Surface waviness due to misfit dislocations

For a constant surface energy density Us = 7, the surface chemical potential X = U — jK is nonuniform in this configuration. The objective is to identify nearby shapes for which the chemical potential is again uniform. 

Let r] x) denote the local perturbation in surface shape, so that the local perturbed film thickness is h + r] x), and assume that the perturbation is also periodic in x with period p. For fluctuations in shape that are small in amplitude compared to p, the fluctuation r] x) is determined by the condition of uniform chemical potential along the surface, which is assured by the differential equation 

The value of the constant incorporated into (8.120) is dictated by the condition of periodicity. The resulting ordinary differential equation is readily solved by numerical methods, with the constants of integration being determined by the constraint of periodicity and conservation of mass. 

If all lengths are normalized by the initial film thickness h, then (8.120) can be written in terms of the three nondimensional parameters 

Results are included here for only two sets of values of these parameters, namely, p = 2, 7=1, 6=1 and p = 8, 7 = 1,6 = 4. Note that the spatial average of elastic strain is the same in the two cases. However, this is accomplished by placing single misfit dislocations at intervals of 2h on the interface in the first case and by placing clusters of four misfit dislocations at intervals of h on the interface in the second case. 

---