Hydrodynamic theories for mixtures of polymers and rodlike liquid crystalline polymers

We develop a hydrodynamic theory for flows of incompressible blends of flexible polymers and rodlike nematic polymers (RNPs) or rodlike nematic liquid crystal polymers (RNLCPs) extending the thermodynamical theory of Muratov and E [J. Chem. Phys. 116, 4723 (2002)] for phase separation kinetics of the blend. We model the flexible polymer molecules in the polymer matrix as Rouse chains and assume the translational diffusion of the molecules is predominantly through the volume fraction of the flexible polymer and the molecules of rodlike nematic liquid crystal polymers. We then (i) derive the translational flux for the rodlike nematic liquid crystal polymers to ensure the incompressibility constraint; (ii) derive the elastic stress tensor, accounting for the contribution from both the rodlike nematic polymer and the flexible polymer matrix, as well as the extra elastic body force due to the nonlocal intermolecular potential for long range molecular interaction; (iii) show that the theory obeys positive entropy production and thereby satisfies the second law of thermodynamics. By applying the gradient expansion technique on the number density function of RNLCPs, we present an approximate, weakly nonlocal theory in differential form in which the intermolecular potential is given by gradients of the number density function of the RNLCP and the volume fraction of the flexible polymer. In the approximate theory, the elastic stress is augmented by an extra stress tensor due to the spatial convection of the macroscopic material point and long range interaction, whose divergence yields the analogous extra elastic body force with respect to the nonlocal intermolecular potential. Finally, we compare the model in steady simple shear with the Doi theory for bulk monodomains of rodlike nematic polymers.