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Anisotropic diffusion and hydrodynamic effects on lamellar relaxation and grain boundary motion in a model of a block copolymer

We consider the effects of anisotropic diffusion and hydrodynamic flows on the relaxation time scales of the lamellar phase of a diblock copolymer. We first extend the two-fluid model of a polymer solution to a block copolymer, and include a tensor mobility for the diffusive relaxation of monomer composition which is consistent with the uniaxial symmetry of the lamellar phase. The resulting equation is coupled to the momentum conservation equation, allowing also for a dissipative stress tensor for a uniaxial fluid. We then study the linear relaxation of weakly perturbed lamellae, and the motion of a tilt grain boundary separating two semi-infinite domains. We find that anisotropic diffusion has a negligible effect on the linear relaxation of the layered phase (in the long wavelenght limit), whereas the introduction of hydrodynamic flows considerably speeds the decay to a rate proportional to $Q^{2}$, where $Q\ll 1$ is the wavenumber of a transverse perturbation to the lamellar phase (diffusive relaxation scales as $Q^{4}$ instead). On the other hand, grain boundary motion is siginificantly affected by anisotropic diffusion because of the coupling between undulation and permeation diffusive modes within the grain boundary region.