A Fractal Simulation Phase Growth Analysis of Spinodal Decomposition

A Fractal Simulation Phase Growth Analysis of Spinodal Decomposition | 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Article A Fractal Simulation Phase Growth Analysis of Spinodal Decomposition Rahul Basu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9062986/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 Spinodal decomposition is a diffusion-controlled phase-separation process that produces complex, interconnected microstructures in multi-component systems. In this study, we investigate the emergence of scale-invariant morphological complexity during spinodal decomposition through phase-field simulations governed by the Cahn–Hilliard equation. Two- and three-dimensional simulations are performed, and the evolving concentration fields are quantitatively characterized using box-counting analysis to determine their effective fractal dimension. Spinodal patterns emerge from continuous concentration fields with diffuse interfaces, in contrast to classical geometric fractals characterized by sharp, self-similar boundaries (e.g., the Koch curve). As such, the fractal dimension reported here measures the multi-scale spatial occupancy and textural complexity of the evolving morphology rather than the Hausdorff dimension of well-defined geometric interfaces. The effective fractal dimension increases systematically with time: from values typical of sparse, filamentary structures in the early stage (~1.3 in 2D) to significantly higher values (~1.7–1.8 in 2D slices; ~2.8 in 3D volume) during late-stage coarsening, reflecting enhanced connectivity and near space-filling behavior. The numerical sensitivity to grid resolution and interface smoothing is carefully assessed, which confirms stable scaling behavior with appropriate refinement. The contrast between spinodal morphology and deterministic geometric fractals in this work clarifies the fundamental difference between exact geometric self-similarity and statistically self-similar patterns driven by thermodynamic free-energy minimization. The results provide robust, quantitative descriptors of interfacial complexity and domain connectivity. This offers practical metrics for the design and optimization of nanostructured materials in applications such as energy storage, catalysis, and microelectronics. Applied Mathematics Artificial Intelligence and Machine Learning Materials Engineering fractals spinodal phase change diffuse interface 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. 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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-9062986","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":602541570,"identity":"9e10d349-4dfd-478f-917a-b8a3f44bea6c","order_by":0,"name":"Rahul Basu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvklEQVRIiWNgGAWjYHCCBIMPFTZAmrHxALFaHhTOOJMG0tJArBbGB5952w6DmcRpMW8/nLiZt+283dr2w0BbamyiCWqROZOWbDjn3O3kbWcSgVqOpeU2ENIiwZCTZvCm7Hay2QGgFsaGw0Ro4X///QcP27lks/MPidUikZBgyNN2wM7sBtG2SDxIMJxxJjnB7AbQlgSi/MKfAIpKO3uz8+kPH3yosSGsBQYSwSoTiFUOAvakKB4Fo2AUjIIRBgDoMEtitoK0FAAAAABJRU5ErkJggg==","orcid":"","institution":"JNTU","correspondingAuthor":true,"prefix":"","firstName":"Rahul","middleName":"","lastName":"Basu","suffix":""}],"badges":[],"createdAt":"2026-03-08 08:48:46","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9062986/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9062986/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104305481,"identity":"41e5ee5e-1682-4ee7-bc8c-6c54b0f975f3","added_by":"auto","created_at":"2026-03-10 09:49:59","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":558963,"visible":true,"origin":"","legend":"","description":"","filename":"onecolPhaseGrowthSpinodalDecomposition.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9062986/v1_covered_694fde15-ce49-4810-9052-6f985ce0a58e.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003e\u003cstrong\u003eA Fractal Simulation Phase Growth Analysis of Spinodal Decomposition\u003c/strong\u003e\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"JNTU","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":"fractals, spinodal, phase change, diffuse interface","lastPublishedDoi":"10.21203/rs.3.rs-9062986/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9062986/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eSpinodal decomposition is a diffusion-controlled phase-separation process that produces complex, interconnected microstructures in multi-component systems. In this study, we investigate the emergence of scale-invariant morphological complexity during spinodal decomposition through phase-field simulations governed by the Cahn–Hilliard equation. Two- and three-dimensional simulations are performed, and the evolving concentration fields are quantitatively characterized using box-counting analysis to determine their effective fractal dimension.\u003c/p\u003e\n\u003cp\u003eSpinodal patterns emerge from continuous concentration fields with diffuse interfaces, in contrast to classical geometric fractals characterized by sharp, self-similar boundaries (e.g., the Koch curve). As such, the fractal dimension reported here measures the multi-scale spatial occupancy and textural complexity of the evolving morphology rather than \u0026nbsp; the Hausdorff dimension of well-defined geometric interfaces. The effective fractal dimension increases systematically with time: from values typical of sparse, filamentary structures in the early stage (~1.3 in 2D) to significantly higher values (~1.7–1.8 in 2D slices; ~2.8 in 3D volume) during late-stage coarsening, reflecting enhanced connectivity and near space-filling behavior.\u003c/p\u003e\n\u003cp\u003eThe numerical sensitivity to grid resolution and interface smoothing is carefully assessed, which confirms stable scaling behavior with appropriate refinement. The contrast between spinodal morphology and deterministic geometric fractals in this work clarifies the fundamental difference between exact geometric self-similarity and statistically self-similar patterns driven by thermodynamic free-energy minimization. The results provide robust, quantitative descriptors of interfacial complexity and domain connectivity. 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