Physical Links: Defining and detecting inter-chain entanglement

Fluctuating filaments, from densely-packed biopolymers to defect lines in structured fluids, are prone to become interlaced and form intricate architectures. Understanding the ensuing mechanical and relaxation properties depends critically on being able to capture such entanglement in quantitative terms. So far, this has been an elusive challenge. Here we introduce the first general characterization of non-ephemeral forms of entanglement in linear curves by introducing novel descriptors that extend topological measures of linking from close to open curves. We thus establish the concept of physical links. This general method is applied to diverse contexts: equilibrated ring polymers, mechanically-stretched links and concentrated solutions of linear chains. The abundance, complexity and space distribution of their physical links gives access to a whole new layer of understanding of such systems and open new perspectives for others, such as reconnection events and topological simplification in dissipative fields and defect lines

Collapsing animals

Lattice animals with fugacities conjugate to the number of independent cycles, or to the number of nearest neighbour contacts, go through a collapse transition at a theta-point at a critical value of the fugacity. We examine the phase diagram of a model which includes both a cycle and a contact fugacity with Monte Carlo methods. Using an underlying cut-and-paste Metropolis algorithm for lattice animals, we implement in the first instance a multiple Markov chain simulation of collapsing animals to estimate the location of the collapse transitions and the values of the crossover exponents associated with these. Secondly, we use umbrella sampling to sample animals over a rectangle in the phase diagram to examine the structure of the phase diagram of these animals

Knotting in stretched polygons

The knotting in a lattice polygon model of ring polymers is examined when a stretching force is applied to the polygon. By examining the incidence of cut-planes in the polygon, we prove a pattern theorem in the stretching regime for large applied forces. This theorem can be used to examine the incidence of entanglements such as knotting and writhing. In particular, we prove that for arbitrarily large positive, but finite, values of the stretching force, the probability that a stretched polygon is knotted approaches 1 as the length of the polygon increases. In the case ofwrithing, we prove that for stretched polygons of length n, and for every function f(n) = o(root n), the probability that the absolute value of the mean writhe is less than f(n) approaches 0 as n -> 8, for sufficiently large values of the applied stretching force

Knot probability of polygons subjected to a force: a Monte Carlo study

We use Monte Carlo methods to study the knot probability of lattice polygons on the cubic lattice in the presence of an external force f. The force is coupled to the span of the polygons along a lattice direction, say the z-direction. If the force is negative polygons are squeezed (the compressive regime), while positive forces tend to stretch the polygons along the z-direction (the tensile regime). For sufficiently large positive forces we verify that the Pincus scaling law in the force-extension curve holds. At a fixed number of edges n the knot probability is a decreasing function of the force. For a fixed force the knot probability approaches unity as 1 - exp(-alpha(0)(f)n + o(n)), where alpha(0)(f) is positive and a decreasing function of f. We also examine the average of the absolute value of the writhe and we verify the square root growth law (known for f = 0) for all values of f

Asymptotics of knotted lattice polygons

We use Monte Carlo methods to investigate the asymptotic behaviour of the number and mean-square radius of gyration of polygons in the simple cubic lattice with fixed knot type. Let p(n)(tau) be the number of n-edge polygons of a fixed knot type tau in the cubic lattice, and let [R-n(2)(tau)] be the mean square radius of gyration of all the polygons counted by p(n)(tau). If we assume that p(n)(tau) similar to n(alpha(tau)-3) mu(tau)(n), where mu(tau) is the growth constant of polygons of knot type tau, and alpha(tau) is the entropic exponent of polygons of knot type tau, then our numerical data are consistent with the relation alpha(tau) = alpha(phi) + N-f, where phi is the unknot and N-f is the number of prime factors of the knot tau. If we assume that [R-n(2)(tau)] similar to A(nu)(tau)n(2 nu(tau)), then our data are consistent with both A(nu)(tau) and nu(tau) being independent of tau. These results support the claims made in Janse van Rensburg and Whittington (1991a J. Phys. A: Math. Gen. 24 3935) and Orlandini er al (1996 J. Phys. A: Math. Gen. 29 L299, 1998 Topology and Geometry in Polymer Science (IMA Volumes in Mathematics and its Applications) (Berlin: Springer))

Self-averaging in random self-interacting polygons

We give a set of conditions under which a system is thermodynamically self-averaging and show that several lattice models of interacting copolymers satisfy these conditions. We prove this result for a general potential which is linear in the numbers of various types of contacts, and show that this includes two potentials which have previously been used in models of random interacting linear copolymers

Thermodynamics and entanglements of walks under stress

We use rigorous arguments and Monte Carlo simulations to study the thermodynamics and the topological properties of self-avoiding walks on the cubic lattice subjected to an external force f. The walks are anchored at one or both endpoints to an impenetrable plane at Z = 0 and the force is applied in the Z-direction. If a force is applied to the free endpoint of an anchored walk, then a model of pulled walks is obtained. If the walk is confined to a slab and a force is applied to the top bounding plane, then a model of stretched walks is obtained. For both models we prove the existence of the limiting free energy for any value of the force and we show that, for compressive forces, the thermodynamic properties of the two models differ substantially. For pulled walks we prove the existence of a phase transition that, by numerical simulation, we estimate to be second order and located at f = 0. By using a pattern theorem for large positive forces we show that almost all sufficiently long stretched walks are knotted. We examine the entanglement complexity of stretched and pulled walks; our numerical results show a sharp reduction with increasing pulling and stretching forces. Finally, we also examine models of pulled and stretched loops. We prove the existence of limiting free energies in these models and consider the knot probability numerically as a function of the applied pulling or stretching force