Flattening of Free Surfaces by Surface Diffusion

Consider a system composed of only one type of atom along with its equilibrium point defects, having an initial surface profile 

The diffusion potential of an atom at the surface is proportional to local surface curvature as demonstrated in Section 3.4. The curvature can be determined from Eq. 14.1 and is a function of x. The local diffusion potential produces boundary conditions for diffusion through the bulk or transport via the vapor phase. For surface diffusion, gradients in the diffusion potential produce fluxes along the surface. 

For surface diffusion, migration is constrained to a thin surface slab of thickness S, as described in Section 9.1. Because the surface is an efficient point-defect source, point defects may be assumed to be at their local equilibrium populations. 

2 Additional material regarding the mathematical descriptions of interfaces is given in Appendix C, particularly Eq. C.9. 

Therefore, combining the force-flux equation (Eq. 2.21) and the diffusion potential at a curved interface (Eq. 3.76), 

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