Microstructure and nanostructure evolution in complex materials is a central issue in materials science, since it is critical in determining the stability and other material properties. The mathematical study of microstructure evolution is challenging, since it entails length scales ranging from atomistic to macroscopic, it is driven by non-equilibrium processes, and often it is influenced by complex mechanical and electrostatic interactions between the various components of the materials.
The goal of this summer school is to introduce graduate students and researchers to the state of the art in the theory of microstructure evolution and dynamics. We will focus primarily on continuum phase field theory, in particular on the Cahn-Hilliard model as well as on some of its many extensions, and its relation to particle methods and atomistic details. The interplay between modeling, analytic theory, and numerical treatment will be emphasized. The mini-courses will also discuss applications of the theory, e.g. in the design of nanostructured electrodes for batteries or polyelectrolyte membranes for fuel cells.