Universal Negative Energetic Elasticity in Polymer Chains: Crossovers among Random, Self-Avoiding, and Neighbor-Avoiding Walks

TL;DR

This paper addresses the origin of negative energetic elasticity in gels by studying two lattice polymer models, the Domb–Joyce (DJ) model and interacting self-avoiding walk (ISAW), across RW–SAW and SAW–NAW crossovers with exact enumeration. It identifies soft-repulsive segment interactions as the microscopic mechanism that yields negative energetic contributions to stiffness and reveals a universal scaling law for the internal energy with exponent $7/4$ that persists across crossovers and end-to-end directions. The approach combines exact enumeration, polynomial reconstructions in chain length, and cross-model analysis to establish universality beyond specific polymers or networks. The findings have significant implications for understanding gel elasticity and guiding the design of solvent-responsive polymer materials.

Abstract

Negative energetic elasticity in gels challenges the conventional understanding of gel elasticity; despite extensive research, a concise explanation remains elusive. In this study, we use the weakly self-avoiding walk (the Domb-Joyce model; DJ model) and interacting self-avoiding walk (ISAW) to investigate the emergence of negative energetic elasticity in polymer chains. Using exact enumeration, we show that both the DJ model and ISAW exhibit negative energetic elasticity, which is caused by effective soft-repulsive interactions between polymer segments. Moreover, we find that a universal scaling law for the internal energy of both models, with a common exponent of $7/4$, holds consistently across both random-walk-self-avoiding-walk and self-avoiding-walk-neighbor-avoiding-walk crossovers. These findings suggest that negative energetic elasticity is a fundamental and universal property of polymer networks and chains.

Figures (6)

• Figure 1: Configurations of 20-step (a) RW $\omega_\mathrm{a}$, (b) SAW $\omega_\mathrm{b}$, and (c) NAW $\omega_\mathrm{c}$ on cubic lattices with endpoints anchored at the origin and $(10,0,0)$. The red concentric circles in $\omega_\mathrm{a}$ indicate a site occupied by two segments, and the red oval in $\omega_\mathrm{b}$ highlights a pair of nearest-neighbor segments. (d) Inclusion relations $\Omega_\mathrm{RW} \supset \Omega_\mathrm{SAW} \supset \Omega_\mathrm{NAW}$, where $\Omega_\mathrm{RW}$, $\Omega_\mathrm{SAW}$, and $\Omega_\mathrm{NAW}$ are the configuration spaces of RWs, SAWs, and NAWs, respectively.
• Figure 2: Interaction energy ($\varDelta E$) in five lattice polymer models. The range of interactions is denoted by $d$; for $d=0$ or $1$, the interaction is between two segments occupying the same site or nearest-neighbor sites, respectively. The RW has no interactions between segments. The SAW introduces a hard-core repulsion ($d=0$) that prohibits segment overlap. The NAW introduces an additional hard-shell repulsion ($d=1$) between nearest-neighbor segments. The DJ model and the ISAW incorporate soft-core ($d=0$) and soft-shell ($d=1$) repulsions, respectively, with interaction strength $\varepsilon\,(>0)$. Varying $\varepsilon$ in these models allows continuous crossovers between the RW, SAW, and NAW.
• Figure 3: Emergence of negative energetic elasticity in RW--SAW and SAW--NAW crossovers with $(n,r)=(20,10)$. The DJ model bridges the RW and SAW (left column), and the ISAW bridges the SAW and NAW (right column). The data for the ISAW are obtained from Ref. ShiraiSakumichi2023. The top panels show the dependence of total stiffness $\beta k$ and the entropic contribution to stiffness $\beta k_S$ on interaction strength $\beta\varepsilon$. The middle panels show the energetic contribution to stiffness $\beta k_U$. The bottom panels show the ratio $-k_U/k$ and its first-order approximation $\tau\beta\varepsilon$ around $\beta\varepsilon=0$. The maximum of $-k_U/k$ occurs around $\beta\varepsilon\simeq 1$ for both models.
• Figure 4: Intuitive interpretation of the emergence of negative energetic elasticity originating from the entropy--energy interplay. (a) The reference state with end-to-end distance $r=r_\mathrm{ref}\,(>0)$ is entropically favorable (top), whereas the stretched state with $r=r_\mathrm{str}\,(>r_\mathrm{ref})$ is energetically favorable (bottom) because it has fewer intersections. This interpretation is consistent with the behavior of (b) the entropy $S_n^\mathrm{DJ}(r,\beta\varepsilon)$ and (c) the internal energy $U_n^\mathrm{DJ}(r,\beta\varepsilon)$, which are monotonically decreasing functions of $r$ for $1 \leq r\leq 20$. Here, we set $\beta\varepsilon=1$ and $n=20$.
• Figure 5: Universal scaling law of the internal energy $U_n(r,\beta\varepsilon)/\varepsilon$ for (a,b) the DJ model across RW--SAW crossover and (c,d) the ISAW across SAW--NAW crossover. In each panel, points correspond to $n=10,11,\ldots,20$ and $2\leq r \leq n-4$; curves represent the four analytic expressions for $U_n(r,\beta\varepsilon)/\varepsilon$ obtained from the polynomials of $W^\mathrm{RW}_{n,m}(r)$ and $W^\mathrm{SAW}_{n,m}(r)$ (see Appendixes \ref{['sec:RW_polynomials']} and \ref{['sec:SAW_polynomials']}) for $r=n-4$ (gray), $n-6$ (orange), $n-8$ (pink), and $n-10$ (red). In (a,c), $\beta\varepsilon=1$; for a fixed $n$, $U_n(r,\beta\varepsilon)/\varepsilon$ monotonically decreases with increasing $r$. In (b,d), $U_n(r,\beta\varepsilon)/\varepsilon$, plotted against the scaled variable $(n-r)^{7/4}/n$, collapses onto master curves with a common scaling exponent $7/4$ for a wide range of interaction strengths ($\beta\varepsilon=0$, $1/4$, $1/2$, $1$, $2$, and $4$). The master curves show a approximately linear behavior in the small $(n-r)^{7/4}/n$ regime.