The Local Lattice Distortion Stability Theorem

The Local Lattice Distortion Stability Theorem | 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PDF Research Article The Local Lattice Distortion Stability Theorem Satish Prajapati This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9425298/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 The design of stable multi-principal element solid solutions, including high-entropy alloys, is hindered by the lack of a predictive, experimentally-agnostic stability criterion. Existing empirical rules based on atomic size mismatch or electronegativity differences fail in up to 40% of cases. Here, we derive a rigorous necessary condition for dynamical stability from first principles. Starting from the Born-Huang stability criterion for crystal lattices, we prove that a substitutional solid solution is stable against spinodal decomposition only if the normalized variance of the local shear modulus satisfies σ2 G/ ¯ G2 ≤ 1/3, where σ2 G is the variance of site-resolved shear moduli {Gi} and ¯ G is their mean. The proof combines three classical results: Popoviciu’s inequality for bounded variables, the Hashin-Shtrikman bounds for composite elasticity, and the Cahn Hilliard spinodal condition. The criterion is universal, chemistry-independent, and computable entirely from first-principles density functional theory without experimental calibration. Validation against fifteen experimentally-characterized alloy systems yields 93% accuracy, significantly outperforming Hume-Rothery rules (62%), Pauling electronegativity (58%), and CALPHAD (74%). The theorem provides a mathematically rigorous framework for high-throughput screening of stable solid solutions and guides the rational design of new alloys. Materials Engineering Materials Theory and Modeling High-Entropy Alloys Solid Solution Stability Local Shear Modulus Popoviciu’s Inequality Hashin-Shtrikman Bounds Cahn-Hilliard Theory 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. 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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-9425298","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":623533377,"identity":"4b6cc1d4-5f6d-4058-b3b8-2bbd96b6055e","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":""}],"badges":[],"createdAt":"2026-04-15 10:15:58","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-9425298/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9425298/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107481071,"identity":"5324df44-5978-4d7e-896b-0639830254a6","added_by":"auto","created_at":"2026-04-22 02:15:45","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2410954,"visible":true,"origin":"","legend":"","description":"","filename":"TheLocalLatticeDistortionStabilityTheorem.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9425298/v1_covered_9982c41a-6b37-400d-802a-3514a0dee361.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eThe Local Lattice Distortion Stability Theorem\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":"High-Entropy Alloys, Solid Solution Stability, Local Shear Modulus, Popoviciu’s Inequality, Hashin-Shtrikman Bounds, Cahn-Hilliard Theory","lastPublishedDoi":"10.21203/rs.3.rs-9425298/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9425298/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe design of stable multi-principal element solid solutions, including high-entropy alloys, is hindered by the lack of a predictive, experimentally-agnostic stability criterion. Existing empirical rules based on atomic size mismatch or electronegativity differences fail in up to 40% of cases. Here, we derive a rigorous necessary condition for dynamical stability from first principles. Starting from the Born-Huang stability criterion for crystal lattices, we prove that a substitutional solid solution is stable against spinodal decomposition only if the normalized variance of the local shear modulus satisfies σ2 G/ ¯ G2 ≤ 1/3, where σ2 G is the variance of site-resolved shear moduli {Gi} and ¯ G is their mean. The proof combines three classical results: Popoviciu’s inequality for bounded variables, the Hashin-Shtrikman bounds for composite elasticity, and the Cahn Hilliard spinodal condition. The criterion is universal, chemistry-independent, and computable entirely from first-principles density functional theory without experimental calibration. Validation against fifteen experimentally-characterized alloy systems yields 93% accuracy, significantly outperforming Hume-Rothery rules (62%), Pauling electronegativity (58%), and CALPHAD (74%). 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