# On the origin of the negative energy-related contribution to the elastic modulus of rubber-like gels

## TL;DR

This paper develops a coarse-grained polymer model to explain the negative energy contribution to the elastic modulus of rubber-like gels, revealing a temperature-dependent correction factor that improves understanding of gel elasticity.

## Contribution

The study introduces an exact free energy expression for a polymer model that captures the negative energy contribution and proposes a modified elastic modulus formula incorporating a temperature-dependent factor.

## Key findings

- The model accurately describes experimental data for tetra-PEG hydrogels.
- The elastic modulus depends on temperature as G(T) ∝ k_B T w(T).
- The correction factor w(T) relates to chain interactions and solvent effects.

## Abstract

We consider a coarse-grained polymer model in order to investigate the origin of a recently discovered negative energy-related contribution to the elastic modulus $G(T)$ of rubber-like gels. From this model, we are able to compute an exact expression for the free energy of the system, which allows us to evaluate a stress-strain relationship that displays a non-trivial dependence on the temperature $T$. We validate our approach through comparisons between the theoretical results and the experimental data obtained for tetra-PEG hydrogels, which indicate that, although simple, the present model works well to describe the experiments. Importantly, our approach unveiled aspects of the experimental analysis which turned out to be different from the conventional entropic and energetic analysis broadly used in the literature. Also, in contrast to the linear dependence predicted by the traditional, {\it i.e.}, purely entropic, models, our results suggest that the general expression of the elastic modulus should be of the form $G(T) \propto k_BT w(T)$, with $w(T)$ being a temperature-dependent correction factor that could be related to the interaction between the chains in the network and the solvent. Accordingly, the correction factor allows the expression found for the elastic modulus to describe both rubber and rubber-like gels.