Scaling in Gel Mechanics: Crossover between Self- and Neighbor-Avoiding Walks

Recent studies on rubber-like polymer gels have highlighted a phenomenon termed "negative energetic elasticity," suggesting that elasticity is not solely derived from entropic contribution. Building on this understanding, we investigate the crossover between the self-avoiding walk (SAW) and the neighbor-avoiding walk (NAW) on a cubic lattice [Phys. Rev. Lett. 130, 148101 (2023)]. This crossover provides insights into the behavior of a single polymer chain within a broader context of polymer gels. Through exact enumerations for steps up to n=20, we clarify the behavior of negative energetic elasticity associated with this crossover. One of the main results of our study is the universal scaling relation between T_U^∞ and the chain slack, represented as n-r. In this context, T_U^∞ emerges as the theoretical counterpart of the experimentally observed T_U, a consistent measure of normalized polymer concentration. This universal scaling exponent sheds light on the mechanics of polymer gels, especially the behavior related to negative energetic elasticity. Our study provides a comprehensive understanding of the mechanics of polymer gels and offers potential directions for future research in gel elasticity.