Rectified polymer translocation induced by solvent assymetry between $cis$ and $trans$ compartments

We report Langevin dynamics simulation studies of translocation of a homopolymer through a nano pore driven by different solvent conditions at either side of the pore. The solvent at the $cis$ compartment is modeled as a ``\textit{good solvent}'' while the solvent at the $trans$ side is modeled as a ``\textit{bad solvent}'' so that the translocated beads of the polymer conforms to a globule and inhibits back translocation from the $trans$ to the $cis$ side. Therefore, the translocating polymer acts like a \textit{Brownian Ratchet}. We study the translocation as a function of the dimensionless quantity $\epsilon/k_BT$, where $\epsilon$ is the strength of the attractive interaction at the $cis$ side, $k_B$ is the Boltzmann constant, and $T$ is the temperature respectively for several chain length $N$. We find that as $N$ gets larger the mean translocation time $\langle \tau \rangle \sim N$ and shows a rather weak dependence on the parameter $\epsilon/k_BT$ . This is consistent with the observation that excepting for the last few monomers, the velocity of the individual monomer s $v(m)$ is roughly constant being independent of the monomer index $m$. We further discuss a plausible physical picture leading to such chain length dependence.