Deformation of a stretched polymer knot

Abstract

The static properties of a knotted polymer under a stretching force f are studied by Monte Carlo simulations. Chain lengths up to $N=82$ and knot types of ${0}_{1},$ ${3}_{1},$ ${4}_{1},$ ${5}_{1},$ ${6}_{1},$ and ${8}_{1}$ are considered. Our simulation data show that the scaling laws proposed by de Gennes and Pincus for a single linear chain under traction force still hold for the knotted type polymers. That is, the average knot size under a force f scales as $〈{R}_{f}〉\ensuremath{\sim}{R}_{F}^{2}f$ at weak tension forces while for strong forces $〈{R}_{f}〉\ensuremath{\sim}{R}_{F}^{1/\ensuremath{\nu}}{f}^{(1/\ensuremath{\nu})\ensuremath{-}1},$ where ${R}_{F}\ensuremath{\sim}{N}^{\ensuremath{\nu}}{p}^{\ensuremath{-}4/15},$ $\ensuremath{\nu}\ensuremath{\approx}3/5$ is the usual self-avoiding walk exponent and p is a topological invariant representing the aspect ratio (length to diameter) of a knotted polymer at its maximum inflated state. Our results also show that the elastic modulus of a knotted polymer is larger compared to an equal-length linear chain. More complex knots are in general stiffer. A simple composite spring model is employed to derive the increase in stiffness of knots relative to the linear chain, and the results agree well with the simulation data. Segregation of the crossings into a small tight region of the knot structure at strong forces is also observed.