The extraordinary importance of self-avoiding behavior in two-dimensional polymers: Insights from large-deviation theory

TL;DR

This work introduces a large-deviation framework to study self-avoiding polymers, revealing that two-dimensional chains exhibit fundamentally non-Gaussian end-to-end statistics due to long-range excluded-volume effects. By linking end-to-end fluctuations to a force-extension equation of state through a rate function, the authors interpolate between small-extension and large-extension regimes using Padé-type forms and validate predictions with Monte Carlo simulations of hard-sphere and discretized worm-like chain models. They show that in 2D, the quadratic and quartic free-energy terms scale as $O(N^{-1})$, making nonlinear elasticity persist at large $N$, while in higher dimensions the elasticity tends to be Gaussian. The approach enables extracting thermodynamic-like information from simulations without sampling extreme rare events and highlights the distinct roles of dimensionality and interaction range in polymer conformations, with potential extensions to more complex polymeric systems.

Abstract

Some recent work pointed out the usefulness of taking a large-deviation perspective when trying to extract anything resembling a macroscopic order parameter from a computer simulation. In this paper we note that the end-to-end distance of polymers is such an order parameter. The presence of long-ranged excluded volume interactions leads to significant qualitative differences between the conformations of two- and three-dimensional polymers, some of which are difficult to quantify in computer simulations of realistic (off-lattice) polymer models. But we show here that phenomena such as the greatly enlarged non-Hooke's-law elasticity present in 2D are straightforward to extract from simulation using a large-deviation framework - even though simulating that nonlinearity is tantamount to simulating a 4th order susceptibility. The large-deviation perspective includes both a set of thermodynamic-like tools suitable for studying finite-size systems and a realization that an accurate description of the system's average behavior needs to be consistent with how improbably large fluctuations would behave in that system. The latter is key because strong correlations are absent in this asymptotic large fluctuation regime, so the regime's far-reaching effects can be analytically incorporated into the analysis of simulation data. That, in turn, allows us to direct the efforts of simulations away from difficult-to-sample rare-event domains. We illustrate this point with two- and three-dimensional Monte Carlo simulations (and exact results) on two models of a single isolated polymer chain: a chain of linked hard spheres, which has long-ranged excluded volume effects, and a discretized worm-like chain, which does not.

Figures (9)

• Figure 1: Equation of state of the discretized worm-like chain model in two dimensions (the 1d classical XY model) at small values of the applied force $f$ and with a monomer-sized persistence length ($K = 1.00$). The dots show the results of Monte Carlo simulations with $N = 1000$, and the line represents our large-deviation theory prediction, Eq. (\ref{['eq2.14']}). In two dimensions, and at small $f$ values, Eqs. (\ref{['eq5.1']}) and (\ref{['eq2.13']}) imply $x/r_F = {\sqrt{N \chi} \over 2} f$. From the simulation for $N=1000$, $\chi (f=0) = 2.60$, so that theory predicts $x/r_F = 25.5 f$, which is very close to the fitted slope of the data ($25.7$). (Analytical predictions from Eq. (\ref{['eq4.2']}) also match these values, yielding $\chi(f = 0) = 2.613$ and a corresponding slope of 25.56.) Data was collected from 10 independent runs, each averaged over 50,000 MC steps after 5,000 steps of equilibration. The error bars are smaller than the symbol size on this plot.
• Figure 2: Equation of state for hard-disk polymers in two dimensions at small values of the applied force $f$. Monte Carlo hard-disk-polymer data (dots) with $N = 1000$ are compared with the linear elastic prediction based on the small $f$ data (dashed line), the 5th order large-deviation theory (Eq. \ref{['eq2.14']}) (lower solid-blue curve), and the 7th order large-deviation theory (Eq. \ref{['eq4.9']}) (upper solid-green curve).
• Figure 3: Equation of state for hard-sphere polymers in three dimensions at small values of the applied force $f$. Monte Carlo hard-sphere polymer data with $N = 1000$ (dots) are compared with the linear elastic prediction based on the small $f$ data (dashed line), and both the $5^{th}$ and $7^{th}$ order large deviation theories (solid lines). All three theoretical curves are essentially indistinguishable at this scale.
• Figure 4: Non-gaussian parameter $\alpha_{NG}$, Eq. (\ref{['eq3.7']}), computed from Monte Carlo simulations of two-dimensional hard-disk polymers over a range of $N$ values. Each data point is from 50 independent runs averaged over 200,000 MC steps after 50,000 equilibration steps.
• Figure 5: Equation of state for freely-jointed polymers in two dimensions. The solid line gives the exact result (Eq. \ref{['eq2.7']}) for the 2d freely-jointed chain, and the dots present the large-deviation theory predictions, Eq. (\ref{['eq2.14']}), with $\chi = 1$ and $\eta = (3-D)/4= 1/4$, equal to their ideal values.