Analytical solution for interdiffusion in multicomponent systems and its application in high entropy alloys

Abstract

In the present work, an analytical solution is derived for the diffusion equation as applicable to a multicomponent diffusion couple with a constant set of interdiffusion coefficients. The multicomponent diffusion equation was first decoupled using similarity transformation of the interdiffusivity matrix. The decoupled equation was then solved for the boundary conditions pertaining to an infinite diffusion couple, in general, containing n diffusing components by using Laplace transform. It is shown that the analytical solution derived here deduces to the forms specifically applicable to binary and ternary systems available in the literature. The analytical solutions were then applied to reproduce the concentration profiles developed in experimental diffusion couples prepared with quaternary Fe–Ni–Co–Cr and quinary Fe–Ni–Co–Cr–Mn alloys. The determination of the sets of 9 quaternary and 16 quinary interdiffusion coefficients in the said couple was done experimentally using Kirladly’s approach and was independent of the closed form solutions derived here. The constant sets of quaternary and quinary interdiffusion coefficients determined at approximately equi-molar compositions in the two systems were used in the present work to reproduce the concentration profiles in these couples, having reasonably large differences in the terminal alloy concentrations. It was found that the concentration profiles calculated using the solutions developed here agree very well with the experimental profiles developed in all three quaternary and all four quinary couples. The excellent agreement between the experimental and theoretical profiles not only validates the closed form solution developed here but also highlights the constancy of interdiffusivities over reasonably large composition ranges in such systems. This is significant as it enables the use of analytical solutions of the diffusion equation for modeling various processes in such concentrated multicomponent alloys.