Testing the Reptation Picture: Topological Constraint from Monomer Dynamics

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 $d_{\mathrm{f}}=1\mathrm{/}\nu$ (where $\nu$ 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.