Four-dimensional polymer collapse II: Pseudo-First-Order Transition in Interacting Self-avoiding Walks

TL;DR

This paper uses Monte Carlo simulations to analyze the collapse transition of four-dimensional interacting self-avoiding walks, revealing a pseudo-first-order transition that scales to a second-order transition in the thermodynamic limit, consistent with Lifshitz theory.

Contribution

It demonstrates that the four-dimensional polymer collapse exhibits a rounded first-order transition that scales away, leading to a second-order transition, aligning with Lifshitz theory predictions.

Findings

Finite-size transition temperatures approach the θ-point as polymer length increases.

The latent heat of the transition decays algebraically with polymer length.

The transition appears first-order in finite systems but becomes second-order in the thermodynamic limit.

Abstract

In earlier work we provided the first evidence that the collapse, or coil-globule, transition of an isolated polymer in solution can be seen in a four-dimensional model. Here we investigate, via Monte Carlo simulations, the canonical lattice model of polymer collapse, namely interacting self-avoiding walks, to show that it not only has a distinct collapse transition at finite temperature but that for any finite polymer length this collapse has many characteristics of a rounded first-order phase transition. However, we also show that there exists a `$\theta$-point' where the polymer behaves in a simple Gaussian manner (which is a critical state), to which these finite-size transition temperatures approach as the polymer length is increased. The resolution of these seemingly incompatible conclusions involves the argument that the first-order-like rounded transition is scaled away in the thermodynamic limit to leave a mean-field second-order transition. Essentially this happens because the finite-size \emph{shift} of the transition is asymptotically much larger than the \emph{width} of the pseudo-transition and the latent heat decays to zero (algebraically) with polymer length. This scenario can be inferred from the application of the theory of Lifshitz, Grosberg and Khokhlov (based upon the framework of Lifshitz) to four dimensions: the conclusions of which were written down some time ago by Khokhlov. In fact it is precisely above the upper critical dimension, which is 3 for this problem, that the theory of Lifshitz may be quantitatively applicable to polymer collapse.