Modelling polycrystalline solidification using phase field theory

We review recent advances made in the phase field modelling of polycrystalline solidification. Areas covered include the development of theory from early approaches that allow for only a few crystal orientations, to the latest models relying on a continuous orientation field and a free energy functional that is invariant to the rotation of the laboratory frame. We discuss a variety of phenomena, including homogeneous nucleation and competitive growth of crystalline particles having different crystal orientations, the kinetics of crystallization, grain boundary dynamics, and the formation of complex polycrystalline growth morphologies including disordered ('dizzy') dendrites, spherulites., fractal-like polycrystalline aggregates, etc. Finally, we extend the approach by incorporating walls, and explore phenomena such as heterogeneous nucleation, particle-front interaction, and solidification in confined geometries (in channels or porous media).