Testing the Reptation Picture: Topological Constraint from Monomer Dynamics

Abstract

The reptation model postulates that entangled polymers slide within a fractal tube. Here we employ a model-independent relation between the zero-displacement probability and the mean-square displacement that applies to time-dependent fractal structures, enabling direct measurement of the fractal dimension $d_\mathrm{f}$ of the geometry experienced by monomer motion. For two-dimensional obstacle arrays and in the slip-link model, $d_\mathrm{f}$ agrees with the reptation prediction $d_\mathrm{f}=1/ν$ (where $ν$ is the Flory exponent). In polymer melts, however, we find $d_\mathrm{f} \approx 2.6$ --- a value close to the fractal dimension of percolation clusters, not the reptation value $d_\mathrm{f}=2$. This contrasts sharply with the reptation picture, in which a Rouse chain slides in a fractal structure with $d_\mathrm{f}=2$, spectral dimension $d_\mathrm{s}=1$, and walk dimension $d_\mathrm{w}=4$; our results point instead to a percolation-like scenario, characterized by $d_\mathrm{f}\approx 2.6$, $d_\mathrm{s}\approx 1.3$, and $d_\mathrm{w}\approx 4$ --- revealing a dynamically emergent, finite-size fractal geometry distinct from the static tube.