Compressed self-avoiding walks in two and three dimensions

TL;DR

This work studies the phase behavior of self-avoiding walks confined between two parallel walls (a slab) in $d=2$ and $d=3$, under compression, and identifies a zero-force critical point separating stretched and compressed phases. The authors develop scaling arguments for the compressed phase and for the critical point, deriving explicit predictions for span and end-to-end-distance exponents, notably $\nu_s$ and $\nu_\perp$ in the compressed regime and $\phi=\nu_d$, $\alpha=2-1/\nu_d$ at criticality. They validate these predictions with extensive flatPERM Monte Carlo simulations, finding in 2D that $\nu_s=\nu_\perp=\nu_d/(\nu_d+1)=3/7$ and $\nu_\parallel=2\nu_d/(\nu_d+1)=6/7$, while in 3D the 2D-inspired relations are approximately but not decisively borne out, with $\nu_s$ close to $\nu_d/(\nu_d+1)$ and $\nu_\parallel$ less clearly matched. The results highlight a continuous, critical transition at zero force and reveal how compression induces quasi-lower-dimensional scaling in 2D and quasi-3D behavior in 3D, offering connections to directed models and higher-dimensional extensions.

Abstract

We consider the phase transition induced by compressing a self-avoiding walk in a slab where the walk is attached to both walls of the slab in two and three dimensions, and the resulting phase once the polymer is compressed. The process of moving between a stretched situation where the walls pull apart to a compressed scenario is a phase transition with some similarities to that induced by pulling and pushing the end of the polymer. However, there are key differences in that the compressed state is expected to behave like a lower dimensional system, which is not the case when the force pushes only on the endpoint of the polymer. We use scaling arguments to predict the exponents both of those associated with the phase transition and those in the compressed state and find good agreement with Monte Carlo simulations.

Figures (5)

• Figure 1: A self-avoiding walk on the square lattice confined to a strip of width $s$ and a positive (negative) force $F$ is applied to the walls to expand (compress) the walk. The walk is terminally attached to the lower surface and touches the upper surface at least once but the endpoint $x_n$ is otherwise free within the strip. The position of $x_n$ gives the components of the endpoint distance $R$.
• Figure 2: Metric quantities for compressed SAWs for (left) $d = 2$ and (right) $d = 3$. From top to bottom are the average extension above the surface $\langle s \rangle$, the average height of the free endpoint, and the average in-plane distance of the free endpoint, each given as a fraction of length $n$. Curves are shown for several different lengths $n = 256, 512, 1024$.
• Figure 3: Specific heat $c_n$ for compressed SAWs, for (a) $d = 2$ and (b) $d = 3$. Curves are shown for several different lengths $n = 256, 512, 1024$, indicating that a peak is forming at the critical point $y = 1$ ($F = 0$).
• Figure 4: Estimates of coefficients of the exact expression for $s_n$ in the compressed phase ($F < 0$), Eq. \ref{['eq:AverageSpanCompressedGeneral']}. The coefficient $A$ of the leading order term behaves as expected, showing simple linear dependence on the known power of $F$, but the coefficient $B$ of the next order term does not. The dashed lines have slopes $(\gamma_1 - 3/2) / (\nu_d + 1)$.
• Figure 5: Exponent estimates for metric exponents $\nu$ as a function of pulling weight $y$, for (a) $d = 2$ and (b) $d = 3$. From bottom to top, the dashed lines mark the values of $\nu_d/(\nu_d + 1), \nu_d, 2\nu_d/(\nu_d + 1)$, where $\nu_d$ is the known exponent for free SAWs, namely $3/4, 0.587\ldots$ for $d = 2, 3$, respectively.