Shuttleworth tension revisited

IF 2.1 4区 化学 Q3 CHEMISTRY, PHYSICAL
Pascal Hecquet
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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":"749 ","pages":"Article 122546"},"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}
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Abstract

On a solid or liquid surface in thermodynamic equilibrium, the Shuttleworth tension is a sum of two pressures (or tensions) of different nature Υ=γ+σxx¯ (we consider only the diagonal component ’xx’). The two pressures are parallel to the surface and are practically located in the surface monolayer. The surface area is A=NSAUL, NS being the number of entities in the surface monolayer and AUL the area unit which is the Lagrangian surface area of one entity. The total energy includes the surface energetic term γA. Its derivative with respect to NS, holding AUL constant, is the tension γP. It is numerically equal to the energy γ. The variation is of chemical nature and discontinuous. The surface monolayer has a chemical potential excess with respect to bulk for one entity (γAUL=Δμ). The other derivative, with respect to AUL, holding NS constant, gives the pressure of ’elastic’ nature σxx¯. It is γ/ϵxx. For a solid, σxx¯ decreases rapidly with the temperature, while γ varies little. For a liquid and when neglecting σxx¯, the literature shows that Υ is reduced to the superficial tension γP. Nevertheless, our simulations on the surface Pt(111) show that σxx¯ and γ 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 1/L, L being the step–step distance. Regardless of the step energy, the step stress also contributes to the equilibrium repulsion between steps.

Abstract Image

重温沙特尔沃思紧张局势
在处于热力学平衡状态的固体或液体表面上,沙特尔沃斯张力是两个不同性质的压力(或张力)之和 Υ=γ+σxx¯ (我们只考虑对角线分量'xx')。这两个压力平行于表面,实际上位于表面单层。表面积为 A=NSAUL,NS 是表面单层中的实体数量,AUL 是面积单位,即一个实体的拉格朗日表面积。总能量包括表面能量项 γA。在 AUL 保持不变的情况下,它相对于 NS 的导数就是张力 γP。它在数值上等于能量 γ,其变化具有化学性质,是不连续的。表面单层的一个实体(γAUL=Δμ)具有相对于主体的化学势过剩。在保持 NS 不变的情况下,相对于 AUL 的另一个导数给出了 "弹性 "压力 σxx¯。它就是 ∂γ/∂ϵxx。对于固体,σxx¯ 随温度迅速降低,而 γ 变化很小。对于液体,当忽略 σxx¯ 时,文献表明 Υ 会减小到表面张力 γP。然而,我们对 Pt(111) 表面的模拟显示,σxx¯ 和 γ 在液态时的数量级相同。因此,当表面弯曲时,拉普拉斯力与两个压力之和成正比。对于被视为平面的表面,表面张力是整个系统内部的偶极力。在临界面上,表面张力会使台阶上的原子收缩,但原子的收缩程度较小,因为阶跃能在两种平衡构型之间没有变化,而这两种平衡构型的不同仅在于是否存在表面张力。阶跃相互作用应力是排斥性的,其变化量为 1/L,L 为阶跃距离。无论阶跃能如何,阶跃应力也会导致阶跃间的平衡排斥。
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来源期刊
Surface Science
Surface Science 化学-物理:凝聚态物理
CiteScore
3.30
自引率
5.30%
发文量
137
审稿时长
25 days
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: • model systems (e.g. in Ultra High Vacuum) under well-controlled reactive conditions • nanoscale science and engineering, including manipulation of matter at the atomic/molecular scale and assembly phenomena • reactivity of surfaces as related to various applied areas including heterogeneous catalysis, chemistry at electrified interfaces, and semiconductors functionalization • phenomena at interfaces relevant to energy storage and conversion, and fuels production and utilization • 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.
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