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Universal negative energetic elasticity in polymer chains: Crossovers among random, self-avoiding, and neighbor-avoiding walks

Nobu C. Shirai1,*,† and Naoyuki Sakumichi2,3,*,‡

  • 1Center for Information Technologies and Networks, Mie University, Tsu, Mie 514-8507, Japan
  • 2Faculty of Social Informatics, ZEN University, Shinjuku, Zushi, Kanagawa 249-0007, Japan
  • 3Department of Chemistry and Biotechnology, The University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan

  • *These authors contributed equally to this work.
  • †Contact author: shirai@cc.mie-u.ac.jp
  • ‡Contact author: sakumichi@gel.t.u-tokyo.ac.jp
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Phys. Rev. E 112, 025406 – Published 4 August, 2025

DOI: https://doi.org/10.1103/pzk3-t3s5

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.

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Corrections

2 February, 2026

Correction: In Eq. (5), variables U and A were transposed and have been fixed; in Table I, a minor formatting change has been made; in Appendix B, a minor typographical error has been fixed.

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Phys. Rev. E 112, 025406– Published 4 August, 2025
  • Received 27 August 2024
  • Revised 14 December 2024
  • Accepted 20 June 2025
  • Corrected 2 February 2026
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