Computer simulations of melts of randomly branching polymers

Abstract

Randomly branching polymers with annealed connectivity are model systems for ring polymers and chromosomes. In this context, the branched structure represents transient folding induced by topological constraints. Here we present computer simulations of melts of annealed randomly branching polymers of 3 ≤ N ≤ 1800 segments in d = 2 and d = 3 dimensions. In all cases, we perform a detailed analysis of the observed tree connectivities and spatial conformations. Our results are in excellent agreement with an asymptotic scaling of the average tree size of R ∼ N, suggesting that the trees behave as compact, territorial fractals. The observed swelling relative to the size of ideal trees, R ∼ N, demonstrates that excluded volume interactions are only partially screened in melts of annealed trees. Overall, our results are in good qualitative agreement with the predictions of Flory theory. In particular, we find that the trees swell by the combination of modified branching and path stretching. However, the former effect is subdominant and difficult to detect in d = 3 dimensions.

Flory theory of randomly branched polymers

“…However, there are other quantities such as entropies 60 and internodal contact probabilities 62,63 , which Flory theory fails to describe even for linear chains 75 . A forthcoming article 76 discusses how to go beyond Flory theory and the Gaussian approximation by analyzing the distribution functions characterising the tree conformations and connectivity.…”

“…To gauge the utility of the approach, its predictions need to be compared to exact solutions 14,44,46,47 , ε -expansions 39 renormalisation group calculations 14,40,41,43,49 , dimensional reduction 45 , series expansions 16,48 exact enumerations 42,50–52,54 , Monte Carlo 53,55–63 and Molecular Dynamics 62 simulations. Tables 1 to 4 list available benchmark results for linear chains and for trees with annealed and with quenched ideal connectivity.…”

“…Here we focus on Flory theories 11,18,33,38 , which provide a unifying description of the rich behaviour of a wide range of tree systems. Our aim is twofold: firstly, we provide a pedagogical, comprehensive review of the theory; secondly, we compare the predictions to available theoretical 14,16,39–49 and computational 50–63 results to gauge the reliability of the approach. We find that predictions of Flory theory are qualitatively excellent.…”

Local loop opening in untangled ring polymer melts: a detailed “Feynman test” of models for the large scale structure

“…Our simulations of lattice-tree melts are described in detail in Refs. [39,51]. Compared to brute-force Molecular Dynamics equilibration the speedup is of the order of ≈ 10 6 for our largest ring sizes [35].…”

“…Systems exhibiting random branching being asymptotically too dense, excluded volume interactions are, thirdly, not fully screened. Lattice tree melts hence exhibit a small amount of swelling with a slightly modified branching statistics [32,39]. These effects becomes relevant for Z r > O(100).…”