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Partitioning law of polymer chains into flexible polymer networks

Haruki Takarai1,*, Takashi Yasuda2,*, Naoyuki Sakumichi1,3,†, and Takamasa Sakai1,‡

  • 1Department of Chemistry and Biotechnology, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
  • 2Faculty of Advanced Life Science, Hokkaido University, Sapporo, Hokkaido 001-0021, Japan
  • 3Faculty of Social Informatics, ZEN University, Shinjuku, Zushi, Kanagawa 249-0007, Japan

  • *These authors contributed equally to this work.
  • †Contact author: sakumichi@gel.t.u-tokyo.ac.jp
  • ‡Contact author: sakai@gel.t.u-tokyo.ac.jp
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Phys. Rev. E 114, 015420 – Published 27 July, 2026

DOI: https://doi.org/10.1103/vvg5-dhy2

Abstract

The equilibrium partitioning of linear polymer chains into flexible polymer networks is governed by intricate entropic constraints arising from the configurational degrees of freedom of both chains and networks; however, a quantitative understanding remains elusive. Using model hydrogels with precisely defined network structures, we experimentally demonstrate a universal law governing the partitioning of linear polymers into flexible polymer networks. We establish a label-free contactless method to measure the partition ratio, based on the increase in the osmotic pressure induced by the partitioning of the external polymer chains. Moreover, we reveal a universal law in which the partition ratio is determined solely by Rg/lcycle, where Rg is the gyration radius of the polymer chain and lcycle≡ξ−1/3 is the characteristic mesh size of the network, defined by the cycle rank ξ, i.e., the number density of elastically effective cycles.

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Phys. Rev. E 114, 015420– Published 27 July, 2026
  • Received 4 June 2025
  • Revised 8 February 2026
  • Accepted 2 June 2026
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