The Effective Diffusivity in Two-Phase Material

Abstract

In this paper we employ Monte Carlo computer simulation to test the extended Maxwell-Garnett expression for the effective diffusivity in a simple model of a two phase material. We determine the effective diffusivity in f.c.c., b.c.c. and s.c. type arrangements of dispersed spheres at variable densities and for cases where the diffusivity in the dispersed phase was less than, and greater than, the diffusivity in the matrix phase. It is shown that the above equation agrees very well with the simulation data for all densities up to where the spheres of the dispersed phase touch and over six orders of magnitude in the ratio of the diffusivity of the dispersed phase to the diffusivity of the matrix phase. Introduction A long-standing problem in the area of diffusion in solids is the determination of accurate expressions for the effective diffusivity Deff in two-phase material given the individual diffusivities in the component phases. We are referring here to diffusion that does not alter the morphology or growth of the two phases during the diffusion time. Microscopic examples might include the tracer diffusion of a host component or an impurity in a stable two-phase alloy, or the (interstitial) permeation of hydrogen through a stable two-phase alloy. At low temperatures we are, of course, likely to encounter principally short-circuit diffusion along the interphase boundaries. At high temperatures, however, we can expect that lattice diffusion will prevail but proceed through each phase at different rates. There will be an overall ‘effective’ bulk diffusivity that is dependent on the relative amounts and the morphology of the two phases. In this paper we will refer to the host or matrix phase as phase 1 and the dispersed phase as phase 2. Much of the older literature on the subject deals with a diffusant in the pore space of an impermeable second phase, usually represented as spheres, see, for example, the review by German [1]. This is a very well-studied special case of diffusion in two-phase material. Maxwell [2] derived the following classic expression relating the effective diffusivity Deff of the diffusant when exploring the pore space: 3 D 2 D 1 eff − = (1) where the pore fraction is given by and D1 is the diffusivity in the absence of the impermeable phase. Eq. 1 was originally derived for the limiting case 1. Neale and Bader [3] derived it once again for an idealized geometric model for use over the entire porosity range. It has subsequently been shown by Hashkin and Shtrikman [4] that Eq. 1 represents the upper bound for Deff/D1 for any isotropic medium for all , even when spheres do not represent the impermeable phase. Using a minimum entropy argument Prager [5] derived the following condition for Deff/D1 for particles of arbitrary shape: Defect and Diffusion Forum Vols. 218-220 (2003) pp. 79-87 online at http://www.scientific.net © (2003) Trans Tech Publications, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 134.148.5.113-01/11/06,04:21:13) − − < 3 1 1 D D 1 eff (2) and for spheres in particular: . 2 1 1 D D 1 eff − − = (3) Bruggemann [6] studied a system where one large sphere is surrounded by a homogeneous distribution of much smaller spheres. Assuming that the system is very dilute in large spheres Bruggemann adjusted the Maxwell result for the limit 1 to give: