Nonparallel solutions of extended nematic polymers under an external field

Acta Mathematica Scientia

2011

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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.

Stability of equilibria of nematic liquid crystalline polymers

Acta Mathematica Scientia

2011

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Elongational perturbations on nematic liquid crystal polymers under a weak shear

2007

Physics of Fluids

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The two-dimensional Smoluchowski equation is employed to study the effect of elongational perturbations on nematic liquid crystal polymers under a weak shear. We use the multiscale asymptotic analysis to show that ͑1͒ when the elongational perturbation is small relative to the weak shear, the orientational probability density function ͑pdf͒ tumbles periodically only in an intermediate range of polymer concentration; outside this intermediate range ͑i.e., for very small and very large polymer concentration͒ the orientational pdf converges to a steady state and there is no tumbling. ͑2͒ When the elongational perturbation is about 20% of the shear rate or larger, the intermediate range of tumbling disappears and the orientational pdf always converges to a steady state regardless of the polymer concentration. Our theoretical predictions are consistent with various earlier results based on the Leslie-Ericksen theory ͓C.

Extendability of Equilibria of Nematic Polymers

Abstract and Applied Analysis

2008

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.