On Concentration-Dependent Solid State Diffusion
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ON CONCENTRATION-DEPENDENT SOLID STATE DIFFUSION Yang-Tse Cheng General Motors Research Laboratories, Warren, Michigan 48090-9055
ABSTRACT Using a master equation approach, we derive a general expression for the diffusion coefficient as a function of concentration-dependent jump rates. When this approach is applied to diffusion in a binary solid, Darken's equation for intrinsic diffusion coefficients is derived together with an expression for self diffusion coefficients which satisfies the semi-empirical Ugaste relationship. This analysis suggests that the Darken term and the self diffusion coefficients are in general related. I. INTRODUCTION Solid state diffusion is an area which has been studied extensively over the past several decades.' -16 The standard macroscopic theory of solid state diffusion is the linear response theory which relates fluxes linearly to the corresponding driving forces. 5 ,10 In the absence of external fields and a temperature gradient, the atomic flux, Ji, of the species i is linearly proportional to the gradient of the chemical potential pji: Ji
-
Lij
,(1)
where the Lij's are phenomenological coefficients which are symmetric (Onsager's theorem): Lij = Lji. It is believed10 that the phenomenological coefficients Lii are kinetic quantities, which in general are functions of thermodynamic variables but are explicitly independent of the driving forces, such as 8tj1/8x. When applied to diffusion in a binary alloy A,-cBc, the linear response theories lead to the expression of the intrinsic 58 0 3 6 diffusion coefficients DA and DB,'- ' " ',° " ln YBL DB = DI(l + 8a--I-n-7 ,
,9lnc(2
(2.)
in which Dý is the self diffusion coefficient of B in a homogeneous alloy A,-c~c and "YB is B's activity coefficient. Eq. (2) is the familiar Darken equation for intrinsic diffusion coefficients." 3 The Darken term, (1 + ln-yB/ 8 1nc), contains the dependence on the activity coefficient 7YB. In the regular solution approximation,14,16 (1 + oln-yB/ 8 lnc) = (1 - 2AH,zi/kBT), where A .Hmiz is the heat of mixing for alloy A,-cBc, kB is Boltzmann's constant, and T is the absolute temperture. Thus, the Darken term depends on the heat of mixing AHjiz. In contrast, D*, which is an explicit function of temperature T, concentration c, and phenomenological constants Lij, does not explicitly depend on -YB or AHmiz in linear response theories.10
Mat. Res. Soc. Symp. Proc. Vol. 230. c 1992 Materials Research Society
Downloaded from https://www.cambridge.org/core. Columbia University Libraries, on 16 Jan 2020 at 02:40:00, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1557/PROC-230-115
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However, a semi-empirical relationship exists between self diffusion coefficients and 7 18 the heat of mixing which was first established by Ugaste and co-workers:1'
InD
-_ (CA
BBkT = lnD*A + cBlnD*B)
'(
AH.
(3))
where DBA, D*B, and Dý axe the self diffusion coefficients for B in pure A, pure B, and AA Bc., respectively. Equation (3) states that the self
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