Estimation of thephenomenological Effective Diffusivity in Nanocrystalline Materials
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Estimation of the Phenomenological Effective Diffusivity in Nanocrystalline Materials Irina V Belova and Graeme E Murch Diffusion in Solids Group, School of Engineering The University of Newcastle, Callaghan, NSW 2308 AUSTRALIA ABSTRACT In this study we construct two new equations to describe the effective diffusivity in nanomaterials where diffusion proceeds via both the grains and the grain boundaries in comparable amounts. We analyze two very simple 2D grain configurations by Monte Carlo methods to test the validity of the equations. In addition, we also test the Hart and the recently extended Maxwell-Garnett equations. It is shown that one of the two new equations provides a very good description of diffusion in the grain models postulated.
INTRODUCTION A long-standing problem in the area of diffusion in solids is the formulation of the effective diffusivity in the presence of both grains and grain boundaries. The effective diffusivity is conceived as the long-time phenomenological diffusivity of particles that explore the assembly of grains and grain boundaries. It is phenomenological in the sense that both the lattice (grain) diffusivity and the grain boundary diffusivity are singlevalued in a given situation. From the standpoint of the usual thin film tracer-source diffusion experiment it is also the diffusivity that is obtained in the Harrison Type-A kinetics region. In this region Λ (= L / (Dlt)1/2) where 2L is the spacing between grain boundaries, Dl is the lattice diffusivity and t is the diffusion time, is much smaller than unity. In physical terms it means that a given diffusing tracer particle will explore a number of grains in the diffusion time. This is probably too stringent a condition: in a recent Monte Carlo study where the grain boundaries were arranged in the usual configuration parallel to the diffusion direction it was shown that the transition between Harrison Type-A kinetics and Harrison Type-B kinetics occurs at Λ ≈ 0.4, i.e. the diffusion length only needs to be comparable to the grain boundary spacing for Harrison Type-A kinetics to be appropriate [1]. In any case, the effective diffusion coefficient Deff in the Harrison Type-A kinetics region is widely assumed to be described by the wellknown Hart equation [2]: Deff = gDgb + (1-g)Dl
(1)
where Dgb is the grain boundary diffusivity and g is the volume fraction occupied by grain boundaries. However, Eqn.1 is strictly only correct in the situation where the grain boundaries are parallel to the diffusion direction; in other words, a diffusing particle has parallel paths available to it in the diffusion direction (these paths being diffusion through the lattice or diffusion along the grain boundaries). In principle, if the diffusing particle W5.5.1 Downloaded from https://www.cambridge.org/core. Teachers College Library - Columbia University, on 19 Sep 2017 at 07:46:28, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1557/PROC-731-W5.5
crosses a grain boundary in the diffu
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