The cholesteric-smectic A transition

Instead of studying the cholesteric-smectic A transition from the analogy with superconductors, we present a direct investigation of the properties of this transition. The transition is of first order and a very simple relationship between the four basic lengths (correlation length, penetration length, layer spacing of the smectic structure, and pitch of the cholesteric structure) exists at the transition temperature. We calculate the wall energy separating a cholesteric region from a smectic region in two different configurations. For one of them, we recover the definition of the type I (positive wall energy) and type II (negative wall energy) smectics A. We do not find a specific value of the Ginzburg-Landau parameter k(c) which gives the distinction between the two types of smectics; k(c) is a function of the product of the perpendicular correlation length and the wave vector of the smectic density modulation. We recover the well known value from the superconductor analogy, k(c) = 1/root 2, only if this product is large and the rotation of the director inside the wall is small. For the second configuration, the wall energy is always positive and there is no analogy with superconductors. Finally, we discuss briefly two different possibilities of mixed phases.