Insertion barrier of a charged polymer ($F(x)=fx-qE{x}^2/2$)

If only the inserted length $x$ feels a uniform inward electric field, its charge experiences force $qEx$, where $q$ is charge per contour length. Integrating gives electric energy $-qE{x}^2/2$. Adding a constant entropic confinement force gives the displayed free-energy profile. Its interior maximum is at $x=f/(qE)$ and has height $f^2/(2qE)$; entry waiting and subsequent directed insertion are distinct times.

With friction $\gamma (x)={\zeta }_{\ell }x$, inward force $qEx$ gives constant entry speed after capture. With the field off, constant outward entropic force gives $x\dot{x}=-f/{\zeta }_{\ell }$ and a quadratic recoil time. Exterior drag, adsorption and the short final end region are omitted. These ideal transport times alone do not establish steady size separation through a whole uniform channel.