{"title":"Shuttleworth tension revisited","authors":"","doi":"10.1016/j.susc.2024.122546","DOIUrl":null,"url":null,"abstract":"<div><p>On a solid or liquid surface in thermodynamic equilibrium, the Shuttleworth tension is a sum of two pressures (or tensions) of different nature <span><math><mrow><mi>Υ</mi><mo>=</mo><mi>γ</mi><mo>+</mo><mover><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow><mo>¯</mo></mover></mrow></math></span> (we consider only the diagonal component ’<span><math><mrow><mi>x</mi><mi>x</mi></mrow></math></span>’). The two pressures are parallel to the surface and are practically located in the surface monolayer. The surface area is <span><math><mrow><mi>A</mi><mo>=</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>S</mi></mrow></msub><msubsup><mrow><mi>A</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>L</mi></mrow></msubsup></mrow></math></span>, <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> being the number of entities in the surface monolayer and <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>L</mi></mrow></msubsup></math></span> the area unit which is the Lagrangian surface area of one entity. The total energy includes the surface energetic term <span><math><mrow><mi>γ</mi><mi>A</mi></mrow></math></span>. Its derivative with respect to <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span>, holding <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>L</mi></mrow></msubsup></math></span> constant, is the tension <span><math><msup><mrow><mi>γ</mi></mrow><mrow><mi>P</mi></mrow></msup></math></span>. It is <em>numerically</em> equal to the energy <span><math><mi>γ</mi></math></span>. The variation is of <em>chemical</em> nature and discontinuous. The surface monolayer has a chemical potential excess with respect to bulk for one entity (<span><math><mrow><mi>γ</mi><mspace></mspace><msubsup><mrow><mi>A</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>L</mi></mrow></msubsup><mo>=</mo><mi>Δ</mi><mi>μ</mi></mrow></math></span>). The other derivative, with respect to <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>U</mi></mrow><mrow><mi>L</mi></mrow></msubsup></math></span>, holding <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>S</mi></mrow></msub></math></span> constant, gives the pressure of <em>’elastic’</em> nature <span><math><mover><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow><mo>¯</mo></mover></math></span>. It is <span><math><mrow><mi>∂</mi><mi>γ</mi><mo>/</mo><mi>∂</mi><msub><mrow><mi>ϵ</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow></math></span>. For a solid, <span><math><mover><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow><mo>¯</mo></mover></math></span> decreases rapidly with the temperature, while <span><math><mi>γ</mi></math></span> varies little. For a liquid and when neglecting <span><math><mover><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow><mo>¯</mo></mover></math></span>, the literature shows that <span><math><mi>Υ</mi></math></span> is reduced to the superficial tension <span><math><msup><mrow><mi>γ</mi></mrow><mrow><mi>P</mi></mrow></msup></math></span>. Nevertheless, our simulations on the surface <span><math><mrow><mi>P</mi><mi>t</mi><mrow><mo>(</mo><mn>111</mn><mo>)</mo></mrow></mrow></math></span> show that <span><math><mover><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub></mrow><mo>¯</mo></mover></math></span> and <span><math><mi>γ</mi></math></span> are of the same order of magnitude for the liquid state. The Laplace force is thus proportional to the sum of two pressures when the surface is curved.</p><p>For a surface considered as flat, the surface tension is a dipole force <em>internal</em> to the whole system. On vicinals, the surface tension tends to contract the atoms on the terrace, but the atoms are less contracted because the step energy does not change between the two equilibrium configurations that differ only by the presence or otherwise of the surface tension. The step–step interaction stress is repulsive and varies as <span><math><mrow><mn>1</mn><mo>/</mo><mi>L</mi></mrow></math></span>, <span><math><mi>L</mi></math></span> being the step–step distance. Regardless of the step energy, the step stress also contributes to the equilibrium repulsion between steps.</p></div>","PeriodicalId":22100,"journal":{"name":"Surface Science","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Surface Science","FirstCategoryId":"92","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0039602824000979","RegionNum":4,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"CHEMISTRY, PHYSICAL","Score":null,"Total":0}
引用次数: 0
Abstract
On a solid or liquid surface in thermodynamic equilibrium, the Shuttleworth tension is a sum of two pressures (or tensions) of different nature (we consider only the diagonal component ’’). The two pressures are parallel to the surface and are practically located in the surface monolayer. The surface area is , being the number of entities in the surface monolayer and the area unit which is the Lagrangian surface area of one entity. The total energy includes the surface energetic term . Its derivative with respect to , holding constant, is the tension . It is numerically equal to the energy . The variation is of chemical nature and discontinuous. The surface monolayer has a chemical potential excess with respect to bulk for one entity (). The other derivative, with respect to , holding constant, gives the pressure of ’elastic’ nature . It is . For a solid, decreases rapidly with the temperature, while varies little. For a liquid and when neglecting , the literature shows that is reduced to the superficial tension . Nevertheless, our simulations on the surface show that and are of the same order of magnitude for the liquid state. The Laplace force is thus proportional to the sum of two pressures when the surface is curved.
For a surface considered as flat, the surface tension is a dipole force internal to the whole system. On vicinals, the surface tension tends to contract the atoms on the terrace, but the atoms are less contracted because the step energy does not change between the two equilibrium configurations that differ only by the presence or otherwise of the surface tension. The step–step interaction stress is repulsive and varies as , being the step–step distance. Regardless of the step energy, the step stress also contributes to the equilibrium repulsion between steps.
期刊介绍:
Surface Science is devoted to elucidating the fundamental aspects of chemistry and physics occurring at a wide range of surfaces and interfaces and to disseminating this knowledge fast. The journal welcomes a broad spectrum of topics, including but not limited to:
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• surface reactivity for environmental protection and pollution remediation
• interactions at surfaces of soft matter, including polymers and biomaterials.
Both experimental and theoretical work, including modeling, is within the scope of the journal. Work published in Surface Science reaches a wide readership, from chemistry and physics to biology and materials science and engineering, providing an excellent forum for cross-fertilization of ideas and broad dissemination of scientific discoveries.