Critical Unmixing of Polymer Solutions

TL;DR

This study uses Monte Carlo simulations to analyze polymer solutions near their critical point, revealing Gaussian chain behavior at large N and scaling laws consistent with mean field theory, with some deviations likely due to finite size effects.

Contribution

It provides detailed simulation evidence showing Gaussian chain conformations at criticality and explores finite size effects and scaling corrections in polymer solutions.

Findings

Chains are Gaussian at the critical point for large N

The temperature difference scales as 1/√N, consistent with mean field predictions

Critical density exhibits non-trivial scaling possibly due to logarithmic corrections

Abstract

We present Monte Carlo simulations of semidilute solutions of long self-attracting chain polymers near their Ising type critical point. The polymers are modeled as monodisperse self-avoiding walks on the simple cubic lattice with attraction between non-bonded nearest neighbors. Chain lengths are up to N=2048, system sizes are up to $2^{21}$ lattice sites and $2.8\times 10^5$ monomers. These simulations used the recently introduced pruned-enriched Rosenbluth method which proved extremely efficient, together with a histogram method for estimating finite size corrections. Our most clear result is that chains at the critical point are Gaussian for large $N$, having end-to-end distances $R\sim\sqrt{N}$. Also the distance $T_\Theta-T_c(N)$ (where $T_\Theta = \lim_{N\to\infty} T_c(N)$) scales with the mean field exponent, $T_\Theta -T_c(N)\sim 1/\sqrt{N}$. The critical density seems to scale with a non-trivial exponent similar to that observed in experiments. But we argue that this is due to large logarithmic corrections. These corrections are similar to the very large corrections to scaling seen in recent analyses of $\Theta$-polymers, and qualitatively predicted by the field theoretic renormalization group. The only serious deviation from this simple global picture concerns the N-dependence of the order parameter amplitudes which disagrees with a minimalistic ansatz of de Gennes. But this might be due to problems with finite size scaling. We find that the finite size dependence of the density of states $P(E,n)$ (where $E$ is the total energy and $n$ is the number of chains) is slightly but significantly different from that proposed recently by several authors.