{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:12:06Z","timestamp":1772118726241,"version":"3.50.1"},"reference-count":94,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T00:00:00Z","timestamp":1709769600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education of the Russian Federation","award":["075-15-2022-311"],"award-info":[{"award-number":["075-15-2022-311"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Casimir\u2013Polder force between spherical nanoparticles and a graphene-coated silica plate is investigated in situations out of thermal equilibrium, i.e., with broken time-reversal symmetry. The response of the graphene coating to the electromagnetic field is described on the basis of first principles of quantum electrodynamics at nonzero temperature using the formalism of the polarization tensor in the framework of the Dirac model. The nonequilibrium Casimir\u2013Polder force is calculated as a function of the mass-gap parameter, the chemical potential of graphene, and the temperature of the graphene-coated plate, which can be both higher or lower than that of the environment. It is shown that the force value increases with the increasing chemical potential, and this increase is more pronounced when the temperature of a graphene-coated plate is lower than that of the environment. The nonequilibrium force also increases with increasing temperature of the graphene-coated plate. This increase is larger when the plate is hotter than the environment. The effect is revealed that the combined impact of the chemical potential, \u03bc, and mass gap, \u0394, of the graphene coating depends on the relationship between \u0394 and 2\u03bc. If 2\u03bc&gt;\u0394, the magnitude of the nonequilibrium force between nanoparticles and a cooled graphene-coated plate becomes much larger than for a graphene coating with \u03bc=0. The physical reasons explaining this effect are elucidated. Possible applications of the obtained results are discussed.<\/jats:p>","DOI":"10.3390\/sym16030320","type":"journal-article","created":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T04:19:02Z","timestamp":1709785142000},"page":"320","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Nonequilibrium Casimir\u2013Polder Force between Nanoparticles and Graphene-Coated Silica Plate: Combined Effect of the Chemical Potential and Mass Gap"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0495-5907","authenticated-orcid":false,"given":"Galina L.","family":"Klimchitskaya","sequence":"first","affiliation":[{"name":"Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, 196140 Saint Petersburg, Russia"},{"name":"Peter the Great Saint Petersburg Polytechnic University, 195251 Saint Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6227-5542","authenticated-orcid":false,"given":"Constantine C.","family":"Korikov","sequence":"additional","affiliation":[{"name":"Huawei Noah\u2019s Ark Lab, Krylatskaya Str. 17, 121614 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8560-998X","authenticated-orcid":false,"given":"Vladimir M.","family":"Mostepanenko","sequence":"additional","affiliation":[{"name":"Central Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, 196140 Saint Petersburg, Russia"},{"name":"Peter the Great Saint Petersburg Polytechnic University, 195251 Saint Petersburg, Russia"},{"name":"Kazan Federal University, 420008 Kazan, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1038\/nmat1849","article-title":"The rise of graphene","volume":"6","author":"Geim","year":"2007","journal-title":"Nat. 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