{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,23]],"date-time":"2026-10-23T20:35:45Z","timestamp":1792787745007,"version":"4.5.0"},"reference-count":28,"publisher":"SAGE Publications","issue":"11","license":[{"start":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T00:00:00Z","timestamp":1766534400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"},{"start":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T00:00:00Z","timestamp":1766534400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"DOI":"10.13039\/501100000038","name":"natural sciences and engineering research council of canada","doi-asserted-by":"publisher","award":["RGPIN-2018-04281, RGPAS-2018-522655"],"award-info":[{"award-number":["RGPIN-2018-04281, RGPAS-2018-522655"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Canada Research Chairs Program","award":["TIER1 2021-00023"],"award-info":[{"award-number":["TIER1 2021-00023"]}],"id":[{"id":"https:\/\/ror.org\/0517h6h17","id-type":"ROR","asserted-by":"crossref"}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Mathematics and Mechanics of Solids"],"published-print":{"date-parts":[[2026,11]]},"abstract":"<jats:p>Disordered biopolymer gels, such as alginate gels, contain disordered amorphous regions as well as ordered regions known as junction zones, the latter playing a paramount role in maintaining the integrity of the gel network. The considerable dimensions of these junction zones and their physical interactions are responsible for distinctive phenomena, such as self-healing, observed in biopolymer gels. Compared to rubber-like materials, where polymer chains are interconnected by chemical cross-links, statistical mechanics modeling of disordered biopolymer gels poses new challenges due to the need to capture the contributions from both the disordered and ordered regions. In the literature, the disordered domains are usually modeled by a collection of random coils, while each junction zone is represented by a rigid rod. Attempt has been made to introduce the two-node coil-rod structure, consisting of a random coil connected to a rigid rod, into classical polymer network models (e.g., the Arruda\u2013Boyce eight-chain model) to predict the mechanical response of the biopolymer gels. Although this approach provides valuable insights and reasonable predictions that agree with experimental data, the interactions between the junction zones and the amorphous regions are significantly simplified. This study aims to extend the two-node coil-rod structure to a four-node coil-rod structure, in which two polymer chains share a junction zone, more explicitly representing chain association induced by physical forces in biopolymer gels. By incorporating statistical mechanics and the phantom network model, the entropy of the system is derived, from which the stress\u2013stretch relationships for the network are obtained. Comparisons are made with the model based on the two-node coil-rod structure, and the conditions for establishing equivalency between the two methods are examined. This work establishes a foundation for developing more advanced network models that incorporates the simultaneous interaction of multiple junction zones, a phenomenon often present in biopolymer gels.<\/jats:p>","DOI":"10.1177\/10812865251390442","type":"journal-article","created":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T09:18:34Z","timestamp":1766567914000},"page":"2599-2631","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Statistical mechanics and phantom network modelling of disordered biopolymer gels"],"prefix":"10.1177","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1164-3733","authenticated-orcid":false,"given":"Hashem","family":"Moosavian","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tian","family":"Tang","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of Alberta, Edmonton, AB, Canada"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2025,12,24]]},"reference":[{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.aag0184"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1723785"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1746626"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1700682"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5096(93)90043-F"},{"key":"e_1_3_4_7_2","doi-asserted-by":"publisher","DOI":"10.1177\/1081286509106441"},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-5096(93)90013-6"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.1177\/108128650100600605"},{"key":"e_1_3_4_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2004.03.011"},{"key":"e_1_3_4_11_2","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1976.0146"},{"key":"e_1_3_4_12_2","doi-asserted-by":"crossref","unstructured":"Ross-Murphy SB. 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