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DYNAMIC PROPERTIES OF GRAIN BOUNDARIES
M. Friesel, I. Manna, W. Gust
To cite this version:
M. Friesel, I. Manna, W. Gust. DYNAMIC PROPERTIES OF GRAIN BOUNDARIES. Journal de Physique Colloques, 1990, 51 (C1), pp.C1-381-C1-390. �10.1051/jphyscol:1990160�. �jpa-00230323�
COLLOQUE DE PHYSIQUE
Colloque Cl, supplkment au nol, Tome 51, janvier 1990
DYNAMIC PROPERTIES OF GRAIN BOUNDARIES
M. FRIESEL, I. MANNA(') and W. GUST
Max-Planck-Institut fur Metallforschung and Institut fur Metallkunde, Seestrasse 75, 0-7000 Stuttgart 1, F.R.G.
Abstract - Dynamic properties of grain boundaries (GBs), as reviewed in the present re- port, include diffusion along stationary and migrating boundaries, various boundary diffu- sion-controlled solid-state processes and mechanism of GB migration. Applicability of dif- ferent GB diffusivity measurement techniques, influence of diffusion direction and mis- orientation angle on diffusivity and extent of agreement between experimental results and theoretical computations have been discussed. Relevant features of discontinuous type of transformations pertaining to GB migration and diffusion have been highlighted. Diffusion data for stationary and migrating boundaries in Ni-In system are presented for a suitable comparison. Finally, a new mechanism has been proposed for the migration of GBs on the ba- sis of some recent experimental results.
1. Introduction - GB diffusion and migration play important roles in a large number of solid-state processes of commercial interest. Mass transport along GBs and migration rate of such boundaries are both strongly dependent on temperature and geometry of the boundary concerned. Systematic studies on GB diffusion and migration require well-characterized bi- crystals for quantitative understanding of the related dynamic processes. Therefore, pre- paration of such oriented bicrystals, either synthetically by diffusion bonding of two single crystals or naturally grown from the melt, constitutes an essential step towards the studies on dynamic properties of boundaries. However, preparation of oriented bi -
crystals is quite tedious and has been successful only in a limited number of metals or alloys.
The present report provides an overview on the recent understanding about the kinetics and mechanism of diffusion a1 ong stationary as well as migrating boundaries with suitable illustrations wherever possible. A short description of the GB diffusion-controlled solid- state transformations has also been included to elucidate the role of GBs in these pro- cesses. Finally, a new mechanism of GB migration has been suggested incorporating some recent experimental observations.
2. Diffusion Along Stationary Grain Boundaries - At temperatures well below the melting point (T,), diffusion proceeds much faster along the GBs than in the matrix. Figure 1 pre- sents a comparative spectrum of the experimental values of self-diffusion coefficients in face-centred cubic (fcc) metals expressed as a function of T,/T, where T is the absolute temperature. It is evident that the coefficient of self-diffusion along GBs (D,) is about seven orders of magnitude higher thav that in the bulk (D) at T,/2. Even at T, the dif- ference between D, and D is as large as three orders of magnitude. It is also important to note that D, is exactly the same as the diffusion coefficient for the liquid state (D,) at T (D, = D, = 2 X l ~ m2/s). Assuming a rational estimate of GB thickness - ~ 6 = 0.5 nm (5 A,), the resultant GB self-diffusivity value (&D,) of 1 X 10-18 m3/s appears to be the upper limit of 6Db in the self-diffusion spectrum of fcc metals. This fast mass transport along GBs plays a key role in a number of metallurgical processes, e.g. discontinuous mode of transformations, sintering, recrystallization, Coble creep, etc. Also, GB diffusion is be1 ieved to be of decisive importance for the long-term stability of microelectronic de- vices. Thus, in addition to the interest as a scientific phenomenon, GB diffusion warrants continued research attention for a better understanding of the above-mentioned solid-state transformations of technological importance.
Quantitative studies on GB diffusion necessitated a number of mathematical model S to be formulated. The first analytical approach was proposed by Fisher in 1951 and subsequently improved for a constant source by Whipple in 1954 and for an instantaneous source by Suzuoka in 1961 [l]. These models envisage GB as a thin, isolated, homogeneous slab of high-diffusivity material having a thickness 6 and joining the two adjacent crystal lat- tices. The diffusant is deposited on the free surface, while the GB plane is supposed to
Q
on
leave from the Dept. of Metallurgical Engineering, Indian Institute of Technology, Kharagpur - 721302, INDIA.Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990160
Cl-382 COLLOQUE DE PHYSIQUE
f c c
Fig.1
Diffusivity spectrum for self-diffusion in fcc metals ex- pressed on a reduced temperature scale [l]. D, D
,
D, and Ds are the biffusion coefficients for the bulk, GB, liquid and surface, respective- ly.be perpendicular to the latter. After the diffusion annealing, the concentration profile determined as a function of the distance from the surface provides the basis for diffusi- vity calculations. The classical solution to the GB diffusion processes in such simplified geometry is customarily expressed as the triple product s6Db, where s is the segregation factor. However, all the constituents in this solution being temperature dependent, indi- vidual estimations of these quantities are not always feasible. For a detailed description of the analytical models and experimental techniques relevant to GB diffusion studies, the reader may refer to Ref. [l].
The sample geometry described above is best realized in well -characterized bicrystals being either grown from the me1 t (natural bicrystal S) or synthesized by diffusion bonding of two single crystals (synthetic bicrystals). In order to estimate s6Db, one of the following experimental parameters is usually measured - (i) the distance (z) from the free surface (along the boundary) corresponding to a particular concentration, (i i ) the angle between the tangent to a particular isoconcentration line and the boundary, or (iii) the average concentration of the diffusant as a function of z. Because of the considerable difficulties and experimental constraints in producing oriented bicrystal S of either type, most of the GB diffusion data are based on experiments with polycrystalline samples ples and thin films. However, it is relevant to point out that s6Db values obtained from sam- ples other than those complying to the GB geometry envisaged in the analytical models may not always be in good agreement with the theoretical predictions.
GBs in the above-mentioned models [l] are treated as a continuum. However, considerable experimental evidences appear to suggest that probably all types of GBs comprise of dis- crete ordered structures. As a consequence, the GB diffusivity is observed to be strongly dependent on the GB misorientation angle and the direction of diffusion relative to the tilt axis of the boundary. The structural unit model [ 2 ] treates the GB as a core of only a few atomic layers, comprising of a uniform array of a single type of structural units.
Each structural unit is defined as a small group.of atoms arranged in a characteristic configuration.
The sectioning methods, in which the average concentration T of the diffusant is deter- mined in a series of thin slices parallel to the free surface as a function of z, may be the most re1 iabl e experimental method concerning studies on GB diffusion. For instance, serial sectioning by microtome or sputtering coupled with radiotracer technique is the most frequently used route of s6Db estimation in both bicrystals and polycrystals. Recent- ly, secondary-ion mass spectrometry (SIMS) has been successfully applied to GB diffusion of the stable isotopes lq51n and "31n in Cu and Ni bicrystals to demonstrate the poten- tiality of SIMS as a method of sbDb estimation even for systems having no suitable radio-
isotopes [3]. It may be noted that the sensitivity and resolution in both SIMS and radio- tracer measurements are quite comparable. In general, s6Db values determined on the basis of serial sectioning technique are independent of the type of the diffusant source and can be obtained , for example, through the Whipple-Le Claire equation [l]:
where t is the annealing time. A logarithmic plot of the measured average concentration
c
versus z 6 l 5 gives a straight line, the slope of which yields the required value of s6Db as per Eq.(l).
Figure 2 reveals the experimentally determined values of the average concentration of Ag radiotracer as a function of z 6 j 5 along two diffusion directions through symmetric tool>
tilt boundaries with a misorientation angle ( B ) of 22.62" in Ag (fcc) bicrystals. It is evident that diffusion proceeds faster along the perpendicular direction to the tilt axis than that along the direction parallel to it. As a consequence, &Db (S = 1 for GB self- diffusion in pure metals) between the above-mentioned directions are significantly dif- ferent (Fig.3). Figure 4 illustrates the expected nature of 6Db dependence on the mis- orientation angle for diffusion only parallel to the tilt axis in a fcc metal obtained ac- cording to the structural unit model 121. Figure 5 presents a similar dependence of 65 on the GB misorientation angle and the direction of self-diffusion in A1 (fcc), predicte8 on the basis of computer-simulated GB structure using static relaxation method [5]. It is quite interesting to note though the materials chosen for comparison are all fcc, still the predicted trend does not always match the experimental observation. For example, the experimental values of 6Db in the direction perpendicular to the tool> tilt axis closely follow the predicted dependence on Q as per Ref. [5] (cf. Figs.3 and 5). On the other hand, such agreement between experimental result and predicted behaviour in the direction parallel to tool> tilt axis is valid only for structural unit model [2] (cf. Figs.3 and 4) while the theoretical prediction as per Ref. [5] (Fig.5) follows a different trend than the experimental observation (Fig.3) in the same diffusion direction. In any case, it can be concluded that the magnitude of &Db is strongly related to the direction of diffusion and GB misorientation.
Parallel studies on the specific mechanism of GB diffusion are also of equal importance as the diffusivity measurements. Diffusion mechanism in the bulk may be correlated to the isotope effect or activation volume for diffusion. Unfortunately, similar correlation of isotope effect with the GB diffusion mechanism may not be precisely valid because the sym- metry conditions required for the isotope effect calculations are not fulfilled for GB diffusion [6]. However, the effect of hydrostatic pressure on GB diffusion ma provide useful information about the GB diffusion mechanism. In fact, GB diffusion of l'
i
Ag in Ag bicrystals as a function of hydrostatic pressure has yielded a value of the activation volume for GB diffusion quite close to the same for lattice diffusion of " O A ~ by vacancy mechanism [7]. This is believed to be a strong evidence in favour of the vacancy mechanism of GB diffusion in Ag [6]. Therefore, the effect of hydrostatic pressure on GB diffusivity in oriented bicrystals is likely to furnish similar clues about GB diffusion mechanism in other systems.3. Diffusion Along Migrating Grain Boundaries
3.1 Discontinuous Precipitation (DP) - As illustrated schematical ly in Fig.6a, precipi ta- tion of a two-phase aggregate, usually lamellar, behind a large-angle GB advancing into a supersaturated solid solution is known as DP. The name 'discontinuous' arises due to the discontinuous changes in lattice parameter and crystal orientation of the matrix phase across the mobile reaction front that provides a short-circuit path of solute transport.
The reaction mechanism of DP necessitates precipitation and concurrent GB migration with the former either as a prerequisite for boundary motion or a resultant of any thermally activated boundary movement [8]. In either case, the local deviation from thermodynamic equilibrium in the wake of boundary migration is postulated to sustain the growth of pre- cipitate colonies. Growth of precipitate colonies may be unidirectional or in opposite di- rections of the initial boundary, resulting into the so-called 'double-seam or S-morpholo- gy' [l]. Among the number of theories for growth kinetics proposed, the simple model of Petermann and Hornbogen [g] appears to be the most widely applicable and generally ac- cepted one. Mathematically, the model is expressed as follows:
where AG is the overall change in Gibbs free energy, X is the interlamellar spacing of the precipitate colony, v is the average growth velocity and R is the gas constant. It may be
Cl-384 COLLOQUE DE PHYSIQUE
noted that the expression presents a very 0 200 pm
convenient way of GB diffusivity estima- tion provided AG is known and X and v are experimentally determinable.
In order to extend the scope of its ap-
Ag I A g *
plicability, Eq.(2) has been modified as
follows [10,11]: 22.62' <001> TiGB
where X, is the average width of the 771
K
a-lamellae and C is the Cahn's parameter ';;
[12]. X, may be derived from the experi-
.z
mentally determined parameter A using the 3
following expression based on lever rule:
Lee-
where xo, xp and xi are the solute con-
3
centrations (expressed in mole fractions) of the supersaturated matrix phase ao, the precipitate phase D and the solute- depleted matrix phase a, respectively.
Combining Eq. (4) with Eq. (3),
1 SSD, = R T X ~ V (X~-X,)~/(-CAG) (xp-xi )'. (5)
For DP xi = X, in Eq. (5), and the Cahn's Diff. 1 <001>
parameter C may be calculated in the fol- l owing way:
O 0
C
200 400 600(24) tanh (4/2) = 0,-x,)/(xo-X,)
,
(6) z6I5 (where X, and X, are the metastable and Fig.2
equilibrium concentrations of the so- Concentration profiles of ' O * A ~ diffusion lute-depleted matrix phase, respective1 y. along a symmetric tilt grain boundary
(TiGB) for diffusion parallel and perpendi-
cular to the tilt axis at 771 K [4]. C is the reciprocal density of the coincident sites relative to the crystal lattice sites.
4 0 I I I I I I I
5:
=
2k 13 175
415
17 1325
I 1
0
(degrees)
Fig.3 Orientation dependence of the GB sel f-diffusivity (&D,) of ' 0 5 ~ g for symmetric TiGBs in the direction parallel and perpendicular to the tilt axis at 771 K [4].
30
V) m \
E F
2 0 -9
- -I-
<001> TiGB -
-
L3 -
(0
10
-
-
-O o
I
3 0 60 9 0
8 (degrees) Fig.5
Orientation dependence of the GB self-diffusivity (6Db) at 523 K For tool> symmetric tilt GBs of Al, calcu- l ated by Bi scondi [ 5 ] .
9 (degrees)
Fig.4 GB self-diffusivity (&Db) as a function of tilt angle B for Cu with the core struc- ture of <001> symmetric tilt GBs according to the structural unit model [ 2 ] .
Fig.6 Schematic descriptions of ( a ) discontinuous precipitation (DP), (b) coarsening (DC)
and (c) dissolution (DD). a. - supersaturated matrix phase, a - solute-depleted ma- trix phase, D - precipitate phase, a, - inhomogeneous matrix formed by DD, and RF - reaction front.
Finally, computation of s6Db through E q . ( 2 ) or E q . ( 5 ) necessitates precise estimation of AG for the process concerned. On the principle of maximum rate of free energy change, the following expression may be derived [Ill:
where 7 is the free energy per unit area of the a / R interface and V, is the molar volume of the alloy.
Cl-386 COLLOQUE DE PHYSIQUE
3.2 Discontinuous Coarsening (DC) - DC is a1 so an example of a moving boundary phenomenon that converts a fine lamellar two-phase aggregate into a coarser one comprising of the same constituents but achieving more near-equilibrium concentrations than the initial aggregate. Figure 6b explains the course of DC schematically. In many cases the reaction is undesirable, especially for the precipitation hardening superalloys, because it amounts to substantial deterioration of the mechanical strength due to the loss of coherency strain. Cahn [l21 has earlier postulated that the equilibrium concentration is not achieved by the matrix phase immediately after DP. On the other hand, the a-lamellae after DC have been reported to attain nearly the equilibrium concentration in a number of systems. Perhaps, the residual supersaturation in the matrix phase after DP adds up to the available driving force derived from the reduction in interfacial area. However, the pre- conditions of a mobile reaction front and mass transport through it distinguish DC from the usual coarsening reactions co,ntrol led by volume diffusion, e.g. Ostwald ripening. It is worth mentioning that the discontinuous characteristics of the DC reaction enables cal- culation of s6D, in a manner similar to DP using Eq.(5) by replacing xi with X, and modi- fying Eq.(6) in the following way:
where X, is the solute concentration of the a-lamellae after DC.
3.3 Discontinuous Dissolution (DD) - Dissolution of a two-phase precipitate colony behind a receding GB is called DD (Fig.6~). Similar to its precipitation counterpart, DD is also characterized by a discontinuous change in composition and orientation across the mobile boundary, which acts as the short-circuit route of mass transport. Discontinuous mode of dissolution predominates the dissolution process if T (even below the equil i brium solvus temperature T, ) and the corresponding D are low enough. Naturally, once the boundary mo- tion is initiated and sufficient driving force via the changes in chemical and interfacial free energies is available, DD is accomplished much more rapidly than the continuous mode of dissolution. This kinetic advantage of DD distinguishes itself as a desirable mode of discontinuous transformation. DD enables repeated use of oriented bicrystals and saves tremendous amount of time and effort. However, equilibrium solute distribution is not attained immediately following the boundary sweeping, and 'ghost images' can be often ob- served in the inhomogeneous matrix swept by the receding boundary. Figure 7 presents an experimental evidence of the ghost images seen after DD in an Fe-Zn alloy.
For s6D, measurements in DD at T I l,,
,
xi = X, in Eq. (5), and Eq. (8) may be modified as follows :(2/JC) tanh (JC/2) = (X, -X, )/(X, -X,) , and at T > T,,
where x3 is the concentration of the inhomogeneous matrix formed by DD replacing the erst- while a-lamellae.
3.4 Diffusion-Induced Grain Boundary Migration (DIGM) - Migration of GBs induced by cer- tain conditions of temperature and chemical potential of the system conducive only for diffusion through GBs is referred as 'DIGM'. GB diffusion of solute atoms into the matrix from an external source (alloying) or diffusing out of solute atoms via GB towards a sink (de-alloying) may both lead to DIGM, provided the corresponding D, >> D. DIGM has several potential scopes of application in materials technology, e.g., low temperature sintering and alloying of metal powders, controlled surface alloying of electronic material S, inter- diffusion studies with thin films, etc.
The overall driving force for DIGM may arise from the change in chemical potential of the matrix in the wake of alloying or de-alloying. It is postulated [2] that diffusion of so- lute and solvent atoms with inequal rates along a GB leads to the GB migration in the di- rection perpendicular to its length by synchronous climb of dislocation ledges. The defect structure of GBs may provide sources or sinks to support this necessary mass transport.
Compressive stresses or porosities generated in the matrix following alloying or de-al- loying and reduction in the average rate of DIGM due to restraints on contraction or ex- pansion normal to the GB plane provide experimental evidences that a net amount of solute is either added or removed by DIGM. Solute-solvent atomic mismatch and coherency strain in the matrix are also important conditions for the occurrence of DIGM. However, ability of the GBs to move does not always ensure DIGM since all the systems known for DP do not ne- cessarily show DIGM [13].
Fig.7
Scanning electron micrograph (SEM) of DD at 886 K for 28 min after D?
at 723 K for 44 h in an Fe - 13.5 at.% Zn alloy. Ghost images (61) re- present the series of the former boundary migration positions during
DD [ll]. OGB
Fig.8 Optical micrographs (LM) showing faceting in diffusion-induced grain boundary mi- gration (DIGM) of Cu(Zn) as a function of depth (z) from the surface after 14 h at 693 K for a C 19a/26.52" <011> tilt GB [15]. OGB is the original position of the GB, Cu is pure copper matrix and Cu-Zn is the solid solution formed by DIGM.
Recent investigations [l41 on DIGM with well-characterized Cu(Ln) bicrystals have demon- strated that the boundary migration rate is strongly dependent on the misorientation angle (cf. Fig.6 in Ref.[l4]). Utilization of DIGM experiments to estimate the important parame- ters concerning GB dynamics has been suitably illustrated in Ref.[l4]. Moreover, some in- teresting morphological features like pronounced faceting in a C 19a GB (Fig.8) indicate that mass transport in DIGM may be closely related to the GB geometry. The faceting strongly depends on the distance (z) from the surface. Complete characterization of such dependence of DIGM on GB geometry and z may furnish important details of solid-state dif- fusion, hitherto unknown.
4. Comparison of Diffusion Along Stationary and Migrating GBs - In general, GB diffusivi- ty measurements assume equilibrium chemical potential of the system implying that the GBs are in stationary (S) condition. In contrast, the discontinuous mode of solid-state trans- formations involve migrating (M) boundaries which may not necessarily be in equilibrium with the system. Further, the precise magnitude of s6Db for migrating GBs is not always available in the literature. However, a suitable comparison for Ni-In system, as an exam- ple out of about 12 systems in which sufficient reliable data for both types of boundaries are available, is presented in Fig.9. It may be noted that s6Db values for In radiotracer diffusion through stationary GBs in Ni have been determined by microtome sectioning tech- nique over a large diffusivity range covering 8 orders of magnitude. These results are compared with the published data for migrating GBs in DP of a Ni - 1.4 at.% In alloy [16].
In a separate experiment, s6D for both migrating and stationary GBs have been estimated also by radiotracer diffusion technique in the same Ni - 1.4 at.% In alloy. It is evident that s6Db values for both stationary and migrating boundaries, irrespective of the method
COLLOQUE DE PHYSIQUE
Fig.9
Arrhenius plot of sdD, to compare GB diffusivi- ty data for stationary (S) and migrating (M) boundaries in Ni-In system [16].
of estimation, are essentially the same. Therefore, it can be inferred that diffusion along migrating GBs is, in general, not 2 to 4 orders of magnitude faster than that along stationary GBs, as was postulated earlier. It is worth mentioning that the reports claim- ing higher diffusivity along migrating GBs than the stationary ones compared GB chemical diffusivity of alloys with GB self-diffusivity of the pure matrix element. In principle, such a comparison is not valid. On the other hand, the GB chemical diffusivity of a dilute alloy may be compared with the GB diffusivity of the solute component in the pure matrix because the diffusion process in the chemical diffusion of a dilute alloy is usually domi- nated by the solute element [l].
5. Grain Boundary Migration - A GB may be conceived as the region interspaced between two adjacent crystals consisting of small structural units of ordered structures. Migration of such interfaces, as any other crystal defect, is feasible under different types of condi- tion or driving force, e.g., stored strain energy, external or internal excess chemical free energy, thermal fluctuation, etc. It is well-known that GBs can be energetically favourable route of mass transport in a number of important solid-state transformations, especially when the GBs are mobile.
Under classical concepts of GB migration, the average velocity is supposed to be constant as long as the driving force remains unaltered. It has been recently proposed [l01 that the overall process of GB migration actually consists of infinitesimally small migration steps, and the average (v) and instantaneuous (v,) rates may differ by several orders of magnitude in each of such individual steps. GB migration in such discrete steps has earlier been reported in interdiffusion of Cu-Ni 1171 and DIGM in Au-Cu [18]. Also, the frequently observed ghost images in DD, indicating a series of earlier position of the migrating GB (cf. Fig:7), also provide an indirect proof of such step-like GB motion.
Similar proof of ghost images in DIGM experiments with Cu(Zn) system (cf. Fig.8) appears to suggest that step-like sequential motion of GBs may be quite a generalized phenomenon when GB migration is accompanied by solute transport, e.g. in any discontinuous transfor- mation. More direct evidences of step-like GB migration has been obtained in the in-situ studies on recrystallization in cold-worked Cu single crystal [l91 and DP in Al-Zn system [20] (Figs.lOa and b). It may be noted that the average values in both these experiments differ by one order of magnitude from the maxima of the instantaneous sequences of stop- and-go nature of GB migration.
1.0 I I I
0.8
-
AI-
15.0 Zn4 3 3 K
-
- In-situ studies o f recrystallization Fig.10a, (RC) in a 35% cold-worked Cu single
crystal at 526 K [l91 (a) and GB migra- tion during DP reaction in an Al- 15 at.% Zn alloy at 433 K 1201 ( b ) . v
-
is the average velocity and v i m a X is the maximum o f the instantaneous rates vi.
v = 1 . 6 x 1 0 - ~ m l s
-
vimax = 1.1 X 10'~ m l s
t (sl
6 . Scopes of Future Work - In view of the present status o f understanding on the dynamic
properties of GBs, future research is solicited in the following directions.
(i) Pre~aration of oriented bicrvstals: Both natural and synthetic bicrystals having well- characterized GB geometry and chemistry are of primary importance for useful studies on GB properties. While the angle of misorientation between the adjoining crystals and the angle of the inclination o f the GB plane define the geometry, chemical identity of the constituents refers to the chemistry of the GB. It is important to know the chemical com- position of the GB material not only in the beginning, but also at the end o f an experi- ment to exclude a possible influence of the undesirable impurities. Besides, preparation of such oriented bicrystals, if possible, is also warranted for alloys to enabTe studies on both phase transformations of discontinuous type and GB dynamics.
Cl-390 COLLOQUE DE PHYSIQUE
(ii) C o m ~ a r i s o n between synthetic and natural bicrvstal oro~erties: It .is essential to verify whether the structure and dynamic properties of the GBs in synthetic and natural bicrystals are precisely the same: Comparison of 6D, values determined for the same type of GBs produced by either technique may also elucidate whether the synthetic bicrystals provide identical GB characteristics as the natural ones.
(i i i) Increased a m 1 ication of hiah Dressure studies: GB diffusion under high hydrostatic pressure is expected to indicate definite clues to the GB diffusion mechanism and enable verification o f the theoretical models on GB structure.
(iv) New a ~ o r o a c h o f modellina and aeneration of reliable data: Development of a 'perfect' analytical model on GB properties necessitates repeated comparison between the results from experiments and computer simulations till the predicted behaviour is enough commensu- rate with the experimental data. Incidentally, majority of such analytical approaches are based on the energy considerations rather than the dynamic properties of the process con- cerned. It may be pointed out that the variation o f energy values as a function of T or B
is usually not more than 50%, while the corresponding limits o f experimental error are a1 - so nearly the same. On the other hand, the dynamic properties are strongly dependent on T and 8, and the changes o f any such property with T or B may be many orders of magnitude larger than the corresponding limits o f experimental uncertainty, which is only within a factor o f 2. Therefore, dynamic properties are expected to provide a more realistic basis for the necessary verification o f the theoretical model S. Perhaps, such continued efforts may enable prediction of G B behaviour even in areas where either reliable experimental da- ta are not available or experimentation is difficult.
(v) In-situ studies: For a better understanding of the mechanism o f GB migration further in-situ observations to evidence the sequence of GB dynamics are required.
(vi) Hiah-resolution studies: For a better correlation between GB structure and dynamic properties, high-resolution transmission electron microscopy on well-characterized GBs for which reliable experimental data for 60, and v are available, is advocated.
Acknowledgements - The financial supports o f the Swedisch National Board for Technological Development (STU) and Deutsche Akademische Austauschdienst (DAAD) are gratefully acknow- ledged by the authors M.F. and I.M., respectively. Finally, the authors would like to thank Prof. Herzig and his group, especially Dr. P. Neuhaus and Mr. J. Sommer, for the success o f radiotracer measurements.
References
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