Temporal Evolution of Two-Phase Microstructures

6 Temporal Evolution of Two-Phase Microstructures Though the calculation given above was both fun and interesting, it suffered from a number of defects. First, the class of intermediate shapes was entirely restrictive 

Diffusional Dynamics. In the present analysis, we consider the explicit treatment of diffusion as the mechanism of interfacial migration, with the rate of such diffusion influenced in turn by the elastic fields implied by a particular particle shape. As a result, the problem of the temporal evolution of particles is posed as a coupled problem in elasticity and mass transport. The particles are given full scope to develop in any way they want. 

Our discussion follows in a conceptual way (with different notation) that of Voorhees et al. (1992). These statements of equilibrium are imposed as follows. 

Mechanical equilibrium is imposed by demanding that the elastic fields satisfy 

We note that just as with our analytic solution for the Eshelby inclusion, the equilibrium equations within the inclusion will have a source term (i.e. an effective body force field) associated with the eigenstrain describing that inclusion. In addition, we require continuity of both displacements, Uj = Uout, and tractions, tj = tout, at the interface between the inclusion and the surrounding matrix. The point of contact between the elastic problem and the diffusion problem is the observation that the interfacial concentration depends upon the instantaneous elastic fields. These interfacial concentrations, in turn, serve as boundary conditions for our treatment of the concentration fields which permits the update of our particle geometries in a way that will be shown below. The concentration at the interface between the inclusion and the matrix may be written as 

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