Stability of a nematic mesophase with internal degrees of freedom Analysis in terms of catastrophe theory

Abstract

A model of a nematic mesophase with internal degrees of freedom is developed using the variational principle by means of catastrophe theory methods. It is shown that the free energy of the model being a function of two order and three control parameters is represented in the form of the superposition of local potentials corresponding to elementary fold and swallow tail catastrophes according to Thom's classification. All of the physical conclusions obtained are a consequence of this result. By means of catastrophe theory a bifurcation set (separatrix) of the model is built which divides the control parameters namely the effective molecular length (∊), their rigidity (γ) and the dimensionless temperature (t) of the system space into six non-intersecting parts each parametrizing qualitatively similar potentials. The lower and upper temperature boundaries of the isotropic liquid and orientationally ordered states, respectively, are determined as sub-sets of the bifurcation set in the control parameter space. On the basis of numerical analysis and catastrophe theory the Maxwell set and all the fundamentally different phase diagrams of the model in coordinates ‘t-γ, t-∊, ∊-γ' have been constructed. It is shown that triple and terminal critical points can be realized in the phase diagrams. Physical critical manifolds of the model in variables ∊-γ-<P ₂> and ∊-γ-x are built (<P ₂> and x are orientational order and conformational disorder parameters, respectively). Analysis of all of the topologically different phase diagrams of the system has been performed.