Molecular Simulation of Tracer Diffusion and Self-Diffusion in Entangled Polymers

Abstract

The dependence of tracer diffusivity ($D_\infty \sim N^{-x_\infty}$), where probe chains move in an environment of infinitely long matrix chains, and self-diffusion coefficient ($D_s \sim N^{-x_s}$), where probe and matrix chains are identical, on the molecular weight of the probe chain $N$ are investigated using three different molecular simulation methods, viz. molecular dynamics, the bond-fluctuation model (BFM) and the slip-spring (SS) model. Experiments indicate $x_s \approx 2.4 \pm 0.2$ over a wide intermediate molecular weight range, and $x_{\infty} \approx 2.0 \pm 0.1$, although the lower molecular weight limit for observing pure reptation in short probes is unclear. These results are partly inconsistent with some tube theories, and older, somewhat underpowered, molecular simulations. Estimating $x_{\infty}$ using brute-force BFM simulations is difficult because it involves large simulation boxes and long trajectories. To overcome this obstacle, an efficient method to estimate $D_\infty$ in which ends of matrix chains are immobilized, is presented and validated. BFM simulations carried out on systems with different probe and matrix chain lengths reveal that $x_s = 2.43 \pm 0.07$, and $x_\infty = 2.24 \pm 0.03$. Over a wider range of molecular weights, probe diffusivities obtained from the more coarse-grained SS model, calibrated with bead-spring molecular dynamics, reveal $x_s > x_\infty$, and $x_\infty > 2$ for weakly and intermediately entangled chains. Tracer diffusivities obtained by artificially switching off constraint release in the SS simulations essentially overlap with probe diffusivities, strongly suggesting that constraint release is primarily responsible for the difference between $x_s$ and $x_\infty$. Nevertheless, both BFM and SS simulations indicate that below a certain chain length threshold, contributions of contour length fluctuations to $D_s$ and $D_\infty$ are important, and result in deviations from pure reptation scaling.