Computation of Trajectories and Displacement Fields in a Three-Dimensional Ternary Diffusion Couple: Parabolic Transform Method

Abstract

The problem of Kirkendall’s trajectories in finite, three- and one-dimensional ternary diffusion couples is studied. By means of the parabolic transformation method, we calculate the solute field, the Kirkendall marker velocity, and displacement fields. The velocity field is generally continuous and can be integrated to obtain a displacement field that is continuous everywhere. Special features observed experimentally and reported in the literature are also studied: (i) multiple Kirkendall’s planes where markers placed on an initial compositional discontinuity of the diffusion couple evolve into two locations as a result of the initial distribution, (ii) multiple Kirkendall’s planes where markers placed on an initial compositional discontinuity of the diffusion couple move into two locations due to composition dependent mobilities, and (iii) a Kirkendall plane that coincides with the interphase interface. The details of the deformation (material trajectories) in these special situations are given using both methods and are discussed in terms of the stress-free strain rate associated with the Kirkendall effect. Our nonlinear transform generalizes the diagonalization method by Krishtal, Mokrov, Akimov, and Zakharov, whose transform of diffusivities was linear.