Mode-coupling theory of the slow dynamics of polymeric liquids: Fractal macromolecular architectures

Recently a mode coupling theory for the dynamics of solutions and melts of entangled linear chain polymers has been developed. We report the extension of this approach to macromolecular architectures different from linear chains. Specifically, this work addresses recent experimental findings on melts of ring shaped polymers, small spherical micro-networks, and linear chains in two dimensions. The mechanical and dielectric response, diffusion, and molecular relaxation times of macromolecules modeled by fractal mass distributions are studied. The distribution is chosen to be Gaussian and then is uniquely determined from the experimentally measured scaling of macromolecular size (R_g) with degree of polymerization (N), i.e., R_g α N^v. The exponent v and the spatial dimension d determine the large N scaling of the transport coefficients and the exponents describing intermediate time anomalous diffusion. Within the theory, entanglement corrections to the single polymer Rouse dynamics are effective for v<2/d only. There, we find D α N^2dv-5 for the diffusion coefficient and that the ratio Dτ_D/R_g² is almost constant, where τ_D is the terminal relaxation time. Using independent input from equilibrium liquid state theories, the magnitude and scaling with macromolecular density and segment length of the dynamical properties is determined. It is also found that macromolecular interpenetration requires progressively higher densities and consequently entanglements become less effective with fractal dimension 1/v approaching the spatial dimension.