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Our results demonstrate that the interchain constraint can increase the knotting probability, but does not alter the scaling relationship between knotting probability and chain length for ring polymers in melts. Having established that, we derived the interchain constraint contribution to the free energy of ring polymers in intra-crossing systems based on the knotting probability and obtained the scaling relationship between the size R and chain length N, i.e., R~N1\/6. And, by calculating the mean-squared radius of gyration of ring polymers in intra-crossing systems, we validated these scaling results. Finally, we analyze the size of knotted ring polymers with different types and compare corresponding scaling exponents for size versus chain lengths of ring polymers with different knotting complexities. These results provide fundamental insights into the static properties of ring polymers in melts.<\/jats:p>","DOI":"10.3390\/polym17070854","type":"journal-article","created":{"date-parts":[[2025,3,25]],"date-time":"2025-03-25T12:18:52Z","timestamp":1742905132000},"page":"854","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Chain Size and Knots of Ring Polymers in All-Crossing and Intra-Crossing Melts"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-9355-3023","authenticated-orcid":false,"given":"Jiangyang","family":"Mo","sequence":"first","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"},{"name":"State Key Laboratory of Polymer Science and Technology, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China"},{"name":"School of Materials Science and Engineering, Shandong University, Jinan 250061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jingqiao","family":"Guo","sequence":"additional","affiliation":[{"name":"Shandong Provincial Engineering Research Center of Novel Pharmaceutical Excipients and Controlled Release Preparations, College of Pharmacy, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xue","family":"Yu","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3957-3722","authenticated-orcid":false,"given":"Jianlei","family":"Yang","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6770-0667","authenticated-orcid":false,"given":"Guodong","family":"Hu","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianhui","family":"Xin","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mengxia","family":"Yan","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Wang","sequence":"additional","affiliation":[{"name":"Shandong Provincial Engineering Research Center of Novel Pharmaceutical Excipients and Controlled Release Preparations, College of Pharmacy, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yongjie","family":"Mo","sequence":"additional","affiliation":[{"name":"Shandong Key Laboratory of Biophysics, Institute of Biophysics, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuxi","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Materials Science and Engineering, Shandong University, Jinan 250061, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lianyong","family":"Wu","sequence":"additional","affiliation":[{"name":"Qilu Synva Pharmaceutical Co., Ltd., Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8323-4018","authenticated-orcid":false,"given":"Yongjin","family":"Ruan","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Polymer Science and Technology, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"204904","DOI":"10.1063\/1.3587137","article-title":"Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. 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