Mean-field theory of phase transitions in liquid-crystalline polymers

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Abstract
We present a self‐consistent‐field structure for the thermodynamic description of concentrated solutions of liquid‐crystal polymers. The polymers are assumed to be locally stiff but capable of curvature over large distances. The formulation of the chain geometry is that of a wormlike polymer, but the final analysis is simpler than the Kratky–Porod form. The phase behavior is analyzed for a cylindrically symmetric mean‐field nematic ordering interaction. As expected, in two dimensions the system undergoes a second‐order phase transition and in three dimensions the nematic transition is first‐order uniaxial. The three relevant physical parameters which describe the phase behavior of the polymers are the molecular mass of the polymers, a measure of the chain stiffness and strength of the nematic interaction. The formulation in this paper is for infinitely long polymers. If the chain stiffness is measured by the bending energy ε, with u as the strength of the hardcore repulsion and U that of the soft interactions, we find the following relationship at the isotropic‐nematic transition:c c β c ε(u+Uβ c )=6. c c is the critical concentration and β c =1/k BT c is the inverse critical temperature. At the transition we find that the order parameter has a universal value of 0.25. We present universal plots for the order parameter and free energy of the system as a function of polymer concentration and temperature.

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