On zigzag shaped diffusion paths in multi-phase diffusion couples

Abstract

Department of Materials Science and Engineering, Royal Institute of Technology,SE-100 44 Stockholm, Sweden(Received June 6, 2000)(Accepted July 24, 2000)Keywords: Diffusion; Multi-phase diffusionIntroductionSome time ago, Hopfe and Morral (1) proposed and investigated a simple model for multicomponentdiffusion in multi-phase diffusion couples. In their model, the multi-phase equilibrium requirementleads to a singular effective diffusion matrix, which in case of a constant diffusivity gives rise topeculiar zigzag shaped diffusion paths. More recently, Chen and Morral (2) investigated the possibleinfluence of a concentration dependent diffusion matrix and they found, when comparing theiranalytical result with numeric calculations performed with the DICTRA software on realistic modelsystems, that the difference was so small that it was of no practical importance. The minute differencebetween the numeric and analytic results was by the authors mainly considered due to the discretisationerrors in the numerical solution. If the numerical calculations are applied to other alloy systems, thefeatures of the diffusion paths are drastically altered and paths like the one in Fig. 1 are obtained. In thispaper, we therefore illustrate how a concentration dependent diffusion matrix may cause significantdeviations from the ideal zigzag shaped diffusion paths.The ModelIn their model Hopfe and Morral consider an idealized diffusion couple that consists of two multi-phasematerials of different composition brought in contact with each other. In their examples they focus onternary Ni-Cr-Al alloys with a matrix phasegand a secondary phase b, the latter in which no long rangediffusion takes place. Instead the bphase serves as point sources or sinks, the composition of which arehomogeneous and in equilibrium with the surrounding gphase in each point in the system. They usethese assumptions to derive a modified diffusion equation for the average composition c# of gand bina differential volume element›c#›t5››xD