Symmetry-breaking transitions in equilibrium shapes of coherent precipitates

We present a general approach for determining the equilibrium shape of isol ated, coherent, misfitting particles by minimizing the sum of elastic and i nterfacial energies using a synthesis of finite element and optimization te chniques. The generality derives from the fact that there is no restriction on the initial or final shape, or on the elastic moduli of the particle an d matrix, or on the nature of misfit. The particle shape is parametrized us ing a set of design variables which are the magnitudes of vectors from a re ference point inside the particle to points on the particle/matrix interfac e. We use a sequential quadratic programming approach to carry out the opti mization. Although this approach can be used to find equilibrium shapes of three-dimensional (3D) particles, we have presented the details of our form ulation for two-dimensional systems under plane strain conditions. For syst ems with cubic elastic anisotropy, the equilibrium shapes and their size de pendence are analyzed within the framework of symmetry-breaking shape trans itions. For systems with dilatational misfit, our results on shape transiti ons are in agreement with those in the literature. For systems with non-dil atational misfit, we obtain a symmetry-breaking shape transition that invol ves a loss of mirror symmetry normal to the x- and y-axes; small particles have this symmetry, while those beyond a critical size do not. With these r esults, we now have a comprehensive picture of symmetry-breaking transition s in two-dimensional systems driven by anisotropy in misfit and elastic mod uli. (C) 2000 Elsevier Science Ltd. All rights reserved.