{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T10:18:03Z","timestamp":1789726683750,"version":"4.0.1"},"reference-count":56,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"name":"The Central Organization Department of China"},{"DOI":"10.13039\/501100002367","name":"Chinese Academy of Sciences","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100002367","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11101276"],"award-info":[{"award-number":["11101276"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91130012"],"award-info":[{"award-number":["91130012"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11271018"],"award-info":[{"award-number":["11271018"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["91230203"],"award-info":[{"award-number":["91230203"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004921","name":"Shanghai Jiao Tong University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004921","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Math."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We derive a set of modified Poisson--Nernst--Planck (PNP) equations for ion transport from the variation of the free energy functional which includes the many-body Coulomb correlation in media of variable dielectric coefficient. The correlation effects are considered through the Debye charging process in which the self energy of an ion is governed by the generalized Debye--H\u00fcckel equation. We develop the asymptotic expansions of the self energy taking the ion radius as the small parameter such that the multiscale model can be solved efficiently by numerical methods. It is shown that the variations of the energy functional give the self-energy-modified PNP equations which satisfy a proper weak formulation. We present numerical results from different asymptotic expansions with a semi-implicit conservative numerical method and investigate the effect of the Coulomb correlation.<\/jats:p>","DOI":"10.1137\/16m110383x","type":"journal-article","created":{"date-parts":[[2018,1,18]],"date-time":"2018-01-18T11:58:51Z","timestamp":1516276731000},"page":"226-245","source":"Crossref","is-referenced-by-count":27,"title":["Modified Poisson--Nernst--Planck Model with Accurate Coulomb Correlation in Variable Media"],"prefix":"10.1137","volume":"78","author":[{"given":"Pei","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xia","family":"Ji","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenli","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,1,18]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.82.1887"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.80.839"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.70.021506"},{"key":"atypb4","unstructured":"P. Markowich, C. Ringhofer, and C. Schimeiser,\n                      Semiconductor\n                      , Springer, New York, 1990."},{"key":"atypb5","first-page":"77","volume":"148","author":"Eisenberg B.","year":"2011","journal-title":"Adv. Chem. Phys."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(03)75095-4"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/S0006-3495(03)75091-7"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.75.021503"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1063\/1.3476262"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.bpj.2011.03.059"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-012-9277-y"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s10884-012-9278-x"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"D. Gillespie,\n                      A review of steric interactions of ions: Why some theories succeed and others fail to account for ion size\n                      , Microfluid. Nanofluid. (2014), pp. 1-22.","DOI":"10.1007\/s10404-014-1489-5"},{"key":"atypb14","first-page":"209","volume":"160","author":"Frydel D.","year":"2016","journal-title":"Adv. Chem. Phys."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.88.011301"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1021\/la5048936"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1063\/1.1520530"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1088\/0953-8984\/14\/46\/313"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.cis.2009.10.001"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.106.046102"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.86.056303"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1021\/jp408330f"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1021\/acs.jpcb.5b11515"},{"key":"atypb24","first-page":"535","volume":"48","author":"Avdeev S. M.","year":"1986","journal-title":"Colloid J. USSR"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.93.062112"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1063\/1.457535"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s101890050023"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1140\/epje\/i2002-10159-0"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2014.07.004"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.81.021501"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1080\/14786444208520813"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1021\/cr60130a002"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1209\/0295-5075\/97\/28010"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2016.v14.n1.a10"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.94.053312"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1063\/1.3489418"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/j.cplett.2015.06.079"},{"key":"atypb38","first-page":"185","volume":"24","author":"Debye P.","year":"1923","journal-title":"Phys. Z."},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1063\/1.4821636"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1063\/1.4904728"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1063\/1.4914170"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.OA-2016-0104"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/s10825-013-0506-3"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2014.02.036"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2014.v12.n4.a9"},{"key":"atypb46","unstructured":"J. P. Hansen and I. R. McDonald,\n                      Theory of Simple Liquids\n                      , Academic Press, Amsterdam, 2006."},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8522(51)90027-X"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.90.013307"},{"key":"atypb49","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1515\/zpch-1926-12312","volume":"123","author":"Gu\u0308ntelberg E.","year":"1926","journal-title":"Z. Phys. Chem."},{"key":"atypb50","doi-asserted-by":"crossref","unstructured":"J. O. Bockris and A. K. N. Reddy,\n                      Modern Electrochemistry: Ionics\n                      , Springer, New York, 1998.","DOI":"10.1007\/b114546"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1145\/1916461.1916464"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1063\/1.4892413"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1016\/S0010-4655(98)00016-2"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1049\/el:19690510"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.74.329"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1088\/0034-4885\/65\/11\/201"}],"container-title":["SIAM Journal on Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M110383X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:53:48Z","timestamp":1787338428000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M110383X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":56,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M110383X"],"URL":"https:\/\/doi.org\/10.1137\/16m110383x","relation":{},"ISSN":["0036-1399","1095-712X"],"issn-type":[{"value":"0036-1399","type":"print"},{"value":"1095-712X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}