On the size of knots in ring polymers

TL;DR

This paper investigates the size and localization properties of knots in ring polymers using Monte Carlo simulations, revealing weak localization in good solvent conditions and delocalization in collapsed regimes, with implications for polymer physics.

Contribution

It introduces two consistent definitions of knot length in polymers and analyzes their statistical properties across different regimes, providing new insights into knot localization behavior.

Findings

Knot length distribution follows a power-law with exponent ~1.25.

In good solvent, knots are weakly localized with length scaling as N^0.75.

In collapsed regime, knots are delocalized with length scaling as N^1.

Abstract

We give two different, statistically consistent definitions of the length l of a prime knot tied into a polymer ring. In the good solvent regime the polymer is modelled by a self avoiding polygon of N steps on cubic lattice and l is the number of steps over which the knot ``spreads'' in a given configuration. An analysis of extensive Monte Carlo data in equilibrium shows that the probability distribution of l as a function of N obeys a scaling of the form p(l,N) ~ l^(-c) f(l/N^D), with c ~ 1.25 and D ~ 1. Both D and c could be independent of knot type. As a consequence, the knot is weakly localized, i.e. <l> ~ N^t, with t=2-c ~ 0.75. For a ring with fixed knot type, weak localization implies the existence of a peculiar characteristic length l^(nu) ~ N^(t nu). In the scaling ~ N^(nu) (nu ~0.58) of the radius of gyration of the whole ring, this length determines a leading power law correction which is much stronger than that found in the case of unrestricted topology. The existence of such correction is confirmed by an analysis of extensive Monte Carlo data for the radius of gyration. The collapsed regime is studied by introducing in the model sufficiently strong attractive interactions for nearest neighbor sites visited by the self-avoiding polygon. In this regime knot length determinations can be based on the entropic competition between two knotted loops separated by a slip link. These measurements enable us to conclude that each knot is delocalized (t ~ 1).