de Cahn – Hilliard equations incorporating elasticity : analysis and comparison to experiments

Quenching a multi-component alloy produces a supersaturated metastable solid which under annealing nucleates small randomly dispersed precipitates. The precipitate morphology then evolves by diffusional mass transport as the multi-phase mixture tends to minimize its energy. During this postphase-transformation morphological evolution, the phase fractions of the matrix and precipitates remain constant. This process is known as coarsening. Coarsening is driven by two contributions to the energy: interfacial and elastic. Given the initial small size of the precipitates, interfacial energy dominates the evolution in the beginning. However, as the particle size increases, the evolution is increasingly and eventually dominantly influenced by elastic energy arising from the difference in lattice parameters (i.e. difference in stress-free strains) between the phases. Consequently, the precipitates tend to align along specific crystallographic directions. Externally applied stresses also contribute to the energy of the system and significantly modify precipitate morphology.