{"id":"https://openalex.org/W2158598889","doi":"https://doi.org/10.1137/080728809","title":"On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn\u2013Hilliard Functional","display_name":"On the Phase Diagram for Microphase Separation of Diblock Copolymers: An Approach via a Nonlocal Cahn\u2013Hilliard Functional","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2158598889","doi":"https://doi.org/10.1137/080728809","mag":"2158598889"},"language":"en","primary_location":{"id":"doi:10.1137/080728809","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080728809","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://pure.tue.nl/ws/files/3209530/Metis218588.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058698689","display_name":"Rustum Choksi","orcid":"https://orcid.org/0000-0003-2419-9520"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Rustum Choksi","raw_affiliation_strings":["Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, BC, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, BC, Canada","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5077512722","display_name":"Mark A. Peletier","orcid":"https://orcid.org/0000-0001-9663-3694"},"institutions":[{"id":"https://openalex.org/I83019370","display_name":"Eindhoven University of Technology","ror":"https://ror.org/02c2kyt77","country_code":"NL","type":"education","lineage":["https://openalex.org/I83019370"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Mark A. Peletier","raw_affiliation_strings":["Department of Mathematics, Eindhoven University of Technology, Eindhoven, The Netherlands"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Eindhoven University of Technology, Eindhoven, The Netherlands","institution_ids":["https://openalex.org/I83019370"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009177807","display_name":"Jim Williams","orcid":"https://orcid.org/0000-0002-1464-2317"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"J. F. Williams","raw_affiliation_strings":["Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, BC, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Simon Fraser University, Burnaby V5A 1S6, BC, Canada","institution_ids":["https://openalex.org/I18014758"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5058698689"],"corresponding_institution_ids":["https://openalex.org/I18014758"],"apc_list":null,"apc_paid":null,"fwci":4.3443,"has_fulltext":true,"cited_by_count":145,"citation_normalized_percentile":{"value":0.95481952,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"69","issue":"6","first_page":"1712","last_page":"1738"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11471","display_name":"Block Copolymer Self-Assembly","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11471","display_name":"Block Copolymer Self-Assembly","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9941999912261963,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12141","display_name":"Fluid Dynamics and Thin Films","score":0.9912999868392944,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/phase-diagram","display_name":"Phase diagram","score":0.7251765727996826},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.588046133518219},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5323634147644043},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.47825518250465393},{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.47111037373542786},{"id":"https://openalex.org/keywords/phase","display_name":"Phase (matter)","score":0.4588312804698944},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.4389035701751709},{"id":"https://openalex.org/keywords/diagram","display_name":"Diagram","score":0.42672401666641235},{"id":"https://openalex.org/keywords/volume-fraction","display_name":"Volume fraction","score":0.42268383502960205},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3964012861251831},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.3388204574584961},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2744033932685852},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.21412071585655212},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.12552985548973083},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.05891674757003784}],"concepts":[{"id":"https://openalex.org/C85906118","wikidata":"https://www.wikidata.org/wiki/Q186693","display_name":"Phase diagram","level":3,"score":0.7251765727996826},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.588046133518219},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5323634147644043},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.47825518250465393},{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.47111037373542786},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.4588312804698944},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.4389035701751709},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.42672401666641235},{"id":"https://openalex.org/C65590680","wikidata":"https://www.wikidata.org/wiki/Q909482","display_name":"Volume fraction","level":2,"score":0.42268383502960205},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3964012861251831},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3388204574584961},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2744033932685852},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.21412071585655212},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.12552985548973083},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.05891674757003784},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/080728809","is_oa":false,"landing_page_url":"https://doi.org/10.1137/080728809","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:pure.tue.nl:openaire_cris_publications/a6df47f4-2736-420c-b494-6068af8f1558","is_oa":true,"landing_page_url":"https://research.tue.nl/en/publications/a6df47f4-2736-420c-b494-6068af8f1558","pdf_url":"https://pure.tue.nl/ws/files/3209530/Metis218588.pdf","source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Choksi, R, Peletier, M A & Williams, J F 2009, 'On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional', SIAM Journal on Applied Mathematics, vol. 69, no. 6, pp. 1712-1738. https://doi.org/10.1137/080728809","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:tue:oai:pure.tue.nl:publications/a6df47f4-2736-420c-b494-6068af8f1558","is_oa":true,"landing_page_url":"https://research.tue.nl/nl/publications/a6df47f4-2736-420c-b494-6068af8f1558","pdf_url":"https://research.tue.nl/nl/publications/a6df47f4-2736-420c-b494-6068af8f1558","source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Applied Mathematics, 69(6), 1712 - 1738. Society for Industrial and Applied Mathematics (SIAM)","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:649865","is_oa":false,"landing_page_url":"http://library.tue.nl/csp/dare/LinkToRepository.csp?recordnumber=649865","pdf_url":null,"source":{"id":"https://openalex.org/S4406923046","display_name":"TU/e Research Portal (Eindhoven University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0036-1399","raw_type":"Article / Letter to the editor"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.298.3817","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.298.3817","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.win.tue.nl/~mpeletie/Research/Papers/ChoksiPeletierWilliams09.pdf","raw_type":"text"},{"id":"pmh:oai:library.tue.nl:649865","is_oa":false,"landing_page_url":"http://repository.tue.nl/649865","pdf_url":null,"source":{"id":"https://openalex.org/S4406923046","display_name":"TU/e Research Portal (Eindhoven University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN:0036-1399","raw_type":"Article / Letter to the editor"}],"best_oa_location":{"id":"pmh:oai:pure.tue.nl:openaire_cris_publications/a6df47f4-2736-420c-b494-6068af8f1558","is_oa":true,"landing_page_url":"https://research.tue.nl/en/publications/a6df47f4-2736-420c-b494-6068af8f1558","pdf_url":"https://pure.tue.nl/ws/files/3209530/Metis218588.pdf","source":{"id":"https://openalex.org/S4406922641","display_name":"TU/e Research Portal","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Choksi, R, Peletier, M A & Williams, J F 2009, 'On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional', SIAM Journal on Applied Mathematics, vol. 69, no. 6, pp. 1712-1738. https://doi.org/10.1137/080728809","raw_type":"info:eu-repo/semantics/publishedVersion"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2158598889.pdf","grobid_xml":"https://content.openalex.org/works/W2158598889.grobid-xml"},"referenced_works_count":54,"referenced_works":["https://openalex.org/W410056061","https://openalex.org/W649818166","https://openalex.org/W651870773","https://openalex.org/W1483014895","https://openalex.org/W1494195777","https://openalex.org/W1499787021","https://openalex.org/W1502620931","https://openalex.org/W1529293474","https://openalex.org/W1553782195","https://openalex.org/W1605954776","https://openalex.org/W1964544422","https://openalex.org/W1965310093","https://openalex.org/W1969558457","https://openalex.org/W1970612242","https://openalex.org/W1975538793","https://openalex.org/W1975899172","https://openalex.org/W1981666484","https://openalex.org/W1986513910","https://openalex.org/W1989545882","https://openalex.org/W1990206922","https://openalex.org/W1990829398","https://openalex.org/W1993278039","https://openalex.org/W1993892019","https://openalex.org/W1999684450","https://openalex.org/W2011228563","https://openalex.org/W2014659302","https://openalex.org/W2026132129","https://openalex.org/W2027319740","https://openalex.org/W2028079037","https://openalex.org/W2032706604","https://openalex.org/W2036437267","https://openalex.org/W2058380015","https://openalex.org/W2067325315","https://openalex.org/W2071291961","https://openalex.org/W2076008129","https://openalex.org/W2076343072","https://openalex.org/W2083583152","https://openalex.org/W2084474041","https://openalex.org/W2086147404","https://openalex.org/W2086587063","https://openalex.org/W2088646368","https://openalex.org/W2093605325","https://openalex.org/W2094443469","https://openalex.org/W2129679542","https://openalex.org/W2132488183","https://openalex.org/W2140577964","https://openalex.org/W2150401053","https://openalex.org/W2151733572","https://openalex.org/W2161138447","https://openalex.org/W2168153553","https://openalex.org/W2341165165","https://openalex.org/W2808010733","https://openalex.org/W2964003705","https://openalex.org/W4230344563"],"related_works":["https://openalex.org/W2967558384","https://openalex.org/W2048454885","https://openalex.org/W2077028260","https://openalex.org/W57680709","https://openalex.org/W2059693824","https://openalex.org/W3194471551","https://openalex.org/W1969464363","https://openalex.org/W4375956809","https://openalex.org/W2022416154","https://openalex.org/W2092912925"],"abstract_inverted_index":{"We":[0],"consider":[1],"analytical":[2],"and":[3,20,67,99,113,174],"numerical":[4,101,170],"aspects":[5],"of":[6,13,31,50,65,116,146,148],"the":[7,42,51,58,81,86,93,105,110,130,144,149],"phase":[8,43,52,111],"diagram":[9],"for":[10,108],"microphase":[11],"separation":[12],"diblock":[14],"copolymers.":[15],"Our":[16],"approach":[17],"is":[18,21,69,85,96],"variational":[19],"based":[22],"upon":[23,37],"a":[24,32,47,63,78,90,114,162],"density":[25],"functional":[26],"theory":[27],"which":[28,40,151],"entails":[29],"minimization":[30],"nonlocal":[33],"Cahn\u2013Hilliard":[34],"functional.":[35],"Based":[36],"two":[38],"parameters":[39],"characterize":[41],"diagram,":[44],"we":[45,56,76,121,138,168],"give":[46],"preliminary":[48],"analysis":[49,66,142],"plane.":[53],"That":[54],"is,":[55],"divide":[57],"plane":[59],"into":[60],"regions":[61],"wherein":[62,80,92,120],"combination":[64],"numerics":[68],"used":[70],"to":[71,159,172],"describe":[72],"minimizers.":[73],"In":[74],"particular":[75],"identify":[77],"regime":[79,91,115],"uniform":[82],"(disordered":[83],"state)":[84],"unique":[87,131],"global":[88,132],"minimizer;":[89],"constant":[94],"state":[95],"linearly":[97],"unstable":[98],"where":[100],"simulations":[102,171],"are":[103,129],"currently":[104],"only":[106],"tool":[107],"characterizing":[109],"geometry;":[112],"small":[117,124],"volume":[118],"fraction":[119],"conjecture":[122],"that":[123],"well-separated":[125],"approximately":[126],"spherical":[127],"objects":[128],"minimizer.":[133],"For":[134,165],"this":[135],"last":[136],"regime,":[137],"present":[139,169],"an":[140],"asymptotic":[141],"from":[143],"point":[145],"view":[147],"energetics":[150],"will":[152],"be":[153],"complemented":[154],"by":[155],"rigorous":[156],"$\\Gamma$-convergence":[157],"results":[158],"appear":[160],"in":[161],"subsequent":[163],"article.":[164],"all":[166],"regimes,":[167],"support":[173],"expand":[175],"on":[176],"our":[177],"findings.":[178]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":11},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":7},{"year":2022,"cited_by_count":10},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":13},{"year":2019,"cited_by_count":5},{"year":2018,"cited_by_count":7},{"year":2017,"cited_by_count":12},{"year":2016,"cited_by_count":11},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":9},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":9}],"updated_date":"2026-05-06T08:25:59.206177","created_date":"2025-10-10T00:00:00"}
