Shape transition for preferred sidewall orientations

Suppose that only certain discrete values of slope of the lateral surface of the island are possible. In this subsection, implications of this assumption are examined for circumstances in which more than one such preferred lateral face orientation exists. The surface energy density of the substrate surface or flat growth surface is 70. Once it is nucleated at a certain aspect ratio, say ai, the volume V of an island continues to increase, due perhaps to deposition or coarsening, but with fixed aspect ratio aj. The surface energy density of the orientation is 71. Within the context of the model of cone-shaped islands, the free energy per unit volume of material in the island is 

The foregoing discussion is based on the assumption that the only possible island shapes are truncated solid cones with any of a discrete set of aspect ratios. The transitions from one to another aspect ratio implied by this simple picture are abrupt. A more realistic situation involving gradual transition might emerge if it is assumed that the islands take on shapes with surface orientation constrained by the slopes ai and 02, but that these orientations can coexist. For example, an intermediate shape might consist 

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