{"id":318,"date":"2017-10-29T11:14:04","date_gmt":"2017-10-29T11:14:04","guid":{"rendered":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/?p=318"},"modified":"2017-11-08T00:32:09","modified_gmt":"2017-11-08T00:32:09","slug":"thermodynamically-consistent-variational-principles-with-applications-to-electrically-and-magnetically-active-systems","status":"publish","type":"post","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2017\/10\/29\/thermodynamically-consistent-variational-principles-with-applications-to-electrically-and-magnetically-active-systems\/","title":{"rendered":"RE Garc\u00eda, CM Bishop, WC Carter “Thermodynamically consistent variational principles with applications to electrically and magnetically active systems” Acta Materialia, 52(1):11-21, 2004."},"content":{"rendered":"

RE Garc\u00eda, CM Bishop, WC Carter “Thermodynamically consistent variational principles with applications to electrically and magnetically active systems<\/a>” Acta Materialia<\/strong>, 52(1):11-21, 2004.<\/p>\n

Abstract<\/h3>\n

We propose a theoretical framework to derive thermodynamically consistent equilibrium equations and kinetic driving forces to describe the time evolution for electrically and magnetically active materials. This procedure starts from the combined statement of the first and second laws of thermodynamics, naturally incorporates Maxwell\u2019s equations, and accommodates the description of continuous phase transformations for conserved and non-conserved order parameters. The kinetics of conserved and non-conserved ordered parameters are introduced, the adequate gradient flow is identified, thus the appropriate kinetics (e.g., Allen\u2013Cahn, Cahn\u2013Hilliard) are derived. Example applications of this theory include the electromechanical fields of piezoelectric materials and the wave equation in the limit of chemically homogeneous solids. Moreover, we derive a thermodynamically consistent set of partial differential equations which describe the transport of charged species in conductive, non-polarizable, magnetizable solids, and in polarizable, electrically insulating, non-magnetizable solids.<\/p>\n","protected":false},"excerpt":{"rendered":"

RE Garc\u00eda, CM Bishop, WC Carter “Thermodynamically consistent variational principles with applications…<\/p>\n

Continue reading “RE Garc\u00eda, CM Bishop, WC Carter “Thermodynamically consistent variational principles with applications to electrically and magnetically active systems” Acta Materialia, 52(1):11-21, 2004.”<\/span>…<\/a><\/div>\n
Continue reading \"RE Garc\u00eda, CM Bishop, WC Carter “Thermodynamically consistent variational principles with applications to electrically and magnetically active systems” Acta Materialia, 52(1):11-21, 2004.\"<\/span>…<\/a><\/div>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_seo_schema_type":"","_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[45],"tags":[6,11,22,48,15,7],"class_list":["post-318","post","type-post","status-publish","format-standard","hentry","category-papers","tag-electrochemistry","tag-ferroelectrics","tag-phase-diagrams","tag-phase-field","tag-properties","tag-thermodynamics","entry"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/peeeSR-58","jetpack_likes_enabled":true,"jetpack-related-posts":[{"id":877,"url":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2020\/12\/10\/j-lund-k-s-n-vikrant-c-m-bishop-w-rheinheimer-r-e-garcia-thermodynamically-consistent-variational-principles-for-charged-interfaces-acta-materialia-205116525-2021\/","url_meta":{"origin":318,"position":0},"title":"J. Lund, K. S. N. Vikrant, C. M. Bishop, W. Rheinheimer, R. E. Garc\u00eda “Thermodynamically Consistent Variational Principles for Charged Interfaces.” Acta Materialia, 205:116525, (2021).","author":"redwing","date":"12\/10\/2020","format":false,"excerpt":"J. Lund, K. S. N. Vikrant, C. M. Bishop, W. Rheinheimer, R. E. Garc\u00eda \"Thermodynamically Consistent Variational Principles for Charged Interfaces.\" Acta Materialia, 205:116525, (2021).\u00a0https:\/\/doi.org\/10.1016\/j.actamat.2020.116525 Abstract A generalized framework that naturally incorporates the free energy contributions of thermochemical, structural, mechanical, and electrical fields is presented to describe the Space Charge\u2026","rel":"","context":"In "Papers"","block_context":{"text":"Papers","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/category\/papers\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":1013,"url":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2026\/08\/01\/ksn-vikrant-re-garcia-electrically-charged-dislocations-in-ionic-ceramics-acta-materialia-121778-2025\/","url_meta":{"origin":318,"position":1},"title":"KSN Vikrant, RE Garc\u00eda “Electrically charged dislocations in ionic ceramics.” Acta Materialia, 121778, 2025.","author":"redwing","date":"08\/01\/2026","format":false,"excerpt":"KSN Vikrant, RE Garc\u00eda \"Electrically charged dislocations in ionic ceramics.\" Acta Materialia, 121778, 2025. https:\/\/doi.org\/10.1016\/j.actamat.2025.121778 Abstract A thermodynamically consistent phase field theory describing coupled electrical, chemical, and mechanical effects on charged dislocations in ionic ceramics is presented. The formulation results in a generalized multiphysical Peach\u2013K\u00f6hler force, establishing a critical electro-chemo-mechanical\u2026","rel":"","context":"In "Papers"","block_context":{"text":"Papers","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/category\/papers\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":1004,"url":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2026\/08\/01\/cm-bishop-ksn-vikrant-re-garcia-ferroelectric-grain-boundary-complexion-transitions-acta-materialia-293-121043-2025\/","url_meta":{"origin":318,"position":2},"title":"CM Bishop, KSN Vikrant, RE Garc\u00eda “Ferroelectric grain boundary complexion transitions.” Acta Materialia 293: 121043, 2025.","author":"redwing","date":"08\/01\/2026","format":false,"excerpt":"CM Bishop, KSN Vikrant, RE Garc\u00eda \"Ferroelectric grain boundary complexion transitions.\" Acta Materialia 293: 121043, 2025. https:\/\/doi.org\/10.1016\/j.actamat.2025.121043 Abstract By starting from equilibrium volumetric and\u00a0interfacial properties\u00a0of\u00a0ferroelectric materials, a thermodynamically-consistent phase field framework incorporating polarization and\u00a0space charge\u00a0for\u00a0polycrystalline\u00a0ferroelectrics is developed. Building on Cahn\u2019s\u00a0critical point\u00a0wetting variational analysis, a series of planar\u00a0[001]\u00a0tilt\u00a0grain boundaries in tetragonal\u2026","rel":"","context":"In "Papers"","block_context":{"text":"Papers","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/category\/papers\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":781,"url":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2018\/10\/26\/oat-matheus-re-garcia-cm-bishop-phase-field-theory-and-coexistence-of-ferroelectric-phases-near-the-morphotropic-phase-boundary-acta-materialia-in-press-oct-2018\/","url_meta":{"origin":318,"position":3},"title":"OA Torres-Matheus, RE Garc\u00eda, CM Bishop. \u201cPhase Coexistence Near the Morphotropic Phase Boundary.\u201d Acta Materialia. 164:577-585, 2019.","author":"redwing","date":"10\/26\/2018","format":false,"excerpt":"OA Torres-Matheus, RE Garc\u00eda, CM Bishop. \u201cPhase \u00a0Coexistence Near the Morphotropic Phase Boundary.\u201d Acta Materialia. 164:577-585, 2019.\u00a0https:\/\/doi.org\/10.1016\/j.actamat.2018.10.041 Abstract A novel multiphase field theory for ferroelectric systems in the vicinity of a polymorphic phase boundary (PPB) is developed by coupling the Landau-Devonshire thermodynamic potentials of the individual phases. The model naturally\u2026","rel":"","context":"In "Papers"","block_context":{"text":"Papers","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/category\/papers\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":921,"url":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2022\/06\/08\/l-d-robinson-k-s-n-vikrant-j-e-blendell-c-a-handwerker-r-e-garcia-interfacial-and-volumetric-melting-regimes-of-sn-nanoparticles-acta-materialia-in-press-2022\/","url_meta":{"origin":318,"position":4},"title":"L.D. Robinson, K.S.N. Vikrant, J.E. Blendell, C.A. Handwerker, R.E. Garc\u00eda “Interfacial and Volumetric Melting Regimes of Sn Nanoparticles.” Acta Materialia. In Press. 2022","author":"redwing","date":"06\/08\/2022","format":false,"excerpt":"L.D. Robinson, K.S.N. Vikrant, J.E. Blendell, C.A. Handwerker, and R.E. Garc\u00eda \"Interfacial and Volumetric Melting Regimes of Sn Nanoparticles.\" Acta Materialia. In Press. 2022.\u00a0https:\/\/doi.org\/10.1016\/j.actamat.2022.118084 Abstract A thermodynamically consistent phase field formulation was developed to describe what has been historically known as the premelted surface layer in Sn nanoparticles. Two interfacial\u2026","rel":"","context":"In "Papers"","block_context":{"text":"Papers","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/category\/papers\/"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/engineering.purdue.edu\/ComputationalMaterials\/wp-content\/uploads\/2022\/06\/1-s2.0-S1359645422004657-ga1_lrg-1.jpg?resize=350%2C200&ssl=1","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/engineering.purdue.edu\/ComputationalMaterials\/wp-content\/uploads\/2022\/06\/1-s2.0-S1359645422004657-ga1_lrg-1.jpg?resize=350%2C200&ssl=1 1x, https:\/\/i0.wp.com\/engineering.purdue.edu\/ComputationalMaterials\/wp-content\/uploads\/2022\/06\/1-s2.0-S1359645422004657-ga1_lrg-1.jpg?resize=525%2C300&ssl=1 1.5x, https:\/\/i0.wp.com\/engineering.purdue.edu\/ComputationalMaterials\/wp-content\/uploads\/2022\/06\/1-s2.0-S1359645422004657-ga1_lrg-1.jpg?resize=700%2C400&ssl=1 2x, https:\/\/i0.wp.com\/engineering.purdue.edu\/ComputationalMaterials\/wp-content\/uploads\/2022\/06\/1-s2.0-S1359645422004657-ga1_lrg-1.jpg?resize=1050%2C600&ssl=1 3x"},"classes":[]},{"id":315,"url":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/2017\/10\/29\/effect-of-charge-separation-on-the-stability-of-large-wavelength-fluctuations-during-spinodal-decomposition\/","url_meta":{"origin":318,"position":5},"title":"CM Bishop, RE Garc\u00eda, WC Carter “Effect of charge separation on the stability of large wavelength fluctuations during spinodal decomposition” \u00a0Acta materialia, 51(6): 1517-1524, 2003.","author":"redwing","date":"10\/29\/2017","format":false,"excerpt":"CM Bishop, RE Garc\u00eda, WC Carter \"Effect of charge separation on the stability of large wavelength fluctuations during spinodal decomposition\" \u00a0Acta materialia, 51(6): 1517-1524, 2003. Abstract A stability analysis of phase separation of charged species by spinodal decomposition is presented. The charge effects introduce a short wave number cutoff for\u2026","rel":"","context":"In "Papers"","block_context":{"text":"Papers","link":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/category\/papers\/"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]}],"_links":{"self":[{"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/posts\/318","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/comments?post=318"}],"version-history":[{"count":4,"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/posts\/318\/revisions"}],"predecessor-version":[{"id":579,"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/posts\/318\/revisions\/579"}],"wp:attachment":[{"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/media?parent=318"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/categories?post=318"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/engineering.purdue.edu\/ComputationalMaterials\/index.php\/wp-json\/wp\/v2\/tags?post=318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}