Chain entanglements. I. Theory

A model of concentrated polymer solution dynamics is described. The forces in a linear generalized Langevin equation for the motion of a probe chain are derived on the assumption that all relaxation of the forces is due to motion of the surrounding matrix. Vicinal chain displacements are classified as viscoelastic deformation, reptation, and minor residual fluctuations. The latter provide a torsional relaxation of the primitive path that minimizes the significance of transverse forces on the probe chain. All displacements of vicinal segments are assumed proportional to the forces that they exert on the probe chain. In response to an external force, the displacement of the probe chain relative to a laboratory frame is increased by viscoelastic deformation of the matrix, but reptative diffusion relative to the deforming matrix is slowed down. The net effect on translational diffusion is negligible if the probe and vicinal chains have the same chain length N, but the friction constant for reptative motion is increased by a factor N1−xs. xs=1/2 if Gaussian conformational statistics applies during the disengagement process, while xs =0.6 if excluded volume statistics applies. The translational friction constant is βp ∼N2, as in reptation theory, but the viscosity is η∼N4−xs . The persistence of entanglements during the translational diffusion of the probe chain across many radii of gyration is rationalized pictorially in terms of correlated reptative motion of the probe and vicinal chains.