Topological Constraints in Directed Polymer Melts

Pablo Serna, Guy Bunin, Adam Nahum

Published 2015-07-08Version 1

Polymers in a melt may be subject to topological constraints, as in the example of unlinked polymer rings. How to do statistical mechanics in the presence of such constraints remains a fundamental open problem. We study the effect of topological constraints on a melt of directed polymers using simulations of a simple quasi-2D model. We find that fixing the global topology of the melt to be trivial changes the polymer conformations drastically. Polymers of length $L$ wander in the transverse direction only by a distance of order $(\ln L)^\zeta$ with $\zeta \simeq 1.5$. This is strongly suppressed in comparison with the Brownian scaling $L^{1/2}$ which holds in the absence of the topological constraint. It is also much less than the prediction $L^{1/4}$ of a mean-field-like `array of obstacles' model: thus we rule out such a model in the present setting. Dynamics are also strongly affected by the constraints, and a tagged monomer in an infinite system performs logarithmically slow subdiffusion. To cast light on the suppression of the strands' wandering, we analyse the topological complexity of subregions of the melt: the complexity is also logarithmically small, and is related to the wandering by a power law. We comment on insights the results give for 3D melts, directed and non-directed.