Simulations on the number of entanglements of a polymer network using knot theory

Topological Analysis and Recovery of Entanglements in Polymer Melts

The viscous flow of polymer chains in dense melts is dominated by topological constraints whenever the single-chain contour length, N, becomes larger than the characteristic scale N_e, defining comprehensively the macroscopic rheological properties of the highly entangled polymer systems. Even though they are naturally connected to the presence of hard constraints like knots and links within the polymer chains, the difficulty of integrating the rigorous language of mathematical topology with the physics of polymer melts has limited somehow a genuine topological approach to the problem of classifying these constraints and to how they are related to the rheological entanglements. In this work, we tackle this problem by studying the occurrence of knots and links in lattice melts of randomly knotted and randomly concatenated ring polymers with various bending stiffness values. Specifically, by introducing an algorithm that shrinks the chains to their minimal shapes that do not violate topological constraints and by analyzing those in terms of suitable topological invariants, we provide a detailed characterization of the topological properties at the intrachain level (knots) and of links between pairs and triplets of distinct chains. Then, by employing the Z1 algorithm on the minimal conformations to extract the entanglement length N_e, we show that the ratio N/N_e, the number of entanglements per chain, can be remarkably well reconstructed in terms of only two-chain links.

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d_f. Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P₀(N) ∝ exp(−N/N₀). However, no such simple considerations can account for the dependence of the knotting length, N₀, on d_f, necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

The formation and structure of Olympic gels

Different methods for creating Olympic gels are analyzed using computer simulations. First ideal reference samples are obtained from freely interpenetrating semi-dilute solutions and melts of cyclic polymers. The distribution of pairwise concatenations per cyclic molecule is given by a Poisson-distribution and can be used to describe the elastic structure of the gels. Several batches of linear chains decorated with different selectively binding groups at their ends are mixed in the "DNA Origami" technique and network formation is realized. While the formation of cyclic molecules follows mean field predictions below overlap of the precursor molecules, an enhanced ring formation above overlap is found that is not explained by mean field arguments. The "progressive construction" method allows to create Olympic gels with a single reaction step from a concentrated mixture of large compressed rings with a low weight fraction of short chains that are below overlap concentration. This method, however, is limited by the difficulty to obtain a sufficiently high degree of polymerization of the large rings.

Dynamical behavior of entanglements by means of the mean standard deviation

In this paper we present our results on the dynamical behavior within the interior parts of an entanglement. Therefore, we analyzed the mean standard deviation s(m) for each monomer of the entangled chains. The calculations were performed at two deformation ratios for three different basic systems-the cross, slant, and along systems-each of them representing a different structural orientation of the entanglement. Additionally, we varied the number of loops l, the distance a of the end monomers, and the inverse temperature beta. Depending on the number of loops l we get different effects. Whereas a one time entangled system rather acts like a chemical cross link, a six times entangled system forms an unoriented entangled area and an orientated residual part.

Rheology and Microscopic Topology of Entangled Polymeric Liquids

The viscoelastic properties of high molecular weight polymeric liquids are dominated by topological constraints on a molecular scale. In a manner similar to that of entangled ropes, polymer chains can slide past but not through each other. Tube models of polymer dynamics and rheology are based on the idea that entanglements confine a chain to small fluctuations around a primitive path that follows the coarse-grained chain contour. Here we provide a microscopic foundation for these highly successful phenomenological models. We analyze the topological state of polymeric liquids in terms of primitive paths and obtain parameter-free, quantitative predictions for the plateau modulus, which agree with experiment for all major classes of synthetic polymers.