Geometrical methods in soft condensed-matter physics

We propose a geometrical picture of understanding the thermodynamic and elastic properties of charged and fuzzy colloidal crystals, by analogy to foams, as well as perform a computational exercise to confirm a new universality class for long polymers with non-trivial topologies. By the foam analogy, we relate the problem of thermodynamic stability to the Kelvin\u27s problem of partitioning space into equal-volume cells of minimal surface area. In particular, we consider the face-centered cubic (FCC), body-centered cubic (BCC) and the β-tungsten (A15) lattices. We write down the free energy of these solid phases directly in terms of geometric and microscopic parameters of the system, and we derive the theoretical phase diagram of an experimental charged colloidal systems [Phys. Rev. Lett. 62, 1524 (1989)]. By considering deformations to the foam cells, we also compute the cubic elastic constants of these three lattices for charged and fuzzy colloids. In the polymer problem, we consider the critical behavior of polymers much longer than their persistence length, with built-in topological constraint in the form of Fuller\u27s relation: Lk = Tw + Wr in a Θ-solvent. We map the problem to the three-dimensional symmetric U( N)-Chern Simons theory as N → 0. To two-loop order, we find a new scaling regime for the topologically constrained polymers, with critical exponents that depend on the chemical potential for writhe which gives way to a fluctuation-induced first-order transition