Pulling Knotted Polymers

TL;DR

This study uses Monte Carlo simulations to analyze the force response of knotted versus unknotted polymers confined between walls, revealing a new length scale associated with knots and their scaling behavior.

Contribution

It introduces a novel interpretation of finite size effects in knotted polymers, identifying a scaling law for the knot size relative to total monomers.

Findings

Unknotted polymers show a universal force scaling with N and R.

Knotted polymers exhibit finite size effects linked to knot size.

Knot size scales as N^0.4, indicating a distinct length scale.

Abstract

We compare Monte Carlo simulations of knotted and unknotted polymers whose ends are connected to two parallel walls. The force $f$ exerted on the polymer is measured as a function of the separation $R$ between the walls. For unknotted polymers of several monomer numbers $N$, the product $fN^\nu$ is a simple function of $R/N^\nu$, where $\nu\simeq 0.59$. By contrast, knotted polymers exhibit strong finite size effects which can be interpreted in terms of a new length scale related to the size of the knot. Based on this interpretation, we conclude that the number of monomers forming the knot scales as $N^t$, with $t=0.4\pm 0.1$.