No abstract

Constitutive equations for entangled polymer melts and solutions are often derived from single chain tube models. Instead of keeping the chain coordinates, these models operate with a coarse-grained description in terms of positions of the tube segments. The dynamics of the tube is then imposed by constraint release, tube renewal at the ends, and deformation by the flow. However, the step of coarse-graining is rarely discussed, and the tube free energy and tube statistics are not derived. Moreover, the microscopic definition of the tube is rarely specified. In this paper we propose to define the tube as a mean path, i.e., a line connecting positions of each monomer averaged over entanglement relaxation time τe . We propose one simple model where such a coarse-graining step can be performed exactly, resulting in a free energy containing the usual Gaussian chain term and an additional bending energy term. This free energy leads to a path in space which is locally smooth and differentiable but has random walk statistics at length scales larger than the tube diameter. This eliminates several problems in previous tube models which use derivatives over contour variables. We then proceed to modify the constitutive equation of Graham et al. (2003) to include the bending energy in constraint release terms. The resulting theory does not contain uncertainties of the original theory and has a clearer and better defined microscopic origin.

Molecular dynamic simulations are reported for the static and dynamic properties of hard sphere fluids in matrices (or media) composed of quenched hard spheres. The effect of fluid and matrix density, matrix structure, and fluid to matrix sphere size ratio on the static and dynamic properties is studied using discontinuous molecular dynamics. The matrix density has a stronger effect on the self-diffusion coefficient than the fluid density, especially at high matrix densities where the geometric constraints due to the quenched spheres are significant. When the ratio of the size of the fluid spheres to that of the matrix spheres is equal to or greater than one, the diffusion increases as the fluid density is increased, at constant total volume fraction. This trend is however reversed if the ratio is smaller than one. Different methods of generating the matrix have a very strong effect on the dynamic properties even though the static correlations are similar. An analysis of the single-chain structure factor of the particle trajectories shows a change in the particle diffusive behavior at different time scales, suggestive of a hopping mechanism, although normal diffusion is recovered at long times. At high matrix densities, there is considerable heterogeneity in the diffusion of the fluid particles. The simulations demonstrate that the correlations in the matrix play a significant role on the diffusion of fluid spheres. For example, the diffusion constant in matrices constructed by different methods can be an order of magnitude different even though the pair correlation functions are almost identical.

No abstract