Unified origin of negative energetic elasticity in a lattice polymer chain: soft self-repulsion and bending stiffness

TL;DR

The paper addresses the origin of negative energetic elasticity in lattice polymers under fixed end-to-end distance by jointly considering Domb–Joyce self-repulsion and a local bending-energy term. It uses exact enumeration of an $n$-step lattice polymer with energy $E(\omega)=\varepsilon m(\omega)+\varepsilon_b b(\omega)$ to compute stiffness $k$ and decomposes it into energetic ($k_U$) and entropic ($k_S$) parts, revealing a robust negative $k_U$ across parameter space and a positive $k_S$. A key contribution is the unification of two previously separate mechanisms of negative energetic elasticity and the demonstration that the bending-only limit maps to Potts/Ising chains, clarifying the microscopic origin of the effect. Additionally, the authors identify a universal internal-energy scaling in the DJ limit, $U/\varepsilon \sim (n-r)^{7/4}/n$, which is progressively disrupted by bending stiffness, providing a diagnostic to distinguish between self-repulsion and bending contributions in real systems.

Abstract

We study a single lattice polymer chain under a fixed end-to-end distance, incorporating both Domb--Joyce (DJ) soft-core self-repulsion between polymer segments and a local bending-energy cost. By decomposing the stiffness into energetic and entropic contributions, we survey the parameter space defined by the self-repulsion strength and bending-energy cost. We find that the energetic contribution to stiffness is negative across the entire explored range, whereas the entropic contribution remains positive. These results unify two previously independent mechanisms of negative energetic elasticity -- solvent-induced self-repulsion and bending stiffness -- and demonstrate that either mechanism alone, as well as their combination, produces the same sign. Beyond this sign-level unification, we analyze the internal-energy scaling and show that, in the absence of the bending-energy term, the DJ (self-repulsion) limit exhibits a robust $(n-r)^{7/4}/n$ scaling collapse. In contrast, the introduction of finite bending stiffness progressively disrupts this scaling, providing an internal-energy-based diagnostic to distinguish between contributions from self-repulsion and bending stiffness.

Figures (4)

• Figure 1: A 20-step random-walk configuration on a cubic lattice with endpoints fixed at the origin $\mathrm{O}$ and $(10,0,0)$ (on-axis constraint). Pink-filled circles indicate chain segments contributing to the bending-energy cost. The inset highlights an on-site overlap ($v_x = 2$), corresponding to two segments occupying the same lattice site, which contributes to the DJ overlap count $m=1$. Local changes in the chain direction contribute to the bend count $b=13$.
• Figure 2: Contour plots of the stiffness decomposition in the model: $\beta k$, $\beta k_U$, $\beta k_S$, and $k_U/k$ across the $(\beta\varepsilon, \beta\varepsilon_b)$ plane for a chain with $(n,r)=(20,10)$. The $x$-axis is displayed on a logarithmic scale. Dotted contours represent lines of constant value in all panels. Negative energetic elasticity is observed throughout the domain, where either $\varepsilon > 0$ or $\varepsilon_b > 0$, as evident in both the $\beta k_U$ and $k_U/k$ panels.
• Figure 3: Illustration of negative energetic elasticity originating from the bending-energy cost ($\varepsilon_b>0$ and $\varepsilon=0$). (a) Schematic comparison of chain conformations at two imposed end-to-end distances: $r_\mathrm{ref}$ for the reference state (top) and $r_\mathrm{str}\,(>r_\mathrm{ref})$ for the stretched state (bottom). Pink-filled circles indicate local bends, i.e., segments contributing to the bending-energy cost. At the larger imposed distance $r_\mathrm{str}$, typical conformations contain fewer bends, so the average bending contribution is reduced relative to $r_\mathrm{ref}$. (b) Entropy $S/k_B$ as a function of $r$, showing that smaller extensions are more entropically favorable ($S_\mathrm{ref}>S_\mathrm{str}$). (c) Internal energy $U/\varepsilon_b$ as a function of $r$, showing that larger extensions are energetically favorable ($U_\mathrm{ref}>U_\mathrm{str}$). Parameters are set to $\varepsilon=0$, $\beta\varepsilon_b=1$, and $n=20$.
• Figure 4: Breakdown of the universal $7/4$-scaling of the internal energy due to bending stiffness. In the left panel, $U/\varepsilon$ is plotted against $n$ at $\beta\varepsilon=1$ for chains with $n=10,11,\ldots,20$ and slacks $n-r=4,6,\ldots,18$; symbol shape represents $n-r$, and color indicates the ratio $\varepsilon_b/\varepsilon$ of the bending penalty to the self-repulsion strength. In the right panel, the same data are plotted against $(n-r)^{7/4}/n$, showing how the scaling with slack is affected by bending. Gray symbols and curves correspond to the DJ model without bending, $\varepsilon_b/\varepsilon=0$, and reproduce the universal $(n-r)^{7/4}/n$ collapse reported in Fig. 5(b) of Ref. ShiraiSakumichi2025. As $\varepsilon_b/\varepsilon$ increases to $2^{-6}$ (pink), $2^{-4}$ (blue), and $2^{-2}$ (orange), the colored data systematically deviate from the gray master curve, demonstrating that bending stiffness progressively disrupts the universal scaling characteristic of pure self-repulsion.