Expressions for the effective diffusivity in materials with interphase boundaries
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N3.14.1
Expressions for the effective diffusivity in materials with interphase boundaries Irina V. Belova and Graeme E. Murch Diffusion in Solids Group, School of Engineering, The University of Newcastle, Callaghan, NSW 2308, AUSTRALIA ABSTRACT We address the problem of calculating the long-time-limit effective diffusivity in stable twophase polycrystalline material. A phenomenological model is used where the high diffusivity interphase boundaries are treated as connected ‘coatings’ of the individual grains. Derivation of expressions for the effective diffusivity with segregation is made along Maxwell lines. Monte Carlo simulation using lattice-based random walks is used to test the validity of the expressions. It is shown that for the case analysed the derived expressions for the effective diffusivity are in very good agreement with simulation results. The equivalent of the Hart equation is also derived. It is shown to be in poor agreement with simulation results. INTRODUCTION Despite its technological importance, the area of interphase boundary atomic transport has received scant attention, both theoretical and experimental, compared with grain boundary diffusion. In the present paper, we are concerned with the calculation of the effective diffusivity Deff within a two-phase material with stable grains. The effective diffusivity is the long-time-limit diffusivity in a material where there are position dependent diffusivities, for example, within the grains of the different phases or within the interphase boundaries. The effective diffusivity is the diffusivity measured in the Harrison Type-A kinetics regime (the diffusion length is greater than the grain size) in a standard tracer diffusion experiment conducted on a polycrystalline two-phase material. For atomic transport in single-phase material, the effective diffusivity is invariably expressed using the well-known Hart Equation [1], (shown here with the inclusion of the solute segregation factor [2,3]: DeffHart = k ( gDb + (1 − g ) Dl / s ) (1) where Db is the grain boundary diffusivity, Dl is the lattice diffusivity, g is the volume fraction of grain boundaries, s is the segregation factor, the ratio of the concentration C of solute in the grain boundaries to the concentration of solute in the grains see, ref. 3, and k is given by: (2) k = (g + (1 − g)/s)-1 Note that s, and therefore k, equal unity for self-diffusion. Eqn. 1 is exact for parallel diffusion paths in the diffusion direction. Recently, it has been shown by the present authors [4,5] that for various closed grain models of single phase microcrystalline and nanocrystalline materials with and without segregation, Eqn. 1 gives a rather poor representation of the effective diffusivity. The modified Maxwell Equation [4-8] gives far better agreement with Monte Carlo results than the Hart Equation. The modified Maxwell Equation is written as:
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D s − Dl kDb 1 − 2 g b 2 Db s + Dl Deff = (3) D s − Dl 1+ g b 2 Db s + Dl The modification of the original Maxwell Equation centres on the i
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