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We perform large scale three-dimensional molecular dynamics simulations of unlinked and unknotted ring polymers diffusing through a background gel, here a three-dimensional cubic lattice. Taking advantage of this architecture, we propose a new method to unambiguously identify and quantify inter-ring threadings (penetrations) and to relate these to the dynamics of the ring polymers. We find that both the number and the persistence time of the threadings increase with the length of the chains, ultimately leading to a percolating network of inter-ring penetrations. We discuss the implications of these findings for the possible emergence of a topological jammed state of very long rings.
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Using molecular dynamics computer simulations we investigate the aging dynamics of a gel. We start from a fractal structure generated by the DLCA-DEF algorithm, onto which we then impose an interaction potential consisting of a short-range attraction as well as a long-range repulsion. After relaxing the system at T=0, we let it evolve at a fixed finite temperature. Depending on the temperature T we find different scenarios for the aging behavior. For T>0.2 the fractal structure is unstable and breaks up into small clusters which relax to equilibrium. For T<0.2 the structure is stable and the dynamics slows down with increasing waiting time. At intermediate and low T the mean squared displacement scales as t^{2/3} and we discuss several mechanisms for this anomalous time dependence. For intermediate T, the self-intermediate scattering function is given by a compressed exponential at small wave-vectors and by a stretched exponential at large wave-vectors. In contrast, for low T it is a stretched exponential for all wave-vectors. This behavior can be traced back to a subtle interplay between elastic rearrangements, fluctuations of chain-like filaments, and heterogeneity.
We analyse the dynamics of different routes to collapse of a constrained polyelectrolyte gel in contact with an ionic bath. The evolution of the gel is described by a model that incorporates non-linear elasticity, Stefan-Maxwell diffusion and interfacial gradient free energy to account for phase separation of the gel. A bifurcation analysis of the homogeneous equilibrium states reveals three solution branches at low ion concentrations in the bath, giving way to only one above a critical ion concentration. We present numerical solutions that capture both the spatial heterogeneity and the multiple time-scales involved in the process of collapse. These solutions are complemented by two analytical studies. Firstly, a phase-plane analysis that reveals the existence of a depletion front for the transition from the highly swollen to the new collapsed equilibrium state. This depletion front is initiated after the fast ionic diffusion has set the initial condition for this time regime. Secondly, we perform a linear stability analysis about the homogeneous states that show that for a range of ion concentrations in the bath, spinodal decomposition of the swollen state gives rise to localized solvent-rich(poor) and, due to the electro-neutrality condition, ion-poor(rich) phases that coarsen on the route to collapse. This dynamics of a collapsing polyelectrolyte gel has not been described before.
In the first part of this work a summary is provided of some recent experiments and theoretical results which are relevant in the research of systems of polymer rings in nontrivial topological conformations. Next, some advances in modeling the behavior of single polymer knots are presented. The numerical simulations are performed with the help of the Wang-Landau Monte Carlo algorithm. To sample the polymer conformation a set of random transformations called pivot moves is used. The crucial problem of preserving the topology of the knots after each move is tackled with the help of two new techniques which are briefly explained. As an application, the results of an investigation of the effects of topology on the thermal properties of polymer knots is reported. In the end, original results are discussed concerning the use of parallelized codes to study polymers knots composed by a large number of segments within the Wang-Landau approach.
Stochastic simulations are used to characterize the knotting distributions of random ring polymers confined in spheres of various radii. The approach is based on the use of multiple Markov chains and reweighting techniques, combined with effective strategies for simplifying the geometrical complexity of ring conformations without altering their knot type. By these means we extend previous studies and characterize in detail how the probability to form a given prime or composite knot behaves in terms of the number of ring segments, $N$, and confining radius, $R$. For $ 50 le N le 450 $ we show that the probability of forming a composite knot rises significantly with the confinement, while the occurrence probability of prime knots are, in general, non-monotonic functions of 1/R. The dependence of other geometrical indicators, such as writhe and chirality, in terms of $R$ and $N$ is also characterized. It is found that the writhe distribution broadens as the confining sphere narrows.
Hydrodynamic interactions as modeled by Multi-Particle Collision Dynamics can dramatically influence the dynamics of fully flexible, ring-shaped polymers in ways not known for any other polymer architecture or topology. We show that steady shear leads to an inflation scenario exclusive to ring polymers, which depends not only on Weissenberg number but also on contour length of the ring. By analyzing velocity fields of the solvent around the polymer, we show the existence of a hydrodynamic pocket which allows the polymer to self-stabilize at a certain alignment angle to the flow axis. This self-induced stabilization is accompanied by transitioning of the ring to a non-Brownian particle and a cessation of tumbling. The ring swells significantly in the vorticity direction, and the horseshoe regions on the stretched and swollen ring are effectively locked in place relative to the rings center-of-mass. The observed effect is exclusive to ring polymers and stems from an interplay between hydrodynamic interactions and topology. Furthermore, knots tied onto such rings can serve as additional stabilization anchors. Under strong shear, the knotted section is pulled tight and remains well-localized while tank-treading from one horseshoe region to the opposite one in sudden bursts. We find knotted polymers of high contour length behave very similarly to unknotted rings of the same contour length, but small knotted rings feature a host of different configurations. We propose a filtering technique for rings and chains based on our observations and suggest that strong shear could be used to tighten knots on rings.
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