We consider a discretized Maier–Saupe model for a polydisperse solution of micelles of axial symmetry. An exact mean-field solution is presented. The phase diagram displays a biaxial phase at small average anisometries and finite dispersion in size, and two opposite uniaxial nematic phases for large average anisometries. Relations to the experimental phase diagram for ternary mixtures are discussed.
We introduce shape fluctuations in a liquid-crystalline system by considering an elementary Maier-Saupe lattice model for a mixture of uniaxial and biaxial molecules. Shape variables are treated in the annealed (thermalized) limit. We analyze the thermodynamic properties of this system in terms of temperature T , concentration c of intrinsically biaxial molecules, and a parameter ∆ associated with the degree of biaxiality of the molecules. At the mean-field level, we use standard techniques of statistical mechanics to draw global phase diagrams, which are shown to display a rich structure, including uniaxial and biaxial nematic phases, a reentrant ordered region, and many distinct multicritical points. Also, we use the formalism to write an expansion of the free energy in order to make contact with the Landau-de Gennes theory of nematic phase transitions.
We analyze the global phase diagram of a Maier-Saupe lattice model with the inclusion of shape-disordered degrees of freedom to mimic a mixture of oblate and prolate molecules (discs and cylinders). In the neighborhood of a Landau multicritical point, solutions of the statistical problem can be written as a Landau-de Gennes expansion for the free energy. If the shape-disordered degrees of freedom are quenched, we confirm the existence of a biaxial nematic structure. If orientational and disorder degrees of freedom are allowed to thermalize, this biaxial solution becomes thermodynamically unstable. Also, we use a two-temperature formalism to mimic the presence of two distinct relaxation times, and show that a slight departure from complete thermalization is enough to stabilize a biaxial nematic phase.
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