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The role of segregation in diffusion-induced grain boundary migration
Acta mater. 49 (2001) 1–11 www.elsevier.com/locate/actamat
THE ROLE OF SEGREGATION IN DIFFUSION-INDUCED GRAIN BOUNDARY MIGRATION A. BROKMAN1*, A. H. KING2 and A. J. VILENKIN1 1
Department of Applied Physics, Graduate School of Applied Science, Hebrew University, 91904 Jerusalem, Israel and 2School of Materials Engineering, Purdue University, West Lafayette, IN 479071289, USA ( Received 15 August 2000; received in revised form 6 September 2000; accepted 11 September 2000 )
Abstract—The problem of grain boundary motion in the diffusion field of a solute is formulated for the case of infinitely fast diffusion along a straight boundary. The steady state solution suggests that (de)alloying occurs by two different modes, namely: the solute diffusion through the stationary boundary to the bulk, or by diffusion-induced grain boundary migration (DIGM). The transition from one mode to another depends on the grain boundary segregation coefficient. The result enables an assessment of the relative importance of different possible driving forces. When the equilibrium concentrations of the bulk solute with the external gas is low, the entropy of mixing is the leading driving force. DIGM does not occur in isotope solution because the solute atom does not segregate to the boundary. Based on this theory, we construct the phase diagram in the plane of the (gas/bulk) equilibrium concentration vs the segregation coefficient, representing the transition from DIGM to alloying via stationary boundaries. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: DIGM; Segregation; Diffusion; Boundary migration
1. INTRODUCTION
Diffusion-induced grain boundary migration (DIGM) was first observed by Rhines and Montgomery in 1938 [1], but has become a subject of curiosity since the late seventies. Since then, many works have been dedicated to the identification of the mechanism, primarily in order to identify the underlying driving force and understand the physical conditions that lead to DIGM. This effort has been reviewed and overviewed extensively [2–4]. The boundary motion may be driven by: entropy of mixing, releasing heat of solution, local curvature or cohesion strain energy. In spite of the long history of DIGM, the role of each of these driving forces remains a subject of debate. For instance, the driving force due to mixing entropy should lead the process since the chemical potential of the solute atom varies logarithmically in the dilute solution (in comparison with a power law for the cohesive term). However, Balluffi and Cahn [5] argued that since DIGM is not expected when the diffusing element is an isotope of the parent material, the entropy of mixing is of no significance. The heat of mixing and the dependence of the boundary energy on the solute atom concentration are sometimes
* To whom all correspondence should be addressed. E-mail address: avnerb얀vms.huji.ac.il (A. Brokman)
excluded since both alloying and dealloying occurs by DIGM in the same binary systems. As a result of the uncertainty in the driving force, the underlying physical conditions are still unresolved. As DIGM represents a kinetic instability, we adopt an unusual strategy that first formulates the kinetic problem for an arbitrary driving force. From the solution of this problem we are able to discuss the relevant interactions that drive DIGM and identify the underlying physical conditions. For instance, we show that the hypothetical isotope experiment will not exhibit DIGM, simply because the isotope does not segregate to the grain boundary, and the entropy of mixing drives DIGM at low pressure of the ambient gas. The present work involves a lengthy and cumbersome algebraic manipulation that may screen the rather simple logic of the calculation. Therefore, we introduce a brief account of the physical rationale of this work. We investigate DIGM in the simplest possible system, which consists of a bi-crystal in a fixed partial pressure of solute atom ambient gas. The bulk solute changes its concentration in order to equilibrate with the chemical potential of the gas, thus causing the alloying or dealloying of the bulk. As the boundary diffusivity is larger than the bulk diffusivity, the first stage of alloying (dealloying) occurs by diffusion through the grain boundary (rather than direct diffusion from the surface to the bulk), and may happen
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 3 2 0 - 7
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BROKMAN et al.: SEGREGATION IN DIGM
through two routes: (a) the solute atom diffuses into (or out of) the stationary grain boundary, and thereafter leaks by diffusion into (or out of) the bulk; and (b) the solute diffuses through a moving grain boundary leaving an alloyed (dealloyed) bulk behind (DIGM). The primary purpose of the present work is to identify the physical conditions, which discriminate between these two modes. The wavy shape of the boundary after DIGM, with islands of alloyed (dealloyed) materials on both sides, suggests that the process of DIGM is initiated by some fluctuation in the boundary shape which becomes unstable against DIGM. Therefore, the boundary segment migrates in the direction of the initial perturbation. This work does not account for the initial growth of the unstable boundary segment, but rather assumes that the boundary moves and identifies the conditions for its motion in a steady state manner. This strategy suggests, without proof, that the steady state motion is an attractor to the general problem of the grain boundary motion, and when a steady state is impossible the boundary is stable against fluctuations. As the steady state is independent of the boundary shape, we consider the simple case of a one-dimensional boundary motion (see also Section 4). With the above assumptions, the problem of DIGM becomes a mathematical problem that governs the motion of the boundary in the field of the diffusing solute atom. It is shown that for a driving force of any origin, the boundary velocity is proportional to the concentration “jump” across the grain boundary plane. The concentration jump with its associated chemical potential drop across the boundary is a necessary condition for solute flux through the boundary. The boundary does not reach a local equilibrium with the bulk as long as the solute fluxes on both of its sides are not balanced to yield zero flux across the boundary. In the case of a steady state motion, the flux, the boundary velocity and the concentration jump are found in this work by solving the self-consistent problem that combines the diffusion equation of the solute atom in the moving coordinate system and (as a necessary boundary condition) the solute conservation in the bi-crystal system. The conditions for DIGM are identified with the solutions that produce a non-zero boundary velocity, and are found to depend on the bulk diffusion coefficient, the segregation coefficient, the grain boundary mobility and the generalized driving force.
2.1. Energetics Consider the two-dimensional bi-crystal consisting of the semi-infinite “Crystal Left” at x⬍X(t) and “Crystal Right” at x>X(t). The two crystals are separated by a straight grain boundary at X(t). The free energy of this system is:
冕
冕 ⬁
X⫺0
E⫽
f(c) dx ⫹
⫺⬁
f(c) dx,
(1)
X⫹0
where f is the free energy per unit of volume depending on the solute atom concentration c. This dependence may consist of an entropy mixing term, elastic cohesive energy, etc. At this point we choose not to limit ourselves to a specific model, and we refer to the relevant interactions in Section 4 below. The driving force for grain boundary motion is: F⫽⫺
dE ⫽ f[c(X ⫹ 0)]⫺f[c(X⫺0)], dX
(2)
and the grain boundary velocity is given within linear transition theory, i.e. v⫽
dy ⫽ mF ⫽ m{f[c(X ⫹ 0)]⫺f[c(X⫺0)]}, dt (3)
where m is the boundary mobility. Suppose that the solute concentration in the grain boundary is in equilibrium with the gas, and c0 is the bulk concentration in equilibrium with the gram boundary, then, in the limit of small deviations from local equilibrium, f[c(X ⫹ 0)]⫺f[c(X⫺0)]⬇
|
df [c(X ⫹ 0) dc c0
(4)
⫺c(X⫺0)].
Thus equation (3) reads: v ⫽ B(c⫺⫺c⫹ ),
(5)
where c ⫹ ⫽ c(X ⫹ 0) and c⫺ ⫽ c(X⫺0) For the sake of simplicity, we assume hereafter that the kinetic coefficient 2. FORMULATION
In the first two parts of this section, we derive some basic concepts which appear in the literature in different contexts. Then, we employ these concepts to the modeling of a moving boundary in the field of the diffusing solute atom.
|
df B ⫽ ⫺m dc c0
is positive (df/dc is to be negative, otherwise the con-
BROKMAN et al.: SEGREGATION IN DIGM
centration is large and the diffusion equation below becomes non-linear). Equation (5) implies that the boundary velocity depends on the concentration jump across the boundary. A similar result was suggested by Brener and Temkin [6].
3
1. v⫽0: in this limit (9) becomes D c1 exp{y0/T}⫺c2 exp{yL/T} L exp{y0/T} ⫹ exp{yL/T} ⫽ h(c1⫺kc2),
J0 ⫽ 2
(10)
2.2. Grain boundary/bulk exchange of solute atom Here we employ some known ideas concerning the diffusion over a potential barrier [7] in order to obtain the flux through a moving potential barrier. The chemical potential of the non-interacting solute atoms (low concentration) is m ⫽ T ln(c) ⫹ y(x), where T is the temperature in Boltzman units, and y varies in space and contains a localized barrier. In the case of a moving barrier, the impurity concentration satisfies the equation ∂tc ⫽ ⫺∂xJ, and in the moving coordinate system (the origin moves with the barrier at a velocity v):
冉
J ⫽ ⫺D⬘ ∂xc ⫹ c
冒冊
dy T ⫺vc dx
(6)
where k ⫽ exp{[yL⫺y0]/T}, and h⫽ 2(D⬘/L)(1/1 ⫹ k) is a kinetic coefficient. We use h as an input parameter, for which we rewrite b: b⫽
vL v 1 ⫽2 ; D⬘ h1⫹k
(11)
J ⫽ ⫺vc2,
(12)
2. v>0, bÀ1 yields:
and the convective part determines the flux; finally: 3. v⬍0, b¿⫺1 yields: J ⫽ ⫺vc1
where D⬘ is a kinetic constant (effective diffusion coefficient) assumed to be independent of x. If the gradient of the potential barrier is localized into a small length L, then J is constant over L, and (6) becomes an ordinary differential equation for c. The flux obtained from this equation with the boundary conditions {c(0) ⫽ c1, c(L) ⫽ c2} is: D⬘
J ⫽ ⫺L
冕 再 exp
冎
再 冋 册
y(x) vx ⫹ dx T D⬘
冋
0
⫺c2 exp
y(0) c1 exp T
(7)
册冎
y(L) vL . ⫹ T D⬘
We simplify (7) by using the following potential model: y(x) ⫽
再
y0, 0⬍x⬍L/2
yL, L/2⬍x⬍L
(8)
for which (7) reads: J⫽v
c1 exp{⫺b/2 ⫹ y0/T}⫺c2 exp{b/2 ⫹ yL/T} , [1⫺exp(⫺b/2)] exp{y0/T} ⫹ [exp(b/2)⫺1] exp{yL/T}
(9)
where b ⫽ vL/D⬘ is the dimensionless velocity of the potential barrier. We can use equation (9) as a boundary condition that enables the formulation of the moving boundary problem with a rather complicated solution. Let us consider the three simple limiting cases:
(13)
The super-positioning of two moving potential barriers with opposite signs is our model for the grain boundary in the next section. 2.3. Moving boundary in the diffusion field Consider a two-dimensionally infinite, thin bi-crystal annealed in an atmosphere of constant chemical potential of the diffusing atom (fixed gas pressure). The impurity atoms condense (or evaporate) on the crystal surface and diffuse into (or out from) the bulk by means of grain boundary and bulk diffusion. At the annealing temperature, the diffusion coefficient of solute along the grain boundary is much larger than the bulk one: DgbÀD. During the time interval h2/Dgb¿t¿h2/D (h being half the bi-crystal thickness), the bulk diffusion across the film and the impurity exchange through the surface are negligible. In this interval of time we simplify the problem by assuming that the grain boundary diffusion coefficient is infinitely large and the boundary solute concentration remains in equilibrium with the ambient gas. When the boundary moves in steady state it (de)alloys the swept bulk to a constant concentration. In the relevant interval of time suggested above, the bulk concentration is not necessarily in equilibrium with the gas, and only when bulk diffusion becomes effective, may the bi-crystal reach full equilibrium. The boundary motion is possible if and only if a net solute flux transferred through the boundary. A necessary condition for a finite flux through the boundary is the non-vanishing difference of chemical potential across the boundary and an associated solute concentration “jump”. Although diffusion across the boundary exists and the width of the boundary is
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BROKMAN et al.: SEGREGATION IN DIGM
small, the concentration jump is maintained for the time scale mentioned above. This phenomenon is prevalent in a wide range of problems concerning boundaries in external fields. The simplest example that demonstrates this effect is the random walk on a one-dimensional lattice with a point defect. The hopping probability into each of the lattice sites is equal, but different for the defect site. It is clear that as long as the external flux in the system is finite, the chemical potential is discontinuous at the defect site. The analogy to the temperature discontinuity across the moving interface, when heat is transferred in external temperature gradient, is also called for (the formal description of heat transfer in the temperature gradient field is similar to the description of solute diffusion in the field of external flux). In the case of DIGM, the unbalanced solute flux into the boundary from both sides and the associated concentration jump, are maintained for any finite diffusivity across the boundary (even very large), as long as the boundary moves. In addition to the insensitivity of the concentration jump to the solute diffusion across the boundary, this phenomenon occurs in the bi-crystal regardless of the actual value of the diffusivity along the grain boundary. In particular, the concentration jump is developed when the diffusion coefficient along the boundary is much larger than that of the bulk, and by assuming infinite boundary diffusivity we do not lose the generality of the phenomenological description. Moreover, finite diffusivity along the boundary would lead to a larger concentration jump (together with enhanced boundary velocity and a difficult mathematical problem, beyond our analytical capability). To conclude, the combination of finite diffusivity across the boundary that conserves the concentration jump, and infinite diffusivity along the boundary that is presented for the sake of mathematical convenience, sets a representative framework to describe DIGM. Suppose that X(t) is the position of the grain boundary. Diffusion away from the boundary into the bulk is governed by the diffusion equation ∂tc ⫽ D∂xxc for both grams separately (i.e. the equation should be solved in the intervals ⫺⬁⬍x⬍X(t) and X(t)⬍x⬍⬁). In the moving coordinate system (x→x⫺vt), the steady state solution of the diffusion equation is: c ⫽ a exp(⫺vx/D) ⫹ b,
where J⫺ is the flux from the left grain into the grain boundary and J⫹ is the flux from the “grain boundary phase” into the right grain. In order to estimate these fluxes, let us consider Fig. 1. In this figure, the boundary is described by a potential well consisting of two opposite barriers of the type considered in the previous section. In accordance with this figure, we substitute into (9) the potentials: y0 ⫽ yg (bulk potential), yL ⫽ ygb (boundary “phase” potential), resulting in: J⫺ ⫽ v
c⫺⫺c0 exp(b) . exp(b/2)⫺1 ⫹ [exp(b)⫺exp(b/2)]/k (17)
where k ⫽ exp{[yg⫺ygb]/T} is the segregation coefficient. In the same way, the flux from the grain boundary into the right grain is (substitute into (9) y0 ⫽ ygb, yL ⫽ yg): J⫹ ⫽ v
c0⫺c⫹ exp(b) . exp(b)⫺exp(b/2) ⫹ [exp(b/2)⫺1]/k (18)
In the case of DIGM under fixed external gas pressure (given c0), equations (14)–(18) yield two algebraic equations with three unknown parameters (v, c⫹ and c⫺). Equation (5) provides an additional relation between these variables, and makes the DIGM problem solvable, as demonstrated in the next section. 3. SOLUTIONS
3.1. Alloying by DIGM Consider the process of alloying by DIGM as presented schematically (in the moving coordinate system) in Fig. 2. Here, the boundary potential well is reduced to a limit of small (inter-atomic spacing) thickness, and the boundary concentration z0, is in equilibrium with the gas (a result of the assumption of infinite grain boundary diffusivity). The grain boundary migrates with a velocity v, leaving a solute
(14)
where a and b are integration constants, and v ⫽ dX/dt ⫽ const. is the velocity of the coordinate system, given by equation (5). Solute atom conservation at the boundary implies: x ⫽ ⫺0: J⫺ ⫽ ⫺D∂xc⫺vc,
(15)
x ⫽ ⫹ 0: J ⫹ ⫽ ⫺D∂xc⫺vc,
(16)
Fig. 1. A potential well model for the grain boundary that combines two potential barriers through which the solute atom diffuses with flux J⫹, J⫺ (see text).
BROKMAN et al.: SEGREGATION IN DIGM
5
Substituting equations (21) and (22) into (5) gives the algebraic equation for b, namely: lb ⫽ f(b),
(23)
where f(b) ⫽
1 (23a) exp(⫺b/2) ⫹ [1⫺exp(⫺b/2)]/k ⫺exp(⫺b)
Here l ⫽ [h(1 ⫹ k)]/2c0B and B is defined in (5). Figure 3 plots the function f(b) for different values of k [Fig. 3(a)] together with a schematic graphical solution of (23). For k>1 the function f(b) increases monotonically from f(0) ⫽ 0 to f(⬁) ⫽ k, as seen in Fig. 3. The first derivative at the origin is:
Fig. 2. Concentration profile for the case of alloying by DIGM: in the moving coordinate system, the concentration at x ⫽ 0 (grain boundary) is in equilibrium with the gas phase and equals z0. The bulk equilibrium concentration is c0(z0 ⫽ kc0, where k is the segregation coefficient). The boundary migrates to the right and leaves an enriched solution behind (x⬍0) with a concentration which depends on its velocity.
f0⬘ ⫽
|
df 3k⫺1 ⫽ . db b ⫽ 0 2k
We can see (Fig. 3) that when l>f0⬘, i.e.
concentration c⫺ behind and a decaying profile of concentration in front, starting with concentration c⫹ near the boundary, and asymptotically reaches the value of the initial concentration, c(⬁) ⫽ 0. The later condition is used as a boundary condition for the bulk diffusion equation and from (14) we have: x>0: c(x) ⫽ c ⫹ exp(⫺vx/D).
(19)
Substituting this result into (16) yields: dc J ⫹ ⫽ ⫺D ⫺vc ⫽ ⫺D(⫺c ⫹ v/D) dx ⫺vc ⫹ ⫽ 0,
(20)
and the steady state flux out of the boundary vanishes. From (18): c ⫹ ⫽ c0 exp(⫺b).
(21)
For x⬍0, equation (14) reads, c(x) ⫽ c⫺ ⫽ const., and (15) yields J⫺ ⫽ ⫺vc⫺, and from (17): 1 c⫺ ⫽ c0 . exp(⫺b/2) ⫹ [1⫺exp(⫺b/2)]/k
(22)
Fig. 3. Graphical presentation of equation (23) (alloying by DIGM): (a) the dependence of f(b)/b on the dimensionless velocity b for different values of k; and (b) graphical solutions of the equation: the straight lines correspond to lb. The intersection of these lines with the curve representing f(b) yields a possible steady state solution; line “1” is the case when equation (23) has the solution v ⫽ 0; line “2” represents multiple steady state solutions; and line “3” represents a single possible (fast) steady state solution at small values of l.
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BROKMAN et al.: SEGREGATION IN DIGM
h>2c0B
3k⫺1 , 2k(1 ⫹ k)
3.2. Dealloying
equation (23) has no solution with non-zero v, and the grain boundary remains stationary. The function f(b) is monotonic, and possesses two inflections (∂2f(b, k)/∂b2 ⫽ 0): one inflection point is too close to the origin to be identified in the scale of Fig. 3. A second inflection is seen in the figure, and becomes pronounced at large values of k. As a result [consult Fig. 3(b)], depending on the specific values of k and l, the solutions to equation (23) consist of two possible classes, namely: (a) a stationary solution; and (b) a steady state solution with one, or two possible nonvanishing velocities. The nature of the transition between these solutions is considered in detail in the discussion below. Here we consider mathematically the two extreme cases:
Figure 4 plots schematically the concentration profile of this case in the moving coordinate system. The initial homogenous concentration is c(⬁) ⫽ c⬁ The chemical potential of the grain boundary is smaller than the initial chemical potential of the bulk solution, and c0⬍c⬁. In this case, (14) becomes: x>0: c(x) ⫽ (c ⫹ ⫺c⬁) exp(⫺vx/D) ⫹ c⬁,
and the flux out of the boundary is found from (16):
冉 冊
dc v J ⫹ ⫺D ⫺vc ⫽ ⫺D(c ⫹ ⫺c⬁) ⫺ ⫺v(c ⫹ ⫺c⬁ dx D ⫹ c⬁) ⫽ ⫺vc⬁.
From (18), we have: 1. Fast alloying (bÀ1): we write the concentrations [equations (21) and (22)], c ⫹ ⫽ 0, and c⫺ ⫽ kc. The latter condition is indicated in Fig. 2 by z0 (z0 ⫽ kc0 is the boundary concentration in equilibrium with the gas phase). Substitution of the latter relations into (5) yields the boundary velocity: v ⫽ B(c⫺⫺c ⫹ ) ⫽ Bkc0, (k>1). 2. Slow alloying (b¿1): an analytical approximation can be realized when f ⬘(0) ⫽ df/db|b ⫽ 0>l, and f ⬘(0)⫺l¿1. In this case, we can expand f(b) near the origin: f(b) ⫽ bf0⬘ ⫹
b2 f ⬙. 2 0
(24)
再
⫺exp(b/2) ⫹
2lk2⫺3k2 ⫹ k , ⫺3k2⫺3k ⫹ 2
exp(b/2)⫺1 k
册冎
(27)
.
Similar to the case of alloying, the solution of (14) for x⬍0 is c⫺ ⫽ const., J⫺ ⫽ ⫺vc⫺, and equation (22) holds for the case of dealloying: 1 c⫺ ⫽ c0 exp(⫺b/2) ⫹ [1⫺exp(⫺b/2)]/k
(28)
Substituting the latter equation and (27) into (5) leads to the following transcendental equation for b: lb ⫽ F(b),
As f0⬙ ⫽ ⫺(3 ⫹ 3/k⫺2/k2)/4, the approximation (24) gives a solution to (23), namely: b ⫽ 2(l⫺f0⬘)/f0⬙ ⫽ 4
冋
c ⫹ ⫽ exp(⫺b) c0 ⫹ c⬁ exp(b)
(29)
where:
In the limit kÀ1 (l ⫽ 3/2⫺e2, e¿1): b ⫽ 4(3⫺ 2λ)/3. Now in the limit b¿1, we obtain c⫹, c⫺ from (21) and (22), namely: c ⫹ ⫽ c0(1⫺b),
c⫺ ⫽
c0 . 1⫺(b/2)(1⫺1/k)
(25)
(26)
Hence, we have found the concentration jump across the boundary, and the associated boundary velocity of the steady state DIGM.
Fig. 4. The same as Fig. 2, but for the case of dealloying by DIGM. The initial bulk concentration c⬁ is diluted by the boundary motion. The ratio c⬁/c0 is larger than one and indicated by “s” in the text.
BROKMAN et al.: SEGREGATION IN DIGM
F(b) ⫽
冉 冊
k exp(b/2) s ⫹ ⫺1 (29a) k⫺1 ⫹ exp(b/2) k k⫺1 exp(⫺b) ⫹ s exp(⫺b/2)⫺s. k
Here s ⫽ c⬁/c0. F is plotted in Fig. 5(a). It is easy to check that F(0) ⫽ 0, and F(⬁) ⫽ k⫺s. Again, we discuss the case of k>1. For small l, two solutions are possible for a moving grain boundary. Analysis is possible in the limits of slow and fast dealloying: 1. Fast dealloying (bÀ1): in this case, (27) yields c ⫹ ⬇c⬁, and (28) becomes c⫺ ⫽ kc0, so that from (5): v ⫽ B(kc0⫺c⬁),
7
2. Slow dealloying (b¿1): from Fig. 5(a) it is clear that slow solution is possible only when F0⬘ ⫽ dF/db|b ⫽ 0 ⫽ l ⫹ e2; (e¿1); , and F0⬙⬍0. In this case, the approximation of the solution of (29) is: b ⫽ 2(l⫺F0⬘)/F0⬙ and (29a) implies b ⫽ 4(2λ ⫹ s⫺3)/s⫺3 (it is easy to see that when kÀ1, s⬍3). In the limit b¿1 we expand the concentrations near the grain boundary in (27): or c ⫹ ⫽ c0(1⫺b) ⫹ c⬁b(k ⫹ 1/2k) ⫹ O(b2), when kÀ1: c ⫹ ⫽ c0 ⫹ b(c⬁/2⫺c0). In the same manner, c⫺ ⫽
冋 冉 冊册
c0 b 1 ⫽ c0 1⫺ ⫺1 1 ⫹ (b/2)[(1/k)⫺1] 2 k ⫹ O(b2)
and l¿k⫺s. and for kÀ1: c⫺ ⫽ c0(1 ⫹ b/2).
Hence, dealloying by DIGM is evident in this example. Numerical results representing the possible DIGM solutions as functions of k, b and l are plotted in Fig. 5(b). The transition from a stationary to a possible DIGM mode of alloying is reflected in the point of minimum k (given l). Below this point a stationary solution is the only possible solution, while above it, DIGM is a possible solution to equation (29). 4. DISCUSSION
Fig. 5. (a) Graphical representation of the solutions to (29) (dealloying); the straight line “1” has no intersection with the curve f(b), and the only possible solution is the one of a stationary grain boundary (b ⫽ 0); line “2” is tangent to f(b) and represents the transition to two possible solutions indicated by the intersections of line “3”. At the transition, a single nonzero velocity solution is possible (as in the case of line “2”). Notice that the asymptotic value of f(b) depends on the initial condition. (b) Numerical solutions of (29) for the case s ⫽ 15. In accordance with the graphical solution, it is seen that for any k and l there are two possible solutions or none. The forbidden velocity zone (slow solutions) is associated with the negative values of f(b) in (a).
When a bi-crystal of a given material (l, k and c⬁ are given) is placed in a gas of the solute atom with a fixed partial pressure (given c0), diffusion occurs to equilibrate the bulk solute with the gas. The dynamics of this process involves either DIGM or diffusion through the stationary grain boundary to the bulk. We showed in Section 2 that a necessary condition for the boundary to move is the temporary development of a solute concentration jump. This condition is obtainable, depending on the segregation kinetics involved. Therefore, the transition from DIGM to stationary (de)alloying depends strongly on the segregation coefficient k. By solving the self-consistent moving boundary problem we showed in Section 3 how the steady state velocity depends on the segregation kinetics [equations (23) and (29)*]. It is the
* In general, equation (29), which describes the dealloying process (s>1), is also valid for the case of alloying (s⬍1). [The limit s ⫽ 0, leading to equation (23), is the one discussed under “alloying” in the text.]
8
BROKMAN et al.: SEGREGATION IN DIGM
purpose of this work to gain an understanding of the transition from the finite velocity in the case of DIGM to the zero velocity in the case of a stationary boundary, which is discussed in detail below. In order to do so, we should first consider the role of the different driving forces in view of the theory. 4.1. Interactions and the “isotope trap” Our analysis demonstrates that steady state DIGM may occur, depending on the magnitude of the driving force, without specifying the origin of this force. A few driving forces are possible: let us first exclude the two driving forces that do not, in themselves, explain the basic phenomena, but rather influence the local velocity of the boundary: 1. Grain boundary curvature. This force can drive the boundary motion in the case of grain growth and in the case of DIGM, may slow the local boundary velocity. As the curvature driven motion is decoupled from the solute atom diffusion, it cannot explain DIGM, but rather modifies the force due to the variation of concentration. 2. The effect of solute atoms on the boundary anisotropy. This may lead to the rotation of the boundary, but will not drive a process with long range translational migration. Our one-dimensional model does not lose itsability to explain the basic dynamics of DIGM byignoring the forces above. The driving forces thatare coupled to the solute atom diffusion and henceessential in the explanation of DIGM include: 3. The decrease in the coherency strain energy that depends on the local concentration. This driving force is frequently used to explain DIGM. For our discussion below we should refer to the fact that to the lowest order, the specific strain energy is quadratic in the concentration. 4. The variation of the bulk free energy due to the chemical mixing. For the ideal substitutional solid solution the specific free energy varies at low concentrations as Tc ln(⍀c) ⫹ O(c) (T being the temperature in Boltzman units, and ⍀ the atomic volume). When the equilibrium concentration c0 is small, the mixing entropy will dominate the driving force. However, we cannot use this conclusion before discussing the role of mixing in the context of the “isotope paradox” which excludes this term as a possible driving force for DIGM. In brief, the isotope paradox (firstly suggested in [5]) notes that the same variation of bulk free energy occurs when one mixes a pure material with an isotope (of the same material as the unalloyed bulk) or a solute (of foreign) atom that forms an ideal solution. As we do not expect DIGM to occur in the diffusing isotope system, the driving force for DIGM in the foreign atom solution must also be of no significance. Our analysis demonstrates that the boundary velocity is proportional to the jump in solute con-
centration across the boundary. This jump depends on the segregation coefficient, and when the latter equals one (no segregation) diffusion smooths out the difference of concentration across the boundary. Since there is no concentration jump in the isotope case, we do not expect the boundary to move, while it certainly can move in the case of a segregating solute atom. In conclusion, the isotope diffuses to the bulk through a stationary grain boundary as suggested, yet the variation of energy due to mixing entropy remains a valid driving force for DIGM in the case of a segregating solute. This example demonstrates that the segregation coefficient should be incorporated into the kinetics, as shown below. 4.2. When does alloying occur by DIGM? In order to demonstrate the implications of the theory, we consider the rather simple case of alloying by DIGM. [The dealloying case is a direct extension of the following problem, using (29) instead of (23). However, the complication due to the extra parameter (namely, s) makes the problem mathematically cumbersome.] With the above discussion, it is clear that when the bulk/gas equilibrium concentration (c0) is small, the variation of the bulk free energy due to mixing entropy provides a leading term in the driving force. For this condition (small equilibrium concentration), we rewrite l: l⫽⫺
D⬘ LmTc0 ln(⍀c0)
(30)
Figure 6 plots l vs the dimensionless velocity b [equation (23)] for different values of the segregation coefficient k. In accordance with the previous section, it is seen that above a critical value of l, a solution with non-zero velocity is impossible and the boundary
Fig. 6. Plot of l vs b for different values of segregation coefficient in the case of alloying. At zero velocity, all curves start at the value l ⫽ 3/2 where their second derivative is negative (not seen in the scale of the plot). Therefore, a narrow interval of l with three possible solutions (number of velocities of given l and k) is evident. In most cases, two non-zero velocity states are possible (below the curve) or a trivial (stationary) solution is obtained (above the curve). At the critical l, a single moving boundary solution is possible.
BROKMAN et al.: SEGREGATION IN DIGM
remains stationary. At the critical point (l ⫽ l∗), a single steady state solution is possible, while below this point two solutions are possible. (The prediction of slow and fast motion in the two-steady-state mode of motion may be associated with the observation of “jerky boundary motion” that has been observed in thin films during DIGM. This idea requires further study since the influence of the thermal groove may also be of importance.) We should also point out that in a small interval of l, i.e. near l ⫽ 3/2 (the value in the limit of zero velocity), three possible solutions are possible due to the existence of two inflections of f(b), discussed above. Figure 7 plots the values of k at the critical points versus the dimensionless boundary velocity. According to equation (30), DIGM occurs when the equilibrium concentration is larger than a critical concentration c0 ⫽ c∗0 given by the transcendental relation: c∗0 ln(⍀c∗0 )⬇⫺
1 , l∗x3c
(31)
where xc ⫽ (D⬘/LmT)⫺1/3 is a coherence length. To demonstrate, we evaluate this length and the critical concentration for the typical DIGM conditions by treating the above relation term by term for some fictitious experimental condition: 앫 L: the length of the solute potential drop-off of the boundary is of the order of a few inter-atomic distances; 앫 m: to our knowledge, there is no experiment which measures the grain boundary mobility m. However, the product of the mobility and the boundary tension (known as the “reduced mobility”) has been deduced from grain growth experiments and
Fig. 7. Numerical result of the value of the segregation coefficient k vs the critical dimensionless velocity (see Fig. 6).
9
varies from 10⫺4 to 10⫺8 cm2/s (depending on the material and the temperature). We use the smaller value below for the reduced mobility. 앫 xc: if the boundary energy is 1000 erg/cm2 and the diffusion coefficient across it at a temperature of 1000 K is 10⫺12 cm2/s (see comment above), the coherence length xc, is of the order of 10⫺7 cm; 앫 l∗: in practice, the segregation coefficient is kÀ1 [8], and may reach the value of 104. In this limit, the curve in Fig. 7 indicates a logarithmic growth of the boundary velocity with the segregation coefficient; 앫 c∗0 : Fig. 8 presents the critical concentration curve that separates the conditions for alloying by DIGM from those of enrichment through the stationary boundary. The numerical values of this curve were calculated from equation (31), for the above values of the different physical parameters. 4.3. One-dimensional steady state and real life In practice, both the diffusion coefficient along the boundary, Dgb, and the diffusion coefficient across the boundary, D⬘, are finite. In order to evaluate the correctness of the one-dimensional model, let us examine the solute distribution, z(y) along the boundary in the case of a steady state motion under the flux found above, but when the diffusion coefficients are finite. In this case. the steady state diffusion equation (with source/sink term) is Dgb∂yyz ⫹ L1 (J⫺⫺J ⫹ ) ⫽ 0 (L being the boundary width). Substituting (18), (19), (21) and (22). we obtain in the limit of large k (alloying case): z(0)⫺z(h) D⬘c0b b/2 ⫽ e . h2 DgbL2
(32)
Fig. 8. Plot of the critical curve c∗0 (k) that separates the possible DIGM (above the curve) from the stationary boundary (below the curve).
10
BROKMAN et al.: SEGREGATION IN DIGM
In order to extract the physical conditions that are consistent with the one-dimensional steady state solution we set a solute homogeneity criteria at the boundary, namely: (z(0)⫺z(h))/z(0)¿1. Substituting, z(0) ⫽ kc0, we obtain from (32) the criteria: D⬘h2 beb/2 ¿1 DgbL2 k
(33)
We are not aware of any measurements of diffusion across the boundary, but since the boundary evolves expansion normal to its plane, we expect that D⬘ will be smaller than the diffusion coefficient along the boundary (Dgb), most probably of the order of the bulk diffusion constant. In the unlikely case when D⬘ and Dgb are of the same order, one should bare in mind that the model might be adequate only when the specimen is very thin or the boundary moves in a slow mode [extract h and b from inequality (33)]. In this case the solution of a moving boundary may imply possible instability of the straight boundary in thick films. In contrast to our analysis, the experimental observation of DIGM does not show a steady state motion of the grain boundary. The motion of the individual boundary involves the enrichment (or dilution) of the bulk material on both sides of the boundary as different segments of the boundary migrate to different directions. Furthermore, with time, this motion slows down and the boundary velocity becomes time dependent. Therefore, a full account of the boundary motion requires the transient analysis of the threedimensional problem, which includes the driving force due to local curvature. This problem is by far more complicated than the present analysis, and should become a future challenge. Besides the remark above, which enables the prediction of DIGM as a phenomenon by means of a curvature-independent model, the steady state solution may also provide the basis for understanding the transition from a stationary to a moving boundary. We view the problem of DIGM as the unstable growth of a boundary perturbation (which reflects the wavy shape observed experimentally). As the role of the boundary curvature is nothing but slowing the boundary motion, we anticipate that the observation of a transition from a stationary to a steady state mode reflects the essential kinetic elements that explain DIGM in “real life”. Although this needs a rigorous proof, we expect that, as in many other mathematical problems involving moving boundary conditions, the steady state solution reflects the existence of an attractor to the transient solution. Thus we believe that the present model is a valid beginning to the formal study of the physical conditions for DIGM to occur. Of particular interest is our finding that segregation plays a crucial role in promoting DIGM. This provides for experimental testing of the theory. Although there have been many experimental studies of DIGM, relatively little work has been done on the solute dis-
tribution in and around the alloyed or dealloyed region. Perhaps the only convincing work is the study of DIGM at low temperatures (200–375°C; 0.35–0.48 Tm) in Au(Cu) by Pan and Balluffi, using a dedicated scanning transmission electron microscope [9]. With this technique, sufficient spatial and chemical resolution is obtained to identify the existence of grain boundary segregation, although the finite probe size tends to convolute the data in such a way that sharp changes of composition are “smeared out”. In thicker specimens, studied with scanning electron microscopy or electron microprobe analysers, the same beam broadening effect completely obviates the possibility of detecting thin segregation layers. In their experiments, Pan and Balluffi identified a very significant composition spike associated with the original boundary position, and a sharp change of composition at the final boundary position. These findings agree well with the conclusions presented here. A large concentration jump is presumably required at the original boundary position to provide sufficient driving force to nucleate DIGM against the restraining forces of surface pinning and grain boundary curvature, when an initial critical bulge is formed. In this case, the magnitude of the concentration jump is largely determined by the driving force required to form a critical nucleus, because the lattice diffusion distances are very small and the boundary concentration (and hence the magnitude of the concentration jump) builds up to the critical value during the early transient stages of the experiment. Once DIGM is nucleated, it can proceed with a smaller composition jump. It remains unclear whether DIGM ends, in this particular case, or if the “final” boundary position is merely set by the termination of the experiment: a different solute profile might be expected if boundary migration had ceased prior to the STEM investigation. At the other extreme, Tashiro and Purdy have observed DIGM in Al(Zn) under high homologous temperature conditions (160–190°C; 0.53–0.56 Tm) that produce large bulk diffusion distances [10]. These conditions would be expected to remove any significant concentration jump between the boundary and the adjacent bulk, unless some segregation occurs. Zhang and Jones have observed segregation of zinc to the grain boundaries of an alloy of Al–0.35 at% Zn, under electron irradiation, at 160°C [11] and the point defect fluxes involved in this case may be similar to those in the experiments of Tashiro and Purdy, so it is not unreasonable to conclude that the conditions set forth in the present theory were also obtained during the Tashiro–Purdy experiments.
REFERENCES 1. Rhines, F. N. and Montgomery, A. M., Nature, 1938, 141, 413. 2. King, A. H., Int. Mater. Rev., 1987, 32, 173. 3. Yoon, D. Y., Int. Mater. Rev., 1995, 40, 149.
BROKMAN et al.: SEGREGATION IN DIGM 4. Cahn, J. W., Fife, P. and Penrose, O., Acta mater., 1997, 45, 4397. 5. Balluffi, R. W. and Cahn, J. W., Acta metall., 1981, 29, 493. 6. Brener, E. A. and Temkin, D. E., Physica status solidi (a), 1990, 120, k7. 7. Chandrasekhar, S., Rev. Mod. Phys., 1943, 15, 1.
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8. Johnson, W. C. and Blakely, J. M. ed. Interfacial Segregation. Am. Soc. for Met., Ohio, 1979. 9. Pan, J. D. and Balluffi, R. W., Acta metall., 1982, 30, 861. 10. Tashiro, K. and Purdy, G. R., Scripta metall., 1983, 17, 455. 11. Zhang, Y. G. and Jones, I. P., J. Nucl. Mater., 1989, 165, 252.
THE ROLE OF SEGREGATION IN DIFFUSION-INDUCED GRAIN BOUNDARY MIGRATION A. BROKMAN1*, A. H. KING2 and A. J. VILENKIN1 1
Department of Applied Physics, Graduate School of Applied Science, Hebrew University, 91904 Jerusalem, Israel and 2School of Materials Engineering, Purdue University, West Lafayette, IN 479071289, USA ( Received 15 August 2000; received in revised form 6 September 2000; accepted 11 September 2000 )
Abstract—The problem of grain boundary motion in the diffusion field of a solute is formulated for the case of infinitely fast diffusion along a straight boundary. The steady state solution suggests that (de)alloying occurs by two different modes, namely: the solute diffusion through the stationary boundary to the bulk, or by diffusion-induced grain boundary migration (DIGM). The transition from one mode to another depends on the grain boundary segregation coefficient. The result enables an assessment of the relative importance of different possible driving forces. When the equilibrium concentrations of the bulk solute with the external gas is low, the entropy of mixing is the leading driving force. DIGM does not occur in isotope solution because the solute atom does not segregate to the boundary. Based on this theory, we construct the phase diagram in the plane of the (gas/bulk) equilibrium concentration vs the segregation coefficient, representing the transition from DIGM to alloying via stationary boundaries. 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: DIGM; Segregation; Diffusion; Boundary migration
1. INTRODUCTION
Diffusion-induced grain boundary migration (DIGM) was first observed by Rhines and Montgomery in 1938 [1], but has become a subject of curiosity since the late seventies. Since then, many works have been dedicated to the identification of the mechanism, primarily in order to identify the underlying driving force and understand the physical conditions that lead to DIGM. This effort has been reviewed and overviewed extensively [2–4]. The boundary motion may be driven by: entropy of mixing, releasing heat of solution, local curvature or cohesion strain energy. In spite of the long history of DIGM, the role of each of these driving forces remains a subject of debate. For instance, the driving force due to mixing entropy should lead the process since the chemical potential of the solute atom varies logarithmically in the dilute solution (in comparison with a power law for the cohesive term). However, Balluffi and Cahn [5] argued that since DIGM is not expected when the diffusing element is an isotope of the parent material, the entropy of mixing is of no significance. The heat of mixing and the dependence of the boundary energy on the solute atom concentration are sometimes
* To whom all correspondence should be addressed. E-mail address: avnerb얀vms.huji.ac.il (A. Brokman)
excluded since both alloying and dealloying occurs by DIGM in the same binary systems. As a result of the uncertainty in the driving force, the underlying physical conditions are still unresolved. As DIGM represents a kinetic instability, we adopt an unusual strategy that first formulates the kinetic problem for an arbitrary driving force. From the solution of this problem we are able to discuss the relevant interactions that drive DIGM and identify the underlying physical conditions. For instance, we show that the hypothetical isotope experiment will not exhibit DIGM, simply because the isotope does not segregate to the grain boundary, and the entropy of mixing drives DIGM at low pressure of the ambient gas. The present work involves a lengthy and cumbersome algebraic manipulation that may screen the rather simple logic of the calculation. Therefore, we introduce a brief account of the physical rationale of this work. We investigate DIGM in the simplest possible system, which consists of a bi-crystal in a fixed partial pressure of solute atom ambient gas. The bulk solute changes its concentration in order to equilibrate with the chemical potential of the gas, thus causing the alloying or dealloying of the bulk. As the boundary diffusivity is larger than the bulk diffusivity, the first stage of alloying (dealloying) occurs by diffusion through the grain boundary (rather than direct diffusion from the surface to the bulk), and may happen
1359-6454/01/$20.00 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 0 ) 0 0 3 2 0 - 7
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BROKMAN et al.: SEGREGATION IN DIGM
through two routes: (a) the solute atom diffuses into (or out of) the stationary grain boundary, and thereafter leaks by diffusion into (or out of) the bulk; and (b) the solute diffuses through a moving grain boundary leaving an alloyed (dealloyed) bulk behind (DIGM). The primary purpose of the present work is to identify the physical conditions, which discriminate between these two modes. The wavy shape of the boundary after DIGM, with islands of alloyed (dealloyed) materials on both sides, suggests that the process of DIGM is initiated by some fluctuation in the boundary shape which becomes unstable against DIGM. Therefore, the boundary segment migrates in the direction of the initial perturbation. This work does not account for the initial growth of the unstable boundary segment, but rather assumes that the boundary moves and identifies the conditions for its motion in a steady state manner. This strategy suggests, without proof, that the steady state motion is an attractor to the general problem of the grain boundary motion, and when a steady state is impossible the boundary is stable against fluctuations. As the steady state is independent of the boundary shape, we consider the simple case of a one-dimensional boundary motion (see also Section 4). With the above assumptions, the problem of DIGM becomes a mathematical problem that governs the motion of the boundary in the field of the diffusing solute atom. It is shown that for a driving force of any origin, the boundary velocity is proportional to the concentration “jump” across the grain boundary plane. The concentration jump with its associated chemical potential drop across the boundary is a necessary condition for solute flux through the boundary. The boundary does not reach a local equilibrium with the bulk as long as the solute fluxes on both of its sides are not balanced to yield zero flux across the boundary. In the case of a steady state motion, the flux, the boundary velocity and the concentration jump are found in this work by solving the self-consistent problem that combines the diffusion equation of the solute atom in the moving coordinate system and (as a necessary boundary condition) the solute conservation in the bi-crystal system. The conditions for DIGM are identified with the solutions that produce a non-zero boundary velocity, and are found to depend on the bulk diffusion coefficient, the segregation coefficient, the grain boundary mobility and the generalized driving force.
2.1. Energetics Consider the two-dimensional bi-crystal consisting of the semi-infinite “Crystal Left” at x⬍X(t) and “Crystal Right” at x>X(t). The two crystals are separated by a straight grain boundary at X(t). The free energy of this system is:
冕
冕 ⬁
X⫺0
E⫽
f(c) dx ⫹
⫺⬁
f(c) dx,
(1)
X⫹0
where f is the free energy per unit of volume depending on the solute atom concentration c. This dependence may consist of an entropy mixing term, elastic cohesive energy, etc. At this point we choose not to limit ourselves to a specific model, and we refer to the relevant interactions in Section 4 below. The driving force for grain boundary motion is: F⫽⫺
dE ⫽ f[c(X ⫹ 0)]⫺f[c(X⫺0)], dX
(2)
and the grain boundary velocity is given within linear transition theory, i.e. v⫽
dy ⫽ mF ⫽ m{f[c(X ⫹ 0)]⫺f[c(X⫺0)]}, dt (3)
where m is the boundary mobility. Suppose that the solute concentration in the grain boundary is in equilibrium with the gas, and c0 is the bulk concentration in equilibrium with the gram boundary, then, in the limit of small deviations from local equilibrium, f[c(X ⫹ 0)]⫺f[c(X⫺0)]⬇
|
df [c(X ⫹ 0) dc c0
(4)
⫺c(X⫺0)].
Thus equation (3) reads: v ⫽ B(c⫺⫺c⫹ ),
(5)
where c ⫹ ⫽ c(X ⫹ 0) and c⫺ ⫽ c(X⫺0) For the sake of simplicity, we assume hereafter that the kinetic coefficient 2. FORMULATION
In the first two parts of this section, we derive some basic concepts which appear in the literature in different contexts. Then, we employ these concepts to the modeling of a moving boundary in the field of the diffusing solute atom.
|
df B ⫽ ⫺m dc c0
is positive (df/dc is to be negative, otherwise the con-
BROKMAN et al.: SEGREGATION IN DIGM
centration is large and the diffusion equation below becomes non-linear). Equation (5) implies that the boundary velocity depends on the concentration jump across the boundary. A similar result was suggested by Brener and Temkin [6].
3
1. v⫽0: in this limit (9) becomes D c1 exp{y0/T}⫺c2 exp{yL/T} L exp{y0/T} ⫹ exp{yL/T} ⫽ h(c1⫺kc2),
J0 ⫽ 2
(10)
2.2. Grain boundary/bulk exchange of solute atom Here we employ some known ideas concerning the diffusion over a potential barrier [7] in order to obtain the flux through a moving potential barrier. The chemical potential of the non-interacting solute atoms (low concentration) is m ⫽ T ln(c) ⫹ y(x), where T is the temperature in Boltzman units, and y varies in space and contains a localized barrier. In the case of a moving barrier, the impurity concentration satisfies the equation ∂tc ⫽ ⫺∂xJ, and in the moving coordinate system (the origin moves with the barrier at a velocity v):
冉
J ⫽ ⫺D⬘ ∂xc ⫹ c
冒冊
dy T ⫺vc dx
(6)
where k ⫽ exp{[yL⫺y0]/T}, and h⫽ 2(D⬘/L)(1/1 ⫹ k) is a kinetic coefficient. We use h as an input parameter, for which we rewrite b: b⫽
vL v 1 ⫽2 ; D⬘ h1⫹k
(11)
J ⫽ ⫺vc2,
(12)
2. v>0, bÀ1 yields:
and the convective part determines the flux; finally: 3. v⬍0, b¿⫺1 yields: J ⫽ ⫺vc1
where D⬘ is a kinetic constant (effective diffusion coefficient) assumed to be independent of x. If the gradient of the potential barrier is localized into a small length L, then J is constant over L, and (6) becomes an ordinary differential equation for c. The flux obtained from this equation with the boundary conditions {c(0) ⫽ c1, c(L) ⫽ c2} is: D⬘
J ⫽ ⫺L
冕 再 exp
冎
再 冋 册
y(x) vx ⫹ dx T D⬘
冋
0
⫺c2 exp
y(0) c1 exp T
(7)
册冎
y(L) vL . ⫹ T D⬘
We simplify (7) by using the following potential model: y(x) ⫽
再
y0, 0⬍x⬍L/2
yL, L/2⬍x⬍L
(8)
for which (7) reads: J⫽v
c1 exp{⫺b/2 ⫹ y0/T}⫺c2 exp{b/2 ⫹ yL/T} , [1⫺exp(⫺b/2)] exp{y0/T} ⫹ [exp(b/2)⫺1] exp{yL/T}
(9)
where b ⫽ vL/D⬘ is the dimensionless velocity of the potential barrier. We can use equation (9) as a boundary condition that enables the formulation of the moving boundary problem with a rather complicated solution. Let us consider the three simple limiting cases:
(13)
The super-positioning of two moving potential barriers with opposite signs is our model for the grain boundary in the next section. 2.3. Moving boundary in the diffusion field Consider a two-dimensionally infinite, thin bi-crystal annealed in an atmosphere of constant chemical potential of the diffusing atom (fixed gas pressure). The impurity atoms condense (or evaporate) on the crystal surface and diffuse into (or out from) the bulk by means of grain boundary and bulk diffusion. At the annealing temperature, the diffusion coefficient of solute along the grain boundary is much larger than the bulk one: DgbÀD. During the time interval h2/Dgb¿t¿h2/D (h being half the bi-crystal thickness), the bulk diffusion across the film and the impurity exchange through the surface are negligible. In this interval of time we simplify the problem by assuming that the grain boundary diffusion coefficient is infinitely large and the boundary solute concentration remains in equilibrium with the ambient gas. When the boundary moves in steady state it (de)alloys the swept bulk to a constant concentration. In the relevant interval of time suggested above, the bulk concentration is not necessarily in equilibrium with the gas, and only when bulk diffusion becomes effective, may the bi-crystal reach full equilibrium. The boundary motion is possible if and only if a net solute flux transferred through the boundary. A necessary condition for a finite flux through the boundary is the non-vanishing difference of chemical potential across the boundary and an associated solute concentration “jump”. Although diffusion across the boundary exists and the width of the boundary is
4
BROKMAN et al.: SEGREGATION IN DIGM
small, the concentration jump is maintained for the time scale mentioned above. This phenomenon is prevalent in a wide range of problems concerning boundaries in external fields. The simplest example that demonstrates this effect is the random walk on a one-dimensional lattice with a point defect. The hopping probability into each of the lattice sites is equal, but different for the defect site. It is clear that as long as the external flux in the system is finite, the chemical potential is discontinuous at the defect site. The analogy to the temperature discontinuity across the moving interface, when heat is transferred in external temperature gradient, is also called for (the formal description of heat transfer in the temperature gradient field is similar to the description of solute diffusion in the field of external flux). In the case of DIGM, the unbalanced solute flux into the boundary from both sides and the associated concentration jump, are maintained for any finite diffusivity across the boundary (even very large), as long as the boundary moves. In addition to the insensitivity of the concentration jump to the solute diffusion across the boundary, this phenomenon occurs in the bi-crystal regardless of the actual value of the diffusivity along the grain boundary. In particular, the concentration jump is developed when the diffusion coefficient along the boundary is much larger than that of the bulk, and by assuming infinite boundary diffusivity we do not lose the generality of the phenomenological description. Moreover, finite diffusivity along the boundary would lead to a larger concentration jump (together with enhanced boundary velocity and a difficult mathematical problem, beyond our analytical capability). To conclude, the combination of finite diffusivity across the boundary that conserves the concentration jump, and infinite diffusivity along the boundary that is presented for the sake of mathematical convenience, sets a representative framework to describe DIGM. Suppose that X(t) is the position of the grain boundary. Diffusion away from the boundary into the bulk is governed by the diffusion equation ∂tc ⫽ D∂xxc for both grams separately (i.e. the equation should be solved in the intervals ⫺⬁⬍x⬍X(t) and X(t)⬍x⬍⬁). In the moving coordinate system (x→x⫺vt), the steady state solution of the diffusion equation is: c ⫽ a exp(⫺vx/D) ⫹ b,
where J⫺ is the flux from the left grain into the grain boundary and J⫹ is the flux from the “grain boundary phase” into the right grain. In order to estimate these fluxes, let us consider Fig. 1. In this figure, the boundary is described by a potential well consisting of two opposite barriers of the type considered in the previous section. In accordance with this figure, we substitute into (9) the potentials: y0 ⫽ yg (bulk potential), yL ⫽ ygb (boundary “phase” potential), resulting in: J⫺ ⫽ v
c⫺⫺c0 exp(b) . exp(b/2)⫺1 ⫹ [exp(b)⫺exp(b/2)]/k (17)
where k ⫽ exp{[yg⫺ygb]/T} is the segregation coefficient. In the same way, the flux from the grain boundary into the right grain is (substitute into (9) y0 ⫽ ygb, yL ⫽ yg): J⫹ ⫽ v
c0⫺c⫹ exp(b) . exp(b)⫺exp(b/2) ⫹ [exp(b/2)⫺1]/k (18)
In the case of DIGM under fixed external gas pressure (given c0), equations (14)–(18) yield two algebraic equations with three unknown parameters (v, c⫹ and c⫺). Equation (5) provides an additional relation between these variables, and makes the DIGM problem solvable, as demonstrated in the next section. 3. SOLUTIONS
3.1. Alloying by DIGM Consider the process of alloying by DIGM as presented schematically (in the moving coordinate system) in Fig. 2. Here, the boundary potential well is reduced to a limit of small (inter-atomic spacing) thickness, and the boundary concentration z0, is in equilibrium with the gas (a result of the assumption of infinite grain boundary diffusivity). The grain boundary migrates with a velocity v, leaving a solute
(14)
where a and b are integration constants, and v ⫽ dX/dt ⫽ const. is the velocity of the coordinate system, given by equation (5). Solute atom conservation at the boundary implies: x ⫽ ⫺0: J⫺ ⫽ ⫺D∂xc⫺vc,
(15)
x ⫽ ⫹ 0: J ⫹ ⫽ ⫺D∂xc⫺vc,
(16)
Fig. 1. A potential well model for the grain boundary that combines two potential barriers through which the solute atom diffuses with flux J⫹, J⫺ (see text).
BROKMAN et al.: SEGREGATION IN DIGM
5
Substituting equations (21) and (22) into (5) gives the algebraic equation for b, namely: lb ⫽ f(b),
(23)
where f(b) ⫽
1 (23a) exp(⫺b/2) ⫹ [1⫺exp(⫺b/2)]/k ⫺exp(⫺b)
Here l ⫽ [h(1 ⫹ k)]/2c0B and B is defined in (5). Figure 3 plots the function f(b) for different values of k [Fig. 3(a)] together with a schematic graphical solution of (23). For k>1 the function f(b) increases monotonically from f(0) ⫽ 0 to f(⬁) ⫽ k, as seen in Fig. 3. The first derivative at the origin is:
Fig. 2. Concentration profile for the case of alloying by DIGM: in the moving coordinate system, the concentration at x ⫽ 0 (grain boundary) is in equilibrium with the gas phase and equals z0. The bulk equilibrium concentration is c0(z0 ⫽ kc0, where k is the segregation coefficient). The boundary migrates to the right and leaves an enriched solution behind (x⬍0) with a concentration which depends on its velocity.
f0⬘ ⫽
|
df 3k⫺1 ⫽ . db b ⫽ 0 2k
We can see (Fig. 3) that when l>f0⬘, i.e.
concentration c⫺ behind and a decaying profile of concentration in front, starting with concentration c⫹ near the boundary, and asymptotically reaches the value of the initial concentration, c(⬁) ⫽ 0. The later condition is used as a boundary condition for the bulk diffusion equation and from (14) we have: x>0: c(x) ⫽ c ⫹ exp(⫺vx/D).
(19)
Substituting this result into (16) yields: dc J ⫹ ⫽ ⫺D ⫺vc ⫽ ⫺D(⫺c ⫹ v/D) dx ⫺vc ⫹ ⫽ 0,
(20)
and the steady state flux out of the boundary vanishes. From (18): c ⫹ ⫽ c0 exp(⫺b).
(21)
For x⬍0, equation (14) reads, c(x) ⫽ c⫺ ⫽ const., and (15) yields J⫺ ⫽ ⫺vc⫺, and from (17): 1 c⫺ ⫽ c0 . exp(⫺b/2) ⫹ [1⫺exp(⫺b/2)]/k
(22)
Fig. 3. Graphical presentation of equation (23) (alloying by DIGM): (a) the dependence of f(b)/b on the dimensionless velocity b for different values of k; and (b) graphical solutions of the equation: the straight lines correspond to lb. The intersection of these lines with the curve representing f(b) yields a possible steady state solution; line “1” is the case when equation (23) has the solution v ⫽ 0; line “2” represents multiple steady state solutions; and line “3” represents a single possible (fast) steady state solution at small values of l.
6
BROKMAN et al.: SEGREGATION IN DIGM
h>2c0B
3k⫺1 , 2k(1 ⫹ k)
3.2. Dealloying
equation (23) has no solution with non-zero v, and the grain boundary remains stationary. The function f(b) is monotonic, and possesses two inflections (∂2f(b, k)/∂b2 ⫽ 0): one inflection point is too close to the origin to be identified in the scale of Fig. 3. A second inflection is seen in the figure, and becomes pronounced at large values of k. As a result [consult Fig. 3(b)], depending on the specific values of k and l, the solutions to equation (23) consist of two possible classes, namely: (a) a stationary solution; and (b) a steady state solution with one, or two possible nonvanishing velocities. The nature of the transition between these solutions is considered in detail in the discussion below. Here we consider mathematically the two extreme cases:
Figure 4 plots schematically the concentration profile of this case in the moving coordinate system. The initial homogenous concentration is c(⬁) ⫽ c⬁ The chemical potential of the grain boundary is smaller than the initial chemical potential of the bulk solution, and c0⬍c⬁. In this case, (14) becomes: x>0: c(x) ⫽ (c ⫹ ⫺c⬁) exp(⫺vx/D) ⫹ c⬁,
and the flux out of the boundary is found from (16):
冉 冊
dc v J ⫹ ⫺D ⫺vc ⫽ ⫺D(c ⫹ ⫺c⬁) ⫺ ⫺v(c ⫹ ⫺c⬁ dx D ⫹ c⬁) ⫽ ⫺vc⬁.
From (18), we have: 1. Fast alloying (bÀ1): we write the concentrations [equations (21) and (22)], c ⫹ ⫽ 0, and c⫺ ⫽ kc. The latter condition is indicated in Fig. 2 by z0 (z0 ⫽ kc0 is the boundary concentration in equilibrium with the gas phase). Substitution of the latter relations into (5) yields the boundary velocity: v ⫽ B(c⫺⫺c ⫹ ) ⫽ Bkc0, (k>1). 2. Slow alloying (b¿1): an analytical approximation can be realized when f ⬘(0) ⫽ df/db|b ⫽ 0>l, and f ⬘(0)⫺l¿1. In this case, we can expand f(b) near the origin: f(b) ⫽ bf0⬘ ⫹
b2 f ⬙. 2 0
(24)
再
⫺exp(b/2) ⫹
2lk2⫺3k2 ⫹ k , ⫺3k2⫺3k ⫹ 2
exp(b/2)⫺1 k
册冎
(27)
.
Similar to the case of alloying, the solution of (14) for x⬍0 is c⫺ ⫽ const., J⫺ ⫽ ⫺vc⫺, and equation (22) holds for the case of dealloying: 1 c⫺ ⫽ c0 exp(⫺b/2) ⫹ [1⫺exp(⫺b/2)]/k
(28)
Substituting the latter equation and (27) into (5) leads to the following transcendental equation for b: lb ⫽ F(b),
As f0⬙ ⫽ ⫺(3 ⫹ 3/k⫺2/k2)/4, the approximation (24) gives a solution to (23), namely: b ⫽ 2(l⫺f0⬘)/f0⬙ ⫽ 4
冋
c ⫹ ⫽ exp(⫺b) c0 ⫹ c⬁ exp(b)
(29)
where:
In the limit kÀ1 (l ⫽ 3/2⫺e2, e¿1): b ⫽ 4(3⫺ 2λ)/3. Now in the limit b¿1, we obtain c⫹, c⫺ from (21) and (22), namely: c ⫹ ⫽ c0(1⫺b),
c⫺ ⫽
c0 . 1⫺(b/2)(1⫺1/k)
(25)
(26)
Hence, we have found the concentration jump across the boundary, and the associated boundary velocity of the steady state DIGM.
Fig. 4. The same as Fig. 2, but for the case of dealloying by DIGM. The initial bulk concentration c⬁ is diluted by the boundary motion. The ratio c⬁/c0 is larger than one and indicated by “s” in the text.
BROKMAN et al.: SEGREGATION IN DIGM
F(b) ⫽
冉 冊
k exp(b/2) s ⫹ ⫺1 (29a) k⫺1 ⫹ exp(b/2) k k⫺1 exp(⫺b) ⫹ s exp(⫺b/2)⫺s. k
Here s ⫽ c⬁/c0. F is plotted in Fig. 5(a). It is easy to check that F(0) ⫽ 0, and F(⬁) ⫽ k⫺s. Again, we discuss the case of k>1. For small l, two solutions are possible for a moving grain boundary. Analysis is possible in the limits of slow and fast dealloying: 1. Fast dealloying (bÀ1): in this case, (27) yields c ⫹ ⬇c⬁, and (28) becomes c⫺ ⫽ kc0, so that from (5): v ⫽ B(kc0⫺c⬁),
7
2. Slow dealloying (b¿1): from Fig. 5(a) it is clear that slow solution is possible only when F0⬘ ⫽ dF/db|b ⫽ 0 ⫽ l ⫹ e2; (e¿1); , and F0⬙⬍0. In this case, the approximation of the solution of (29) is: b ⫽ 2(l⫺F0⬘)/F0⬙ and (29a) implies b ⫽ 4(2λ ⫹ s⫺3)/s⫺3 (it is easy to see that when kÀ1, s⬍3). In the limit b¿1 we expand the concentrations near the grain boundary in (27): or c ⫹ ⫽ c0(1⫺b) ⫹ c⬁b(k ⫹ 1/2k) ⫹ O(b2), when kÀ1: c ⫹ ⫽ c0 ⫹ b(c⬁/2⫺c0). In the same manner, c⫺ ⫽
冋 冉 冊册
c0 b 1 ⫽ c0 1⫺ ⫺1 1 ⫹ (b/2)[(1/k)⫺1] 2 k ⫹ O(b2)
and l¿k⫺s. and for kÀ1: c⫺ ⫽ c0(1 ⫹ b/2).
Hence, dealloying by DIGM is evident in this example. Numerical results representing the possible DIGM solutions as functions of k, b and l are plotted in Fig. 5(b). The transition from a stationary to a possible DIGM mode of alloying is reflected in the point of minimum k (given l). Below this point a stationary solution is the only possible solution, while above it, DIGM is a possible solution to equation (29). 4. DISCUSSION
Fig. 5. (a) Graphical representation of the solutions to (29) (dealloying); the straight line “1” has no intersection with the curve f(b), and the only possible solution is the one of a stationary grain boundary (b ⫽ 0); line “2” is tangent to f(b) and represents the transition to two possible solutions indicated by the intersections of line “3”. At the transition, a single nonzero velocity solution is possible (as in the case of line “2”). Notice that the asymptotic value of f(b) depends on the initial condition. (b) Numerical solutions of (29) for the case s ⫽ 15. In accordance with the graphical solution, it is seen that for any k and l there are two possible solutions or none. The forbidden velocity zone (slow solutions) is associated with the negative values of f(b) in (a).
When a bi-crystal of a given material (l, k and c⬁ are given) is placed in a gas of the solute atom with a fixed partial pressure (given c0), diffusion occurs to equilibrate the bulk solute with the gas. The dynamics of this process involves either DIGM or diffusion through the stationary grain boundary to the bulk. We showed in Section 2 that a necessary condition for the boundary to move is the temporary development of a solute concentration jump. This condition is obtainable, depending on the segregation kinetics involved. Therefore, the transition from DIGM to stationary (de)alloying depends strongly on the segregation coefficient k. By solving the self-consistent moving boundary problem we showed in Section 3 how the steady state velocity depends on the segregation kinetics [equations (23) and (29)*]. It is the
* In general, equation (29), which describes the dealloying process (s>1), is also valid for the case of alloying (s⬍1). [The limit s ⫽ 0, leading to equation (23), is the one discussed under “alloying” in the text.]
8
BROKMAN et al.: SEGREGATION IN DIGM
purpose of this work to gain an understanding of the transition from the finite velocity in the case of DIGM to the zero velocity in the case of a stationary boundary, which is discussed in detail below. In order to do so, we should first consider the role of the different driving forces in view of the theory. 4.1. Interactions and the “isotope trap” Our analysis demonstrates that steady state DIGM may occur, depending on the magnitude of the driving force, without specifying the origin of this force. A few driving forces are possible: let us first exclude the two driving forces that do not, in themselves, explain the basic phenomena, but rather influence the local velocity of the boundary: 1. Grain boundary curvature. This force can drive the boundary motion in the case of grain growth and in the case of DIGM, may slow the local boundary velocity. As the curvature driven motion is decoupled from the solute atom diffusion, it cannot explain DIGM, but rather modifies the force due to the variation of concentration. 2. The effect of solute atoms on the boundary anisotropy. This may lead to the rotation of the boundary, but will not drive a process with long range translational migration. Our one-dimensional model does not lose itsability to explain the basic dynamics of DIGM byignoring the forces above. The driving forces thatare coupled to the solute atom diffusion and henceessential in the explanation of DIGM include: 3. The decrease in the coherency strain energy that depends on the local concentration. This driving force is frequently used to explain DIGM. For our discussion below we should refer to the fact that to the lowest order, the specific strain energy is quadratic in the concentration. 4. The variation of the bulk free energy due to the chemical mixing. For the ideal substitutional solid solution the specific free energy varies at low concentrations as Tc ln(⍀c) ⫹ O(c) (T being the temperature in Boltzman units, and ⍀ the atomic volume). When the equilibrium concentration c0 is small, the mixing entropy will dominate the driving force. However, we cannot use this conclusion before discussing the role of mixing in the context of the “isotope paradox” which excludes this term as a possible driving force for DIGM. In brief, the isotope paradox (firstly suggested in [5]) notes that the same variation of bulk free energy occurs when one mixes a pure material with an isotope (of the same material as the unalloyed bulk) or a solute (of foreign) atom that forms an ideal solution. As we do not expect DIGM to occur in the diffusing isotope system, the driving force for DIGM in the foreign atom solution must also be of no significance. Our analysis demonstrates that the boundary velocity is proportional to the jump in solute con-
centration across the boundary. This jump depends on the segregation coefficient, and when the latter equals one (no segregation) diffusion smooths out the difference of concentration across the boundary. Since there is no concentration jump in the isotope case, we do not expect the boundary to move, while it certainly can move in the case of a segregating solute atom. In conclusion, the isotope diffuses to the bulk through a stationary grain boundary as suggested, yet the variation of energy due to mixing entropy remains a valid driving force for DIGM in the case of a segregating solute. This example demonstrates that the segregation coefficient should be incorporated into the kinetics, as shown below. 4.2. When does alloying occur by DIGM? In order to demonstrate the implications of the theory, we consider the rather simple case of alloying by DIGM. [The dealloying case is a direct extension of the following problem, using (29) instead of (23). However, the complication due to the extra parameter (namely, s) makes the problem mathematically cumbersome.] With the above discussion, it is clear that when the bulk/gas equilibrium concentration (c0) is small, the variation of the bulk free energy due to mixing entropy provides a leading term in the driving force. For this condition (small equilibrium concentration), we rewrite l: l⫽⫺
D⬘ LmTc0 ln(⍀c0)
(30)
Figure 6 plots l vs the dimensionless velocity b [equation (23)] for different values of the segregation coefficient k. In accordance with the previous section, it is seen that above a critical value of l, a solution with non-zero velocity is impossible and the boundary
Fig. 6. Plot of l vs b for different values of segregation coefficient in the case of alloying. At zero velocity, all curves start at the value l ⫽ 3/2 where their second derivative is negative (not seen in the scale of the plot). Therefore, a narrow interval of l with three possible solutions (number of velocities of given l and k) is evident. In most cases, two non-zero velocity states are possible (below the curve) or a trivial (stationary) solution is obtained (above the curve). At the critical l, a single moving boundary solution is possible.
BROKMAN et al.: SEGREGATION IN DIGM
remains stationary. At the critical point (l ⫽ l∗), a single steady state solution is possible, while below this point two solutions are possible. (The prediction of slow and fast motion in the two-steady-state mode of motion may be associated with the observation of “jerky boundary motion” that has been observed in thin films during DIGM. This idea requires further study since the influence of the thermal groove may also be of importance.) We should also point out that in a small interval of l, i.e. near l ⫽ 3/2 (the value in the limit of zero velocity), three possible solutions are possible due to the existence of two inflections of f(b), discussed above. Figure 7 plots the values of k at the critical points versus the dimensionless boundary velocity. According to equation (30), DIGM occurs when the equilibrium concentration is larger than a critical concentration c0 ⫽ c∗0 given by the transcendental relation: c∗0 ln(⍀c∗0 )⬇⫺
1 , l∗x3c
(31)
where xc ⫽ (D⬘/LmT)⫺1/3 is a coherence length. To demonstrate, we evaluate this length and the critical concentration for the typical DIGM conditions by treating the above relation term by term for some fictitious experimental condition: 앫 L: the length of the solute potential drop-off of the boundary is of the order of a few inter-atomic distances; 앫 m: to our knowledge, there is no experiment which measures the grain boundary mobility m. However, the product of the mobility and the boundary tension (known as the “reduced mobility”) has been deduced from grain growth experiments and
Fig. 7. Numerical result of the value of the segregation coefficient k vs the critical dimensionless velocity (see Fig. 6).
9
varies from 10⫺4 to 10⫺8 cm2/s (depending on the material and the temperature). We use the smaller value below for the reduced mobility. 앫 xc: if the boundary energy is 1000 erg/cm2 and the diffusion coefficient across it at a temperature of 1000 K is 10⫺12 cm2/s (see comment above), the coherence length xc, is of the order of 10⫺7 cm; 앫 l∗: in practice, the segregation coefficient is kÀ1 [8], and may reach the value of 104. In this limit, the curve in Fig. 7 indicates a logarithmic growth of the boundary velocity with the segregation coefficient; 앫 c∗0 : Fig. 8 presents the critical concentration curve that separates the conditions for alloying by DIGM from those of enrichment through the stationary boundary. The numerical values of this curve were calculated from equation (31), for the above values of the different physical parameters. 4.3. One-dimensional steady state and real life In practice, both the diffusion coefficient along the boundary, Dgb, and the diffusion coefficient across the boundary, D⬘, are finite. In order to evaluate the correctness of the one-dimensional model, let us examine the solute distribution, z(y) along the boundary in the case of a steady state motion under the flux found above, but when the diffusion coefficients are finite. In this case. the steady state diffusion equation (with source/sink term) is Dgb∂yyz ⫹ L1 (J⫺⫺J ⫹ ) ⫽ 0 (L being the boundary width). Substituting (18), (19), (21) and (22). we obtain in the limit of large k (alloying case): z(0)⫺z(h) D⬘c0b b/2 ⫽ e . h2 DgbL2
(32)
Fig. 8. Plot of the critical curve c∗0 (k) that separates the possible DIGM (above the curve) from the stationary boundary (below the curve).
10
BROKMAN et al.: SEGREGATION IN DIGM
In order to extract the physical conditions that are consistent with the one-dimensional steady state solution we set a solute homogeneity criteria at the boundary, namely: (z(0)⫺z(h))/z(0)¿1. Substituting, z(0) ⫽ kc0, we obtain from (32) the criteria: D⬘h2 beb/2 ¿1 DgbL2 k
(33)
We are not aware of any measurements of diffusion across the boundary, but since the boundary evolves expansion normal to its plane, we expect that D⬘ will be smaller than the diffusion coefficient along the boundary (Dgb), most probably of the order of the bulk diffusion constant. In the unlikely case when D⬘ and Dgb are of the same order, one should bare in mind that the model might be adequate only when the specimen is very thin or the boundary moves in a slow mode [extract h and b from inequality (33)]. In this case the solution of a moving boundary may imply possible instability of the straight boundary in thick films. In contrast to our analysis, the experimental observation of DIGM does not show a steady state motion of the grain boundary. The motion of the individual boundary involves the enrichment (or dilution) of the bulk material on both sides of the boundary as different segments of the boundary migrate to different directions. Furthermore, with time, this motion slows down and the boundary velocity becomes time dependent. Therefore, a full account of the boundary motion requires the transient analysis of the threedimensional problem, which includes the driving force due to local curvature. This problem is by far more complicated than the present analysis, and should become a future challenge. Besides the remark above, which enables the prediction of DIGM as a phenomenon by means of a curvature-independent model, the steady state solution may also provide the basis for understanding the transition from a stationary to a moving boundary. We view the problem of DIGM as the unstable growth of a boundary perturbation (which reflects the wavy shape observed experimentally). As the role of the boundary curvature is nothing but slowing the boundary motion, we anticipate that the observation of a transition from a stationary to a steady state mode reflects the essential kinetic elements that explain DIGM in “real life”. Although this needs a rigorous proof, we expect that, as in many other mathematical problems involving moving boundary conditions, the steady state solution reflects the existence of an attractor to the transient solution. Thus we believe that the present model is a valid beginning to the formal study of the physical conditions for DIGM to occur. Of particular interest is our finding that segregation plays a crucial role in promoting DIGM. This provides for experimental testing of the theory. Although there have been many experimental studies of DIGM, relatively little work has been done on the solute dis-
tribution in and around the alloyed or dealloyed region. Perhaps the only convincing work is the study of DIGM at low temperatures (200–375°C; 0.35–0.48 Tm) in Au(Cu) by Pan and Balluffi, using a dedicated scanning transmission electron microscope [9]. With this technique, sufficient spatial and chemical resolution is obtained to identify the existence of grain boundary segregation, although the finite probe size tends to convolute the data in such a way that sharp changes of composition are “smeared out”. In thicker specimens, studied with scanning electron microscopy or electron microprobe analysers, the same beam broadening effect completely obviates the possibility of detecting thin segregation layers. In their experiments, Pan and Balluffi identified a very significant composition spike associated with the original boundary position, and a sharp change of composition at the final boundary position. These findings agree well with the conclusions presented here. A large concentration jump is presumably required at the original boundary position to provide sufficient driving force to nucleate DIGM against the restraining forces of surface pinning and grain boundary curvature, when an initial critical bulge is formed. In this case, the magnitude of the concentration jump is largely determined by the driving force required to form a critical nucleus, because the lattice diffusion distances are very small and the boundary concentration (and hence the magnitude of the concentration jump) builds up to the critical value during the early transient stages of the experiment. Once DIGM is nucleated, it can proceed with a smaller composition jump. It remains unclear whether DIGM ends, in this particular case, or if the “final” boundary position is merely set by the termination of the experiment: a different solute profile might be expected if boundary migration had ceased prior to the STEM investigation. At the other extreme, Tashiro and Purdy have observed DIGM in Al(Zn) under high homologous temperature conditions (160–190°C; 0.53–0.56 Tm) that produce large bulk diffusion distances [10]. These conditions would be expected to remove any significant concentration jump between the boundary and the adjacent bulk, unless some segregation occurs. Zhang and Jones have observed segregation of zinc to the grain boundaries of an alloy of Al–0.35 at% Zn, under electron irradiation, at 160°C [11] and the point defect fluxes involved in this case may be similar to those in the experiments of Tashiro and Purdy, so it is not unreasonable to conclude that the conditions set forth in the present theory were also obtained during the Tashiro–Purdy experiments.
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BROKMAN et al.: SEGREGATION IN DIGM 4. Cahn, J. W., Fife, P. and Penrose, O., Acta mater., 1997, 45, 4397. 5. Balluffi, R. W. and Cahn, J. W., Acta metall., 1981, 29, 493. 6. Brener, E. A. and Temkin, D. E., Physica status solidi (a), 1990, 120, k7. 7. Chandrasekhar, S., Rev. Mod. Phys., 1943, 15, 1.
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8. Johnson, W. C. and Blakely, J. M. ed. Interfacial Segregation. Am. Soc. for Met., Ohio, 1979. 9. Pan, J. D. and Balluffi, R. W., Acta metall., 1982, 30, 861. 10. Tashiro, K. and Purdy, G. R., Scripta metall., 1983, 17, 455. 11. Zhang, Y. G. and Jones, I. P., J. Nucl. Mater., 1989, 165, 252.