Role of instabilities in determination of the shapes

The shapes of growing crystals are determined by an interplay of complex processes that include transport of energy and matter through bulk phases, capillarity-related processes that determine local equilibrium conditions at the crystal-nutrient interface, and non-equilibrium kinetic processes that take place locally to that interface. A mathematical description of crystal growth results in a free boundary problem (the free boundary being the location and shape of the interface as a function of time) for thermal and compositional fields, governed by partial differential equations and subject to boundary conditions on the free boundary and at the extremities of the system. This free boundary problem is not only difficult to solve but often displays multiple solutions. One attempts to select from among the possible solutions by invoking principles of dynamic stability. By treating the above three processes in pairs, we show that the interplay of transport and interface kinetics leads to a consideration of facet instability; the interplay of diffusive transport and capillarity to morphological instability; and the interplay of capillarity and interface kinetics to a consideration of corner instability and to interesting modifications of the classical work of Frank that describes the evolution of crystal growth shapes under conditions for which capillary effects are negligible. Theories that treat all three processes simultaneously are rare, but numerical computations based on the boundary integral technique and the phase field model are beginning to yield interesting results, especially with respect to dendritic growth.