Fokker-Planck-Equation Approach to Flow Alignment in Liquid Crystals

Abstract

A nonlinear inhomogeneous relaxation equation for the alignment tensor is derived from a Fokker-Planck-equation for the orientational distribution function where torques exerted by a molecular field and by the gradient of the flow velocity are taken into account. Phenomenological coefficients characterizing the flow alignment in the isotropic and in the nematic phases are related to molecular parameters.Theoretical results for the flow alignment in the isotropic and nematic phases of liquid crystals are presented. The point of departure is a FokkerPlanck equation for the orientational distribution function. The inter-particle correlation is taken into account through an internal molecular field. An equation of motion for the alignment tensor is derived. It is similar to a phenomenological equation previously obtained 1 within the framework of irreversible thermodynamics. Now, however, the characteristic coefficients are related to "molecular" parameters and, in the nematic phase, to the magnitude of the order parameter. Applications to the isotropic phase (pretransitional behavior), to the nematic phase (dependence of the flow alignment angle on the order parameter), and the interrelation between flow alignment in both phases are discussed. Fokker-Planck EquationA liquid composed of (effectively) axisymmetric particles is considered. The unit vector parallel to the figure axis of a particle is denoted by 11. The average ( W) of a function W -W(M) is given by (W) = f T / d 2 u where / = /(/, 11) is the orientational probability distribution function normalized according to //d 2 « = l.The alignment associated with the birefringence in the isotropic phase and also with the spontaneous order in the nematic phase is the second rank tensor a=(^>) with & = Here TTT indicates the symmetric traceless part of a tensor. A convenient choice for the numerical factor £ is £ = Vty . This implies = 5. In the special case of an unaxial alignment, a can be written as 2L=V\a tin where the director II is a spacefixed unit vector. The scalar order parameter a is given by a = ]/ § £ (P 2 (n-u)), P 2 (z) = f (x 2 -I) ; S= (P 2 ) is the order parameter used by Maier and Saupe 2 . An equation of motion for a will be derived from a Fokker-Planck-equation for / to be stated below.With the help of the differential operator JL = MX 3/3u, the Fokker-Planck-equation assumed to govern / is written asThe second term describes the change of / due to an average angular velocity co of the molecules. In the following, co = g rot V is used where V is the average flow velocity of the liquid. The positive quantity w is an orientational relaxation coefficient (often referred to as orientational diffusion coefficient). The term in Eq. (1) containing the tensor F = 7 1-1 7 1 * a + (6 w)x G, G = vr ,is due to the torques exerted on the particles by an internal molecular field and by the velocity gradient G; T is the temperature of the liquid. The characteristic temperature T* is defined such that the energy Ji associated with the molecular alignment is ...

DC-conductivity testing combined with photometry for measuring fibre orientations in SFRC

“…Mesoscopic continuum theory introduces new variables as, for example, the mesoscopic distribution function, which in our case is a statistical orientation distribution function. It was developed about 50 years ago for LCs (rod-like particles) [33,34,35]. The orientation distribution function (ODF) defines a probability of finding a rod-like particle between the given angles on a sphere.…”

Stability of equilibria of nematic liquid crystalline polymers

“…Since both (q (1) 1 , q (1) 2 ) and (q (2) 1 , q (2) 2 ) satisfy (22), Δρ(m) must satisfy (19), which means…”

“…One of the most famous kinetic models for polymer liquid crystals was developed by Doi and Edwards [6] in 1980s and Hess [19]. The crucial idea of the Doi theory is to model the polymeric molecules as a suspension of rigid rodlike nematogens and then describe the ensemble with an orientational probability distribution function.…”

Inertial Manifolds for a Smoluchowski Equation on the Unit Sphere

“…The Smoluchowski equation was first proposed in the works of Doi [10] and Hess [17] as a dynamical model for nematic liquid crystalline polymers. In this paper, we study a version of the equation in which the interaction with the ambient flow is neglected in which case the equation is a gradient system with the free energy as the Lyapunov functional.…”