Landau Theory of Complex Ordered Phases

Abstract

Recently, a number of spherical packing phases belonging to the class of Frank-Kasper (F-K) phases have been observed in a wide range of soft matter systems, including block copolymers, ionic surfactants, liquid crystalline dendrimers, and giant surfactants. Although their emergence in such systems has been conjectured to be due to a competition between mesodomain sphericity and incompressibility, we lack a description of a precise and general mechanism underlying the formation of F-K phases in soft matter systems. In this work, we consider the two most common F-K phases found in soft matter systems, the σ and A15 phases, and study their stability in the context of a well-known Landau model known as the Landau-Brazovskii model. This model has been applied to systems ranging from block copolymers to liquid crystals. We find that the phase behavior of the Landau-Brazovskii model is controlled only by two parameters, rather than by three parameters, as was suggested by previous works. We also find that the Landau-Brazovksii phase diagram contains regions in which the σ or A15 phase is the most stable among a set of candidate phases. The fact that such a simple model can predict these complex phases provides some insight into the question of why the occurrence of the Frank-Kasper phases in soft matter is so widespread.