Notes on Diffusion with Volume Change

We explore some fundamental aspects of diffusion in a binary (chemical species A and B) system. The ultimate objective is to better understand the relationship among the various treatments of a one-dimensional diffusion couple when there is volume change on mixing. We begin by examining the relationships among diffusion fluxes that are true irrespective of the nature of constitutive equations. We then introduce constitutive equations and relate the various diffusivities therein. By means of an explicit transformation along the lines suggested by den Broeder [1], we transform the Eulerian description for a one dimensional diffusion couple with volume change to a Lagrangian description, thus eliminating the explicit appearance of a convective term. The resulting equation can be solved by the BoltzmannMatano method [2,3]. We deduce by purely analytical means the formulae of Wagner [4] for the diffusivity and for the location of the Matano interface. We develop explicit formulae for the movement of this Wagner-Matano interface with respect to the ends of the diffusion couple. We show that the ends of the couple and the Wagner-Matano interface move with respect to one another by amounts that are proportional to √ t where t is time. We then relate this solution to an alternative description by Sauer and Freise [5] and show explicitly the relationship of the Wagner-Matano interface to the two Matano planes of Sauer and Freise which are fixed with respect to either end of the diffusion couple.