A hydrodynamic theory for solutions of nonhomogeneous nematic liquid crystalline polymers with density variations

1 I n t r o d u c t i o n Liquid crystals of variety of molecular configurations may form the nematic phase, in which an orientational order, but no translational order, exists [1, 2]. These include the two drastically different configurations: rodlike and discotic liquid crystals. Most of the hydrodynan~ical theories formulated for liquid crystal materials are based on the rodlike molecules, whic theo tals, base Edw brac logi high thou liqu uid TR liqu a ph som to b a m [31] rigo In rigi be mer bee har pha Tak ior ing solu 1 h include the well known Leslie-Ericksen (LE) ry[3], suitable to low molar weight liquid crysthe Doi kinetic theory[4] and a variety of tensor d theories such as the Hand's theory[5], Beris and ards' (BE) theory formulated through Poisson kets[6], and Tsuji and Rey's (TR) phenomenocal theory[7], etc., perceived to be applicable to molar weight liquid crystalline polymers. Algh the LE theory was first developed for rodlike id crystals, it has also been applied to discotic liqcrystals [8, 9]. Recently, Singh and Rey used the theory to model homogeneous flows of discotic id crystalline polymers by reversing the sign of enomenological "shape parameter" and showed e promising results[10]. This approach appears e not only convenient, but also reasonable from olecular point of view. The theory developed in aimed at addressing the concerns and provide a rous justification for the convenient practice. the theory, the LCP molecules are modeled as d spheroids of equal size so that the theory could used to model a variety of configurations of polyic liquid crystal molecules. This approach has n undertaken by several pioneers in the past. Isia studied the effect of the spheroidal shape on the se transition behavior of colloidal solutions[Ill. serman-Krozer and Ziabicki studied the behavof polymer solutions in a velocity field by treatpolymer molecules as rigid ellipsoids in dilute tions[12]. In Helfrich's molecular theory for neCopyright © 2002 by ASME m e t t g t atic liquid crystals, the molecules are treated as qually and rigidly oriented ellipsoids[13]. In an effort o address the relationship between the Doi kinetic heory and the Leslie-Ericksen theory, Kuzuu and Doi eneralized the Doi theory for homogeneous LCPs o account for the finite aspect ratio of spheroidal molecules[14] and gave the Leslie viscosity coefficients in terms of the uniaxial order parameter and a few physical parameters in the molecular theory, including the aspect ratio of the spheroid. Baalss and Hess also treated liquid crystal molecules as spheroids in their liquid crystal theory[15]. Baalss and Hess' theory predicts the liquid crystal is always flow aligning which has since been proven to be limited since tumbling has been observed in many LC flows. On the other hand, the Kuzuu and Doi theory handles both flow aligning and tumbling at different aspect ratios and polymer concentrations. The theory developed in [31] extends the Kuzuu and Doi theory to flowing systems of nonhomogeneous liquid crystalline polymers by considering the long range elastic interaction through an extended anisotropic intermolecular potential. It also generalizes the existing Marrucci-Greco theory to a variety of spheroidal LCP configurations through a shape parameter. However, the translational diffusion was neglected in the study for highlighting the effect of the molecular shape and the anisotropic elasticity. As we all know, however, that the spatial nonhomogeneous structure of LCPs is also correlated to the translational diffusion of LCP molecules. So, for completeness, a theory for nonhomogeneous LCPs must also account for the translational diffusion. This paper aims at adding the effect of the translational diffusion to the previous theory to explore the impact of the translational diffusion to the, intermolecular potential, Smoluchowski equation and the stress tensor. The rest of the paper consists of the derivation of the intermolecular potential, the Smoluchowski equation, the elastic stress tensor and the proof of the second law of thermodynamic theory. 2 K i n e t i c t h e o r y for L C P s o f s p h e r o i d a l m o l e c u l e s We first extend the intermolecular potential developed in [31] that accounts for the intermediate to long range molecular interaction for liquid crystalline polymers of the spheroidal configuration with finite aspect ratios and derive one of its approximations through the gradient expansion of the number density function (defined below) [16] Then, we extend the Smoluchowski equation in the Doi kinetic theory for rodlik of the fusion the vi theory isothe