Driven polymer translocation through a nanopore: a manifestation of anomalous diffusion

Abstract

We study the translocation dynamics of a polymer chain threaded through a nanopore by an external force. By means of diverse methods (scaling arguments, fractional calculus and Monte Carlo simulation) we show that the relevant dynamic variable, the translocated number of segments $s(t)$, displays an {\em anomalous} diffusive behavior even in the {\em presence} of an external force. The anomalous dynamics of the translocation process is governed by the same universal exponent $α= 2/(2ν+2 - γ_1)$, where $ν$ is the Flory exponent and $γ_1$ - the surface exponent, which was established recently for the case of non-driven polymer chain threading through a nanopore. A closed analytic expression for the probability distribution function $W(s, t)$, which follows from the relevant {\em fractional} Fokker - Planck equation, is derived in terms of the polymer chain length $N$ and the applied drag force $f$. It is found that the average translocation time scales as $τ\propto f^{-1}N^{\frac{2}α -1}$. Also the corresponding time dependent statistical moments, $< s(t) > \propto t^α$ and $< s(t)^2 > \propto t^{2α}$ reveal unambiguously the anomalous nature of the translocation dynamics and permit direct measurement of $α$ in experiments. These findings are tested and found to be in perfect agreement with extensive Monte Carlo (MC) simulations.