Exactly Solvable Models for Self-Assembly

Doctoral thesis, 2014

The field of self-assembly studies the spontaneous formation of order from preexisting components. It holds the promise of fabricating tomorrows materials and devices not by traditional methods, but by designing building blocks that will act as supramolecular ``atoms'' and form the desired structures without external input. Theoretical understanding in self-assembly has previously been achieved mainly through approximative and simulation-based methods. In contrast, this thesis shows that a range of many-particle models of interest in self-assembly can be mapped onto exactly solvable models from statistical mechanics. For such models the behavior of interest, be it ground states or the full partition function, can be obtained through analytical calculations, without approximations. The thesis consists of an introductory text and seven appended research papers. It introduces two analytical tools for use in self-assembly. First, the exactly solved spherical spin model is generalized to arbitrary isotropic interactions, various geometries, and multiple particle types. The resulting models are shown to admit exact solutions for their ground states and these turn out to predict pattern formation in corresponding systems. The theory is developed in Papers I, II, and VI, and used in Paper VII to design patchy colloids for self-assembly. Second, an exact design method for lattice self-assembly is presented in Papers III-V. Given a target lattice structure, it produces an isotropic potential that can be proven to have the desired lattice as its ground state. The method is used to design potentials that in Monte Carlo simulations cause self-assembly into various two- and three-dimensional lattices. In Paper V a method for simplifying designed potentials is discussed.