Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 3 Part 3 pot

556 32 Grain-Boundary Diffusion Fig. 32.2. Low-angle tilt boundary after Burgers [19] Low-angle twist boundaries can be considered as a planar network of screw dislocations [23]. As the misorientation Θ between the two grains increases, the strain fields of dislocations progressively cancel each other so that γ increases slower than Eq. (32.2) predicts. In general, when the misorientation exceeds 10 to 15 degrees the dislocation spacing is so small that the dislocation cores overlap. It is then impossible to identify individual dislocations. At this stage, the grain-boundary energy is almost independent of misorientation. High-angle boundaries contain large areas of poor fit and have a relatively open structure (Fig. 32.3). The bonds between the atoms are broken or highly distorted and consequently the grain-boundary energy is relatively high. Correlations between the macroscopic parameters of grain boundaries and their energy have been explored by atomistic computer simulations. For a review, we refer the reader to the already mentioned anthology of Wolf and Yip [22]. The Fig. 32.3. Random high-angle grain boundary (schematic) 32.2 Grain Boundaries 557 grain-boundary energy plays a central rˆole in grain-boundary diffusion and in segregation of foreign atoms to the boundary. As a rule of thumb, high-angle grain-boundary energies are often found to be about one third of the energy of the free surface. In low-angle boundaries, however, most of the atoms fit very well into both lattices so that there is little free volume and the interatomic bonds are only slightly distorted. The regions of poor fit are restricted to dislocation cores. 32.2.2 Special High-Angle Boundaries Not all high-angle boundaries have an open disordered structure. Special high-angle boundaries have significantly lower energies than random high- angle boundaries. Special boundaries occur at particular misorientations of the grains and orientations of the boundary plane which allow the adjoin- ing lattices to fit together with relatively little distortion of the interatomic bonds. The simplest special high-angle boundary is the boundary between twins. If the boundary is parallel to the twinning plane, the atoms in the boundary fit perfectly into both grains. The result is a coherent twin boundary illus- trated in Fig. 32.4. In fcc metals the twinning plane is a close-packed {111} plane. Twin orientations in fcc metals correspond to a misorientation of 70.2 degrees around a 110-axis. A coherent twin boundary is a symmetric tilt boundary between the twin-related crystals. The atoms in such a boundary are essentially in undistorted positions and the energy of a coherent twin boundary is very low in comparison to the energy of a random high-angle boundary. If the twin boundary does not lie exactly parallel to the twinning plane the atoms do not fit perfectly into each grain and the boundary energy is higher. Such boundaries are denoted as incoherent twin boundaries. The energy of a twin boundary is very sensitive to the orientation of the grain-boundary plane. If the boundary energy is plotted as a function of the boundary orien- tation (right part of Fig. 32.4) a sharp cusped minimum is obtained at the position of the coherent boundary. Low grain-boundary energies are also found for other large-angle bound- aries. A two-dimensional example is shown in Fig. 32.5. This is a symmetrical Fig. 32.4. A coherent twin boundary (left). Twin-boundary energy γ as a function of the orientation φ of the grain-boundary plane (right) 558 32 Grain-Boundary Diffusion Fig. 32.5. A special large-angle boundary according to Gleiter [24] tilt grain boundary between grains with a misorientation of 38.2 degrees. The boundary atoms fit rather well into both grains leaving little free volume. Moreover, a small group of atoms is repeated at regular intervals along the boundary. High-resolution transmission electron microscopy (HREM) can be used to resolve the atomic structure of a grain boundary (see, e.g., [26]). Interfaces suitable for HREM are tilt boundaries, whose tilt axis coincides with a low- index zone axis. Figure 32.6 shows the HREM micrograph of a (113) [113] symmetric tilt boundary in a gold bicrystal. This grain boundary is periodic and several grain-boundary units along the boundary can be identified. This image also illustrates that the grain-boundary width δ (see below) is of the order of 0.5 nm. Fig. 32.6. A high-resolution transmission electron microscope image of a (113)[113] symmetric tilt boundary in gold according to Wolf and Merkle [25] 32.3 Diffusion along an Isolated Boundary (Fisher Model) 559 32.3 Diffusion along an Isolated Boundary (Fisher Model) Most of the mathematical treatment of grain-boundary diffusion is based on the model first proposed by Fisher [7]. The grain boundary is represented by a semi-infinite, uniform, and isotropic slab of high diffusivity embedded in a low-diffusivity isotropic crystal (Fig. 32.7). The grain boundary is de- scribed by two physical parameters: the grain-boundary width δ and the grain-boundary diffusivity D gb . The latter of course depends on the grain- boundary structure discussed above. It is usually much larger than the lattice diffusivity D in the adjoining grains, i.e. D gb  D. The grain-boundary width is of the order of an interatomic distance. δ ≈ 0.5 nm is a widely accepted value (see above). In a tracer diffusion experiment a layer of tracer atoms (either self- or foreign atoms) is deposited at the surface. Then, the specimen is annealed at constant temperature T for some time t. During the annealing treatment the labeled atoms diffuse into the specimen in two ways: (i) by lattice diffusion directly into the grains and (ii) much faster along the grain boundary. Atoms which diffuse along the grain boundary eventually leave it and continue their diffusion path in the grains, thus giving rise to a lattice diffusion zone around the grain boundary. The total concentration of the diffuser in the specimen is the result of two contributions: a concentration c, established Fig. 32.7. Fisher’s model of an isolated grain boundary. D: lattice diffusivity, D gb : diffusivity in the grain boundary, δ: grain-boundary width 560 32 Grain-Boundary Diffusion either directly by in-diffusion from the source or by leaking out from the grain boundary and the concentration inside the grain boundary, c gb . Mathematically, this diffusion problem can be described by applying Fick’s second law to diffusion inside the grains and inside the grain-boundary slab. For composition-independent diffusivities we have: ∂c ∂t = D  ∂ 2 c ∂y 2 + ∂ 2 c ∂z 2  for | y |≥ δ/2 , ∂c gb ∂t = D gb  ∂ 2 c gb ∂y 2 + ∂ 2 c gb ∂z 2  for | y |<δ/2 . (32.3) In Eqs. (32.3) the coordinate system was chosen in such a way that the xz-plane is the symmetry plane of the grain boundary. Then, the concentra- tion field depends on the variables y and z. Continuity of the concentrations and of the diffusion fluxes across the interfaces between grain-boundary and grains require the following boundary conditions: c(±δ/2,z,t)=c gb (±δ/2,z,t) (32.4) and D  ∂c(y,z, t) ∂y  |y|=δ/2 = D gb  ∂c gb (y,z, t) ∂y  |y|=δ/2 . (32.5) These conditions apply for self-diffusion. In the case of foreign atom diffusion, grain-boundary segregation requires a modification, which is discussed later in this chapter. Since the grain-boundary width is very small (δ ≈ 0.5 nm) and D gb  D, one can simplify the problem (see, e.g., [8]) and arrive at the following set of two coupled equations: ∂c ∂t = D  ∂ 2 c ∂y 2 + ∂ 2 c ∂z 2  for | y |≥ δ/2 , (32.6) ∂c gb ∂t = D gb ∂ 2 c gb ∂z 2 + 2D δ  ∂c ∂y  y=δ/2 for | y |<δ/2 . (32.7) The first equation represents direct diffusion from the source into the lat- tice. In the second equation, the first term on the right-hand side represents the concentration change due to diffusion in the grain boundary. The sec- ond term describes the concentration change due to leakage of the diffusing species through the ‘walls’ of the grain-boundary slab into the grains. The mathematical problem reduces to the solution of Eqs. (32.6) and (32.7) after suitable initial and boundary conditions have been chosen. It is convenient to introduce normalised variables, which correspond to the spatial coordinates y, z and to time t, respectively: 32.3 Diffusion along an Isolated Boundary (Fisher Model) 561 ξ ≡ y −δ/2 √ Dt , η ≡ z √ Dt , β ≡ (∆ − 1)δ 2 √ Dt ≈ δD gb 2D √ Dt . (32.8) In these abbreviations ∆ ≡ D gb D (32.9) is a dimensionless parameter, which equals the ratio of grain-boundary and lattice diffusivity. In physical terms, the variable ξ accounts for the extent of lateral lattice diffusion from the grain boundary into the grains. The quantity η accounts for the influence of direct lattice diffusion from the source into the grains; the smaller η the stronger is this influence. Whereas the physical meaning of ξ and η are obvious this is according to the author’s experience rather less so for the parameter β, also called the Le Claire parameter.Itis a measure of the extent to which grain-boundary diffusion is enhanced relative to lattice diffusion. Loosely speaking, one can consider β as the ratio of the transport capacity inside the grain-boundary slab, c gb D gb δ, to the transport capacity along the grain-boundary fringe, cD √ Dt, which has a width √ Dt. As we shall see below, diffusion profiles in bi- or polycrystals usually consist of a near-surface part dominated by lattice diffusion and a deep pen- etrating grain-boundary tail. Grain-boundary tails of the concentration field tend to level out as β decreases. Then, it becomes more difficult to reveal the influence of enhanced diffusion along grain boundaries in experiments. A question in this context is, what are the optimum conditions for the deter- mination of grain-boundary diffusivities? The quantity β is relevant for this question. This can be seen from Fig. 32.8, in which isoconcentration contours are plotted for various values of β. The dotted line corresponds to the limit- ing case, D gb = D, for which preferential grain-boundary diffusion is absent. The isoconcentration contours illustrate that the penetration of the diffuser along the grain boundary is much greater than anywhere else in the crystal. The larger the value of β, the more pronounced is the lateral diffusion fringe along the grain boundary. For an accurate determination of D gb from section- ing experiments (see below) β must be at least 10. The annealing conditions must be chosen accordingly. The solution for diffusion along an isolated grain-boundary slab embedded in a crystal can be written as follows: c(ξ,η,β)=c 1 (η)+c 2 (ξ,η,β) (32.10) in the grains and c(η)=c gb (η) (32.11) 562 32 Grain-Boundary Diffusion Fig. 32.8. Isoconcentration contours for various values of the Le Claire parameter β inside the boundary. In Eq. (32.10) the first term represents in-diffusion into the grains from the external source. The second term represents the leaking- out contribution from the grain boundary. The direct grain-boundary con- tribution, c gb , can be neglected when √ Dt  δ; studies of the direct grain- boundary diffusion require √ Dt < δ. These distinctions are also related to the kinetic regimes B and C of diffusion in polycrystals, discussed later in this chapter. Constant Source Solution: Let us at first consider the case of a constant source (also called infinite or inexhaustible source), with the diffuser concen- tration kept constant at the surface and zero everywhere inside the sample at the beginning. The initial and boundary conditions are: c(y, 0,t)=c 0 for t>0 , c(y, z, 0) = 0 for z>0 , (32.12) c(y, ∞, 0) = 0 . c o is the concentration of the diffuser at the surface in the source. An approximate solution of the diffusion problem formulated in Eqs. (32.6), (32.7), and (32.12) was given already by Fisher [7]. An exact solution has been worked out three years later by Whipple [27] using the Fourier-Laplace transformation method (see Chap. 3). We shall not go through the long and rather tedious mathematical exercise of deriving it. A transparent derivation of this solution can be found, e.g., in a textbook by Adda and Philib- ert [28]. The first term of Eq. (32.10) is a complementary error function c 1 = c 0 erfc(η/2) (32.13) 32.3 Diffusion along an Isolated Boundary (Fisher Model) 563 and represents direct in-diffusion into the grains from the inexhaustible source. The second term in Eq. (32.10) represents the leakage contribution from the grain boundary into the grains. It is given by c 2 (ξ,η,β)= c 0 η 2 √ π ∆  1 exp(−η 2 /4σ) σ 3/2 erfc  1 2  ∆ − 1 ∆ −σ  1/2  ξ + σ −1 β   dσ, (32.14) where σ is an integration variable. Note that the time variable is included in η andalsointheLeClaireparameterβ. At a fixed temperature β ∝ 1/ √ t, i.e. β decreases with increasing time (see Eqs. 32.8). Instantaneous Source (or Thin-Film) Solution: For an instantaneous source initial and boundary conditions are expressed by: c(y, z, 0) = Mδ(z), c(y, z, 0) = 0 for z>0, c(y, ∞,t)=0, ∂c(y,z, t) ∂z | z=0 =0. (32.15) δ(z) is the Dirac delta function and M the amount of diffuser deposited per unit area. This surface condition entails that the initial layer is completely consumed during the diffusion experiment. An exact solution of the diffusion problem formulated in Eqs. (32.6), (32.7), and (32.15) has been worked out by Suzuoka [29, 30], using the method of Fourier-Laplace transforms (see Chap. 3). The first term in Eq. (32.10) is c 1 (η)= M √ πDt exp  − η 2 4  . (32.16) It describes lattice in-diffusion into the grains from a thin-film source. The second term in Eq. (32.10) represents the leakage contribution from the grain boundary: c 2 (ξ,η,β)= M √ πDt ∆  1  η 2 4σ − 1 2  exp(−η 2 /4σ) σ 3/2 erfc  1 2  ∆ −1 ∆ − σ  1/2  ξ + σ −1 β   dσ. (32.17) A comparison between the constant source solution, Eq. (32.14), and the instantaneous source solution, Eq. (32.17), shows that the latter can be ob- tained from Eq. (32.14) by a transformation through the operator − √ Dt ∂/∂η 564 32 Grain-Boundary Diffusion Fig. 32.9. Concentration contours for constant source (left) and a thin-film source solutions (right) for an arbitrary value of β = 50 according to Suzuoka [30] and replacing c 0 by M . Furthermore, in the case of the instantaneous source solution it can be shown that ∞  0 c 2 (ξ,η,β)dη =0. (32.18) A consequence of this equation is that the total amount of diffuser is given by the volume diffusion term c 1 (η), thus establishing that the total amount M of diffuser is conserved. Figure 32.9 shows a comparison between the two types of diffusion sources for an arbitrarily chosen value of β = 50. For the thin-layer source the grain- boundary term c 2 is negative near the surface, indicating that in the near- surface region the crystal is supplying diffusing material to the grain bound- ary. The reason is that the source concentration decreases much more rapidly at the grain boundary than anywhere else. Thus, in the near-surface region the grain boundary behaves as a ‘sink for the diffuser’. Beyond a certain depth the grain-boundary behaviour changes to that of a ‘source for the dif- fuser’, since then the direct volume diffusion from the source is negligible. In contrast, for an inexhaustible source the contribution c 2 is always pos- itive. This implies that the grain boundary behaves as source of diffuser, irrespective of whether the near-surface region or the deeper regions are con- sidered. Average Concentrations in Thin Layers: Average concentrations are of prime interest for the analysis of grain-boundary diffusion experiments, which 32.3 Diffusion along an Isolated Boundary (Fisher Model) 565 are carried out by the radiotracer technique (see Chap. 13). Let us therefore discuss an expression for the average concentration in a thin layer, ¯c,atsome depth z (Fig. 32.7). The total amount in a thin section between z − ∆z/2 and z +∆z/2 and parallel to the free surface is given by the integral ¯c = 1 L x L∆z +L x /2  −L x /2 +L/2  −L/2 z+∆z/2  z−∆z/2 [c(y,z, t)+c gb (y,z, t)] dxdydz (32.19) L x and L are the dimensions of the bicrystal along the x− and y−axes, re- spectively. The quantity L x L∆z is the section volume. For sake of simplicity, let us assume that the grain boundary lies in the center of the bicrystal and that the section is so thin that the concentration along the z−axis remains constant within a section. Then, the average concentration ¯c is obtained by ¯c = 1 L +L/2  −L/2 [c(y,z, t)+c gb (y,z, t)] dy. (32.20) Furthermore, since c is an even function of y and c gb practically constant within the boundary, we obtain ¯c(z, t)= δ L ¯c gb (z,t)+ 2 L L/2  δ/2 c(y, z, t)dy. (32.21) Let us for the moment neglect the amount of diffuser, c gb , inside the bound- ary 3 . Then, we have ¯c(z, t)= 2 L L/2  δ/2 c(y, z, t)dy. (32.22) We know from Eq. (32.10) that the concentration field has two contributions, where c 1 represents bulk diffusion and c 2 is the grain-boundary leakage contri- bution given either by the thin-film solution (32.17) or by the constant-source solution (32.14). Since c 1 is constant in the xy-plane, the bulk diffusion con- tribution to the average concentration equals c 1 and we have ¯c(z, t)=c 1 (z,t)+ 2 L L/2  δ/2 c 2 (y,z, t)dy. (32.23) 3 The same assumption is made in the next section for to type B kinetics in polycrystals. In type C kinetics the direct grain-boundary contributions is dom- inating. [...]... diagrams like Fig 32 .14 as D (−∂¯/∂z 6/5 )−5 /3 c sDgb δ = 1 .32 2 (32 .42) t for constant-source conditions and sDgb δ = 1 .30 8 D (−∂¯/∂z 6/5 )−5 /3 c t (32 . 43) for instantaneous-source conditions If the lattice diffusion coefficient D is known from independent measurements, the triple product can be determined In the case of self-diffusion the product Dgb δ is obtained Equations (32 .42) and (32 . 43) require that... condition Eq (32 .4) simply expresses the continuity of concentration across the grain/grain-boundary interfaces This assumption must be modified for solute atoms because they can segregate into the boundary Accoording to Gibbs [31 ] segregation can be taken into account by introducing the segregation factor s The matching condition then reads sc(±δ/2, z, t) = cgb (±δ/2, z, t) (32 .33 ) Equation (32 .33 ) rests... self-diffusion, Hart [39 ] proposed an effective diffusivity for dislocated crystals Modified for diffusion in polycrystals an approximate expression for the effective diffusivity is given by 32 .4 Diffusion Kinetics in Polycrystals 569 Fig 32 .10 Illustration of the type A, B, and C diffusion regimes in a polycrystal according to Harrisons classification [38 ] Def f = gDgb + (1 − g)D (32 .35 ) In Eq (32 .35 ) the quantity... Then, Eq (32 .37 ) reduces to Def f = sgDgb + (1 − sg)D (32 .39 ) This equation was suggested already in 1960 by Mortlock [40] For conventional polycrystals, say with d > 50 µm, we have δ/d ≈ 10−5 Then, even for large s-values sg 1 is fulfilled and Eq (32 .39 ) simplifies further: Def f ≈ sgDgb + D (32 .40) For a more detailed discussion of effective diffusion especially in nanomaterials we refer to Chap 34 and... is described by c2 of Sect 32 .3 and is con¯ sidered in what follows As already mentioned, one is usually interested in the case ∆ ≡ Dgb /D 1 with β remaining finite (see Eqs 32 .8 and 32 .9) The variations of c2 com¯ puted from Eqs (32 .26) and (32 .28) for a value of β = 100 (with ∆ = 2 × 106 √ and Dt =10 µm) are displayed in Fig 32 .12 For the instantaneous source the negative part of c2 near the source... ∆ η2 −2 σ M c2 (η, β) = √ ¯ L π exp(−η 2 /4σ) σ 3/ 2 1 ∆−σ ∆−1 1/2 exp(−Y 2 ) √ − Y erfcY dσ , π where Y = σ−1 2β (32 .26) 1/2 ∆−σ ∆−1 (32 .27) Similarly, using the constant-source solution of Eq (32 .14) gives √ c0 Dt 2η √ c2 (η, β) = ¯ L π ∆ η2 −2 σ exp(−η 2 /4σ) σ 3/ 2 1 ∆−σ ∆−1 1/2 exp(−Y 2 ) √ − Y erfcY dσ (32 .28) π The factor 1/L in Eqs (32 .26) and (32 .28) is important For a bicrystal with dimensions... Chap.29) References 1 2 3 4 5 6 7 8 R.S Barnes, Nature 166, 1 032 (1950) A.D Le Claire Philos Mag 42, 468 (1951) R.E Hoffman, D Turnbull, J Appl Phys 22, 634 – 639 (1951) W.C Roberts-Austen, Phil Trans Roy Soc A 187 (1896) 38 3–414 J Groh, G von Hevesy, Ann Physik 63, 85–92 (1920) J Groh, G von Hevesy, Ann Physik 65, 216–222 (1921) J.C Fisher, J Appl Phys 22, 74 (1951) I Kaur, Y Mishin, W Gust, Fundamentals of... (32 .29) 0 In Eq (32 .29) λ represents the grain-boundary length per unit area on the sample surface exposed to the diffuser 32 .3 Diffusion along an Isolated Boundary (Fisher Model) 567 For a bicrystal with a width L normal to the grain-boundary, Λ and λ are given by 1 L and λ = (32 .30 ) Λ= √ L 2 Dt For an array of uniformly spaced grain-boundaries with spacing ds we have ds Λ= √ 2 Dt and λ = 1 , ds (32 .31 )... c0 Dt/L for a constant source (i) The parameter β defined by Le Claire [36 ] β= sDgb δ √ = α∆ 2D Dt (32 .44) must be large enough (in practice β > 10) In the case of self-diffusion, we have s = 1 and β is given by Eq (32 .8) (ii) The parameter sδ α= √ (32 .45) 2 Dt must be small enough, in practice α < 0.1 Equations (32 .42) and (32 . 43) are relations that are often used in the analysis of grain-boundary diffusion... Claire parameter β is smaller than 104 the numerical constants in Eq (32 .42) and (32 . 43) are slightly different An analysis of the various solutions has been presented by Le Claire [36 ] A detailed practice-oriented discussion for various β ranges can be found in the textbook of Kaur, Mishin, and Gust [8] We note that Eqs (32 .42) and (32 . 43) only provide the triple-product sDgb δ or Dgb δ for self-diffusion . slopes of diagrams like Fig. 32 .14 as sD gb δ =1 .32 2  D t (−∂¯c/∂z 6/5 ) −5 /3 (32 .42) for constant-source conditions and sD gb δ =1 .30 8  D t (−∂¯c/∂z 6/5 ) −5 /3 (32 . 43) for instantaneous-source. Accoording to Gibbs [31 ] segregation can be taken into account by introducing the segregation factor s. The matching condition then reads sc(±δ/2,z,t)=c gb (±δ/2,z,t) (32 .33 ) Equation (32 .33 ) rests on. Philib- ert [28]. The first term of Eq. (32 .10) is a complementary error function c 1 = c 0 erfc(η/2) (32 . 13) 32 .3 Diffusion along an Isolated Boundary (Fisher Model) 5 63 and represents direct in-diffusion