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

Abstract

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 dipoledipole 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.

From kinetic to fluid models of liquid crystals by the moment method

“…As η increases, G ηAΩ shows increasing order evidenced by the increase of the order parameter S 2 . Now, we have the following This proposition is a consequence of the result of Wang and Hoffman [56] which will be recalled in Section 4. With this, we formulate the following conjecture, which has been verified in dimension n = 2 [28], n = 3 [1,29,47,48] and n = 4 [31].…”

“…Indeed, we anticipate that only stable equilibria can lead to a long time dynamics described by hydrodynamic equations. First, we should note that local equilibria are known in any dimension n [56] (see also [15,28,49] for the case n = 2 and [1,14,29,47,48,61,63] for the case n = 3). However, the stability of these equilibria is not known for general dimension n but only for n = 2 [28], n = 3 [1,29,47,48] and n = 4 [31].…”

“…To complete the characterization of the equilibria, we need to solve the compatibility equation (78). As pointed out above, this has been done in any dimension n in [56] (see also [49] for n = 2 and [1,29,47,48,63] for n = 3). This result is summarized without proof in the following lemma Lemma 4.4 (Final characterization of the equilibria [56]).…”

“…As pointed out above, this has been done in any dimension n in [56] (see also [49] for n = 2 and [1,29,47,48,63] for n = 3). This result is summarized without proof in the following lemma Lemma 4.4 (Final characterization of the equilibria [56]). Let f be an an equilibrium.…”

“…1b). In all cases, the equation ρ n (λ) = ρ has a solution if and only if ρ ≥ ρ * and this solution is unique in the case n = 2 while, in the case n ≥ 3, there are two solutions if ρ ∈ (ρ * , ρ n (0)], and one solution if ρ ∈ {ρ * } ∪ (ρ n (0), ∞) (see [56] for details).…”