206 CHAPTER 8. DIFFUSION IN CRYSTALS The hysteresis loop will therefore appear as a line of negligible width and slope 1/SR as in Fig. 8.24a. Negligible internal friction therefore occurs. U U U Figure 8.24: Frequency dependence of anelastic behavior. (a) w~ << 1. (b) WT >> 1. (c) WT = 1. (b) When wt >> 1, EI = Suu0 and €2 =O (8.172) The hysteresis loop will therefore appear again as a line of negligible width but with a larger slope, as in Fig. 8.246. Negligible internal friction occurs. (c) When wt = 1, (8.173) The hysteresis loop will therefore appear as in Fig. 8.24~. The slope of the dashed line is (8.174) SR - SU and EZ = ~u, SR + SU = 2uo 1 - uo El (SRfSU)/2 _- and the width of the loop at u = 0 is 2EZ = (SR - sU)Uo (8.175) Also, because the strain lags behind the stress, the direction of traversal of the loop must be as indicated. In this situation, maximum internal friction occurs. 8.22 Describe in detail how to determine the diffusivity of C in b.c.c. Fe using a torsion pendulum. Include all of the necessary equations. See Section 8.3.1 and Fig. 8.8, where C atoms in sites 1, 2, and 3 expand the crystal preferentially along z, y, and z, respectively. Solution. Using a torsion pendulum, find the anelastic relaxation time, T, by measuring the frequency of the Debye peak, up, and applying the relation W~T = 1. Having T, the relationship between T and the C atom jump frequency r is found by using the procedure to find this relationship for the split-dumbbell interstitial point defects in Exercise 8.5. Assume the stress cycle shown in Fig. 8.16 and consider the anelastic relaxation that occurs just after the stress is removed. A C atom in a type 1 site can jump into two possible nearest-neighbor type 2 sites or two possible type 3 sites. Therefore, - dcl = -4r/cl + 2r’c2 + 2r/c3 dt (8.176) Because c1 + c2 + c3 = ctot = constant, (8.177) EXERCISES 207 which may be integrated to obtain 8.23 [ c;] [ c;] -6r’t ci(t) - - = ~(0) - - =e (8.178) The relaxation time is then T = 1/(6l?), and because the total jump frequency is r = 4l?, T = 2/(3r). According to Eq. 7.52, DZ = rr2/6 because f = 1, and because r = a/2, DI = ra2/24. Substituting for r, (8.179) Finally, insert the experimentally determined value of T into Eq. 8.179 to obtain DI Under equilibrium conditions in a stressed b.c.c. Fe crystal, interstitial C atoms are generally unequally distributed among the three types of sites iden- tified in Fig. 8.8b. This occurs because the C atoms in sites 1, 2, and 3 in Fig. 8.8b expand the crystal preferentially along the 2, y, and z directions, respectively. These directions are oriented differently in the stress field, and the C atoms in the various types of sites therefore have different interaction energies with the stress field. In the absence of applied stress, this effect does not exist and all sites are populated equally. In Exercise 8.22 it was shown that when the stress on an equilibrated specimen is suddenly released, the re- laxation time for the nonuniformly distributed C atoms to achieve a random distribution, T, is T = 2/(3r), where r is the total jump frequency of a C atom in the unstressed crystal. Show that when stress is suddenly applied to an unstressed crystal, the relax- ation time for the randomly distributed C atoms to assume the nonrandom distribution characteristic of the stressed state is again T = 2/(3r). Assume the energy-level system for the specimen shown in Fig. 8.25. Write the kinetic equations for the rates of change of the concentrations of the interstitials in the various types of sites and solve them subject to the appropriate initial and final conditions. Assume that the barri- ers to the jumping interstitials shown in Fig. 8.25 are distorted by the differences in the site energies (indicated in Fig. 8.21). Figure 8.25: atom in sites 1, 2, or 3 illustrated in Fig. 8.8. Energy-level diagram for a stressed b.c.c. specimen containing an interstitial Solution. Let c1, c2, and c3 be the concentrations of interstitials occupying sites of types 1, 2, and 3, respectively. Also, c1 +c2 +c3 = ctot = constant. Since an interstitial 208 CHAPTER 8: DIFFUSION IN CRYSTALS in a given type of site can jump into two sites of each other type, dci dt dc2 dt - = - 2 (rL + rL3 + rL) c1 + 2 (rL1 - rL) c2 + 2r’3+1~t0t - = - 2 (rL1 + rL3 + rL2) c2 + 2 (rL - r;,2) c1 + 2r$-2~tot (8.180) If the barrier to the jump of an interstitial between two sites of differing energy is deformed as indicated in Fig. 8.21, the information given in Fig. 8.25 may be used to derive expressions for the various jump rates that appear in the coefficients of Eq. 8.180. Neglecting small differences in the entropies of activation in the presence and absence of stress, and expanding Boltzmann factors of the form exp[-U,,,/(kT)] to first order so that exp[-U,-,/(kT)] = 1 + Ut-J/(kT), r;-2 = r’ (1 - w) = r/ (1 - w) r;+l = r’ (1 + W-J rLl = r’ (1 + w) r;,3 = r’ (1 - h) = r’ (1 + - h) r;,, = r’ (1 - + *) 2kT 2kT 2kT (8.181) where I?’ is the jump rate between any two adjacent sites in the absence of stress. Equa- tion 8.180 is a pair of simultaneous linear first-order equations with constant coefFicients. The initial and final conditions are c1 (m) = ci‘ c2(m) = c;q (8.182) where el( ) and c2(m) are the final equilibrium concentrations reached at long times in the presence of the applied stress. In view of the symmetry of Eqs. 8.180, we try Ctot Cl(0) = c2(0) = - 3 Cl(t) = (f - ‘p> e-k‘t + .;‘ c2(t) = (f - c;~) e-“lt + c‘lq (8.183) which satisfy the conditions in Eq. 8.182. Direct substitution shows that Eqs. 8.183 indeed satisfy Eqs. 8.180 when higher-order terms involving products of the small quan- tities Ut-j/(kT) are neglected and k‘ = 6r’ c‘lq = - I+-+-) Ul-2 u1-3 3 ( 3kT 3kT (8.184) 1 2u1+2 3kT + 3kT This shows that relaxation to the equilibrium distribution occurs exponentially with a relaxation time T = 1/(6r’). Since = 4r’, where r is the total jump frequency in the unstressed crystal, 7 = 2/(3r). Finally, the equilibrium concentrations obtained in Eqs. 8.184 from the kinetic equations agree with those obtained using equilibrium statistical mechanics. In the three-level system in Fig. 8.25, the occupation probability for level 1 is Since c1 = ctotpl, the result for CI is the same as that given by Eq. 8.184. Similar agreement is obtained for c2. CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS Experiments demonstrate that along crystal imperfections such as dislocations, in- ternal interfaces, and free surfaces, diffusion rates can be orders of magnitude faster than in crystals containing only point defects. These line and planar defects pro- vide short-circuit diffusion paths, analogous to high-conductivity paths in electrical systems. Short-circuit diffusion paths can provide the dominant contribution to diffusion in a crystalline material under conditions described in this chapter. 9.1 THE DIFFUSION SPECTRUM IN IMPERFECT CRYSTALS Rapid diffusion along line and planar crystal imperfections occurs in a thin region centered on the defect core. For a dislocation, the region is cylindrical, roughly two interatomic distances in diameter, and includes the “bad material” in the dislocation core.’ For a grain boundary, the region is a thin slab, roughly two interatomic distances thick, including the bad material in the grain boundary core. For a free surface, this region is the first few atomic layers of the material at the surface. These regions are very thin in comparison to the usual diffusional transport distances. To model the diffusion due to these imperfections, we replace them by thin slabs or cylinders of effective thickness, 6, possessing effective diffusivities which are much larger than the diffusivity in the adjoining crystalline material. Table 9.1 lists the Bad material is disordered material in which the regular atomic structure characteristic of the crystalline state no longer exists. Good bulk material is free of line or planar imperfections. Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 209 Copyright @ 2005 John Wiley & Sons, Inc. 210 CHAPTER 9 DIFFUSION ALONG CRYSTAL IMPERFECTIONS Table 9.1: Notation for Short-circuit Diffusivities DD (undissoc) diffusivity along an undissociated dislocation core (i.e., a cylin- der, or a “pipe” of diameter, 6) DD (dissoc) diffusivity along a dissociated dislocation core (i.e., a cylinder, or a “pipe” of diameter, 6) DB diffusivity along a grain boundary (i.e., a slab of thickness, 6) DS DXL diffusivity along a free surface (i.e., a slab of thickness, 6) diffusivity in a bulk crystal free of line or planar imperfections DL diffusivity in a liquid notation to be used to describe the diffusivities in various regions of crystalline materials containing line and planar imperfections. Figure 9.1 presents self-diffusivity data for *DD (dissoc), *DD (undissoc), *DB, *Ds, *DxL, and *DL, for f.c.c. metals on a single Arrhenius plot. With the excep- tion of the surface diffusion data, the data are represented by ideal straight-line Arrhenius plots, which would be realistic if the various activation energies were constants (independent of temperature). However, the data are not sufficiently accurate or extensive to rule out some possible curvature, at least for the grain boundary and dislocation curves, as discussed in Section 9.2.3. Dislocations, grain boundaries, and surfaces can possess widely differing struc- tures, and these structural variations affect their diffusivities to significant degrees. If the defective core region is less dense or “looser” than defect-free material, or if a defect possesses structurally “open” channels in its core structure, transport will generally be more rapid along the defect, particularly in the open directions. Some grain boundary structures can be represented by dislocation arrays, and their boundary diffusivity can be modeled in terms of transport along the grain-boundary 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Reduced temperature, T,,, lT Figure 9.1: Master Arrhenius plot of *DxL, *DD(dissoc), *D”(undissoc), *DB, *Ds, and *DL characteristic of f.c.c. metals. Data for various f.c.c. metals have been normalized by using a reduced reciprocal temperature scale, (l/T)/(l/Tm) = T,/T. All diffusivities were derived from experimental data by assuming that all 6 = 0.5 nm. From Gjostein [I]. 9.1: THE DIFFUSION SPECTRUM 211 dislocation cores. General grain boundary structures cannot support discrete local- ized dislocations but, nevertheless, still act as short-circuit diffusion paths. Short-circuit diffusion along grain boundaries has been studied extensively via experiments and modeling. Because diffusion along dislocations and crystal sur- faces is comparatively less well characterized, particular attention is paid to grain- boundary transport in this chapter. However, briefer discussions of diffusion along dislocations and free surfaces are also presented. To describe the effects of grain-boundary structure on boundary diffusion, it is necessary to review briefly some important aspects of boundary structure. Addi- tional details appear in Appendix B. It takes a minimum of five geometric pa- rameters to define a crystalline interface. Three describe the crystal/crystal mis- orientation: e.g., two to specify the axis about which one crystal is rotated with respect to the other, and one for the rotation angle. The remaining two parameters define the inclination of the plane along which the crystals abut at the interface.2 If the interface is a free surface, just two parameters are required to specify the surface's inclination (unit normal). Crystal symmetries determine special values of the parameters at which the interfacial energies take on extreme values. Depending on the specific nature of a system with interfaces, some of the parameters may be constrained and others free to vary as the system seeks a lower-energy state. Small-angle grain boundaries have crystal misorientations less than about 15" and consist of regular arrays of discrete dislocations (Le., where the cores are sep- arated by regions of defect-free material). As the crystal misorientation across the boundary increases beyond about 15", the dislocation spacing becomes so small that the cores overlap and the boundary becomes a continuous slab of bad mate- rial; these are called large-angle boundaries. Large-angle boundaries can be further classified into singular boundaries, vicinal boundaries, and general b~undaries.~ An interface is regarded as singular with respect to a degree of freedom if it is at a local minimum in energy with respect to changes in that degree of freedom. It is therefore stable against changes in that degree of freedom. A vicinal interface is an interface that deviates from being singular by a rela- tively small variation of one or more of its geometric parameters from their singular- interface values. A vicinal interface can therefore minimize its energy by adopting a fit-misfit structure consisting of patches of the nearby minimum-energy singular interface delineated by arrays of discrete interfacial dislocations or steps as illus- trated in Figs. B.4 and B.9. These line defects serve to accommodate the relatively small deviations of the vicinal interfaces from the singular interfaces. A general interface is not energy-minimized with respect to any of its degrees of freedom, and is far from any singular-interface values of the parameters that set its degrees of freedom. Such an interface cannot reduce its energy by adopting a fit-misfit structure (as in the vicinal case) and therefore cannot support localized dislocations or steps. Two examples serve to clarify these distinctions: Example 1 The tilt grain boundary in Fig. B.4a is singular with respect to its tilt angle.4 The boundary in Fig. B.4c is vicinal to the singular boundary 2Additional variables may be required, such as three that specify a relative translation of one crystal with respect to the other. 3Similar terminology is used for classification of free-surface structure. 4See Appendix B for descriptions of tilt, twist, and mixed grain boundaries. 212 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS with respect to its tilt angle. It consists of patches of the singular boundary delineated by dislocations that accommodate the change in tilt angle. Example 2 A surface corresponding to the patch of light-colored atoms in Fig. B.l is singular with respect to its inclination about an axis parallel to the surface steps in the figure. The stepped surface in Fig. B.l is vicinal to such a flat surface and consists of patches of the flat surface delineated by steps that accommodate the change in surface inclination. Because the structure of general large-angle grain boundaries is usually less reg- ular and rigid than that of singular or vicinal boundaries, its activation energies for diffusion are typically lower and the diffusivities correspondingly higher. The diffusion rate along small-angle grain boundaries is generally lower than along large- angle grain boundaries and, indeed, approaches DxL as the crystal misorientation approaches zero. This is due to two factors: first, the diffusion rate along the bad material in dislocation cores is about the same as, or lower than, that along large-angle grain boundary cores (see Fig. 9.1); second, because small-angle grain boundaries consist of periodic arrays of lattice dislocations at discrete spacings that approach infinity as the crystal misorientation approaches zero, the density of fast- diffusion paths is smaller in small-angle boundaries than in large-angle boundaries. Figure 9.2 presents diffusivity data for a series of tilt boundaries as a function of the misorientation tilt angle. The structures of these boundaries vary considerably as the misorientation changes. In the central part of the plot, the minima occur at crystal misorientations (values of Q) corresponding to singular and vicinal boundaries. The ends of the plot (where the crystal misorientation approaches zero) correspond to small-angle boundaries, and the diffusivities are correspondingly low. The regions centered around the maxima in Fig. 9.2 correspond to general grain boundaries. Polycrystalline materials not subjected to special processing conditions possess mainly general boundaries; the grain-boundary data in Fig. 9.1 are for general boundaries that have fairly similar diffusivities and can therefore be described reasonably well by average normalized values. s4/ ;; I\ 0 N 0 I\ I - X 11 4 I I 60 120 180 6 (degrees) Figure 9.2: Grain-boundary diffusivity of Zn along the tilt axes of (1101 symmetric tilt grain boundaries in A1 as a function of tilt angle, 8. From Interfaces in Cvystalline Materials by A.P. Sutton and R.W. Balluffi (1995). Reprinted by permission of Oxford University Press. Data from I. Herbeuval and M. Biscondi [2]. 9.1: THE DIFFUSION SPECTRUM 213 The wide range of diffusivity magnitudes evident in the diffusivity spectrum in Fig. 9.1 may be expected intuitively; as the atomic environment for jumping becomes progressively less free, the jump rates, r, decrease accordingly in the sequence rS > rB x rD(undissoc) > rD(dissoc) > rXL. The activation energies for these diffusion processes consistently follow the reverse behavior, ES < EB FZ ED(undissoc) < ED(dissoc) < ExL (9.1) The diffusivity in free surfaces is larger than that in general grain boundaries, which is about the same as that in undissociated dislocations. Furthermore, the diffusivity in undissociated dislocations is greater than that in dissociated dislocations, which is greater than that in the cry~tal:~ *Ds > *DB M *DD(undissoc) > *DD(dissoc) > *DxL (94 Free-surface and grain-boundary diffusivities in metals at 0.5Tm are seven to eight orders of magnitude larger than crystal diffusivities. Provided that defects are present at sufficiently high densities, significant amounts of mass transport can occur in crystals at 0.5Tm via surface and grain-boundary diffusion even though the cross-sectional area through which the diffusional flux occurs is relatively very small. As the temperature is lowered further, the ratio of diffusivities becomes larger and short-circuit diffusion assumes even greater importance. Generally, similar behavior is found in ionically bonded crystals, as shown in Fig. 9.3. N E v 8 M 0 - -1 2 -1 4 -1 6 -1 8 -20 -22 6 8 10 12 1041~ (~-1) Figure 9.3: Self-diffusivities of 0 and Ni on their respective sublattices in a NiO sin le crystal free of significant line imperfections and along grain boundaries in a polycrystal. &e grain-boundary diffusivities of both Ni and 0 in the oxide semiconductor NiO are very much greater than corresponding crystal diffusivities. From Atkinson [3]. There are many situations, particularly at low temperatures, where short-circuit diffusion along grain boundaries and free surfaces is the dominant mode of diffu- sional transport and therefore controls important kinetic phenomena in materials; 5We discuss diffusion along dislocations and free surfaces in Sections 9.3 and 9.4. 214 CHAPTER 9: DIFFUSION ALONG CRYSTAL IMPERFECTIONS several examples are discussed in Sections 9.2 and 9.4. Similar conclusions hold for dislocation diffusional short-circuiting, although to a lesser degree because of the relatively small cross sections of the high-diffusivity pipes. 9.2 DIFFUSION ALONG GRAIN BOUNDARIES 9.2.1 In a polycrystal containing a network of grain boundaries, atoms may migrate in both the grain interiors and the grain boundary slabs [4]. They may jump into or out of boundaries during the time available, and spend various lengths of time jumping in the grains and along the boundaries. Widely different situations may occur, depending upon such variables as the grain size, the temperature, the diffusion time, and whether the boundary network is stationary or moving. For example, as the grain size is reduced and more boundaries become available, the overall diffusion will be enhanced due to the relatively fast diffusion along the boundaries. At elevated temperatures where the ratio of the boundary diffusivity to the crystal diffusivity is lower than at low temperatures (Fig. 9.1), the importance of the boundary diffusion will be diminished. At very long diffusion times, the distance each atom diffuses will be relatively large, and each atom will be able to sample a number of grains and grain boundaries. If the boundaries are moving, an atom in a grain may be overrun by a moving boundary and be able to diffuse rapidly in the boundary before being deposited back into crystalline material behind the moving boundary. Consider first the relatively simple case where the boundaries are stationary and each diffusing atom is able to diffuse both in the grains and along at least several grain boundaries during the diffusion time available. This will occur whenever the diffusion distance in the grains during the diffusion time t is significantly larger than the grain size [i.e., approximately when the condition *DXLt > s2 (where s is the grain size) is satisfied]. For each atom, the fraction of time spent diffusing in grain boundaries is then equal to the ratio of the number of atomic sites that exist in the grain boundaries over the total number of atomic sites in the specimen [5]. This fraction is q x 36/s: for each atom, the mean-square displacement due to diffusion along grain boundaries is then *DBqt, and the mean-square displacement in the grains is *DxL(l - q)t. The total mean-square displacement is then the sum of these quantities, which can be written Regimes of Grain-Boundary Short-circuit Diffusion in a Polycrystal (*D)t =* DxL(l - q)t +*DBqt (9.3) and because q << 1, (*D) = *DxL + (36/s)*DB *DLt > s2 (9.4) The quantity (*D) is the average effective diffusivity, which describes the overall dif- fusion in the system. The diffusion in the system therefore behaves macroscopically as if bulk diffusion were occurring in a homogeneous material possessing a uniform diffusivity given by Eq. 9.4. The situation is illustrated schematically in Fig. 9.4a, and experimental data for diffusion of this type are shown in Fig. 9.5. This diffu- sion regime is called the multiple-boundary daffusion regime since the diffusion field 9 2 GRAIN BOUNDARIES 215 B regime (b) boundary region C regime (c) core only Figure 9.4: The A. B, and C regimes for self-diffusion in polycrystal with stationary grain boundaries according to Harrison [6]. The tracer atoms are diffusing into the semi- infinite specimen from the surface located along the top of each figure. Regions of relatively high tracer concentration are shaded. (a) Regime A: the diffusion length in the grains is considerably longer than the average grain size. (b) Regime B: the diffusion length in the grains is significant but smaller than the grain size. (c) Regime C: the diffusion length in the grains is negligible, but significant diffusion occurs along the grain boundaries. In all figures, preferential penetration within the grain boundaries is too narrow to be depicted. overlaps multiple boundaries. Note that in Fig. 9.4a, fast grain-boundary diffusion will cause preferential diffusion to occur along the narrow grain-boundary cores beyond the main diffusion front, but the number of atoms will be relatively small and this effect cannot be depicted. 900°C 700°C 500°C 350°C -12 I I I I I I I I 1 .o 1.2 1.4 1.6 lOOO/T (K-I) Figure 9.5: Values of the average self-diffusivity. (*D). in single- and polycrystalline silver. At lower temperatures. pain-boundary diffusion makes significant contributions to the overall measured average di usivity in the polycrystal. From Turnbull [7]. [...]... energies of the interstitial sites follow a Gaussian distribution around a mean value, good agreement was obtained between the model and experiment The increase of DI -1 1 -1 3 -5 -4 -3 -2 -1 log P Figure 10.5: Logarithm of the diffusivity of H in amorphous PdsoSizo as a function of the H concentration probability at different temperatures Points are experimental data The curves are the predictions of the... J.L Bocquet, G Brebec, and Y Limoge Diffusion in metals and alloys In R.W Cahn and P Haasen, editors, Physical Metallurgy, pages 53 5 -6 68 North-Holland, Amsterdam, 2nd edition, 19 96 EXERCISES 9.1 In a Type-A regime, short-circuit grain-boundary self-diffusion can enhance the effective bulk self-diffusivity according to Eq 9.4 A density of lattice dislocations distributed throughout a bulk single crystal... 15(8):95 1-9 56, 1981 15 R.W Balluffi On measurements of self diffusion rates along dislocations in f.c.c metals Phys Status Solidi, 42(1):1 1-3 4, 1970 16 R.E Reed-Hill and R Abbaschian Boston, 1992 Physical Metallurgy Principles PWS-Kent, 17 Y.K Ho and P.L Pratt Dislocation pipe diffusion in sodium chloride crystals Radiat Eff., 75~18 3-1 92, 1983 18 P Shewmon Diffusion in Solids The Minerals, Metals and Materials. .. 57(7):119 1-1 199, 1 961 7 D Turnbull Grain boundary and surface diffusion In J.H Holloman, editor, A t o m Movements, pages 12 9-1 51, Cleveland, OH, 1951 American Society for Metals Special Volume of ASM 8 J.W Cahn and R.W Balluffi Diffusional mass-transport in polycrystals containing stationary or migrating grain boundaries Scripta Metall Mater., 13 (6) :49 9-5 02, 1979 9 I Kaur and W Gust Fundamentals of Grain... space before the displaced first particle returns, a diffusive-type jump will have occurred Diffusion therefore occurs as a result of the redistribution of the free volume that occurs at essentially constant energy because of the flatness of the interatomic potentials According to the kinetic theory of gases, the self-diffusivity of a hard-sphere gas is given by *DG = (2/5)(u)L, where (u) the average... Kramer, editors, Materials Science and Technology-A Comprehensive Treatment, volume 11, pages 29 5-3 37, Wienheim, Germany, 1994 VCH Publishers 4 A.P Sutton and R.W Balluffi Interfaces in Crystalline Materials Oxford University Press, Oxford, 19 96 5 E.W Hart On the role of dislocations in bulk diffusion Acta Metall., 5(10):597, 1957 6 L.G Harrison Influence of dislocations on diffusion kinetics in solids... [22] Figure 10 .6 plots the tracer diffusivity data for a number of solute species in glassy Ni80Zr50 as a function of their metallic radius The diffusivity increases rapidly as the metallic radius decreases The relatively rapid diffusion of the small atoms in this case may result from the fact that they diffuse by the interstitial mechanism [lo, 181 -1 9 - -2 0 Ti h N E v P 21 g - -2 2 I -2 3 Metallic... becomes cz L ( z l , y l , t l ) = -exp X 1 k and, finally, Eq 9.13 becomes [ (A) - [- (A)”* YI] Yl] [1 - e r f I) $- (9.23) (9.24) (9.25) 9.4 As described in Section 9.2.2, grain-boundary diffusion rates in the Type-C diffusion regime can be measured by the surface-accumulation method illustrated in Fig 9.12 Assume that the surface diffusion is much faster than the grain-boundary diffusion and that the... 9 .6 The asymmetric small-angle tilt boundary in Fig B.5a consists of an array of parallel edge dislocations running parallel to the tilt axis During diffusion they will act as fast diffusion “pipes.” Show that fast self-diffusion along this boundary parallel to the tilt axis can be described by an overall boundary diffusivity, lr sin 4 cos 4 e (9.29) *DB(para) = - *DD6 4 b where b is the magnitude of. .. 41(1):13 3-1 41, 1993 13 R.W Balluffi Grain boundary diffusion mechanisms in metals In G.E Murch and A S Nowick, editors, Diffusion in Crystalline Solids, pages 31 9-3 77, Orlando, FL, 1984 Academic Press EXERCISES 225 14 R.W Balluffi, T Kwok, P.D Bristowe, A Brokman, P.S Ho, and S Yip Deter- mination of the vacancy mechanism for grain-boundary self-diffusion by computer simulation Scripta Metall Mater., . because of their low two-dimensional symmetry. When the surface structure consists of parallel rows of closely spaced atoms, separated by somewhat larger inter-row distances, diffusion is usually. dislocations and crystal sur- faces is comparatively less well characterized, particular attention is paid to grain- boundary transport in this chapter. However, briefer discussions of diffusion. constrained and others free to vary as the system seeks a lower-energy state. Small-angle grain boundaries have crystal misorientations less than about 15" and consist of regular arrays