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Phase Refining
The free energy of mixing can cause phase separation. The phase boundary energy causes coarsening. The following example demonstrates how the elastic energy in a film-substrate system can cause refining.
Say the film can form two domains, A and B. Domain A has a larger lattice constant than the substrate and would be under compression on the substrate. Domain B has a smaller lattice constant than the substrate and would be under tension on the substrate.
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When they alternate on the substrate, the two kinds of domains compensate: the elastic energy of the polydomain film on the substrate is lower than a monodomain film on the substrate. The mismatch in the polydomain still causes a fringe elastic field in the substrate, whose depth scales with the domain size.
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When the domain size is refined, the fringe field depth is reduced, and so is the elastic energy. Consequently, the elastic energy in the film-substrate composite tends to refine the domains. The refinement, however, adds more domain wall areas, which increases the total domain wall energy. That is, the domain wall energy tends to coarsen the domains. The elastic energy is mediated by a long range field. The domain wall energy is mediated by short range interactions among atoms at the domain wall. The two competing actions—refining and coarsening—can select an equilibrium domain size.
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The above example shows that the elasticity in a film-substrate system can cause refining, where the film forms multiple domains. On the other hand, it should be noted that bulk elastic misfit does not cause refining. Consider two bulk phases that have different atomic lattice constants. If the phase boundaries are coherent, the lattice constant misfit induces an elastic field. For simplicity, assume that both phases have cubic atomic lattices, with lattice constants  and  , respectively. The misfit strain is  . Let E be an elastic modulus. The elastic energy per unit volume scales as  , invariant with the average particle size.
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