Molecular theory of the entanglement effect of rodlike polymers. Two alternative approaches

Two alternative approaches are presented for evaluating the rotational diffusion constant D_r of entangled rodlike polymers in a semidilute solution. One is a quantitative refinement of the Doi-Edwards tube model, which gave for the first time a qualitative expression of D_r. To evaluate the mean square of a step angular displacement of the rodlike molecule, we use a computer simulation for Poisson line processes, while the mean lifetime of the tube is calculated by use of the first exit time of a stochastic process. The result is in good agreement with experimental data so far obtained. The other approach is a new Green function formulation on the dynamics of entangled rods; we regard the obstructions by neighbouring rods as a sequence of perturbation elements exerted on the free diffusion of the test rod. Incorporating the effects of the sequential perturbation one by one, we obtain an expression of D_r for rods in solutions in a wide range of concentrations from dilute to semidilute. In the semidilute region, this reduces to the result of the tube model.