A unified view on the rotational symmetry of equilibiria of nematic polymers, dipolar nematic polymers, and polymers in higher dimensional space

Abstract

We study equilibrium states of the Smoluchowski equation for rigid, rod-like polymer ensembles. We start with several cases in the three dimensional space: a) nematic polymers where the only intermolecular interaction is the excluded volume effect, modeled using the Maier-Saupe potential, b) dipolar nematic polymers where the intermolecular interaction consists of the dipole- dipole potential and the Maier-Saupe potential, c) dipolar nematic polymers in the presence of a stretching elongational flow, and d) nematic polymers in higher dimensional space. For each of the cases a), b) and c), it has been established separately with various mathematical manipulations that all stable equilibrium states have rotational symmetry. In this study, we present a unified view of the rotational symmetry of cases a), b) and c). Specifically, in cases a), b) and c), the rotational symmetry is determined by a key inequality. The inequality, once established for case a), is extended elegantly to cases b) and c). Furthermore, this inequality is used in case d) to establish the rotational symmetry of equilibrium states of nematic polymers in higher dimensional space. In three dimensional space, rotational symmetry simply means axisymmetry. In higher dimensional space, rotational symmetry is more complex in structure. For example, in four dimensional space, rotational symmetry may be around a one dimensional sub-space (i.e., axisymmetry) or it may be around a two dimensional sub-space. Nevertheless, the rotational symmetry significantly simplifies the classification of equilibrium states. We calculate and present phase diagrams of nematic polymers in higher dimensional spaces.