Variational Models for Self-Assembly and Pattern Formation: Analysis and Computation

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Self-assembly, a process whereby a disordered system of components forms an organized, structured pattern solely as a consequence of component interactions, is both ubiquitous in nature and important for the synthesis of many designer materials. Many systems which exhibit self-assembly and pattern formation may be viewed as energy-driven; therefore, one can successfully capture and analyze the phenomena via minimization of some, often postulated, free energy defined over all possible configurations. The resulting variational problems involve highly non-trivial energy landscapes and are mathematically and computationally very challenging. In this proposal we will study three such variational models (and connections between them) from the point of view of mathematical analysis and computation. A common theme will be the focus on nonlocal problems and global techniques, i.e. those whose vision extends beyond the local study of critical points and the analysis of their stability. ***The first variational model is a nonlocal perturbation of Coulombic-type to the well-known Ginzburg-Landau/Cahn-Hilliard free energy. It is the simplest model for self-assembly of diblock copolymers. The rich and complex energy landscape, particularly in three space dimensions, has many metastable states, and rigorous analytical results on the structure of minimizers are scarce. We will combine analytical information about the functional with numerical methods to gain insight into the energy landscape, and access states of lower energy. For example, we will develop methods for assessing whether or not a particular computed metastable state is close to a global minimizer. ***The second variational model is a purely geometric and finite-dimensional paradigm for self-assembly whereby one generalizes the notion of centroidal Voronoi tessellations for point generators to general shapes. Using a level set formulation, we a priori fix the geometry for the component structures, and consider self-assembly entirely dictated by distance functions. We have recently produced a novel fast algorithm for the simulation of such self-assemblies of rigid shapes (generalized centroidal Voronoi tessellations) and will now use this method to empirically explore the non-convex energy landscape. We will also extend our distance function formulation to produce a simple algorithm for the packing of general shapes. ***The third model is based upon functionals defined over densities and measures consisting of an interaction kernel which is repulsive at short range but attractive at long ranges. These functionals are directly connected to a class of self-assembly/aggregation models which recently have received much attention. We will explore questions of existence and properties of global and local minimizers.**