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<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" article-type="other" dtd-version="1.2" xml:lang="en"><front><journal-meta><journal-id journal-id-type="publisher-id">Frontier Materials &amp; Technologies</journal-id><journal-title-group><journal-title xml:lang="en">Frontier Materials &amp; Technologies</journal-title><trans-title-group xml:lang="ru"><trans-title>Frontier Materials &amp; Technologies</trans-title></trans-title-group></journal-title-group><issn publication-format="print">2782-4039</issn><issn publication-format="electronic">2782-6074</issn><publisher><publisher-name xml:lang="en">Togliatti State University</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="publisher-id">172</article-id><article-id pub-id-type="doi">10.18323/2782-4039-2021-4-27-38</article-id><article-categories><subj-group subj-group-type="toc-heading" xml:lang="en"><subject>Articles</subject></subj-group><subj-group subj-group-type="toc-heading" xml:lang="ru"><subject>Статьи</subject></subj-group><subj-group subj-group-type="article-type"><subject></subject></subj-group></article-categories><title-group><article-title xml:lang="en">Resiadual stress relaxation in decahedral particles through the formation of a central spherical void</article-title><trans-title-group xml:lang="ru"><trans-title>Релаксация остаточных напряжений в декаэдрических частицах путем образования центральной сферической поры</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-4363-8242</contrib-id><name-alternatives><name xml:lang="en"><surname>Krasnitsky</surname><given-names>Stanislav A.</given-names></name><name xml:lang="ru"><surname>Красницкий</surname><given-names>Станислав Андреевич</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>PhD (Physics and Mathematics), senior researcher</p></bio><bio xml:lang="ru"><p>кандидат физико-математических наук, старший научный сотрудник</p></bio><email>krasnitsky@inbox.ru</email><xref ref-type="aff" rid="aff1"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-4116-4821</contrib-id><name-alternatives><name xml:lang="en"><surname>Kolesnikova</surname><given-names>Anna L.</given-names></name><name xml:lang="ru"><surname>Колесникова</surname><given-names>Анна Львовна</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Doctor of Sciences (Physics and Mathematics), leading researcher</p></bio><bio xml:lang="ru"><p>доктор физико-математических наук, ведущий научный сотрудник</p></bio><xref ref-type="aff" rid="aff2"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-0727-6352</contrib-id><name-alternatives><name xml:lang="en"><surname>Gutkin</surname><given-names>Mikhail Yu.</given-names></name><name xml:lang="ru"><surname>Гуткин</surname><given-names>Михаил Юрьевич</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Doctor of Sciences (Physics and Mathematics), chief researcher</p></bio><bio xml:lang="ru"><p>доктор физико-механических наук, главный научный сотрудник</p></bio><xref ref-type="aff" rid="aff2"/></contrib><contrib contrib-type="author"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0003-3738-408X</contrib-id><name-alternatives><name xml:lang="en"><surname>Romanov</surname><given-names>Aleksey E.</given-names></name><name xml:lang="ru"><surname>Романов</surname><given-names>Алексей Евгеньевич</given-names></name></name-alternatives><address><country country="RU">Russian Federation</country></address><bio xml:lang="en"><p>Doctor of Sciences (Physics and Mathematics), Professor</p></bio><bio xml:lang="ru"><p>доктор физико-математических наук, профессор</p></bio><xref ref-type="aff" rid="aff3"/></contrib></contrib-group><aff-alternatives id="aff1"><aff><institution xml:lang="en">Peter the Great St. Petersburg Polytechnic University, St. Petersburg (Russia)</institution></aff><aff><institution xml:lang="ru">Санкт-Петербургский политехнический университет Петра Великого, Санкт-Петербург (Россия)</institution></aff></aff-alternatives><aff-alternatives id="aff2"><aff><institution xml:lang="en">Institute of Problems of Mechanical Engineering of the Russian Academy of Sciences, St. Petersburg (Russia)</institution></aff><aff><institution xml:lang="ru">Институт проблем машиноведения Российской академии наук, Санкт-Петербург (Россия)</institution></aff></aff-alternatives><aff-alternatives id="aff3"><aff><institution xml:lang="en">ITMO University, St. Petersburg (Russia)</institution></aff><aff><institution xml:lang="ru">Университет ИТМО, Санкт-Петербург (Россия)</institution></aff></aff-alternatives><pub-date date-type="pub" iso-8601-date="2021-12-30" publication-format="electronic"><day>30</day><month>12</month><year>2021</year></pub-date><issue>4</issue><issue-title xml:lang="en"/><issue-title xml:lang="ru"/><fpage>27</fpage><lpage>38</lpage><history><date date-type="received" iso-8601-date="2021-12-30"><day>30</day><month>12</month><year>2021</year></date><date date-type="accepted" iso-8601-date="2021-12-30"><day>30</day><month>12</month><year>2021</year></date></history><permissions><ali:free_to_read xmlns:ali="http://www.niso.org/schemas/ali/1.0/"/></permissions><self-uri xlink:href="https://vektornaukitech.ru/jour/article/view/172">https://vektornaukitech.ru/jour/article/view/172</self-uri><abstract xml:lang="en"><p>Small metal particles with a body-centered crystal lattice (BCC) often take the form of polyhedrons with fifth-order symmetry axes such as the icosahedron, decahedron, and pentagonal prism. The quintic symmetry axes, forbidden by the traditional crystallography laws, cause inhomogeneous elastic stress and strain in these particles. Under certain conditions, these stress and strain could relax through the change in the particle structure: the formation of partial and perfect dislocations, misfit layers, and the nucleation of cracks and voids. Within the quasi-equilibrium energy approach, the authors proposed a theoretical model of residual stress relaxation in decahedral particles due to the formation of a central spherical void. The explicit analytical expressions for energies of solid and hollow decahedral particles are found. The elastic energy of a hollow decahedral particle is defined as the work spent on the nucleation of a positive wedge disclination with the power ω≈0.0163 rad (≈7°20') in the elastic spherical shell under its own stress field. The authors determined the change in the surface energy due to the formation of a void considering the influence of the relaxation effect of the first coordination sphere surrounding the vacancy on the particle volume change. The energy change of decahedral particles during the formation of a spherical void is calculated and the optimal and critical parameters of this process are determined. The study shows that there some critical radius of a particle, if reached the formation of the central spherical void becomes energetically favorable. Moreover, the study shows that a pore germ will grow until it reaches a certain optimal size corresponding to the greatest change in the system energy. The numerical calculations correspond with experimental observations of unstable voids in the rather small silver and gold decahedral particles with the diameter of 30–40 nm and stable voids in relatively large copper decahedral particles with the diameter of ~1 μm.</p></abstract><trans-abstract xml:lang="ru"><p>Малые металлические частицы с объемно-центрированной кристаллической решеткой (ОЦК) часто принимают форму многогранников с осями симметрии пятого порядка, таких как икосаэдр, декаэдр и пятиугольная призма<italic>. </italic>Оси симметрии пятого порядка, запрещенные законами классической кристаллографии, вызывают в таких частицах неоднородные упругие напряжения и деформации. При некоторых условиях эти напряжения и деформации могут релаксировать за счет изменения структуры частицы, а именно образования частичных и полных дислокаций, слоев несоответствия, зарождения трещин и пор.<italic> </italic>В рамках квазиравновесного энергетического подхода предложена теоретическая модель, описывающая релаксацию неоднородных упругих напряжений и деформаций в декаэдрических частицах за счет формирования центральной сферической полости. Найдены явные аналитические выражения для энергий сплошных и полых декаэдрических частиц. Упругая энергия полой декаэдрической частицы определена как работа, затраченная на зарождение в упругой сферической оболочке положительной клиновой дисклинации мощностью <italic>ω</italic>≈0,0163 рад (≈7°20<italic>'</italic>) в поле ее собственных напряжений. Изменение поверхностной энергии за счет формирования поры определено с учетом влияния эффекта релаксации первой координационный сферы, окружающей вакансию, на изменение объема частицы. Определено изменение энергии декаэдрических частиц при образовании сферической поры, установлены оптимальные и критические параметры этого процесса. Показано, что существует некоторый критический радиус частицы, при достижении которого формирование центральной сферической поры становится энергетически выгодным. Кроме того, показано, что зародыш поры будет расти, пока не достигнет некоторого оптимального размера, соответствующего наибольшему изменению энергии системы. Численные расчеты согласуются с экспериментальными наблюдениями нестабильных пор в относительно малых серебряных и золотых декаэдрических частицах диаметром 30–40 нм и стабильных пор в относительно больших медных декаэдрических частицах диаметром ~1 мкм.</p></trans-abstract><kwd-group xml:lang="en"><kwd>hollow decahedral particles</kwd><kwd>residual stress relaxation</kwd><kwd>void formation</kwd><kwd>spherical void</kwd></kwd-group><kwd-group xml:lang="ru"><kwd>полые декаэдрические частицы</kwd><kwd>релаксация остаточных напряжений</kwd><kwd>формирование пор</kwd><kwd>сферическая пора</kwd></kwd-group><funding-group><funding-statement xml:lang="en">The work was financially supported by the Russian Science Foundation (Grant No. 19-19-00617). The paper was written on the reports of the participants of the X International School of Physical Materials Science (SPM-2021), Togliatti, September 13–17, 2021.</funding-statement><funding-statement xml:lang="ru">Работа выполнена при финансовой поддержке Российского научного фонда (грант № 19-19-00617). Статья подготовлена по материалам докладов участников X Международной школы «Физическое материаловедение» (ШФМ-2021), Тольятти, 13–17 сентября 2021 года.</funding-statement></funding-group></article-meta></front><body></body><back><ref-list><ref id="B1"><label>1.</label><citation-alternatives><mixed-citation xml:lang="en">Genç A., Patarroyo J., Sancho-Parramon J., Bastús N.G., Puntes V., Arbiol, J. Hollow metal nanostructures for enhanced plasmonics: synthesis, local plasmonic properties and applications. Nanophotonics, 2016, vol. 6, no. 1, pp. 193–213. 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