Phenomenological Model for Smectic C Phases With Separate Tilt Angles for the Layer Thickness and the Optical Axis

Abstract

A simple, phenomenological model is given which may describe the smectic A-smectic C transition in the case where the molecules make an appreciable angle with the layer normal in both the A and C phases. X-ray experiments by several workers on a number of compounds indicate that this case may be quite common. The model is based on an earlier suggestion that the essential feature in these A to C transitions is the onset of long range order in the molecular tilt directions. The free energy is expressed as a function of the layer tilt angle ε (which determines the layer thickness) and the ratio r = α/ε, where α is the optical tilt angle. The quantity r (0 ≤ r ≤ 1) is a measure of the order of the molecular tilt directions. When ε is eliminated (which corresponds to considering the system at fixed pressure instead of fixed layer thickness), it is found that the transition can be either first or second order, depending on the model parameters. For the first order transition the approximate temperature variation r ∼ (γ ΔT/D′)^1/4, for r ² in the range (Δr)² ≪ r ² ≤ 1/4, is obtained—where Δr is the discontinuity in r. D′ ∼ 5/12, γ is a parameter, ΔT = T _o - T, and T _o is the transition temperature. The 1/4 power in this result leads to a very rapid increase of r and α near the transition. Data of Johnson and Saupe suggests γ ∼ 1/10°K⁻¹. For the second order transition we obtain r ∼ (γΔT/2Ψ)^1/2 for r ² ≪ Ψ, and r ∼ (γT/D′)^1/4 for r ² ≪ Ψ, where Ψ is a parameter. The model predicts a strong discontinuity in the dilation modulus, with the ratio:

Such a discontinuity has been deduced by Johnson and Saupe for two compounds from experimental data on undulations. The latent heat per mole L for the first order transition is given by L = γT _o(Δr)² RT_o (R = gas constant). Assuming ε ∼ 15°. T _o ∼ 350°K and γ ∼ 1/10°K⁻¹, the discontinuity Δα in the optical tilt angle can be estimated from this relation as Δα ∼ 0.5 × (L/20)^1/2, where Δα is in degrees and L in cal/mol.