Stability of equilibria of nematic liquid crystalline polymers

Abstract

We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability with respect to perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is below a critical value b * (6.7314863965), the only equilibrium state is isotropic and it is stable; when b * < b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable oblate state.

Phase Transitions and Macroscopic Limits in a BGK Model of Body-Attitude Coordination

“…In particular, since (5) is also a free energy in the Fokker-Planck case, this analysis shows the instability of equilibria of the Fokker-Planck of the form ρ M D when D is of one of the unstable equilibria of (28a) described in Theorem 8. This technique is similar to the one which was used in Zhou and Wang (2011) in the case of 3D polymers. However, it does not provide global or local convergence of the solution of the Fokker-Planck equation towards an equilibrium.…”

“…The links between this topological structure and the Riemannian structure (detailed in Sect. 3 and Appendix B) will be the key to reduce the problem to a form that shares structural properties with the models of nematic alignment of polymers, studied in a completely different context to model liquid crystals (Han et al 2015;Wang and Hoffman 2008;Zhou and Wang 2011;Ball and Majumdar 2010;Ball 2017). These two worlds will be formally linked through the isomorphism between SO 3 (R) and the group of unit quaternions detailed in Sect.…”

“…4, we will describe, depending on the density, all the possible equilibria of the system. We will use the tools developed to mathematically study the alignment of polymers (Wang and Hoffman 2008;Zhou and Wang 2011). In Sect.…”

Phase transitions and macroscopic limits in a BGK model of body-attitude coordination

“…In particular, the rotation group SO 3 (R) is a compact Lie group, endowed with a Haar measure. The links between this topological structure and the Riemannian structure (detailed in Section 3 and Appendix B) will be the key to reduce the problem to a form that shares structural properties with the models of nematic alignment of polymers, studied in a completely different context to model liquid crystals [1,2,32,55,57]. These two worlds will be formally linked through the isomorphism between SO 3 (R) and the group of unit quaternions detailed in Section 4.2 and Appendix A.…”

“…In Section 4, we will describe, depending on the density, all the possible equilibria of the system. We will use the tools developed to mathematically study the alignment of polymers [55,57]. In Section 5 we will describe the asymptotic behaviour of the system and in particular which equilibria are attained, leading to a self-organised dynamics or not.…”