We present an analytical model to study translocation of Gaussian polymers across a cylindrical channel of nanometric size with a chemical potential inside the channel.
Results show that polymer conformational entropy generates an entropic M-like free energy barrier for translocation.
The presence of a small negative chemical potential reduces the entropic free energy barrier rendering the translocation time to follow a power law $\tau = A N^{\nu}$ as function of polymer size $N$.
Power law exponents $\nu$ found here in varying the channel radius $R$, run from 1.525 to 1.999 for unforced translocation, and from 1.594 to 2.006 for translocation with small chemical potentials when $R=\SI{1}{\nano\meter}$.
Presence of large negative chemical potentials generate a free energy well rendering the translocation time to follow an exponential growth $\tau = A e^{\alpha N}$.
We show existence of a negative chemical potential $\mu_c$ that minimizes the translocation time due to an interplay of conformational entropy and channel-polymer interactions.
The translocation time as function of channel length $L$ grows exponentially as $\tau = A e^{cL}$, it runs from milliseconds up to decades in the range of lengths found in biological channels.
Interestingly, small negative chemical potentials approaching $\mu_c$ overcome the effect of large channel lengths reducing the translocation time below seconds.
Translocation speeds $<v(N)>$ show a maximum of micrometers per second then it decays with polymer size and channel length, the characteristic decay $<v(N)>\sim N^{-1}$ has been observed in previous experiments of voltage-driven DNA translocation.