Critical Elastic Coefficient of Liquid Crystals and Hysteresis

P. G. de Gennes predicted the analogies between the effect of the elastic coefficients to liquid crystals and the effect of applied magnetic fields to superconductors, and predicted that all elastic coefficients diverge to infinity at smectic-C to nematic transition. One would expect quantitative comparison in the analogies. In the case of equal elastic coefficients (K ₁ = K ₂ = K ₃ = K), we define the critical value K ^c of the elastic coefficients and make comparison of it with the upper critical magnetic field H _{C 3} for type II superconductors. We classify the smectic liquid crystals into subcritical, critical and supercritical cases according to the Ginzburg-Landau parameter κ, the wave number q and the boundary value of the director at the surface. We show that in the subcritical case the liquid crystal does not undergo phase transition; and in the supercritical case both phase transition and hysteresis occur. The prediction of de Gennes is true in the critical case where μ _π (u ₀, q) = κ ² and K ^c = + ∞.