Elasticity of smectic liquid crystals with in-plane orientational order and dispiration asymmetry

The Nelson-Peliti formulation of the elasticity theory of isolated fluid membranes with orientational order emphasizes the interplay between geometry, topology, and thermal fluctuations. Fluid layers of lamellar liquid crystals such as smectic-C, hexatic smectics, and smectic-C∗ are endowed with in-plane orientational order. We extend the Nelson-Peliti formulation so as to bring these smectics within its ambit. Using the elasticity theory of smectics-C∗, we show that positive and negative dispirations (topological defects in Smectic-C∗ liquid crystals) with strengths of equal magnitude have disparate energies - a result that is amenable to experimental tests.