Crossover to entangled dynamics in polymer solutions and melts

A statistical dynamical theory of the crossover from unentangled Rouse dynamics to entangled behavior is constructed for chain polymer solutions and melts. Both time and spatial crossovers in long chain fluids, and the degree of polymerization crossover for short polymers, are treated. The analysis is based on a microscopic theory of the perturbative dynamical corrections to Rouse theory arising from chain connectivity and intermolecular excluded volume forces. The dependence of crossover properties such as the plateau shear modulus and entanglement time and length scale on solution density, solvent quality, and chain statistical segment length are derived by combining the dynamical theory with equilibrium liquid state integral equation methods. Scaling relations are obtained which appear to be in general accord with most experiments on both solutions and melts. The physical origin of the predicted scaling behaviors is the fractional power law temporal decay of the entanglement friction memory function on intermediate time scales, and power law reduced density dependence of the equilibrium force correlations. The theory is also applied to compute the dependence of the chain normal mode relaxation times on polymer density and chain length. Favorable qualitative comparisons with recent neutron spin echo experiments are made.