Final Report Summary - PHASEFIELD (Approximations to dynamic density functional theory - phase field simulations on atomic scales) Shapes of complex geometry can be found in our natural environment, in man-made objects and materials, and in biological systems. Examples are snowflakes, crack patterns, material microstructures, solid tumors or vein networks in plant leaves. Such shapes have a huge practical importance since there is often a direct link between geometry and physical properties or biological functions. The time evolution of these shapes can be described using complex free boundary problems. Phase field models which approximate these problems have seen an explosive growth in recent years, particularly in the study of solidification and solid state phenomena. Most phase field models employ a similar approach: one or more phenomenological order parameter equations are coupled to one or more diffusion fields. Dynamical evolution of these fields are constructed by gradient flows of a phenomenological free energy, which lead to systems of nonlinear parabolic partial differential equations. The method has been applied to a wide variety of both physical and biological phenomena including alloy solidification, commensurate-incommensurate transitions, spherulites in polymer crystallization, electromigration on crystal surfaces, crack propagation, dislocation dynamics, self-assembly of nanostructures in epitaxial films, meandering and bunching of steps on vicinal surfaces, electrostatically induced pattern formation in thin polymer films, raft formation and pearling instability in membranes and vesicles, reaction-diffusion in cells and on cell membranes, and heart dynamics. One of the main challenges in the field is to construct models which encompass the complexity of practically relevant materials or biological systems to make quantitatively accurate predictions but on the other side are mathematically simple enough to be solved on physically realistic time and length scales. A limitation of the phase field models to achieve this goal is that that they are formulated in terms of fields which are spatially uniform in a single phase region in equilibrium. This precludes phenomena that arise from the periodic symmetries inherent in crystalline phases, including elastic and plastic deformation, anisotropy and multiple grain orientations. A way around this problem has been to couple the order parameter fields with auxiliary fields describing such features as the density of dislocations, continuum stress and strain fields and the crystal grain orientation, which makes the model very complicated and contradicts the goal to have a simple enough model to be solved on scales of practical use. A recent innovation has been the phase field crystal model introduced by Elder et al. Instead of a phenomenological phase field variable the quantity of interest is the atomic number density. This formulation has made it possible to incorporate the kinetics of phase transformations with properties of solids that arise due to their periodic structure. This includes elastic strain, topological defects, vacancy diffusion and polycrystalline grain boundary interactions. The appealing feature of the phase field crystal model is its connection with classical density functional theory. This makes it possible to develop different classes of phase field crystal models whose form and parameters are microscopically motivated. Another advantage of phase field crystal models is the potential of calculating certain material parameters that enter classical phase field models and to directly derive new thermodynamically consistent phase field type models from fundamental microscopic theories, which self-consistently incorporate atomic-scale features. Such an approach will have several important applications in microstructure modeling. The objectives of the joint exchange program were the● verification of the approximations made to derive the phase field crystal model from classical density functional theory ● development of efficient numerical tools to solve the higher order non linear phase field crystal model on time and spatial scales of interest ● derivation of multiscale models which combine the phase field crystal and classical phase field models ● extention of the phase field crystal methodology to new fields, e.g. in microfluidics and biophysics ● application of the model to industrial problems in materials science, nanotechnology and biology. Besides the application in industrial problems, all objectives could be achieved. This today allow to solve atomistic models on diffusive time scales, orders of magnitude larger than possible with molecular dynamic simulations. The phase field crystal model is today well established and has a large impact on microstructure modeling. This could only be achieved by interdisciplinary research, including theoretical physics, mathematics, computer science and engineering. The joint exchange program allowed to directly combine state-of-the-art modeling with advanced numerical algorithms, the latest developments in high-performance-computing and important applications.
