Participating Network Members: J.W. Barrett (Imperial) [JWB], C.M. Elliott (Sussex) [CME], D. Kay (Sussex) [DAK], R. Nürnberg (Imperial) [RN], P. Plechac (Warwick) [PP], V. M. Styles (Sussex) [VMS].

Modelling morphological surface evolution and growth and its role in the understanding of microstructure is fundamental in material science. Typically, surface energy determines the laws of evolution for grain boundaries and the interfaces between phases. These naturally lead to curvature laws of motion, which in the canonical cases of grain boundary motion and surface diffusion are characterised by motion by mean curvature and the surface Laplacian of the mean curvature. Generalisations can include field equations in the bulk for heat transfer and elastic effects, anisotropic surface energies and multiphases. The models lead to complicated interface motion which involve topological change, many interfaces and triple junctions etc. The complexity of these problems has led to a burgeoning increase of research activity in phase field models in which sharp interfaces become diffuse interfaces spread over a transition region of width epsilon, say, separating values of a phase field or order parameter. One can take a rational thermodynamic perspective in deriving phase field models or view them as yielding computationally attractive methods for the sharp interface. To exploit the full power of the phase field approach for computing complex interface problems one needs an understanding of the mathematical structure of the equations in order to derive spatial and temporal discretisations, which model the surface energy, the temporal decrease in energy, and the diffuse interface. Furthermore, the resolution of the transition region is crucial for meaningful quantitative computations. Adaptivity involving fine meshes in the transition regions are a necessity.

The groups in Imperial and Sussex are at the forefront of the mathematical research in computational phase field models and their applications. In particular, the double (or multi-) obstacle phase field potential pioneered by these groups has had a significant impact on the success of phase field computations. Much work needs to be done to extend these techniques problems involving anisotropy, elastic energy and multi-species diffusion. In particular, we mention the degenerate double obstacle Cahn-Hilliard phase field systems and their use in approximating surface diffusion connected to electro-migration of voids and the formation of cracks [1].

Phase field equations of Allen-Cahn and Cahn-Hilliard type and with long-range interactions involving integro-differential operators have been derived from stochastic atomistic Ising models [3]. It is intended that this network will facilitate the interaction of the PDE based work of the Imperial/Sussex nodes with the stochastic coarse graining work of PP in Warwick. There is also the potential for developing computational tools in micromagnetics linking PDE based Ginzburg-Landau models to stochastic lattice models [2].

Diffuse interface approaches have also been introduced via Cahn-Hilliard Navier-Stokes fluids to approximate interfaces between immiscible fluids [4]. Part of the future research in this network will be to explore the approximation of these free boundaries for fluids using double obstacle Cahn-Hillard system coupled to the Navier-Stokes equations. Some recent progress on the analysis of discretisation has been made in Cambridge during the Computational Challenges in PDEs program by JWB and CME.

DAK and research students in Sussex are developing multigrid codes for dealing with Cahn-Hilliard systems, which are notoriously difficult to solve efficiently. There is also the possibility of developing multi-grid algorithms for stochastic lattice models in collaboration with the Warwick group and relating aspects of Theme I and III. The research of RN and VMS is strongly related to the applications of phase field models to electro-migration of voids, thin films and diffusion induced grain boundary motion.