Statistical derivation of basic equations of diffusional kinetics in alloys with application to the description of diffusion of carbon in austenite

Abstract

Basic equations of diffusional kinetics in alloys are statistically derived using the master equation approach. To describe diffusional transformations in substitution alloys, we derive the "quasi-equilibrium" kinetic equation which generalizes its earlier versions by taking into account possible "interaction renormalization" effects. For the interstitial alloys Me-X, we derive the explicit expression for the diffusivity D of an interstitial atom X which notably differs from those used in previous phenomenological treatments. This microscopic expression for D is applied to describe the diffusion of carbon in austenite basing on some simple models of carbon-carbon interaction. The results obtained enable us to make certain conclusions about the real form of these interactions, and about the scale of the "transition state entropy" for diffusion of carbon in austenite.

Master equation approach to the theory of diffusion in alloys and calculations of enhancement factors for tracer solvent and tracer solute diffusion in FCC alloys

“…Basic equations of the master equation approach for the diffusional kinetics of substitution alloys have been derived earlier [17,18]. Below we present the necessary relations from Ref.…”

“…Below we present the necessary relations from Ref. [18]. We consider a substitution alloy with (m+1) components p ′ which include host atoms denoted by index h, solute atoms denoted by Greek letters α, β, λ, µ, ν, and vacancies denoted by v. Latin letters p, q, r will denote all kinds of atoms, both h and α, while Greek letters ρ, σ, τ will denote both solute atoms α and vacancies v, thus the whole set p ′ can be written either as {p, v} or as {h, ρ} .…”

“…The saddle point energy ÊSP pi,vj in (7) depends in general on the atomic configuration near the ij bond. We describe this dependence by the pairwise interaction model [13,18] and write this energy as follows:…”

“…Here E p h is the saddle point energy for a p-species atom in the pure host metal, the parameter ∆ λl p,ij (to be called the "saddle-point interaction") describes changes in this energy due to a possible substitution of a host atom in site l by a λ-species solute atom, while ε λl p,ij and ε hl p,ij are microscopic parameters which can be calculated using either ab initio [13] or model [18] calculations. The most general expression for the probability P in (6) can be written as [10,17]…”

“…non-pairwise effective interactions will be discussed elsewhere. Then after some manipulations described in detail in [18], Eqs. ( 13) can be written similarly to Eqs.…”