MODELING OFGRAIN BOUNDARY DIFFUSION INBULK NANOSTRUCTUREDMATERIALS

5.1. Introduction

Nanostructured materials have become increasinglyimportant both in fundamental and applied researchbecause of their outstanding functional and struc-tural properties; see, e.g. [97-106]. Of special inter-est, from both fundamental and application view-points, is the diffusion behavior of bulknanocrystalline materials which, in general, is dif-ferent from that of conventional coarse-grained poly-crystals. Thus, following [1,81,98, 107-109], bulk

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Fe3Si

FeAlNiAl

Fe

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FeAl

Ni

material mean temp.of ordering

diffusor

650 K

600 K450 K

A2 D03

A2 B2

nanocrystalline materials exhibit the anomalouslyenhanced diffusional properties. For instance, theboundary diffusion coefficients in bulk nanocrystallinematerials fabricated by high-pressure compactionand severe plastic deformation methods are severalorders of magnitude larger than those in conventionalpolycrystalline materials with the same chemicalcomposition [1, 81, 97,107-109]. With these experi-mental data, the anomalously fast diffusion is treatedas the phenomenon inherent to only nanocrystallinematerials and is attributed to their specific struc-tural and behavioral peculiarities [1, 97]. At the sametime, however, there are experimental data indicat-ing that the boundary diffusion coefficients in densebulk nanocrystalline materials are lower than thosemeasured in [107, 81,108,109] and similar to theboundary diffusion coefficients in conventionalcoarse-grained polycrystals or a little higher [75-78,80]. These data form a basis for the viewpointthat the atomic diffusion in nanocrystalline materi-als is similar to that in conventional coarse-grainedpolycrystals. In doing so, the difference in thediffusivities between nanocrystalline and coarse-grained materials is treated to be related to only thedifference in the volume fraction of the grain bound-ary phase. Thus, in general, there are controversialexperimental data and theoretical representationson diffusion processes in nanocrystalline materials.Nevertheless, mechanically synthesized nanocrys-talline bulk materials are definitely recognized toexhibit the enhanced diffusion properties comparedto those of nanocrystalline materials fabricated bynon-mechanical (more equilibrium than mechanical)methods and those of coarse-grained polycrystals;

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see [81,108,109] and experimental results reportedin this review. The experimentally documented phe-nomenon in question can be naturally explained asthat occurring due to the deformation-induced non-equilibrium state of grain boundaries, in which casethe grain boundary diffusivity highly increases. Inthis part of review we will consider theoretical mod-els of diffusion processes in nanocrystalline materi-als with the special attention being paid to the roleof non-equilibrium grain boundaries in the diffusionenhancement occurring in mechanically synthe-sized bulk nanocrystalline materials.

5.2. Theoretical models of diffusion innanocrystalline materials. Generalaspects

Theoretical models of diffusion in nanocrystallinematerials can be divided into the three following basiccategories: (1) Models that describe grain bound-ary diffusion in conventional coarse-grained polycrys-tals and are directly extended to the situation withnanocrystalline materials, with nano-scale effectsneglected. (2) Models focusing on the specific fea-tures of diffusion in nanocrystalline materials in thegeneral situation, taking into account the nano-scaleeffects and neglecting the influence of preparationtechnologies on the structure and the diffusion prop-erties of nanocrystalline materials. In short, suchmodels deal with the “nanostructure-diffusion”relationship. (3) Models focusing on the structuraland behavioral peculiarities of grain boundaries innanocrystalline materials, taking into account thepeculiarities of their fabrication. In short, such mod-els are concerned with the “preparation-nanostructure-diffusion” relationship. Theoreticalmodels of the first type and their application to adescription of diffusion in nanocrystalline materialshave been reviewed in detail by Larikov [72]. Suchmodels are, in particular, those [110-114] describ-ing spatial distributions of diffusing species withinand near grain boundaries in polycrystals, models[115-123] focusing on microscopic mechanisms ofgrain boundary diffusion and models [123-131] deal-ing with diffusion-assisted processes (diffusionalcreep, diffusion-induced grain boundary migration,transformations of grain boundary dislocations anddisclinations, etc.) in polycrystals. Now let us turnto a brief discussion of models describing theenhanced diffusion in nanocrystalline materials inthe general situation, with the specific features oftheir preparation being neglected. In papers[132,133] it has been demonstrated that nano-

crystalline materials exhibit much larger variety ofpossible kinetic diffusion regimes compared tocoarse-grained polycrystals. In particular, togetherwith conventional A-, B- and C-regimes occurring inpolycrystals [114], the additional diffusion kineticregimes have been distinguished, depending on thegrain size, diffusion temperature and time, the grainboundary segregation level in the case of impuritydiffusion, and other parameters [133]. Also, as notedby Gleiter [1], the anomalously fast diffusion innanocrystalline materials is related to the three fol-lowing factors: (i) The diffusion in nanocrystallinematerials is essentially enhanced due to the exist-ence of high-density network of grain boundary junc-tion tubes which commonly are characterized bymore increased diffusion rate than grain boundariesthemselves. (The highly enhanced diffusivity of triplejunctions of grain boundaries is experimentally iden-tified in polycrystals; see [134-138] and discussionin review article [139] concerning grain boundaryjunctions.) (ii) Rigid body relaxation of grain bound-aries that occurs via relative translational displace-ments of their adjacent crystallites and reduces theboundary free volume is hampered in nanocrystallinematerials. This is related to the fact that rigid bodyrelaxation of the various boundaries surroundingevery nanocrystallite require its different displace-ments due to their different atomic structure. (iii) Ingrain boundaries of nanocrystalline materials theconcentration of impurities that often reduce bound-ary diffusivity is lower than that in grain boundariesin conventional polycrystals. Let us discuss fac-tors (i)-(iii) in terms of Arrhenius formula for diffusivityin solids. The coefficient D of self diffusion occur-ring via transfer of vacancies1 is given by the follow-ing Arrhenius relationship (e.g., [122,140]):

D DkT

f m= −+�

��0

exp ,ε ε

(5.1)

where D0 denotes the constant dependent on pa-

rameters of the ideal crystalline lattice, k is theBoltzmann’s constant, T is the absolute tempera-ture, ε

f and ε

m the energies of formation and migra-

tion of vacancies, respectively. In nanocrystallinematerials (characterized by extremely high volumefraction of the grain boundary phase) self diffusionprocesses occur mostly via transfer of grain bound-

1 In general, diffusion processes in crystals occur viatransfer of point defects of different types. However,

self diffusion occurring via transfer of vacancies is mosteffective; it is characterized by the largest coefficient ofdiffusion, e.g.[122,140].

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ary vacancies. In these circumstances, all the fac-tors (i)-(iii) discussed by Gleiter induce the charac-teristic boundary free volume to be increased and,as a corollary, both formation and migration of bound-ary vacancies to be facilitated in nanocrystallinematerials as compared to those in conventional poly-crystals. In terms of Arrhenius formula (5.1), the dif-fusion coefficient D is increased in nanocrystallinematerials, because their characteristic energies, ε

f

and εm, are decreased due to factors (i)-(iii) dis-

cussed in paper [1].Now let us discuss theoretical models describ-

ing the enhanced diffusion in nanocrystalline mate-rials fabricated at highly non-equilibrium conditions(say, by severe plastic deformation methods). Fol-lowing [141-144], together with factors (i)-(iii) dis-cussed by Gleiter [1], the existence and transfor-mations of ensembles of grain boundary disloca-tions and disclinations strongly enhance diffusionprocesses in nanocrystalline materials fabricated athighly non-equilibrium conditions. The effects of suchdefects on diffusion are briefly as follows [143,144]:(a) Disorderedly distributed grain boundary disloca-

tions and disclinations are formed in nano-crystalline materials fabricated at highly non-equilibrium conditions. Such dislocations anddisclinations re-arrange to annihilate or to formmore ordered, low-energy configurations, in whichcase they move to new positions. Climb of (grainboundary) dislocations is accompanied by gen-eration of new point defects [140,145] – vacan-cies and interstitial atoms – serving as newcarriers of diffusion. In this situation, generationof new vacancies occurs under the action of thedriving force related to a release of the elasticenergy of “non-equilibrium” grain boundary dislo-cations during their transformations. The actionof the driving force in question facilitates diffu-sion processes in nanocrystalline materials. Theeffect discussed is quantitatively described byformula (5.1) with the sum ε

f+ε

m being replaced

by εf -W

v+ε

m, where W

v (>0) is the energy release

due to the climb of a grain boundary dislocation,per one vacancy emitted by the dislocation.

(b) Cores of grain boundary dislocations anddisclinations are characterized by excess freevolume, in which case their presence in grainboundaries increases the total free volume thatcharacterizes the grain boundary phase innanocrystalline materials. This decreases valuesof ε

f and ε

m and, therefore, enhances diffusion

processes occurring by vacancy mechanism.(c) Dilatation stress fields of grain boundary dislo-

cations and disclinations influence migration of

vacancies. More precisely, in spirit of the theoryof diffusion in stressed solids [146], the elasticinteraction between vacancies and grain bound-ary dislocations and disclinations ischaracterized by the energy:

εσ

int= − ii

V∆

3, (5.2)

where σii denotes the sum of dilatation com-

ponents (say, components σxx

, σyy

, σzz, written in

(x,y,z) – coordinates) of stress fields created bygrain boundary defects, and ∆V the excess freevolume associated with a vacancy. To minimizeε

int, vacancies migrate to regions where

compressive stresses exist. This vacancy mi-gration contributes to the enhanced diffusion innanocrystalline materials. The effect discussedis quantitatively described by formula (5.1) withthe sum ε

f+ ε

m being replaced by the sum

εf+ ε

m+ ε

int.

Thus, the factor (a) gives rise to the diffusionenhancement owing to the influence of grain bound-ary dislocations on the generation of vacancies. Thefactors (b) and (c) facilitate the vacancy migrationand therefore, cause the diffusion enhancement innanocrystalline materials. According to the theo-retical analysis [143], the most essentialcontribution to the diffusion enhancement is due tothe dislocation climb in grain boundaries (see fac-tor (a)). With this taken into account, in next sec-tion we, following [143,144], will consider the effectsof grain boundary dislocations on diffusion innanocrystalline materials, with focusing on the climbof grain boundary dislocations composing dipoleconfigurations.

5.3. Climb of grain boundarydislocations and diffusion innanocrystalline materials

Nanocrystalline materials are effectively fabricatedby mechanical methods at highly non-equilibriumconditions (see, e.g., [105,147] and this review), inwhich case “non-equilibrum” defect structures areformed at grain boundaries. In particular, grainboundaries in such materials contain not only geo-metrically necessary dislocations (that is, grainboundary dislocations associated with the meanboundary misorientation, e.g., [123]), but also theso called “excess” grain boundary dislocations (thatprovide local deviations of the boundarymisorientation from its mean value and commonlyare chaotically distributed along a grain boundary)

��

(Fig. 5.1a). In doing so, owing to highly non-equilib-rium conditions of synthesis, the geometrically nec-essary dislocations at grain boundaries in mechani-cally synthesized nanocrystalline materials, in gen-eral, are disorderedly displaced relative to their “equi-librium” positions (Fig. 5.1a). During some relax-ation time interval after mechanical synthesis of ananocrystalline sample, transformations of the en-semble of grain boundary dislocations occur lead-ing to a release of its elastic energy. In doing so,the “excess” dislocations with opposite Burgersvectors move (climb and glide) to each other andannihilate, and the geometrically necessary dislo-cations move (climb and glide) to their equilibriumpositions (Fig. 5.1b).

The climb of grain boundary dislocations duringthe relaxation time interval is accompanied by bothemission and absorption of point defects: vacan-cies and interstitial atoms (Fig. 5.2). In these cir-cumstances, emission of vacancies (“take-off” ofvacancies from dislocation cores and theirconsequent displacement into the surrounding grainboundary phase, see Fig. 5.2a) is more intensive

�� +�#�� #�� <

�E�#��E�#��

�� ó��

WGb d

rZ

d( ) ( )

( )lnλ λ

π ν

λ= ≈

−+

��

�

22 1

2

0

(5.3)

for λ>2a. Here Wd(λ) denotes the energy of disloca-

tion having λ as the screening length of its stressfield, d the dislocation length, ±b the dislocationBurgers vectors, G the shear modulus, ν the Pois-son ratio, r

0 the dislocation core radius, Z the fac-

tor taking into account the contribution of the dislo-cation core into energy W.

Climb of one of the dislocations composing thedipole configuration by a, the mean interatomic dis-tance in the grain boundary, (Figs. 5.3a and b)releases the dipole energy by value of ∆W(λ)=W(λ)-W(λ-a) and results in emission of d/a vacancies. Inthis situation, ~εf , the energy of formation of onevacancy at the climb of the dislocations thatcompose the dipole configuration (Figs. 5.3a andb), is given as:

~ ( )ε ε λνf f W= − , (5.4)

where εf is the energy of formation of one vacancy

in the grain boundary phase in the absence of dislo-cations, and W

v(λ) is the decrease of the disloca-

tion dipole energy W due to emission of one va-cancy, related to the climb of the dislocationscomposing the dipole:

a

dW

a

dW W aν λ λ λ λ( ) ( ) ( ( ) ( ))= = − −∆ (5.5)

for λ > 2a, from (5.4) and (5.5) we find

Gb a

aν λ π ν

λ

λ( )

( )ln=

− −

2

2 1. (5.6)

For λ ≤ 2a, stress fields of the dislocationscomposing the dipole configuration, are localizedmostly in dislocation cores, in which case the dis-

Fig. 5.3. Climb and annihilation of grain boundarydislocations composing dipole configuration.

location energy is highly influenced by factor Z. Thefurther climb of the dislocation towards each otherrepresents, in fact, the process of their annihilationresulting in emission of 2(d/a) vacancies and de-creasing the dipole energy from its value

aGb a Z

( )(ln )

( )λ

π ν= ≈

+

−2

2

2 1

2

at λ = 2a to zero that

corresponds to λ = 0 (annihilation). As a corollary,for λ ≤ 2a, the dipole energy decreases by value of

Gb a Zν λ π ν( )

(ln )

( )≈

+

−

2 2

4 1 (5.7)

due to emission of one vacancy, which accompa-nies the dislocation annihilation.

The dependence of Wv on λ/a, given by formulae

(5.6) and (5.7), is shown in Fig. 5.4, for the followingcharacteristic values of parameters: G=50 GPa, a≈ 0.3nm, b ≈ a/3, Z ≈ 1, and n ≈ 1/3. From Fig. 5.4it follows that emission of vacancies is facilitatedmore intensively with decrease of the distance λbetween the dislocations composing the dipoleconfiguration.

Let us discuss the effect of dislocation-climb-induced emission of vacancies on diffusion innanocrystalline materials. The coefficient of diffu-sion occurring via vacancy mechanism (which com-monly is most effective) in the absence of “excess”boundary defects is given by formula (5.1), wherefactor exp(-ε

f/kT) characterizes equilibrium concen-

tration of vacancies (when dislocation climb and di-latation stresses do not influence vacancy forma-tion). The vacancy concentration in vicinity of theclimbing dislocations (Fig. 5.3) is higher then theequilibrium concentration, because vacancy forma-tion is facilitated due to the climb of dislocations.The effect in question is quantitatively described bythe following change of the vacancy formation energy:ε

f→ ~εf = εf - Wν and by the corresponding local

change of diffusion coefficient: D → D*, where D* invicinity of the climbing dislocations (Fig. 5.3) is givenas:

D DW

kT

* exp= ��

ν. (5.8)

With the dependence Wv(λ) (see Fig. 5.4) taken

into consideration and factor exp(Wv /kT) averagedon λ (ranging from 0 to 10a), we find that the diffu-sion coefficient in vicinity of the climbing disloca-tions is D* ≈ 300·D.

In nanocrystalline materials during relaxation timeinterval (after synthesis at highly non-equilibrium

��

conditions, say, after fabrication by severe plasticdeformation method) almost all grain boundariescontain non-equilibrium defect structures, in particu-lar, “excess” dislocations whose annihilationenhances diffusion. In this situation, D* plays therole of grain boundary diffusion coefficient innanocrystalline materials fabricated at highly non-equilibrium conditions. To summarize, our estima-tions allow us to conclude that the climb of grainboundary dislocations (Fig. 5.3) essentially enhancesdiffusion processes in nanocrystalline materials (inparticular, nanocrystalline materials fabricated bysevere plastic deformation), causing increase of thegrain boundary diffusion coefficient by about 2 ordersof magnitude. Thus, the climb of grain boundary dis-locations is capable of strongly contributing to theexperimentally observed (see [81,107-109])anomaously fast diffusion in nanocrystalline materi-als during some relaxation time interval after theirsynthesis at highly non-equilibrium conditions.

Similar effects on the diffusion enhancement arecaused by climb transformations of complicatedlyarranged boundary dislocation structures (say, dis-location walls) in nanocrystalline bulk materials andfilms [148]. In doing so, misfit stresses (which com-monly play the important role in processes occur-ring in single-, poly- and nanocrystalline films, e.g.,[149-158]) essentially affect transformations of grainboundary defects and, therefore, the diffusion pro-cesses in nanocrystalline films [148].

Following [141-143], the elastic interactions be-tween the diffusing species and grain boundary dis-

��+��%��λF�

locations and disclinations (see point (c) in section5.2) facilitate migration of vacancies and impurities,causing the diffusion enhancement in vicinities ofgrain boundaries. In this situation, the diffusion rateis increased by factor M ~1-5 in vicinities of grainboundaries. The effect discussed is small comparedto that related to the climb of grain boundary dislo-cations.

5.3. Concluding remarks

Thus, there are controversial experimental data andtheoretical representations on diffusion processesin nanocrystalline materials. At the same time, dif-fusion in mechanically synthesized bulk nano-crystalline materials is definitely recognized as thestrongly enhanced diffusion compared to that incoarse-grained polycrystals and nanocrystallinematerials fabricated by non-mechanical (more equi-librium than mechanical) methods. This experimen-tally documented phenomenon is naturally explainedas that occurring due to the deformation-inducednon-equilibrium state of grain boundaries, in whichcase mechanically generated grain boundary de-fects strongly enhance the boundary diffusivity.

6. RELATED PHENOMENA (GRAINBOUNDARY PHASE PHENOMENA,SUPERPLASTICITY) INNANOSTRUCTURED MATERIALS

The properties of modern materials, especially thoseof superplastic, nanocrystalline or composite ma-terials, depend critically on the properties of inter-nal interfaces such as grain boundaries and inter-phase boundaries (IBs). All processes which canchange the properties of GBs and IBs affect drasti-cally the behaviour of polycrystalline metals andceramics [159]. GB phase transitions are one ofthe important examples of such processes [160].Recently, the lines of GB phase transitions beganto appear in the traditional bulk phase diagrams [160-165]. The addition of these equilibrium lines to thebulk phase diagrams ensures an adequate descrip-tion of polycrystalline materials, particularly theirdiffusion permeability, deformation behaviour and theevolution of the microstructure.

In this work GB wetting, prewetting andpremelting phase transitions are discussed.Consider the contact between a bicrystal and a liquidphase L. If the GB energy σ

GB is lower than the energy

of two solid/liquid interfaces 2σSL, the GB is notcompletely wetted and the contact angle θ > 0. Ifσ

GB > 2σ

SL the GB is wetted completely by the liquid

�� ó��

phase and θ = 0. If the temperature dependenciesσ

GB(T) and 2σ

SL(T) intersect, then the GB wetting

phase transition proceeds at the temperature Tw of

their intersection. The contact angle θ decreasesgradually with increasing temperature down to zeroat T

w. At T > T

w the contact angle θ = 0. The tie line

of the GB wetting phase transition appears at Tw

inthe two-phase region (S+L) of the bulk phase dia-gram. Above this tie line GBs with an energy σ

GB

cannot exist in equilibrium with the liquid phase.The liquid phase forms a layer separating thecrystals. Correct investigations of the GB phasetransitions were perfomed using metallic bicrystalswith individual tilt GBs in the Al–Sn [161], Cu–In[162], Al–Pb–Sn [161,166,167], Al–Ga, Al–Sn–Ga[168,169], Cu–Bi [163,170,171], Fe–Si–Zn [172-175], Mo–Ni [176], W–Ni [177] and Zn–Sn [165]systems. The tie-lines of the GB wetting phase tran-sition were constructed basing on the experimentaldata [161,162,165-177]. The difference in the GBwetting phase transition temperature was experi-mentally revealed for GBs with different energies[162,166]. The precize measurements of the tem-perature dependence of the contact angle revealedalso that the GB wetting phase transition is of thefirst order [166].

The deformation behavior of metals in the semisolid state has been extensively investigated fromthe viewpoint of rheological flow [179-180]. Thesestudies have shown that the viscosity of the semi-solid metals depends on the volume fraction andmorphology of the solid phase and the shear strainrate. In addition, the deformation behavior in a semi-solid state at the early stages of melting has beeninvestigated by compressive creep tests [181-184].Vaandrager and Pharr [182] showed that the defor-mation mechanism in a semi-solid state at the earlystages of melting is grain boundary sliding accom-modated by cavitation in a liquid phase for the coppercontaining a liquid bismuth. This deformation mecha-nism in the semi-solid state at the early stages ofmelting appears to be dif

diffusion and related phenomena in bulk ...the grain boundary diffusion in nanostructured materials...

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