938 Macromolecules zyxwvu 1991,24, 938-941 Friction Tensor of Flexible Polymer Chains in Nematic Liquid Crystals Shi-Qing Wang Department zyxwvut of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio zyxwvuts 44106 Received March 12, 1990; Revised Manuscript Received August 13, 1990 ABSTRACT: Hydrodynamic properties of a flexible polymer chain in low molecular weight liquid crystals (LC) have been studied. The starting point is the derivation of the hydrodynamic Oseen tensor for aniso- tropic fluids that satisfy a modified Navier-Stokes equation. The nematics of LC breaks the spatial isotropy and leads to a necessary introduction of the friction tensor zyxwvu Z for polymers. The general expression for 2: is obtained as a formal perturbation expansion within the Kirkwood-Riseman theory, and the component corresponding to the polymer diffusive motion along the nematic direction is evaluated explicitly by using the renormalization group method and by considering the smallness of one of the Leslie coefficients of the nematic solvent. The result depends on the LC Miesowicz viscosities in a nontrivial way, yet the molecular weight dependence is still universal. I. Introduction Over the past several decades both equilibrium and transport properties of polymers in isotropic solvents have been extensively studied. Now many books that review experimental results as well as classical and modern theories of polymer solutions. The ideas of scaling laws2 and renormalization group theories4have been particularly successful in analyzing and rationalizing experimentally observed universal behaviors of uncharged linear flexible polymers in isotropic solutions. For polyelectrolytes, stiff chains, and polymers with complex architectures or chemical combinations (copolymers, random branched polymers, etc), their statistical mechanical theories are highly nontrivial even in the infinite-dilution limit. On the other hand, the non-Newtonian problem of a single linear flexible chain in a strong simple shear flow has not been settled despite enormous efforts made in the past 30 years starting with Rouse5 and Zimm.6 Furthermore, except for a few cases7-l0 little attention has been paid to polymer chains in solution with liquid crystals (LC) as anisotropic solvents. As more people are beginning to work on polymeric liquid crystals, it becomes important to obtain some basic understanding of conformational and dynamic properties of macromolecules in nematic LC. Conformations of flexible polymer chains in LC depend on microscopic interactions between the polymeric mono- mers and anisotropic molecules of LC. A molecular-level statistical mechanical theory of such interactions is not available, and how they change or invalidate scaling relations between equilibrium properties and the molecular weight is unknown. Thus it remains to be shown how an increase in the nematic strength would lead to a coil- collapse phase transition in the flexible polymer immersed in the anisotropic solvent. It is interesting and useful to ask whether there exist any good nematic solvents for flexible polymers in the sense that polymers take an expanded averaged shape in those good solvents. There is also the important question of how linear flexible polymers contribute to the elasticity of liquid crystals, i.e., how the Frank elastic constants depend upon the molecular weight and polymer concentration. Answers to these questions will undoubtedly increase our under- standing of polymers in their liquid crystalline phase. Rather than looking for solutions to the above- mentioned equilibrium problems of polymer-nematics, here we would like to study the dynamics of polymers in LC. We begin by giving a derivation of the modified 0024-9297/91/2224-0938$02.50/0 hydrodynamic Oseen tensor for an anisotropic fluid. Then the hydrodynamic interaction (HI) among polymer seg- ments is incorporated to evaluate the tensorial friction for the polymer as a function of the polymericmolecular weight and Leslie viscosities of the nematic liquid crystal. It is found that the dynamic exponent is unaffected by the solvent anisotropy although the HI parameter becomes a highly complicated function of the Miesowicz viscosity coefficients. The existence of a HI fixed point derived from our renormalization group analysis indicates the universality of dynamic properties of polymers in a low molecular weight liquid crystal. This suggests that the modification of Miesowicz viscosity coefficients of the liquid crystal dispersion due to flexible polymers should also be universal and can be looked at in a similar way. We will perform such an evaluation of these Miesowicz viscosities in a future publication, while the present work is merely a beginning of a series of investigations into the viscoelastic properties of liquid crystal dispersions of polymers. In section 11,the Leslie viscous stress tensor is introduced into the Stokes equation and the fluid propagator of this modified Navier-Stokes equation is derived as the new Oseen type hydrodynamic tensor. The Kirkwood-Rise- man theory for polymer dynamics in nematics is then described in terms of the "anisotropic" Oseen tensor in section 111. The friction tensor of flexible polymers in liquid crystals is defined, and one of its components is evaluated to first order in hydrodynamic interactions within the scheme of a renormalization group theory. The paper ends with a brief discussion in section IV. 11. Hydrodynamic Interactions in Anisotropic Fluids Simple fluids are normally composed of isotropic molecules and their properties are therefore independent of any specific direction(s). The isotropy of normal liquids allows for a convenient description of the fluid motion via the well-known Navier-Stokes equation for the velocity field. In particular, for incompressible fluids the viscous term involves the square of the Laplace differential operator. Such a mathematical structure leads directly to the simplistic long-range zyxw 1/R dependence of hydro- dynamic interactions between two objects R distance apart. The purpose of this section is to study the motion of an- isotropic fluids in terms of their fluid mechanical equation. Below we first write down the equation of motion, and 0 1991 American Chemical Society