Stress relaxation by grain boundary diffusion

Suppose that a mass transport process actually takes place in response to this configurational force. Mass flow along the free surface in the positive x—direction is represented by the flux jh x, t) and mass flow along the grain boundary in the positive y—direction is represented by jhivA)- No other transport processes are active. The objective of this example is to arrive at estimates of the fluxes by appeal to the functional (9.51) underlying the variational principle stated earlier in this section. 

In constructing an admissible mass flux field, it is assumed that the free surface of the grain remains essentially flat, corresponding to a relatively high surface energy compared to the interfacial energy of the grain boundary. 

The unstrained film thickness at time t is h t), where h 0) = ho- It is also assumed that the surface flux is S5rmmetric with respect to the center of the grain, with the flux being toward the nearer grain boundary at all points on the surface. A surface flux consistent with these characteristics is 

At the junction of the grain boundary and the free surface, mass conservation requires that the fluxes are related according to 

Under the assumptions adopted, the chemical potentials are spatially uniform over both the free surface and the grain boundary. The free surface chemical potential is denoted by Xh x,t) and is determined according to (8.8), while the grain boundary chemical potential is denoted by Xh y,t) and is determined according to (8.14). In terms of instantaneous grain dimensions, these are 

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The particular evolution phenomena in material systems considered in this chapter include the transition from a nominally flat surface to a wavy surface in a stressed solid, the spontaneous growth of epitaxial islands due to deposition of a material on a substrate with lattice mismatch, stress relaxation by grain boundary diffusion, the role of stress in altering compositional variations in solid solutions, and stress-assisted diffusion in the presence of an electric field or electromigration. 

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