Reptation and Contour-Length Fluctuations in Melts of Linear Polymers

We present an analytical theory of stress relaxation in monodisperse linear polymer melts that contains contributions from both reptation and contour-length fluctuations, modeled as in our previous work on arm retraction in star polymers. Our approach resolves two long-standing problems with reptation theory: it predicts a zero-shear viscosity $\eta$ scaling as $\eta \sim N^{3.4}$ over a broad range in chain length $N$ before reaching an asymptotic $N^3$ dependence, and a power law ${\omega }^{-\alpha }$ in the dynamic loss modulus $G^{\prime \prime }\left(\omega \right)$ with $0<\mathrm{}\alpha <1\mathrm{}/4$ depending on chain length, in agreement with experiment.