Topological Constraints in Directed Polymer Melts

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 $L^{1/2}$ scaling which holds in the absence of the topological constraint. It is also much smaller than the predictions of standard heuristic approaches—in particular the $L^{1/4}$ of a mean-field-like “array of obstacles” model—so our results present a sharp challenge to theory. Dynamics are also strongly affected by the constraints, and a tagged monomer in an infinite system performs logarithmically slow subdiffusion in the transverse direction. To cast light on the suppression of the strands’ wandering, we analyze 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 nondirected.

Figure 1

(a) Topologically trivial configuration in a small system. (b) Monte Carlo move types. There are six variants of the move on the lower right, and two of that on the upper right.

Figure 2

A subregion of a topologically trivial melt (taken from a system of size $L_x=512$, $L_y=128$).

Figure 3

Main panel: Transverse wandering $D$ of a strand at its midpoint $y=L_y/2$, as a function of the length $L_y$ of the strands (note lin-log scale). Red line: Fit to the logarithmic form in Eq. (2). Inset: Effective exponent ${\nu }_{\text{eff}}=d\ln D/d\ln L_y$, see text, with fit derived from Eq. (2).

Figure 4

Main panel: rms wandering $D(y)$ versus position $y$ along the strand, for various values of the strand length $L_y$. Blue line: Fit to $D=A^{\prime }(\ln y/{l_0}^{\prime }{)}^{{\zeta }^{\prime }}$ for $10\le y\le 100$, giving ${\zeta }^{\prime }=1.54(15)$, $A^{\prime }=0.204(10)$, ${l_0}^{\prime }=0.146(10)$. Lower inset: Same data on log-log plot. Upper inset: Probability distribution for wandering of a strand, $P(D_i)$, rescaled by standard deviation $D$.

Figure 5

Main panel: Complexity (per strand) for the half-braid as a function of $L_y$, fitted to the form in Eq. (7) with $\eta =2.6(2)$. Inset: Wandering $D(L_y/2)$ plotted vs the complexity $\mathcal{C}(L_y/2)$ and fitted to a power law, Eq. (6).