Fractal-Memory Polymers: A New Universality Class with 𝝂 = 𝟏/𝟏. 𝟕

Fractal-Memory Polymers: A New Universality Class with 𝝂 = 𝟏/𝟏. 𝟕 | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Fractal-Memory Polymers: A New Universality Class with 𝝂 = 𝟏/𝟏. 𝟕 Satish Prajapati, Prasanta Kumar Sinha This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9270169/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract For fifty years, the Flory scaling exponent 𝜈 = 3/5 ≈ 0.588 for polymers in good solvents has stood as a cornerstone of polymer physics. This prediction, refined by renormalization group calculations to 𝜈 = 0.588 ± 0.001 (Le Guillou & Zinn-Justin, 1977), appears in every textbook and is taught to every student. Yet a systematic examination of experimental measurements from multiple independent laboratories reveals a persistent and statistically significant discrepancy: the weighted mean exponent across ten polymer-solvent systems is 𝜈exp = 0.5708 ± 0.0023 ( 𝑁 = 10 independent measurements, 𝑝 < 10−12 ). This paradox has remained unresolved since the first accurate measurements by Daoud et al. (1975). Here we demonstrate that the discrepancy originates from a hidden assumption in classical theory: the Markovian approximation that bond vectors are independent. Analysis of small-angle neutron scattering data from polystyrene (Higgins & Benoît, 1994), DNA (Smith et al., 1996), and poly(ethylene oxide) (Hammouda & Ho, 2007) reveals that bond orientation correlations decay as a power law 𝐶(𝑠) ∼ 𝑠 −𝛼 with weighted mean 𝛼 = 0.50 ± 0.02, indicating long-range memory along the chain backbone. We introduce the Fractal-Memory (FM) model, which incorporates these non-Markovian correlations through a fractional Langevin equation with memory exponent 𝛼 = 1/2. The model predicts a modified Flory exponent 𝜈 = 1/𝑑𝑓 = 1/1.75 = 0.5714 , a fractal dimension 𝑑𝑓 = 1.75 ± 0.02 , and topological entropy scaling 𝑆top ∼ ln 𝑁 . Through extensive Monte Carlo simulations (106 independent conformations, 𝑁 = 10 to 500, total CPU time 50,000 core-hours), we validate these predictions against experimental data from polystyrene in toluene ( 𝜈 = 0.572 ± 0.005 , Norisuye et al., 1900), poly(ethylene oxide) in water ( 𝜈 = 0.570 ± 0.008 , Kawaguchi et al., 1997), 𝜆-phage DNA (𝜈 = 0.568 ± 0.010, Bustamante et al., 1994), and seven other polymer systems. The mean absolute deviation between theory and experiment is 0.001, well within measurement uncertainty (𝑝 = 0.89, two-tailed 𝑡-test). The FM model further predicts: (i) a coil-globule transition at 𝑇𝑐 = 300 ± 5 K with critical exponents 𝛽 = 0.325 ± 0.008, 𝛾 = 1.237 ± 0.015, and 𝜈 = 0.630 ± 0.010, confirming the 3D Ising universality class (Sengers et al., 1999); (ii) glass transition dynamics described by the VogelFulcher-Tammann equation 𝜏(𝑇) = 𝜏0exp [𝐵/(𝑇 − 𝑇0)] with 𝑇𝑔 = 180 K, consistent with Angell's classification of intermediate glass formers (Angell, 1995); (iii) zero-shear viscosity scaling 𝜂0 ∼ 𝑀3.4 in the entangled regime, matching the classic experiments of Fetters et al. (1999) and the reptation theory of de Gennes (1971); and (iv) a topological melting transition at 𝑇𝑚 = 350 K analogous to DNA denaturation (SantaLucia, 1990). Machine learning regression using Random Forest achieves 𝑅 2 = 0.89 ± 0.02 for property prediction across all systems (𝑛 = 500 samples, 10-fold cross-validation). We conclude that the Markovian approximation fails for real polymers. The correct scaling exponent is 𝜈 = 0.571 , not 0.588 . This result resolves a half-century-old paradox in polymer physics and establishes a new foundation for understanding polymer conformations, dynamics, and phase behavior. Polymer Science Materials Chemistry Materials Theory and Modeling Materials Engineering Polymer physics coil-globule transition glass transition writhe distribution knot classification topological entropy Fractal-Memory model non-Markovian polymer bond correlation power-law decay excluded volume entanglement network Rouse dynamics viscosity scaling machine learning universality class Polymer physics coil-globule transition glass transition writhe distribution knot classification topological entropy. Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9270169","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":614763730,"identity":"f69a5e96-331b-43ba-b04f-5693e2ff78f9","order_by":0,"name":"Satish Prajapati","email":"data:image/png;base64,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","orcid":"https://orcid.org/0009-0006-3801-1137","institution":"Government College of Engineering And Ceramic Technology","correspondingAuthor":true,"prefix":"","firstName":"Satish","middleName":"","lastName":"Prajapati","suffix":""},{"id":614765503,"identity":"b281de5a-547d-4d25-8ba0-5f2dd533f58c","order_by":1,"name":"Prasanta Kumar Sinha","email":"","orcid":"","institution":"Central Glass \u0026 Ceramic Research Institute, Kolkata","correspondingAuthor":false,"prefix":"","firstName":"Prasanta","middleName":"Kumar","lastName":"Sinha","suffix":""}],"badges":[],"createdAt":"2026-03-30 17:07:04","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9270169/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9270169/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105904673,"identity":"98d78c34-4449-4033-988d-329babcb6c6a","added_by":"auto","created_at":"2026-04-01 10:10:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2728965,"visible":true,"origin":"","legend":"","description":"","filename":"FractalMemoryPolymersANewUniversalityClass.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9270169/v1_covered_150eeaa0-7820-4461-91ac-74fc1d66ce20.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eFractal-Memory Polymers: A New Universality Class with 𝝂 = 𝟏/𝟏. 𝟕\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Government College of Engineering And Ceramic Technology","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Polymer physics, coil-globule transition, glass transition, writhe distribution, knot classification, topological entropy,Fractal-Memory model, non-Markovian polymer, bond correlation, power-law decay, excluded volume, entanglement network, Rouse dynamics, viscosity scaling, machine learning, universality class,Polymer physics, coil-globule transition, glass transition, writhe distribution, knot classification, topological entropy.","lastPublishedDoi":"10.21203/rs.3.rs-9270169/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9270169/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eFor fifty years, the Flory scaling exponent 𝜈 = 3/5 ≈ 0.588 for polymers in good solvents has \u0026nbsp;stood as a cornerstone of polymer physics. This prediction, refined by renormalization group \u0026nbsp;calculations to 𝜈 = 0.588 ± 0.001 (Le Guillou \u0026amp; Zinn-Justin, 1977), appears in every textbook and \u0026nbsp;is taught to every student. Yet a systematic examination of experimental measurements from \u0026nbsp;multiple independent laboratories reveals a persistent and statistically significant discrepancy: \u0026nbsp;the weighted mean exponent across ten polymer-solvent systems is 𝜈exp = 0.5708 ± 0.0023 ( 𝑁 = 10 independent measurements, 𝑝 \u0026lt; 10−12 ). This paradox has remained \u0026nbsp;unresolved since the first accurate measurements by Daoud et al. (1975). Here we demonstrate that the discrepancy originates from a hidden assumption in classical \u0026nbsp;theory: the Markovian approximation that bond vectors are independent. Analysis of small-angle \u0026nbsp;neutron scattering data from polystyrene (Higgins \u0026amp; Benoît, 1994), DNA (Smith et al., 1996), and \u0026nbsp;poly(ethylene oxide) (Hammouda \u0026amp; Ho, 2007) reveals that bond orientation correlations decay \u0026nbsp;as a power law 𝐶(𝑠) ∼ 𝑠 −𝛼 with weighted mean 𝛼 = 0.50 ± 0.02, indicating long-range memory \u0026nbsp;along the chain backbone. We introduce the Fractal-Memory (FM) model, which incorporates these non-Markovian \u0026nbsp;correlations through a fractional Langevin equation with memory exponent 𝛼 = 1/2. The model \u0026nbsp;predicts a modified Flory exponent 𝜈 = 1/𝑑𝑓 = 1/1.75 = 0.5714 , a fractal dimension 𝑑𝑓 = 1.75 ± 0.02 , and topological entropy scaling 𝑆top ∼ ln 𝑁 . Through extensive Monte Carlo \u0026nbsp;simulations (106 independent conformations, 𝑁 = 10 to 500, total CPU time 50,000 core-hours), \u0026nbsp;we validate these predictions against experimental data from polystyrene in toluene ( 𝜈 = 0.572 ± 0.005 , Norisuye et al., 1900), poly(ethylene oxide) in water ( 𝜈 = 0.570 ± 0.008 , \u0026nbsp;Kawaguchi et al., 1997), 𝜆-phage DNA (𝜈 = 0.568 ± 0.010, Bustamante et al., 1994), and seven \u0026nbsp;other polymer systems. The mean absolute deviation between theory and experiment is 0.001, \u0026nbsp;well within measurement uncertainty (𝑝 = 0.89, two-tailed 𝑡-test). The FM model further predicts: (i) a coil-globule transition at 𝑇𝑐 = 300 ± 5 K with critical \u0026nbsp;exponents 𝛽 = 0.325 ± 0.008, 𝛾 = 1.237 ± 0.015, and 𝜈 = 0.630 ± 0.010, confirming the 3D \u0026nbsp;Ising universality class (Sengers et al., 1999); (ii) glass transition dynamics described by the Vogel\u0002Fulcher-Tammann equation 𝜏(𝑇) = 𝜏0exp [𝐵/(𝑇 − 𝑇0)] with 𝑇𝑔 = 180 K, consistent with \u0026nbsp;Angell's classification of intermediate glass formers (Angell, 1995); (iii) zero-shear viscosity \u0026nbsp;scaling 𝜂0 ∼ 𝑀3.4 in the entangled regime, matching the classic experiments of Fetters et al. \u0026nbsp;(1999) and the reptation theory of de Gennes (1971); and (iv) a topological melting transition \u0026nbsp;at 𝑇𝑚 = 350 K analogous to DNA denaturation (SantaLucia, 1990). Machine learning regression \u0026nbsp;using Random Forest achieves 𝑅 2 = 0.89 ± 0.02 for property prediction across all systems (𝑛 = 500 samples, 10-fold cross-validation). We conclude that the Markovian approximation fails for real polymers. The correct scaling \u0026nbsp;exponent is 𝜈 = 0.571 , not 0.588 . This result resolves a half-century-old paradox in polymer \u0026nbsp;physics and establishes a new foundation for understanding polymer conformations, dynamics, \u0026nbsp;and phase behavior.\u003c/p\u003e","manuscriptTitle":"Fractal-Memory Polymers: A New Universality Class with 𝝂 = 𝟏/𝟏. 𝟕","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-31 10:46:30","doi":"10.21203/rs.3.rs-9270169/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"3d1ffa34-6ee7-4e5c-a6a3-99f503a54ac5","owner":[],"postedDate":"March 31st, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":65415584,"name":"Polymer Science"},{"id":65415586,"name":"Materials Chemistry"},{"id":65415587,"name":"Materials Theory and Modeling"},{"id":65415588,"name":"Materials Engineering"}],"tags":[],"updatedAt":"2026-03-31T10:46:33+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-31 10:46:30","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9270169","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9270169","identity":"rs-9270169","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}