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class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>In terms of force</span> </div> </a> <ul id="toc-In_terms_of_force-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_terms_of_energy" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#In_terms_of_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>In terms of energy</span> </div> </a> <ul id="toc-In_terms_of_energy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Effects" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Effects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Effects</span> </div> </a> <button aria-controls="toc-Effects-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Effects subsection</span> </button> <ul id="toc-Effects-sublist" class="vector-toc-list"> <li id="toc-Water" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Water"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Water</span> </div> </a> <ul id="toc-Water-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surfactants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surfactants"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Surfactants</span> </div> </a> <ul id="toc-Surfactants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_curvature_and_pressure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surface_curvature_and_pressure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Surface curvature and pressure</span> </div> </a> <ul id="toc-Surface_curvature_and_pressure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Floating_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Floating_objects"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Floating objects</span> </div> </a> <ul id="toc-Floating_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liquid_surface" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liquid_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Liquid surface</span> </div> </a> <ul id="toc-Liquid_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contact_angles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contact_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Contact angles</span> </div> </a> <ul id="toc-Contact_angles-sublist" class="vector-toc-list"> <li id="toc-Special_contact_angles" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Special_contact_angles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6.1</span> <span>Special contact angles</span> </div> </a> <ul id="toc-Special_contact_angles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Liquid_in_a_vertical_tube" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liquid_in_a_vertical_tube"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>Liquid in a vertical tube</span> </div> </a> <ul id="toc-Liquid_in_a_vertical_tube-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Puddles_on_a_surface" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Puddles_on_a_surface"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.8</span> <span>Puddles on a surface</span> </div> </a> <ul id="toc-Puddles_on_a_surface-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Breakup_of_streams_into_drops" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Breakup_of_streams_into_drops"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.9</span> <span>Breakup of streams into drops</span> </div> </a> <ul id="toc-Breakup_of_streams_into_drops-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gallery" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gallery"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.10</span> <span>Gallery</span> </div> </a> <ul id="toc-Gallery-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Thermodynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Thermodynamics</span> </div> </a> <button aria-controls="toc-Thermodynamics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Thermodynamics subsection</span> </button> <ul id="toc-Thermodynamics-sublist" class="vector-toc-list"> <li id="toc-Thermodynamic_theories_of_surface_tension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamic_theories_of_surface_tension"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Thermodynamic theories of surface tension</span> </div> </a> <ul id="toc-Thermodynamic_theories_of_surface_tension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Thermodynamics_of_bubbles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Thermodynamics_of_bubbles"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Thermodynamics of bubbles</span> </div> </a> <ul id="toc-Thermodynamics_of_bubbles-sublist" class="vector-toc-list"> <li id="toc-Influence_of_temperature" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Influence_of_temperature"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.1</span> <span>Influence of temperature</span> </div> </a> <ul id="toc-Influence_of_temperature-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Influence_of_solute_concentration" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Influence_of_solute_concentration"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.2</span> <span>Influence of solute concentration</span> </div> </a> <ul id="toc-Influence_of_solute_concentration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Influence_of_particle_size_on_vapor_pressure" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Influence_of_particle_size_on_vapor_pressure"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2.3</span> <span>Influence of particle size on vapor pressure</span> </div> </a> <ul id="toc-Influence_of_particle_size_on_vapor_pressure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Methods_of_measurement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Methods_of_measurement"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Methods of measurement</span> </div> </a> <ul id="toc-Methods_of_measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Values" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Values"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Values</span> </div> </a> <button aria-controls="toc-Values-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Values subsection</span> </button> <ul id="toc-Values-sublist" class="vector-toc-list"> <li id="toc-Data_table" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Data_table"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Data table</span> </div> </a> <ul id="toc-Data_table-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_tension_of_water" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surface_tension_of_water"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Surface tension of water</span> </div> </a> <ul id="toc-Surface_tension_of_water-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surface_tension_of_seawater" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surface_tension_of_seawater"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Surface tension of seawater</span> </div> </a> <ul id="toc-Surface_tension_of_seawater-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Explanatory_notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Explanatory_notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Explanatory notes</span> </div> </a> <ul id="toc-Explanatory_notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Surface tension</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 66 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-66" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">66 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%88%D8%AA%D8%B1_%D8%B3%D8%B7%D8%AD%D9%8A" title="توتر سطحي – Arabic" lang="ar" hreflang="ar" data-title="توتر سطحي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Tensi%C3%B3n_superficial" title="Tensión superficial – Asturian" lang="ast" hreflang="ast" data-title="Tensión superficial" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%83%E0%A6%B7%E0%A7%8D%E0%A6%A0%E0%A6%9F%E0%A6%BE%E0%A6%A8" title="পৃষ্ঠটান – Bangla" lang="bn" hreflang="bn" data-title="পৃষ্ঠটান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Pi%C3%A1u-b%C4%ABn_tiong-la%CC%8Dt" title="Piáu-bīn tiong-la̍t – Minnan" lang="nan" hreflang="nan" data-title="Piáu-bīn tiong-la̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%B5%D0%B2%D0%B0%D0%B5_%D0%BD%D0%B0%D1%86%D1%8F%D0%B6%D1%8D%D0%BD%D0%BD%D0%B5" title="Паверхневае нацяжэнне – Belarusian" lang="be" hreflang="be" data-title="Паверхневае нацяжэнне" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%8A%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%BE_%D0%BD%D0%B0%D0%BF%D1%80%D0%B5%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Повърхностно напрежение – Bulgarian" lang="bg" hreflang="bg" data-title="Повърхностно напрежение" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Povr%C5%A1inski_napon" title="Površinski napon – Bosnian" lang="bs" hreflang="bs" data-title="Površinski napon" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Tensi%C3%B3_superficial" title="Tensió superficial – Catalan" lang="ca" hreflang="ca" data-title="Tensió superficial" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%C3%87%D0%B8%D0%B9%D0%B5%D0%BB_%D0%BA%D0%B0%D1%80%C4%83%D0%BD%D0%B0%D0%B2%C4%95" title="Çийел карăнавĕ – Chuvash" lang="cv" hreflang="cv" data-title="Çийел карăнавĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Povrchov%C3%A9_nap%C4%9Bt%C3%AD" title="Povrchové napětí – Czech" lang="cs" hreflang="cs" data-title="Povrchové napětí" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Overfladesp%C3%A6nding" title="Overfladespænding – Danish" lang="da" hreflang="da" data-title="Overfladespænding" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Oberfl%C3%A4chenspannung" title="Oberflächenspannung – German" lang="de" hreflang="de" data-title="Oberflächenspannung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pindpinevus" title="Pindpinevus – Estonian" lang="et" hreflang="et" data-title="Pindpinevus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%80%CE%B9%CF%86%CE%B1%CE%BD%CE%B5%CE%B9%CE%B1%CE%BA%CE%AE_%CF%84%CE%AC%CF%83%CE%B7" title="Επιφανειακή τάση – Greek" lang="el" hreflang="el" data-title="Επιφανειακή τάση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tensi%C3%B3n_superficial" title="Tensión superficial – Spanish" lang="es" hreflang="es" data-title="Tensión superficial" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Surfaca_tensio" title="Surfaca tensio – Esperanto" lang="eo" hreflang="eo" data-title="Surfaca tensio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gainazal-tentsio" title="Gainazal-tentsio – Basque" lang="eu" hreflang="eu" data-title="Gainazal-tentsio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D8%B4%D8%B4_%D8%B3%D8%B7%D8%AD%DB%8C" title="کشش سطحی – Persian" lang="fa" hreflang="fa" data-title="کشش سطحی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Tension_superficielle" title="Tension superficielle – French" lang="fr" hreflang="fr" data-title="Tension superficielle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Teannas_dromchla" title="Teannas dromchla – Irish" lang="ga" hreflang="ga" data-title="Teannas dromchla" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Tensi%C3%B3n_superficial" title="Tensión superficial – Galician" lang="gl" hreflang="gl" data-title="Tensión superficial" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%91%9C%EB%A9%B4%EC%9E%A5%EB%A0%A5" title="표면장력 – Korean" lang="ko" hreflang="ko" data-title="표면장력" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A0_%E0%A4%A4%E0%A4%A8%E0%A4%BE%E0%A4%B5" title="पृष्ठ तनाव – Hindi" lang="hi" hreflang="hi" data-title="पृष्ठ तनाव" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Povr%C5%A1inska_napetost" title="Površinska napetost – Croatian" lang="hr" hreflang="hr" data-title="Površinska napetost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tegangan_permukaan" title="Tegangan permukaan – Indonesian" lang="id" hreflang="id" data-title="Tegangan permukaan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tensione_superficiale" title="Tensione superficiale – Italian" lang="it" hreflang="it" data-title="Tensione superficiale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%AA%D7%97_%D7%A4%D7%A0%D7%99%D7%9D" title="מתח פנים – Hebrew" lang="he" hreflang="he" data-title="מתח פנים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B3%87%E0%B2%B2%E0%B3%8D%E0%B2%AE%E0%B3%88_%E0%B2%8E%E0%B2%B3%E0%B3%86%E0%B2%A4" title="ಮೇಲ್ಮೈ ಎಳೆತ – Kannada" lang="kn" hreflang="kn" data-title="ಮೇಲ್ಮೈ ಎಳೆತ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B5%D1%82%D1%82%D1%96%D0%BA_%D0%BA%D0%B5%D1%80%D1%96%D0%BB%D1%83" title="Беттік керілу – Kazakh" lang="kk" hreflang="kk" data-title="Беттік керілу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Tansyon_sifas" title="Tansyon sifas – Haitian Creole" lang="ht" hreflang="ht" data-title="Tansyon sifas" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Virsmas_spraigums" title="Virsmas spraigums – Latvian" lang="lv" hreflang="lv" data-title="Virsmas spraigums" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Pavir%C5%A1iaus_%C4%AFtemptis" title="Paviršiaus įtemptis – Lithuanian" lang="lt" hreflang="lt" data-title="Paviršiaus įtemptis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://hu.wikipedia.org/wiki/Fel%C3%BCleti_fesz%C3%BClts%C3%A9g" title="Felületi feszültség – Hungarian" lang="hu" hreflang="hu" data-title="Felületi feszültség" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D1%80%D1%88%D0%B8%D0%BD%D1%81%D0%BA%D0%B8_%D0%BD%D0%B0%D0%BF%D0%BE%D0%BD" title="Површински напон – Macedonian" lang="mk" hreflang="mk" data-title="Површински напон" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A4%E0%B4%B2%E0%B4%AC%E0%B4%B2%E0%B4%82" title="പ്രതലബലം – Malayalam" lang="ml" hreflang="ml" data-title="പ്രതലബലം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Tegangan_permukaan" title="Tegangan permukaan – Malay" lang="ms" hreflang="ms" data-title="Tegangan permukaan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%93%D0%B0%D0%B4%D0%B0%D1%80%D0%B3%D1%8B%D0%BD_%D1%82%D0%B0%D1%82%D0%B0%D0%BB%D1%86%D0%B0%D0%BB" title="Гадаргын таталцал – Mongolian" lang="mn" hreflang="mn" data-title="Гадаргын таталцал" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Oppervlaktespanning" title="Oppervlaktespanning – Dutch" lang="nl" hreflang="nl" data-title="Oppervlaktespanning" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AA%E0%A5%83%E0%A4%B7%E0%A5%8D%E0%A4%A0_%E0%A4%A4%E0%A4%A8%E0%A4%BE%E0%A4%B5" title="पृष्ठ तनाव – Nepali" lang="ne" hreflang="ne" data-title="पृष्ठ तनाव" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%A8%E9%9D%A2%E5%BC%B5%E5%8A%9B" title="表面張力 – Japanese" lang="ja" hreflang="ja" data-title="表面張力" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Overflatespenning" title="Overflatespenning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Overflatespenning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Overflatespenning" title="Overflatespenning – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Overflatespenning" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Sirt_taranglik" title="Sirt taranglik – Uzbek" lang="uz" hreflang="uz" data-title="Sirt taranglik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%A4%E0%A8%B9%E0%A9%80_%E0%A8%95%E0%A8%B8%E0%A8%BC%E0%A8%AE%E0%A8%95%E0%A9%B1%E0%A8%B8%E0%A8%BC" title="ਸਤਹੀ ਕਸ਼ਮਕੱਸ਼ – Punjabi" lang="pa" hreflang="pa" data-title="ਸਤਹੀ ਕਸ਼ਮਕੱਸ਼" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Napi%C4%99cie_powierzchniowe" title="Napięcie powierzchniowe – Polish" lang="pl" hreflang="pl" data-title="Napięcie powierzchniowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Tens%C3%A3o_superficial" title="Tensão superficial – Portuguese" lang="pt" hreflang="pt" data-title="Tensão superficial" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ro.wikipedia.org/wiki/Tensiune_superficial%C4%83" title="Tensiune superficială – Romanian" lang="ro" hreflang="ro" data-title="Tensiune superficială" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%BE%D1%81%D1%82%D0%BD%D0%BE%D0%B5_%D0%BD%D0%B0%D1%82%D1%8F%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5" title="Поверхностное натяжение – Russian" lang="ru" hreflang="ru" data-title="Поверхностное натяжение" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Tensioni_sip%C3%ABrfaq%C3%ABsor" title="Tensioni sipërfaqësor – Albanian" lang="sq" hreflang="sq" data-title="Tensioni sipërfaqësor" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B4%E0%B7%98%E0%B7%82%E0%B7%8A%E0%B6%A8%E0%B7%92%E0%B6%9A_%E0%B6%86%E0%B6%AD%E0%B6%AD%E0%B7%92%E0%B6%BA" title="පෘෂ්ඨික ආතතිය – Sinhala" lang="si" hreflang="si" data-title="පෘෂ්ඨික ආතතිය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Surface_tension" title="Surface tension – Simple English" lang="en-simple" hreflang="en-simple" data-title="Surface tension" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Povrchov%C3%A9_nap%C3%A4tie" title="Povrchové napätie – Slovak" lang="sk" hreflang="sk" data-title="Povrchové napätie" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Povr%C5%A1inska_napetost" title="Površinska napetost – Slovenian" lang="sl" hreflang="sl" data-title="Površinska napetost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Povr%C5%A1inski_napon" title="Površinski napon – Serbian" lang="sr" hreflang="sr" data-title="Površinski napon" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Povr%C5%A1inski_napon" title="Površinski napon – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Površinski napon" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Pintaj%C3%A4nnitys" title="Pintajännitys – Finnish" lang="fi" hreflang="fi" data-title="Pintajännitys" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ytsp%C3%A4nning" title="Ytspänning – Swedish" lang="sv" hreflang="sv" data-title="Ytspänning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%AA%E0%AE%B0%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81_%E0%AE%87%E0%AE%B4%E0%AF%81%E0%AE%B5%E0%AE%BF%E0%AE%9A%E0%AF%88" title="மேற்பரப்பு இழுவிசை – Tamil" lang="ta" hreflang="ta" data-title="மேற்பரப்பு இழுவிசை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A4%E0%B0%B2%E0%B0%A4%E0%B0%A8%E0%B1%8D%E0%B0%AF%E0%B0%A4" title="తలతన్యత – Telugu" lang="te" hreflang="te" data-title="తలతన్యత" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A3%E0%B8%87%E0%B8%95%E0%B8%B6%E0%B8%87%E0%B8%9C%E0%B8%B4%E0%B8%A7" title="แรงตึงผิว – Thai" lang="th" hreflang="th" data-title="แรงตึงผิว" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Y%C3%BCzey_gerilimi" title="Yüzey gerilimi – Turkish" lang="tr" hreflang="tr" data-title="Yüzey gerilimi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B2%D0%B5%D1%80%D1%85%D0%BD%D0%B5%D0%B2%D0%B8%D0%B9_%D0%BD%D0%B0%D1%82%D1%8F%D0%B3" title="Поверхневий натяг – Ukrainian" lang="uk" hreflang="uk" data-title="Поверхневий натяг" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%A9c_c%C4%83ng_b%E1%BB%81_m%E1%BA%B7t" title="Sức căng bề mặt – Vietnamese" lang="vi" hreflang="vi" data-title="Sức căng bề mặt" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%A1%A8%E9%9D%A2%E5%BC%A0%E5%8A%9B" title="表面张力 – Wu" lang="wuu" hreflang="wuu" data-title="表面张力" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A1%A8%E9%9D%A2%E5%BC%B5%E5%8A%9B" title="表面張力 – Cantonese" lang="yue" hreflang="yue" data-title="表面張力" data-language-autonym="粵語" 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print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the short story by James Blish, see <a href="/wiki/Surface_Tension_(short_story)" title="Surface Tension (short story)">Surface Tension (short story)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Cleanup_rewrite plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Crystal_Clear_app_kedit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/40px-Crystal_Clear_app_kedit.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/60px-Crystal_Clear_app_kedit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Crystal_Clear_app_kedit.svg/80px-Crystal_Clear_app_kedit.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may need to be rewritten</b> to comply with Wikipedia's <a href="/wiki/Wikipedia:Manual_of_Style" title="Wikipedia:Manual of Style">quality standards</a>.<span class="hide-when-compact"> <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Surface_tension&action=edit">You can help</a>. The <a href="/wiki/Talk:Surface_tension" title="Talk:Surface tension">talk page</a> may contain suggestions.</span> <span class="date-container"><i>(<span class="date">June 2019</span>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:RainDrops1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/RainDrops1.jpg/260px-RainDrops1.jpg" decoding="async" width="260" height="173" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/RainDrops1.jpg/390px-RainDrops1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/RainDrops1.jpg/520px-RainDrops1.jpg 2x" data-file-width="5472" data-file-height="3648" /></a><figcaption>Rain water flux from a canopy. Among the forces that govern drop formation: surface tension by <a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">cohesion</a>, <a href="/wiki/Van_der_Waals_force" title="Van der Waals force">Van der Waals force</a>, <a href="/wiki/Plateau%E2%80%93Rayleigh_instability" title="Plateau–Rayleigh instability">Plateau–Rayleigh instability</a>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv/220px--Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="165" data-durationhint="11" data-mwtitle="Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="640" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv" type="video/ogg; codecs="theora"" data-width="640" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="192" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="480" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f8/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv/Cutting_a_water_droplet_using_a_superhydrophobic_knife_on_superhydrophobic_surfaces.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="480" data-height="360" /></video></span><figcaption>Surface tension and <a href="/wiki/Hydrophobicity" class="mw-redirect" title="Hydrophobicity">hydrophobicity</a> interact in this attempt to cut a <a href="/wiki/Water_droplet" class="mw-redirect" title="Water droplet">water droplet</a>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Surface_tension_experimental_demonstration.ogv/220px--Surface_tension_experimental_demonstration.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="48" data-mwtitle="Surface_tension_experimental_demonstration.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Surface_tension_experimental_demonstration.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/5/50/Surface_tension_experimental_demonstration.ogv" type="video/ogg; codecs="theora, opus"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/5/50/Surface_tension_experimental_demonstration.ogv/Surface_tension_experimental_demonstration.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>Surface tension experimental demonstration with soap</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></th></tr><tr><td class="sidebar-image"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1856f88def2056f28ed27c7d31180a6240820ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.874ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}}}"></span><div class="sidebar-caption"><a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Laws</div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Conservations</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Conservation_of_mass" title="Conservation of mass">Mass</a></li> <li><a href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">Momentum</a></li> <li><a href="/wiki/Conservation_of_energy" title="Conservation of energy">Energy</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;font-weight:normal;"> Inequalities</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Clausius%E2%80%93Duhem_inequality" title="Clausius–Duhem inequality">Clausius–Duhem (entropy)</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Solid_mechanics" title="Solid mechanics">Solid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Deformation_(physics)" title="Deformation (physics)">Deformation</a></li> <li><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a> <ul><li><a href="/wiki/Linear_elasticity" title="Linear elasticity">linear</a></li></ul></li> <li><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></li> <li><a href="/wiki/Hooke%27s_law" title="Hooke's law">Hooke's law</a></li> <li><a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">Stress</a></li> <li><a href="/wiki/Strain_(mechanics)" title="Strain (mechanics)">Strain</a> <ul><li><a href="/wiki/Finite_strain_theory" title="Finite strain theory">Finite strain</a></li> <li><a href="/wiki/Infinitesimal_strain_theory" title="Infinitesimal strain theory">Infinitesimal strain</a></li></ul></li> <li><a href="/wiki/Compatibility_(mechanics)" title="Compatibility (mechanics)">Compatibility</a></li> <li><a href="/wiki/Bending" title="Bending">Bending</a></li> <li><a href="/wiki/Contact_mechanics" title="Contact mechanics">Contact mechanics</a> <ul><li><a href="/wiki/Frictional_contact_mechanics" title="Frictional contact mechanics">frictional</a></li></ul></li> <li><a href="/wiki/Material_failure_theory" title="Material failure theory">Material failure theory</a></li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Fluid" title="Fluid">Fluids</a></th></tr><tr><td class="sidebar-content"> <div class="wraplinks"> <ul><li><a href="/wiki/Hydrostatics" title="Hydrostatics">Statics</a> <b>·</b> <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Dynamics</a></li> <li><a href="/wiki/Archimedes%27_principle" title="Archimedes' principle">Archimedes' principle</a> <b>·</b> <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Poiseuille equation</a> <b>·</b> <a href="/wiki/Pascal%27s_law" title="Pascal's law">Pascal's law</a></li> <li><a href="/wiki/Viscosity" title="Viscosity">Viscosity</a> <ul><li>(<a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian</a> <b>·</b> <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian</a>)</li></ul></li> <li><a href="/wiki/Buoyancy" title="Buoyancy">Buoyancy</a> <b>·</b> <a href="/wiki/Mixing_(process_engineering)" title="Mixing (process engineering)">Mixing</a> <b>·</b> <a href="/wiki/Pressure" title="Pressure">Pressure</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Liquid" title="Liquid">Liquids</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Adhesion" title="Adhesion">Adhesion</a></li> <li><a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></li> <li><a href="/wiki/Chromatography" title="Chromatography">Chromatography</a></li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion (chemistry)</a></li> <li><a class="mw-selflink selflink">Surface tension</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Gas" title="Gas">Gases</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Atmosphere" title="Atmosphere">Atmosphere</a></li> <li><a href="/wiki/Boyle%27s_law" title="Boyle's law">Boyle's law</a></li> <li><a href="/wiki/Charles%27s_law" title="Charles's law">Charles's law</a></li> <li><a href="/wiki/Combined_gas_law" class="mw-redirect" title="Combined gas law">Combined gas law</a></li> <li><a href="/wiki/Fick%27s_law" class="mw-redirect" title="Fick's law">Fick's law</a></li> <li><a href="/wiki/Gay-Lussac%27s_law" title="Gay-Lussac's law">Gay-Lussac's law</a></li> <li><a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></th></tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Rheology" title="Rheology">Rheology</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Viscoelasticity" title="Viscoelasticity">Viscoelasticity</a></li> <li><a href="/wiki/Rheometry" title="Rheometry">Rheometry</a></li> <li><a href="/wiki/Rheometer" title="Rheometer">Rheometer</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-style:italic;"> <a href="/wiki/Smart_fluid" title="Smart fluid">Smart fluids</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electrorheological_fluid" title="Electrorheological fluid">Electrorheological</a></li> <li><a href="/wiki/Magnetorheological_fluid" title="Magnetorheological fluid">Magnetorheological</a></li> <li><a href="/wiki/Ferrofluid" title="Ferrofluid">Ferrofluids</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Robert_Boyle" title="Robert Boyle">Boyle</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a></li> <li><a href="/wiki/Jacques_Charles" title="Jacques Charles">Charles</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Adolf_Eugen_Fick" title="Adolf Eugen Fick">Fick</a></li> <li><a href="/wiki/Joseph_Louis_Gay-Lussac" title="Joseph Louis Gay-Lussac">Gay-Lussac</a></li> <li><a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Graham</a></li> <li><a href="/wiki/Robert_Hooke" title="Robert Hooke">Hooke</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Claude-Louis_Navier" title="Claude-Louis Navier">Navier</a></li> <li><a href="/wiki/Walter_Noll" title="Walter Noll">Noll</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">Stokes</a></li> <li><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Continuum_mechanics" title="Template:Continuum mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Continuum_mechanics" title="Template talk:Continuum mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Continuum_mechanics" title="Special:EditPage/Template:Continuum mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Surface tension</b> is the tendency of <a href="/wiki/Liquid" title="Liquid">liquid</a> surfaces at rest to shrink into the minimum <a href="/wiki/Surface_area" title="Surface area">surface area</a> possible. Surface <a href="/wiki/Tension_(physics)" title="Tension (physics)">tension</a> is what allows objects with a higher density than water such as <a href="/wiki/Razor_blade" class="mw-redirect" title="Razor blade">razor blades</a> and insects (e.g. <a href="/wiki/Gerridae" title="Gerridae">water striders</a>) to float on a water surface without becoming even partly submerged. </p><p>At liquid–air interfaces, surface tension results from the greater attraction of liquid molecules to each other (due to <a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">cohesion</a>) than to the molecules in the air (due to <a href="/wiki/Adhesion" title="Adhesion">adhesion</a>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>There are two primary mechanisms in play. One is an inward force on the surface molecules causing the liquid to contract.<sup id="cite_ref-usgs_2-0" class="reference"><a href="#cite_note-usgs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-berry_3-0" class="reference"><a href="#cite_note-berry-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Second is a tangential force parallel to the surface of the liquid.<sup id="cite_ref-berry_3-1" class="reference"><a href="#cite_note-berry-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> This <i>tangential</i> force is generally referred to as the surface tension. The net effect is the liquid behaves as if its surface were covered with a stretched elastic membrane. But this analogy must not be taken too far as the tension in an elastic membrane is dependent on the amount of deformation of the membrane while surface tension is an inherent property of the liquid<i>–</i>air or liquid<i>–</i>vapour interface.<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Because of the relatively high attraction of water molecules to each other through a web of <a href="/wiki/Hydrogen_bond" title="Hydrogen bond">hydrogen bonds</a>, water has a higher surface tension (72.8 <a href="/wiki/Millinewton" class="mw-redirect" title="Millinewton">millinewtons</a> (mN) per meter at 20 °C) than most other liquids. Surface tension is an important factor in the phenomenon of <a href="/wiki/Capillary_action" title="Capillary action">capillarity</a>. </p><p>Surface tension has the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimension</a> of <a href="/wiki/Force" title="Force">force</a> per unit <a href="/wiki/Length" title="Length">length</a>, or of <a href="/wiki/Energy" title="Energy">energy</a> per unit <a href="/wiki/Area" title="Area">area</a>.<sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The two are equivalent, but when referring to energy per unit of area, it is common to use the term <a href="/wiki/Surface_energy" title="Surface energy">surface energy</a>, which is a more general term in the sense that it applies also to <a href="/wiki/Solid" title="Solid">solids</a>. </p><p>In <a href="/wiki/Materials_science" title="Materials science">materials science</a>, surface tension is used for either <a href="/wiki/Surface_stress" title="Surface stress">surface stress</a> or <a href="/wiki/Surface_energy" title="Surface energy">surface energy</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Causes">Causes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=1" title="Edit section: Causes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Wassermolek%C3%BCleInTr%C3%B6pfchen.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wassermolek%C3%BCleInTr%C3%B6pfchen.svg/220px-Wassermolek%C3%BCleInTr%C3%B6pfchen.svg.png" decoding="async" width="220" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wassermolek%C3%BCleInTr%C3%B6pfchen.svg/330px-Wassermolek%C3%BCleInTr%C3%B6pfchen.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Wassermolek%C3%BCleInTr%C3%B6pfchen.svg/440px-Wassermolek%C3%BCleInTr%C3%B6pfchen.svg.png 2x" data-file-width="512" data-file-height="519" /></a><figcaption>Diagram of the cohesive forces on molecules of a liquid</figcaption></figure> <p>Due to the <a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">cohesive forces</a>, a molecule located away from the surface is pulled equally in every direction by neighboring liquid molecules, resulting in a net force of zero. The molecules at the surface do not have the <i>same</i> molecules on all sides of them and therefore are pulled inward. This creates some <a href="/wiki/Internal_pressure" title="Internal pressure">internal pressure</a> and forces liquid surfaces to contract to the minimum area.<sup id="cite_ref-usgs_2-1" class="reference"><a href="#cite_note-usgs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>There is also a tension parallel to the surface at the liquid-air interface which will resist an external force, due to the cohesive nature of water molecules.<sup id="cite_ref-usgs_2-2" class="reference"><a href="#cite_note-usgs-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-berry_3-2" class="reference"><a href="#cite_note-berry-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The forces of attraction acting between molecules of the same type are called cohesive forces, while those acting between molecules of different types are called adhesive forces. The balance between the cohesion of the liquid and its adhesion to the material of the container determines the degree of <a href="/wiki/Wetting" title="Wetting">wetting</a>, the <a href="/wiki/Contact_angle" title="Contact angle">contact angle</a>, and the shape of <a href="/wiki/Meniscus_(liquid)" title="Meniscus (liquid)">meniscus</a>. When cohesion dominates (specifically, adhesion energy is less than half of cohesion energy) the wetting is low and the meniscus is convex at a vertical wall (as for mercury in a glass container). On the other hand, when adhesion dominates (when adhesion energy is more than half of cohesion energy) the wetting is high and the similar meniscus is concave (as in water in a glass). </p><p>Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of water tend to be pulled into a spherical shape by the imbalance in cohesive forces of the surface layer. In the absence of other forces, drops of virtually all liquids would be approximately spherical. The spherical shape minimizes the necessary "wall tension" of the surface layer according to <a href="/wiki/Young%E2%80%93Laplace_equation" title="Young–Laplace equation">Laplace's law</a>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Water_droplet_lying_on_a_damask.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Water_droplet_lying_on_a_damask.jpg/220px-Water_droplet_lying_on_a_damask.jpg" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Water_droplet_lying_on_a_damask.jpg/330px-Water_droplet_lying_on_a_damask.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Water_droplet_lying_on_a_damask.jpg/440px-Water_droplet_lying_on_a_damask.jpg 2x" data-file-width="4310" data-file-height="3312" /></a><figcaption>Water droplet lying on a <a href="/wiki/Damask" title="Damask">damask</a>. Surface tension is high enough to prevent seeping through the textile</figcaption></figure> <p>Another way to view surface tension is in terms of energy. A molecule in contact with a neighbor is in a lower state of energy than if it were alone. The interior molecules have as many neighbors as they can possibly have, but the boundary molecules are missing neighbors (compared to interior molecules) and therefore have higher energy. For the liquid to minimize its energy state, the number of higher energy boundary molecules must be minimized. The minimized number of boundary molecules results in a minimal surface area.<sup id="cite_ref-white_5-0" class="reference"><a href="#cite_note-white-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> As a result of surface area minimization, a surface will assume a smooth shape. </p> <div class="mw-heading mw-heading2"><h2 id="Physics">Physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=2" title="Edit section: Physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Physical_units">Physical units</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=3" title="Edit section: Physical units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface tension, represented by the symbol <i><a href="/wiki/Gamma" title="Gamma">γ</a></i> (alternatively <i><a href="/wiki/Sigma" title="Sigma">σ</a></i> or <i><a href="/wiki/T" title="T">T</a></i>), is measured in <a href="/wiki/Force" title="Force">force</a> per <a href="/wiki/Length" title="Length">unit length</a>. Its <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> unit is <a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a> per meter but the <a href="/wiki/Cgs" class="mw-redirect" title="Cgs">cgs</a> unit of <a href="/wiki/Dyne" title="Dyne">dyne</a> per centimeter is also used. For example,<sup id="cite_ref-MIT1_6-0" class="reference"><a href="#cite_note-MIT1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =1~\mathrm {\frac {dyn}{cm}} =1~\mathrm {\frac {erg}{cm^{2}}} =1~\mathrm {\frac {10^{-7}\,m\cdot N}{10^{-4}\,m^{2}}} =0.001~\mathrm {\frac {N}{m}} =0.001~\mathrm {\frac {J}{m^{2}}} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">n</mi> </mrow> <mrow> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">m</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> </mrow> <mrow> <mi mathvariant="normal">c</mi> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>7</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> <mo>⋅</mo> <mi mathvariant="normal">N</mi> </mrow> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.001</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">m</mi> </mfrac> </mrow> <mo>=</mo> <mn>0.001</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">J</mi> <msup> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =1~\mathrm {\frac {dyn}{cm}} =1~\mathrm {\frac {erg}{cm^{2}}} =1~\mathrm {\frac {10^{-7}\,m\cdot N}{10^{-4}\,m^{2}}} =0.001~\mathrm {\frac {N}{m}} =0.001~\mathrm {\frac {J}{m^{2}}} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f84d74bdd99c878b58613a7963c1831d66e898a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.73ex; height:6.176ex;" alt="{\displaystyle \gamma =1~\mathrm {\frac {dyn}{cm}} =1~\mathrm {\frac {erg}{cm^{2}}} =1~\mathrm {\frac {10^{-7}\,m\cdot N}{10^{-4}\,m^{2}}} =0.001~\mathrm {\frac {N}{m}} =0.001~\mathrm {\frac {J}{m^{2}}} .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=4" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Surface_growing.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Surface_growing.png/220px-Surface_growing.png" decoding="async" width="220" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Surface_growing.png/330px-Surface_growing.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f7/Surface_growing.png 2x" data-file-width="431" data-file-height="247" /></a><figcaption>This diagram illustrates the force necessary to increase the surface area. This force is proportional to the surface tension.</figcaption></figure> <p>Surface tension can be defined in terms of force or energy. </p> <div class="mw-heading mw-heading4"><h4 id="In_terms_of_force">In terms of force</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=5" title="Edit section: In terms of force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface tension <span class="texhtml mvar" style="font-style:italic;">γ</span> of a liquid is the force per unit length. In the illustration on the right, the rectangular frame, composed of three unmovable sides (black) that form a "U" shape, and a fourth movable side (blue) that can slide to the right. Surface tension will pull the blue bar to the left; the force <span class="texhtml mvar" style="font-style:italic;">F</span> required to hold the movable side is proportional to the length <span class="texhtml mvar" style="font-style:italic;">L</span> of the immobile side. Thus the ratio <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>F</i></span><span class="sr-only">/</span><span class="den"><i>L</i></span></span>⁠</span></span> depends only on the intrinsic properties of the liquid (composition, temperature, etc.), not on its geometry. For example, if the frame had a more complicated shape, the ratio <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>F</i></span><span class="sr-only">/</span><span class="den"><i>L</i></span></span>⁠</span></span>, with <span class="texhtml mvar" style="font-style:italic;">L</span> the length of the movable side and <span class="texhtml mvar" style="font-style:italic;">F</span> the force required to stop it from sliding, is found to be the same for all shapes. We therefore define the surface tension as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {F}{2L}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {F}{2L}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d340bee2abaca021b1d07a90e80f4d870b4190db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.589ex; height:5.176ex;" alt="{\displaystyle \gamma ={\frac {F}{2L}}.}"></span> The reason for the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> is that the film has two sides (two surfaces), each of which contributes equally to the force; so the force contributed by a single side is <span class="texhtml"><i>γL</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>F</i></span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span>. </p> <div class="mw-heading mw-heading4"><h4 id="In_terms_of_energy">In terms of energy</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=6" title="Edit section: In terms of energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface tension <span class="texhtml mvar" style="font-style:italic;">γ</span> of a liquid is the ratio of the change in the energy of the liquid to the change in the surface area of the liquid (that led to the change in energy). This can be easily related to the previous definition in terms of force:<sup id="cite_ref-MIT_Non-Newtonian_7-0" class="reference"><a href="#cite_note-MIT_Non-Newtonian-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> if <span class="texhtml mvar" style="font-style:italic;">F</span> is the force required to stop the side from <i>starting</i> to slide, then this is also the force that would keep the side in the state of <i>sliding at a constant speed</i> (by Newton's Second Law). But if the side is moving to the right (in the direction the force is applied), then the surface area of the stretched liquid is increasing while the applied force is doing work on the liquid. This means that increasing the surface area increases the energy of the film. The work done by the force <span class="texhtml mvar" style="font-style:italic;">F</span> in moving the side by distance <span class="texhtml">Δ<i>x</i></span> is <span class="texhtml"><i>W</i> = <i>F</i>Δ<i>x</i></span>; at the same time the total area of the film increases by <span class="texhtml">Δ<i>A</i> = 2<i>L</i>Δ<i>x</i></span> (the factor of 2 is here because the liquid has two sides, two surfaces). Thus, multiplying both the numerator and the denominator of <span class="texhtml"><i>γ</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>F</i></span><span class="sr-only">/</span><span class="den"><i>L</i></span></span>⁠</span></span> by <span class="texhtml">Δ<i>x</i></span>, we get <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ={\frac {F}{2L}}={\frac {F\Delta x}{2L\Delta x}}={\frac {W}{\Delta A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>L</mi> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>W</mi> <mrow> <mi mathvariant="normal">Δ</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ={\frac {F}{2L}}={\frac {F\Delta x}{2L\Delta x}}={\frac {W}{\Delta A}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e78f375d81d0721c98c1461718220e7d0da55e4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.148ex; height:5.509ex;" alt="{\displaystyle \gamma ={\frac {F}{2L}}={\frac {F\Delta x}{2L\Delta x}}={\frac {W}{\Delta A}}.}"></span> This work <span class="texhtml mvar" style="font-style:italic;">W</span> is, by the <a href="/wiki/Potential_Energy#Work_and_potential_energy" class="mw-redirect" title="Potential Energy">usual arguments</a>, interpreted as being stored as potential energy. Consequently, surface tension can be also measured in SI system as joules per square meter and in the <a href="/wiki/Centimetre_gram_second_system_of_units" class="mw-redirect" title="Centimetre gram second system of units">cgs</a> system as <a href="/wiki/Erg" title="Erg">ergs</a> per cm<sup>2</sup>. Since <a href="/wiki/Minimum_total_potential_energy_principle" title="Minimum total potential energy principle">mechanical systems try to find a state of minimum potential energy</a>, a free droplet of liquid naturally assumes a spherical shape, which has the minimum surface area for a given volume. The equivalence of measurement of energy per unit area to force per unit length can be proven by <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>.<sup id="cite_ref-s_z_8-0" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div style="clear:right;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Effects">Effects</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=7" title="Edit section: Effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Water">Water</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=8" title="Edit section: Water"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several effects of surface tension can be seen with ordinary water: </p> <div><ol style="list-style-type:upper-alpha"><li>Beading of rain water on a waxy surface, such as a leaf. Water <a href="/wiki/Hydrophobic_effect" title="Hydrophobic effect">adheres weakly</a> to wax and strongly to itself, so water clusters into drops. Surface tension gives them their near-spherical shape, because a sphere has the smallest possible <a href="/wiki/Surface_area_to_volume_ratio" class="mw-redirect" title="Surface area to volume ratio">surface area to volume ratio</a>.</li><li>Formation of <a href="/wiki/Drop_(liquid)" title="Drop (liquid)">drops</a> occurs when a mass of liquid is stretched. The animation (below) shows water adhering to the faucet gaining mass until it is stretched to a point where the surface tension can no longer keep the drop linked to the faucet. It then separates and surface tension forms the drop into a sphere. If a stream of water were running from the faucet, the stream would break up into drops during its fall. Gravity stretches the stream, then surface tension pinches it into spheres.<sup id="cite_ref-MIT5_9-0" class="reference"><a href="#cite_note-MIT5-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li><li>Flotation of objects denser than water occurs when the object is nonwettable and its weight is small enough to be borne by the forces arising from surface tension.<sup id="cite_ref-white_5-1" class="reference"><a href="#cite_note-white-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, <a href="/wiki/Gerridae" title="Gerridae">water striders</a> use surface tension to walk on the surface of a pond in the following way. The nonwettability of the water strider's leg means there is no attraction between molecules of the leg and molecules of the water, so when the leg pushes down on the water, the surface tension of the water only tries to recover its flatness from its deformation due to the leg. This behavior of the water pushes the water strider upward so it can stand on the surface of the water as long as its mass is small enough that the water can support it. The surface of the water behaves like an elastic film: the insect's feet cause indentations in the water's surface, increasing its surface area<sup id="cite_ref-MIT3_10-0" class="reference"><a href="#cite_note-MIT3-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and tendency of minimization of surface curvature (so area) of the water pushes the insect's feet upward.</li><li>Separation of oil and water (in this case, water and liquid wax) is caused by a tension in the surface between dissimilar liquids. This type of surface tension is called "interface tension", but its chemistry is the same.</li><li><a href="/wiki/Tears_of_wine" title="Tears of wine">Tears of wine</a> is the formation of drops and rivulets on the side of a glass containing an alcoholic beverage. Its cause is a complex interaction between the differing surface tensions of water and <a href="/wiki/Ethanol" title="Ethanol">ethanol</a>; it is induced by a combination of surface tension modification of water by <a href="/wiki/Ethanol" title="Ethanol">ethanol</a> together with ethanol <a href="/wiki/Evaporating" class="mw-redirect" title="Evaporating">evaporating</a> faster than water.</li></ol></div> <ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 162px"> <div class="thumb" style="width: 160px;"><span typeof="mw:File"><a href="/wiki/File:Dew_2.jpg" class="mw-file-description" title="A. Water beading on a leaf"><img alt="A. Water beading on a leaf" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dew_2.jpg/240px-Dew_2.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dew_2.jpg/360px-Dew_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Dew_2.jpg/480px-Dew_2.jpg 2x" data-file-width="2784" data-file-height="2088" /></a></span></div> <div class="gallerytext"><b>A.</b> Water beading on a leaf</div> </li> <li class="gallerybox" style="width: 92px"> <div class="thumb" style="width: 90px;"><span typeof="mw:File"><a href="/wiki/File:Water_drop_animation_enhanced_small.gif" class="mw-file-description" title="B. Water dripping from a tap"><img alt="B. Water dripping from a tap" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Water_drop_animation_enhanced_small.gif/135px-Water_drop_animation_enhanced_small.gif" decoding="async" width="90" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/bd/Water_drop_animation_enhanced_small.gif 1.5x" data-file-width="180" data-file-height="240" /></a></span></div> <div class="gallerytext"><b>B.</b> Water dripping from a tap</div> </li> <li class="gallerybox" style="width: 149.33333333333px"> <div class="thumb" style="width: 147.33333333333px;"><span typeof="mw:File"><a href="/wiki/File:WaterstriderEnWiki.jpg" class="mw-file-description" title="C. Water striders stay at the top of liquid because of surface tension"><img alt="C. Water striders stay at the top of liquid because of surface tension" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/WaterstriderEnWiki.jpg/221px-WaterstriderEnWiki.jpg" decoding="async" width="148" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1a/WaterstriderEnWiki.jpg/332px-WaterstriderEnWiki.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1a/WaterstriderEnWiki.jpg/442px-WaterstriderEnWiki.jpg 2x" data-file-width="500" data-file-height="407" /></a></span></div> <div class="gallerytext"><b>C.</b> <a href="/wiki/Water_strider" class="mw-redirect" title="Water strider">Water striders</a> stay at the top of liquid because of surface tension</div> </li> <li class="gallerybox" style="width: 74px"> <div class="thumb" style="width: 72px;"><span typeof="mw:File"><a href="/wiki/File:1990s_Mathmos_Astro.jpg" class="mw-file-description" title="D. Lava lamp with interaction between dissimilar liquids: water and liquid wax"><img alt="D. Lava lamp with interaction between dissimilar liquids: water and liquid wax" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/1990s_Mathmos_Astro.jpg/108px-1990s_Mathmos_Astro.jpg" decoding="async" width="72" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f2/1990s_Mathmos_Astro.jpg/162px-1990s_Mathmos_Astro.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f2/1990s_Mathmos_Astro.jpg/216px-1990s_Mathmos_Astro.jpg 2x" data-file-width="900" data-file-height="1500" /></a></span></div> <div class="gallerytext"><b>D.</b> <a href="/wiki/Lava_lamp" title="Lava lamp">Lava lamp</a> with interaction between dissimilar liquids: water and liquid wax</div> </li> <li class="gallerybox" style="width: 74px"> <div class="thumb" style="width: 72px;"><span typeof="mw:File"><a href="/wiki/File:Wine_legs_shadow.jpg" class="mw-file-description" title="E. Photo showing the "tears of wine" phenomenon."><img alt="E. Photo showing the "tears of wine" phenomenon." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Wine_legs_shadow.jpg/108px-Wine_legs_shadow.jpg" decoding="async" width="72" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/Wine_legs_shadow.jpg/162px-Wine_legs_shadow.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/Wine_legs_shadow.jpg/216px-Wine_legs_shadow.jpg 2x" data-file-width="862" data-file-height="1438" /></a></span></div> <div class="gallerytext"><b>E.</b> Photo showing the "<a href="/wiki/Tears_of_wine" title="Tears of wine">tears of wine</a>" phenomenon. </div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="Surfactants">Surfactants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=9" title="Edit section: Surfactants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface tension is visible in other common phenomena, especially when <a href="/wiki/Surfactant" title="Surfactant">surfactants</a> are used to decrease it: </p> <ul><li><a href="/wiki/Soap_bubble" title="Soap bubble">Soap bubbles</a> have very large surface areas with very little mass. Bubbles in pure water are unstable. The addition of surfactants, however, can have a stabilizing effect on the bubbles (see <a href="/wiki/Marangoni_effect" title="Marangoni effect">Marangoni effect</a>). Surfactants actually reduce the surface tension of water by a factor of three or more.</li> <li><a href="/wiki/Emulsion" title="Emulsion">Emulsions</a> are a type of colloidal dispersion in which surface tension plays a role. Tiny droplets of oil dispersed in pure water will spontaneously coalesce and phase separate. The addition of surfactants reduces the interfacial tension and allow for the formation of oil droplets in the water medium (or vice versa). The stability of such formed oil droplets depends on many different chemical and environmental factors.<div style="clear:right;" class=""></div></li></ul> <div class="mw-heading mw-heading3"><h3 id="Surface_curvature_and_pressure">Surface curvature and pressure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=10" title="Edit section: Surface curvature and pressure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CurvedSurfaceTension.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/CurvedSurfaceTension.png/220px-CurvedSurfaceTension.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/CurvedSurfaceTension.png/330px-CurvedSurfaceTension.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/66/CurvedSurfaceTension.png 2x" data-file-width="385" data-file-height="184" /></a><figcaption>Surface tension forces acting on a tiny (differential) patch of surface. <span class="texhtml mvar" style="font-style:italic;">δθ<sub>x</sub></span> and <span class="texhtml mvar" style="font-style:italic;">δθ<sub>y</sub></span> indicate the amount of bend over the dimensions of the patch. Balancing the tension forces with pressure leads to the <a href="/wiki/Young%E2%80%93Laplace_equation" title="Young–Laplace equation">Young–Laplace equation</a></figcaption></figure> <p>If no force acts normal to a tensioned surface, the surface must remain flat. But if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. In order for the surface tension forces to cancel the force due to pressure, the surface must be curved. The diagram shows how surface curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the resulting equation is known as the <a href="/wiki/Young%E2%80%93Laplace_equation" title="Young–Laplace equation">Young–Laplace equation</a>:<sup id="cite_ref-cwp_11-0" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p=\gamma \left({\frac {1}{R_{x}}}+{\frac {1}{R_{y}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>=</mo> <mi>γ</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p=\gamma \left({\frac {1}{R_{x}}}+{\frac {1}{R_{y}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13fd97fd027b6bbaa6604c3935136eaf1d6db1b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.537ex; height:6.176ex;" alt="{\displaystyle \Delta p=\gamma \left({\frac {1}{R_{x}}}+{\frac {1}{R_{y}}}\right)}"></span> where: </p> <ul><li><span class="texhtml">Δ<i>p</i></span> is the pressure difference, known as the <a href="/wiki/Laplace_pressure" title="Laplace pressure">Laplace pressure</a>.<sup id="cite_ref-Physics_and_Chemistry_of_Interfaces_12-0" class="reference"><a href="#cite_note-Physics_and_Chemistry_of_Interfaces-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li><span class="texhtml mvar" style="font-style:italic;">γ</span> is surface tension.</li> <li><span class="texhtml mvar" style="font-style:italic;">R<sub>x</sub></span> and <span class="texhtml mvar" style="font-style:italic;">R<sub>y</sub></span> are <a href="/wiki/Radius_of_curvature_(mathematics)" class="mw-redirect" title="Radius of curvature (mathematics)">radii of curvature</a> in each of the axes that are parallel to the surface.</li></ul> <p>The quantity in parentheses on the right hand side is in fact (twice) the <a href="/wiki/Mean_curvature" title="Mean curvature">mean curvature</a> of the surface (depending on normalisation). Solutions to this equation determine the shape of water drops, puddles, menisci, soap bubbles, and all other shapes determined by surface tension (such as the shape of the impressions that a <a href="/wiki/Water_strider" class="mw-redirect" title="Water strider">water strider</a>'s feet make on the surface of a pond). The table below shows how the internal pressure of a water droplet increases with decreasing radius. For not very small drops the effect is subtle, but the pressure difference becomes enormous when the drop sizes approach the molecular size. (In the limit of a single molecule the concept becomes meaningless.) </p> <table class="wikitable" style="float:center; clear:right;"> <caption><span class="texhtml">Δ<i>p</i></span> for water drops of different radii at <a href="/wiki/Standard_conditions_for_temperature_and_pressure" class="mw-redirect" title="Standard conditions for temperature and pressure">STP</a> </caption> <tbody><tr> <th style="width:120px;">Droplet radius </th> <td style="width:120px;">1 mm </td> <td style="width:120px;">0.1 mm </td> <td style="width:120px;">1 <a href="/wiki/Micrometre" title="Micrometre">μm</a> </td> <td style="width:120px;">10 <a href="/wiki/Nanometer" class="mw-redirect" title="Nanometer">nm</a> </td></tr> <tr> <th><span class="texhtml">Δ<i>p</i></span> (<a href="/wiki/Atmosphere_(unit)" class="mw-redirect" title="Atmosphere (unit)">atm</a>) </th> <td>0.0014 </td> <td>0.0144 </td> <td>1.436 </td> <td>143.6 </td></tr></tbody></table> <div style="clear:right;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Floating_objects">Floating objects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=11" title="Edit section: Floating objects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Flotation_of_flexible_objects" title="Flotation of flexible objects">Flotation of flexible objects</a></div><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Surface_Tension_Diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Surface_Tension_Diagram.svg/220px-Surface_Tension_Diagram.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Surface_Tension_Diagram.svg/330px-Surface_Tension_Diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6a/Surface_Tension_Diagram.svg/440px-Surface_Tension_Diagram.svg.png 2x" data-file-width="277" data-file-height="139" /></a><figcaption>Cross-section of a needle floating on the surface of water. <span class="texhtml"><i>F</i><sub>w</sub></span> is the weight and <span class="texhtml"><i>F</i><sub>s</sub></span> are surface tension resultant forces.</figcaption></figure> <p>When an object is placed on a liquid, its weight <span class="texhtml"><i>F</i><sub>w</sub></span> depresses the surface, and if surface tension and downward force become equal then it is balanced by the surface tension forces on either side <span class="texhtml"><i>F</i><sub>s</sub></span>, which are each parallel to the water's surface at the points where it contacts the object. Notice that small movement in the body may cause the object to sink. As the angle of contact decreases, surface tension decreases. The horizontal components of the two <span class="texhtml"><i>F</i><sub>s</sub></span> arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up<sup id="cite_ref-white_5-2" class="reference"><a href="#cite_note-white-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> to balance <span class="texhtml"><i>F</i><sub>w</sub></span>. The object's surface must not be wettable for this to happen, and its weight must be low enough for the surface tension to support it. If <span class="texhtml"><i>m</i></span> denotes the mass of the needle and <span class="texhtml"><i>g</i></span> acceleration due to gravity, we have </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mathrm {w} }=2F_{\mathrm {s} }\sin \theta \quad \Leftrightarrow \quad mg=2\gamma L\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mspace width="1em" /> <mo stretchy="false">⇔</mo> <mspace width="1em" /> <mi>m</mi> <mi>g</mi> <mo>=</mo> <mn>2</mn> <mi>γ</mi> <mi>L</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mathrm {w} }=2F_{\mathrm {s} }\sin \theta \quad \Leftrightarrow \quad mg=2\gamma L\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638a3fde8cc3efd05bc72a0e9436ff7f717b1447" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.511ex; height:2.676ex;" alt="{\displaystyle F_{\mathrm {w} }=2F_{\mathrm {s} }\sin \theta \quad \Leftrightarrow \quad mg=2\gamma L\sin \theta }"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Liquid_surface">Liquid surface</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=12" title="Edit section: Liquid surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Povr%C5%A1inska_napetost_milnica.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Povr%C5%A1inska_napetost_milnica.jpg/220px-Povr%C5%A1inska_napetost_milnica.jpg" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Povr%C5%A1inska_napetost_milnica.jpg/330px-Povr%C5%A1inska_napetost_milnica.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Povr%C5%A1inska_napetost_milnica.jpg/440px-Povr%C5%A1inska_napetost_milnica.jpg 2x" data-file-width="901" data-file-height="599" /></a><figcaption>Minimal surface</figcaption></figure> <p>To find the shape of the <a href="/wiki/Minimal_surface" title="Minimal surface">minimal surface</a> bounded by some arbitrary shaped frame using strictly mathematical means can be a daunting task. Yet by fashioning the frame out of wire and dipping it in soap-solution, a locally minimal surface will appear in the resulting soap-film within seconds.<sup id="cite_ref-s_z_8-1" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>The reason for this is that the pressure difference across a fluid interface is proportional to the <a href="/wiki/Mean_curvature" title="Mean curvature">mean curvature</a>, as seen in the <a href="/wiki/Young%E2%80%93Laplace_equation" title="Young–Laplace equation">Young–Laplace equation</a>. For an open soap film, the pressure difference is zero, hence the mean curvature is zero, and minimal surfaces have the property of zero mean curvature. </p> <div class="mw-heading mw-heading3"><h3 id="Contact_angles">Contact angles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=13" title="Edit section: Contact angles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Contact_angle" title="Contact angle">Contact angle</a></div> <p>The surface of any liquid is an interface between that liquid and some other medium.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup> The top surface of a pond, for example, is an interface between the pond water and the air. Surface tension, then, is not a property of the liquid alone, but a property of the liquid's interface with another medium. If a liquid is in a container, then besides the liquid/air interface at its top surface, there is also an interface between the liquid and the walls of the container. The surface tension between the liquid and air is usually different (greater) than its surface tension with the walls of a container. And where the two surfaces meet, their geometry must be such that all forces balance.<sup id="cite_ref-s_z_8-2" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cwp_11-1" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <table style="float:right;"> <tbody><tr> <td><figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SurfTensionContactAngle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/bb/SurfTensionContactAngle.png" decoding="async" width="255" height="192" class="mw-file-element" data-file-width="255" data-file-height="192" /></a><figcaption>Forces at contact point shown for contact angle greater than 90° (left) and less than 90° (right)</figcaption></figure> </td></tr></tbody></table> <p>Where the two surfaces meet, they form a <a href="/wiki/Contact_angle" title="Contact angle">contact angle</a>, <span class="texhtml mvar" style="font-style:italic;">θ</span>, which is the angle the tangent to the surface makes with the solid surface. Note that the angle is measured <i>through the liquid</i>, as shown in the diagrams above. The diagram to the right shows two examples. Tension forces are shown for the liquid–air interface, the liquid–solid interface, and the solid–air interface. The example on the left is where the difference between the liquid–solid and solid–air surface tension, <span class="texhtml"><i>γ</i><sub>ls</sub> − <i>γ</i><sub>sa</sub></span>, is less than the liquid–air surface tension, <span class="texhtml"><i>γ</i><sub>la</sub></span>, but is nevertheless positive, that is </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {la} }>\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>></mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {la} }>\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10bbeabb8f3bd7ebee9433dddeb7410ccb14fb75" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.364ex; height:2.676ex;" alt="{\displaystyle \gamma _{\mathrm {la} }>\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }>0}"></span> </p><p>In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point, known as <a href="/wiki/Mechanical_equilibrium" title="Mechanical equilibrium">equilibrium</a>. The horizontal component of <span class="texhtml"><i>f</i><sub>la</sub></span> is canceled by the adhesive force, <span class="texhtml"><i>f</i><sub>A</sub></span>.<sup id="cite_ref-s_z_8-3" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {A} }=f_{\mathrm {la} }\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {A} }=f_{\mathrm {la} }\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8726f2aea99a0675c87003763ff052c15ad34c97" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.074ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {A} }=f_{\mathrm {la} }\sin \theta }"></span> </p><p>The more telling balance of forces, though, is in the vertical direction. The vertical component of <span class="texhtml"><i>f</i><sub>la</sub></span> must exactly cancel the difference of the forces along the solid surface, <span class="texhtml"><i>f</i><sub>ls</sub> − <i>f</i><sub>sa</sub></span>.<sup id="cite_ref-s_z_8-4" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {ls} }-f_{\mathrm {sa} }=-f_{\mathrm {la} }\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {ls} }-f_{\mathrm {sa} }=-f_{\mathrm {la} }\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d29e6a0a513b998d8fe11715d85f6b9c45ecce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.692ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {ls} }-f_{\mathrm {sa} }=-f_{\mathrm {la} }\cos \theta }"></span> </p> <table class="toccolours" border="1" style="float: right; clear: right; margin: 0 0 1em 1em; border-collapse: collapse;"> <caption><b>Some liquid–solid contact angles</b><sup id="cite_ref-s_z_8-5" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </caption> <tbody><tr style="text-align:center; background:#c0c0f0;"> <th>Liquid </th> <th>Solid </th> <th>Contact<br />angle </th></tr> <tr> <td><a href="/wiki/Water" title="Water">water</a> </td> <td rowspan="6"> <table cellpadding="0" cellspacing="0" border="0"> <tbody><tr style="background:#f8f8f8;"> <td>soda-lime glass </td></tr> <tr style="background:#f8f8f8;"> <td>lead glass </td></tr> <tr style="background:#f8f8f8;"> <td><a href="/wiki/Fused_quartz" title="Fused quartz">fused quartz</a> </td></tr></tbody></table> </td> <td rowspan="6" style="text-align:center;">0° </td></tr> <tr> <td><a href="/wiki/Ethanol" title="Ethanol">ethanol</a> </td></tr> <tr> <td><a href="/wiki/Diethyl_ether" title="Diethyl ether">diethyl ether</a> </td></tr> <tr> <td><a href="/wiki/Carbon_tetrachloride" title="Carbon tetrachloride">carbon tetrachloride</a> </td></tr> <tr> <td><a href="/wiki/Glycerol" title="Glycerol">glycerol</a> </td></tr> <tr> <td><a href="/wiki/Acetic_acid" title="Acetic acid">acetic acid</a> </td></tr> <tr> <td rowspan="2"><a href="/wiki/Water" title="Water">water</a> </td> <td>paraffin wax </td> <td style="text-align:center;">107° </td></tr> <tr> <td>silver </td> <td style="text-align:center;">90° </td></tr> <tr> <td rowspan="3"><a href="/wiki/Iodomethane" title="Iodomethane">methyl iodide</a> </td> <td>soda-lime glass </td> <td style="text-align:center;">29° </td></tr> <tr> <td>lead glass </td> <td style="text-align:center;">30° </td></tr> <tr> <td>fused quartz </td> <td style="text-align:center;">33° </td></tr> <tr> <td><a href="/wiki/Mercury_(element)" title="Mercury (element)">mercury</a> </td> <td>soda-lime glass </td> <td style="text-align:center;">140° </td></tr></tbody></table> <p>Since the forces are in direct proportion to their respective surface tensions, we also have:<sup id="cite_ref-cwp_11-2" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }=-\gamma _{\mathrm {la} }\cos \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }=-\gamma _{\mathrm {la} }\cos \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a312987f3a91d845f86f224d051e0f53d7b072" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.887ex; height:2.676ex;" alt="{\displaystyle \gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }=-\gamma _{\mathrm {la} }\cos \theta }"></span> </p><p>where </p> <ul><li><span class="texhtml"><i>γ</i><sub>ls</sub></span> is the liquid–solid surface tension,</li> <li><span class="texhtml"><i>γ</i><sub>la</sub></span> is the liquid–air surface tension,</li> <li><span class="texhtml"><i>γ</i><sub>sa</sub></span> is the solid–air surface tension,</li> <li><span class="texhtml mvar" style="font-style:italic;">θ</span> is the contact angle, where a concave <a href="/wiki/Meniscus_(liquid)" title="Meniscus (liquid)">meniscus</a> has contact angle less than 90° and a convex meniscus has contact angle of greater than 90°.<sup id="cite_ref-s_z_8-6" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul> <p>This means that although the difference between the liquid–solid and solid–air surface tension, <span class="texhtml"><i>γ</i><sub>ls</sub> − <i>γ</i><sub>sa</sub></span>, is difficult to measure directly, it can be inferred from the liquid–air surface tension, <span class="texhtml"><i>γ</i><sub>la</sub></span>, and the equilibrium contact angle, <span class="texhtml mvar" style="font-style:italic;">θ</span>, which is a function of the easily measurable advancing and receding contact angles (see main article <a href="/wiki/Contact_angle" title="Contact angle">contact angle</a>). </p><p>This same relationship exists in the diagram on the right. But in this case we see that because the contact angle is less than 90°, the liquid–solid/solid–air surface tension difference must be negative: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {la} }>0>\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>></mo> <mn>0</mn> <mo>></mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {la} }>0>\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8612b6e2da34ace6678044232c8d4440eaad3a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.364ex; height:2.676ex;" alt="{\displaystyle \gamma _{\mathrm {la} }>0>\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Special_contact_angles">Special contact angles</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=14" title="Edit section: Special contact angles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Observe that in the special case of a water–silver interface where the contact angle is equal to 90°, the liquid–solid/solid–air surface tension difference is exactly zero. </p><p>Another special case is where the contact angle is exactly 180°. Water with specially prepared <a href="/wiki/Polytetrafluoroethylene" title="Polytetrafluoroethylene">Teflon</a> approaches this.<sup id="cite_ref-cwp_11-3" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Contact angle of 180° occurs when the liquid–solid surface tension is exactly equal to the liquid–air surface tension. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {la} }=\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }>0\qquad \theta =180^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>></mo> <mn>0</mn> <mspace width="2em" /> <mi>θ</mi> <mo>=</mo> <msup> <mn>180</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {la} }=\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }>0\qquad \theta =180^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e27b5e38d6bc6fff1c77c208d5d965dd139205d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.74ex; height:2.843ex;" alt="{\displaystyle \gamma _{\mathrm {la} }=\gamma _{\mathrm {ls} }-\gamma _{\mathrm {sa} }>0\qquad \theta =180^{\circ }}"></span> </p> <div style="clear:right;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Liquid_in_a_vertical_tube">Liquid in a vertical tube</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=15" title="Edit section: Liquid in a vertical tube"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Capillary_action" title="Capillary action">Capillary action</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:HgBarometer.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/HgBarometer.gif/130px-HgBarometer.gif" decoding="async" width="130" height="238" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cd/HgBarometer.gif 1.5x" data-file-width="162" data-file-height="297" /></a><figcaption>Diagram of a <a href="/wiki/Mercury_(element)" title="Mercury (element)">mercury</a> <a href="/wiki/Barometer" title="Barometer">barometer</a></figcaption></figure> <p>An old style <a href="/wiki/Mercury_(element)" title="Mercury (element)">mercury</a> <a href="/wiki/Barometer" title="Barometer">barometer</a> consists of a vertical glass tube about 1 cm in diameter partially filled with mercury, and with a vacuum (called <a href="/wiki/Evangelista_Torricelli" title="Evangelista Torricelli">Torricelli</a>'s vacuum) in the unfilled volume (see diagram to the right). Notice that the mercury level at the center of the tube is higher than at the edges, making the upper surface of the mercury dome-shaped. The center of mass of the entire column of mercury would be slightly lower if the top surface of the mercury were flat over the entire cross-section of the tube. But the dome-shaped top gives slightly less surface area to the entire mass of mercury. Again the two effects combine to minimize the total potential energy. Such a surface shape is known as a convex meniscus. </p><p>We consider the surface area of the entire mass of mercury, including the part of the surface that is in contact with the glass, because mercury does not adhere to glass at all. So the surface tension of the mercury acts over its entire surface area, including where it is in contact with the glass. If instead of glass, the tube was made out of copper, the situation would be very different. Mercury aggressively adheres to copper. So in a copper tube, the level of mercury at the center of the tube will be lower than at the edges (that is, it would be a concave meniscus). In a situation where the liquid adheres to the walls of its container, we consider the part of the fluid's surface area that is in contact with the container to have <i>negative</i> surface tension. The fluid then works to maximize the contact surface area. So in this case increasing the area in contact with the container decreases rather than increases the potential energy. That decrease is enough to compensate for the increased potential energy associated with lifting the fluid near the walls of the container. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:CapillaryAction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/CapillaryAction.svg/130px-CapillaryAction.svg.png" decoding="async" width="130" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/CapillaryAction.svg/195px-CapillaryAction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/CapillaryAction.svg/260px-CapillaryAction.svg.png 2x" data-file-width="512" data-file-height="758" /></a><figcaption>Illustration of capillary rise and fall. Red=contact angle less than 90°; blue=contact angle greater than 90°</figcaption></figure> <p>If a tube is sufficiently narrow and the liquid adhesion to its walls is sufficiently strong, surface tension can draw liquid up the tube in a phenomenon known as <a href="/wiki/Capillary_action" title="Capillary action">capillary action</a>. The height to which the column is lifted is given by <a href="/wiki/Jurin%27s_law" title="Jurin's law">Jurin's law</a>:<sup id="cite_ref-s_z_8-7" class="reference"><a href="#cite_note-s_z-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {2\gamma _{\mathrm {la} }\cos \theta }{\rho gr}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mrow> <mi>ρ</mi> <mi>g</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {2\gamma _{\mathrm {la} }\cos \theta }{\rho gr}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830f221522d6ce6ea1c72dfdc5c0ccbe62d7b984" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.128ex; height:6.009ex;" alt="{\displaystyle h={\frac {2\gamma _{\mathrm {la} }\cos \theta }{\rho gr}}}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">h</span> is the height the liquid is lifted,</li> <li><span class="texhtml"><i>γ</i><sub>la</sub></span> is the liquid–air surface tension,</li> <li><span class="texhtml mvar" style="font-style:italic;">ρ</span> is the density of the liquid,</li> <li><span class="texhtml mvar" style="font-style:italic;">r</span> is the radius of the capillary,</li> <li><span class="texhtml mvar" style="font-style:italic;">g</span> is the acceleration due to gravity,</li> <li><span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle of contact described above. If <span class="texhtml mvar" style="font-style:italic;">θ</span> is greater than 90°, as with mercury in a glass container, the liquid will be depressed rather than lifted.</li></ul> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Puddles_on_a_surface">Puddles on a surface</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=16" title="Edit section: Puddles on a surface"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SurfTensionEdgeOfPool.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/SurfTensionEdgeOfPool.png/220px-SurfTensionEdgeOfPool.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/SurfTensionEdgeOfPool.png/330px-SurfTensionEdgeOfPool.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e4/SurfTensionEdgeOfPool.png 2x" data-file-width="374" data-file-height="238" /></a><figcaption>Profile curve of the edge of a puddle where the contact angle is 180°. The curve is given by the formula:<sup id="cite_ref-cwp_11-4" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-x_{0}={\frac {1}{2}}H\cosh ^{-1}\left({\frac {H}{h}}\right)-H{\sqrt {1-{\frac {h^{2}}{H^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>H</mi> <msup> <mi>cosh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>H</mi> <mi>h</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-x_{0}={\frac {1}{2}}H\cosh ^{-1}\left({\frac {H}{h}}\right)-H{\sqrt {1-{\frac {h^{2}}{H^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54b23785650ae8dbc42351d65eaeb688a9e53418" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.384ex; height:7.509ex;" alt="{\displaystyle x-x_{0}={\frac {1}{2}}H\cosh ^{-1}\left({\frac {H}{h}}\right)-H{\sqrt {1-{\frac {h^{2}}{H^{2}}}}}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle H=2{\sqrt {{\gamma }/{g\rho }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>ρ</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle H=2{\sqrt {{\gamma }/{g\rho }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c64536752ff4e12d873f76be75df6901b99dd70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.391ex; height:3.343ex;" alt="{\textstyle H=2{\sqrt {{\gamma }/{g\rho }}}}"></span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Exploring_new_continents_1200728.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Exploring_new_continents_1200728.JPG/220px-Exploring_new_continents_1200728.JPG" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Exploring_new_continents_1200728.JPG/330px-Exploring_new_continents_1200728.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Exploring_new_continents_1200728.JPG/440px-Exploring_new_continents_1200728.JPG 2x" data-file-width="1600" data-file-height="1029" /></a><figcaption>Small puddles of water on a smooth clean surface have perceptible thickness.</figcaption></figure> <p>Pouring mercury onto a horizontal flat sheet of glass results in a <a href="/wiki/Puddle#Physics" title="Puddle">puddle</a> that has a perceptible thickness. The puddle will spread out only to the point where it is a little under half a centimetre thick, and no thinner. Again this is due to the action of mercury's strong surface tension. The liquid mass flattens out because that brings as much of the mercury to as low a level as possible, but the surface tension, at the same time, is acting to reduce the total surface area. The result of the compromise is a puddle of a nearly fixed thickness. </p><p>The same surface tension demonstration can be done with water, lime water or even saline, but only on a surface made of a substance to which water does not adhere. Wax is such a substance. Water poured onto a smooth, flat, horizontal wax surface, say a waxed sheet of glass, will behave similarly to the mercury poured onto glass. </p><p>The thickness of a puddle of liquid on a surface whose contact angle is 180° is given by:<sup id="cite_ref-cwp_11-5" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=2{\sqrt {\frac {\gamma }{g\rho }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>γ</mi> <mrow> <mi>g</mi> <mi>ρ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=2{\sqrt {\frac {\gamma }{g\rho }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5962c766a80d28d8d130fab307802c8ae5f947c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.078ex; height:6.343ex;" alt="{\displaystyle h=2{\sqrt {\frac {\gamma }{g\rho }}}}"></span> </p><p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">h</span> is the depth of the puddle in centimeters or meters.</li> <li><span class="texhtml mvar" style="font-style:italic;">γ</span> is the surface tension of the liquid in dynes per centimeter or newtons per meter.</li> <li><span class="texhtml mvar" style="font-style:italic;">g</span> is the acceleration due to gravity and is equal to 980 cm/s<sup>2</sup> or 9.8 m/s<sup>2</sup></li> <li><span class="texhtml mvar" style="font-style:italic;">ρ</span> is the density of the liquid in grams per cubic centimeter or kilograms per cubic meter</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Surface_tension.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Surface_tension.svg/220px-Surface_tension.svg.png" decoding="async" width="220" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Surface_tension.svg/330px-Surface_tension.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Surface_tension.svg/440px-Surface_tension.svg.png 2x" data-file-width="1300" data-file-height="270" /></a><figcaption>Illustration of how lower contact angle leads to reduction of puddle depth</figcaption></figure> <p>In reality, the thicknesses of the puddles will be slightly less than what is predicted by the above formula because very few surfaces have a contact angle of 180° with any liquid. When the contact angle is less than 180°, the thickness is given by:<sup id="cite_ref-cwp_11-6" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\sqrt {\frac {2\gamma _{\mathrm {la} }\left(1-\cos \theta \right)}{g\rho }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>2</mn> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>g</mi> <mi>ρ</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\sqrt {\frac {2\gamma _{\mathrm {la} }\left(1-\cos \theta \right)}{g\rho }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92bf43e4ffb5ae577c1dd2d07f2e7fbdf6cc459" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.91ex; height:7.509ex;" alt="{\displaystyle h={\sqrt {\frac {2\gamma _{\mathrm {la} }\left(1-\cos \theta \right)}{g\rho }}}.}"></span> </p><p>For mercury on glass, <span class="texhtml"><i>γ</i><sub>Hg</sub></span> = 487 dyn/cm, <span class="texhtml"><i>ρ</i><sub>Hg</sub></span> = 13.5 g/cm<sup>3</sup> and <span class="texhtml mvar" style="font-style:italic;">θ</span> = 140°, which gives <span class="texhtml"><i>h</i><sub>Hg</sub></span> = 0.36 cm. For water on paraffin at 25 °C, <span class="texhtml mvar" style="font-style:italic;">γ</span> = 72 dyn/cm, <span class="texhtml mvar" style="font-style:italic;">ρ</span> = 1.0 g/cm<sup>3</sup>, and <span class="texhtml mvar" style="font-style:italic;">θ</span> = 107° which gives <span class="texhtml"><i>h</i><sub>H<sub>2</sub>O</sub></span> = 0.44 cm. </p><p> The formula also predicts that when the contact angle is 0°, the liquid will spread out into a micro-thin layer over the surface. Such a surface is said to be fully wettable by the liquid.</p><div style="clear:right;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Breakup_of_streams_into_drops">Breakup of streams into drops</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=17" title="Edit section: Breakup of streams into drops"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dripping_faucet_2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Dripping_faucet_2.jpg/170px-Dripping_faucet_2.jpg" decoding="async" width="170" height="227" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Dripping_faucet_2.jpg/255px-Dripping_faucet_2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Dripping_faucet_2.jpg/340px-Dripping_faucet_2.jpg 2x" data-file-width="1704" data-file-height="2272" /></a><figcaption>Breakup of an elongated stream of water into droplets due to surface tension.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Plateau%E2%80%93Rayleigh_instability" title="Plateau–Rayleigh instability">Plateau–Rayleigh instability</a></div> <p>In day-to-day life all of us observe that a stream of water emerging from a faucet will break up into droplets, no matter how smoothly the stream is emitted from the faucet. This is due to a phenomenon called the <a href="/wiki/Plateau%E2%80%93Rayleigh_instability" title="Plateau–Rayleigh instability">Plateau–Rayleigh instability</a>,<sup id="cite_ref-cwp_11-7" class="reference"><a href="#cite_note-cwp-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> which is entirely a consequence of the effects of surface tension. </p><p>The explanation of this instability begins with the existence of tiny perturbations in the stream. These are always present, no matter how smooth the stream is. If the perturbations are resolved into <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> components, we find that some components grow with time while others decay with time. Among those that grow with time, some grow at faster rates than others. Whether a component decays or grows, and how fast it grows is entirely a function of its wave number (a measure of how many peaks and troughs per centimeter) and the radii of the original cylindrical stream. </p> <div class="mw-heading mw-heading3"><h3 id="Gallery">Gallery</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=18" title="Edit section: Gallery"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-traditional center"> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:UnstableLiquidSheet.jpg" class="mw-file-description" title="Breakup of a moving sheet of water bouncing off of a spoon."><img alt="Breakup of a moving sheet of water bouncing off of a spoon." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/UnstableLiquidSheet.jpg/160px-UnstableLiquidSheet.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/UnstableLiquidSheet.jpg/240px-UnstableLiquidSheet.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/UnstableLiquidSheet.jpg/320px-UnstableLiquidSheet.jpg 2x" data-file-width="1280" data-file-height="960" /></a></span></div> <div class="gallerytext">Breakup of a moving sheet of water bouncing off of a spoon.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:SurfaceTension.jpg" class="mw-file-description" title="Photo of flowing water adhering to a hand. Surface tension creates the sheet of water between the flow and the hand."><img alt="Photo of flowing water adhering to a hand. Surface tension creates the sheet of water between the flow and the hand." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/SurfaceTension.jpg/120px-SurfaceTension.jpg" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/SurfaceTension.jpg/180px-SurfaceTension.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/SurfaceTension.jpg/240px-SurfaceTension.jpg 2x" data-file-width="1000" data-file-height="1000" /></a></span></div> <div class="gallerytext">Photo of flowing water adhering to a hand. Surface tension creates the sheet of water between the flow and the hand.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Ggb_in_soap_bubble_1.jpg" class="mw-file-description" title="A soap bubble balances surface tension forces against internal pneumatic pressure."><img alt="A soap bubble balances surface tension forces against internal pneumatic pressure." src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ggb_in_soap_bubble_1.jpg/121px-Ggb_in_soap_bubble_1.jpg" decoding="async" width="121" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ggb_in_soap_bubble_1.jpg/181px-Ggb_in_soap_bubble_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Ggb_in_soap_bubble_1.jpg/242px-Ggb_in_soap_bubble_1.jpg 2x" data-file-width="1536" data-file-height="1524" /></a></span></div> <div class="gallerytext">A soap bubble balances surface tension forces against internal <a href="/wiki/Pneumatic" class="mw-redirect" title="Pneumatic">pneumatic</a> <a href="/wiki/Pressure" title="Pressure">pressure</a>.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:2006-01-15_coin_on_water.jpg" class="mw-file-description" title="Surface tension prevents a coin from sinking: the coin is indisputably denser than water, so it must be displacing a volume greater than its own for buoyancy to balance mass."><img alt="Surface tension prevents a coin from sinking: the coin is indisputably denser than water, so it must be displacing a volume greater than its own for buoyancy to balance mass." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/2006-01-15_coin_on_water.jpg/92px-2006-01-15_coin_on_water.jpg" decoding="async" width="92" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/2006-01-15_coin_on_water.jpg/139px-2006-01-15_coin_on_water.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/2006-01-15_coin_on_water.jpg/185px-2006-01-15_coin_on_water.jpg 2x" data-file-width="672" data-file-height="872" /></a></span></div> <div class="gallerytext">Surface tension prevents a coin from sinking: the coin is indisputably denser than water, so it must be displacing a volume greater than its own for <a href="/wiki/Buoyancy" title="Buoyancy">buoyancy</a> to balance mass.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:3_Moeda_(5).jpg" class="mw-file-description" title="An aluminium coin floats on the surface of the water at 10 °C. Any extra weight would drop the coin to the bottom."><img alt="An aluminium coin floats on the surface of the water at 10 °C. Any extra weight would drop the coin to the bottom." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/3_Moeda_%285%29.jpg/160px-3_Moeda_%285%29.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/3_Moeda_%285%29.jpg/240px-3_Moeda_%285%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/3_Moeda_%285%29.jpg/320px-3_Moeda_%285%29.jpg 2x" data-file-width="1600" data-file-height="1200" /></a></span></div> <div class="gallerytext">An aluminium coin floats on the surface of the water at 10 °C. Any extra weight would drop the coin to the bottom.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Dscn3156-daisy-water_1200x900.jpg" class="mw-file-description" title="A daisy. The entirety of the flower lies below the level of the (undisturbed) free surface. The water rises smoothly around its edge. Surface tension prevents water from displacing the air between the petals and possibly submerging the flower."><img alt="A daisy. The entirety of the flower lies below the level of the (undisturbed) free surface. The water rises smoothly around its edge. Surface tension prevents water from displacing the air between the petals and possibly submerging the flower." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Dscn3156-daisy-water_1200x900.jpg/160px-Dscn3156-daisy-water_1200x900.jpg" decoding="async" width="160" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Dscn3156-daisy-water_1200x900.jpg/240px-Dscn3156-daisy-water_1200x900.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Dscn3156-daisy-water_1200x900.jpg/320px-Dscn3156-daisy-water_1200x900.jpg 2x" data-file-width="1200" data-file-height="900" /></a></span></div> <div class="gallerytext">A daisy. The entirety of the flower lies below the level of the (undisturbed) free surface. The water rises smoothly around its edge. Surface tension prevents water from displacing the air between the petals and possibly submerging the flower.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Surface_Tension_01.jpg" class="mw-file-description" title="A metal paper clip floats on water. Several can usually be carefully added without overflow of water."><img alt="A metal paper clip floats on water. Several can usually be carefully added without overflow of water." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Surface_Tension_01.jpg/160px-Surface_Tension_01.jpg" decoding="async" width="160" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Surface_Tension_01.jpg/240px-Surface_Tension_01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Surface_Tension_01.jpg/320px-Surface_Tension_01.jpg 2x" data-file-width="2967" data-file-height="1938" /></a></span></div> <div class="gallerytext">A metal paper clip floats on water. Several can usually be carefully added without overflow of water.</div> </li> <li class="gallerybox" style="width: 195px"> <div class="thumb" style="width: 190px; height: 150px;"><span typeof="mw:File"><a href="/wiki/File:Paperclip_floating_on_water_(with_%27contour_lines%27).jpg" class="mw-file-description" title="A metal paperclip floating on water. A grille in front of the light has created the 'contour lines' which show the deformation in the water surface caused by the metal paper clip."><img alt="A metal paperclip floating on water. A grille in front of the light has created the 'contour lines' which show the deformation in the water surface caused by the metal paper clip." src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Paperclip_floating_on_water_%28with_%27contour_lines%27%29.jpg/160px-Paperclip_floating_on_water_%28with_%27contour_lines%27%29.jpg" decoding="async" width="160" height="104" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Paperclip_floating_on_water_%28with_%27contour_lines%27%29.jpg/240px-Paperclip_floating_on_water_%28with_%27contour_lines%27%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Paperclip_floating_on_water_%28with_%27contour_lines%27%29.jpg/320px-Paperclip_floating_on_water_%28with_%27contour_lines%27%29.jpg 2x" data-file-width="400" data-file-height="261" /></a></span></div> <div class="gallerytext">A metal paperclip floating on water. A grille in front of the light has created the 'contour lines' which show the deformation in the water surface caused by the metal paper clip.</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Thermodynamics">Thermodynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=19" title="Edit section: Thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Thermodynamic_theories_of_surface_tension">Thermodynamic theories of surface tension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=20" title="Edit section: Thermodynamic theories of surface tension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">J.W. Gibbs</a> developed the thermodynamic theory of capillarity based on the idea of surfaces of discontinuity.<sup id="cite_ref-gibbseq_15-0" class="reference"><a href="#cite_note-gibbseq-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Gibbs considered the case of a sharp mathematical surface being placed somewhere within the microscopically fuzzy physical interface that exists between two homogeneous substances. Realizing that the exact choice of the surface's location was somewhat arbitrary, he left it flexible. Since the interface exists in thermal and chemical equilibrium with the substances around it (having temperature <span class="texhtml mvar" style="font-style:italic;">T</span> and chemical potentials <span class="texhtml"><i>μ</i><sub>i</sub></span>), Gibbs considered the case where the surface may have excess energy, excess entropy, and excess particles, finding the natural free energy function in this case to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U-TS-\mu _{1}N_{1}-\mu _{2}N_{2}\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U-TS-\mu _{1}N_{1}-\mu _{2}N_{2}\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c55ecae42d6f3545cddbf9fa1ddf11db448da4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.302ex; height:2.676ex;" alt="{\displaystyle U-TS-\mu _{1}N_{1}-\mu _{2}N_{2}\cdots }"></span>, a quantity later named as the <a href="/wiki/Grand_potential" title="Grand potential">grand potential</a> and given the symbol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Gibbs_Model.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Gibbs_Model.tif/lossless-page1-220px-Gibbs_Model.tif.png" decoding="async" width="220" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Gibbs_Model.tif/lossless-page1-330px-Gibbs_Model.tif.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Gibbs_Model.tif/lossless-page1-440px-Gibbs_Model.tif.png 2x" data-file-width="967" data-file-height="655" /></a><figcaption>Gibbs' placement of a precise mathematical surface in a fuzzy physical interface.</figcaption></figure> <p>Considering a given subvolume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> containing a surface of discontinuity, the volume is divided by the mathematical surface into two parts A and B, with volumes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a2e1fa8988238b508039e3eb3753d61e317c0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle V_{\text{A}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7c764c5be0d40dc4702e6f836ce2ceeb3f8b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.751ex; height:2.509ex;" alt="{\displaystyle V_{\text{B}}}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=V_{\text{A}}+V_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=V_{\text{A}}+V_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a8a8381fbc948dce93052c0c06465de58cd67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.297ex; height:2.509ex;" alt="{\displaystyle V=V_{\text{A}}+V_{\text{B}}}"></span> exactly. Now, if the two parts A and B were homogeneous fluids (with pressures <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dcec12a7c3d0476dfc339ab0dc8ec438d4bc1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.724ex; height:2.009ex;" alt="{\displaystyle p_{\text{A}}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0166a55453130e77eec754bf58fc663ada4a73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.655ex; height:2.009ex;" alt="{\displaystyle p_{\text{B}}}"></span>) and remained perfectly homogeneous right up to the mathematical boundary, without any surface effects, the total grand potential of this volume would be simply <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d7a81c6cfc1f51befaf2d165924e1fd24b8a57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.419ex; height:2.509ex;" alt="{\displaystyle -p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}}"></span>. The surface effects of interest are a modification to this, and they can be all collected into a surface free energy term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{\text{S}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{\text{S}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8911f2a43b3b296eec0b92cfbf4d104e7a8eca96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.824ex; height:2.509ex;" alt="{\displaystyle \Omega _{\text{S}}}"></span> so the total grand potential of the volume becomes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =-p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}+\Omega _{\text{S}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =-p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}+\Omega _{\text{S}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af5f76734d0eb9cf9c2ec16c8052043c508629ec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.507ex; height:2.509ex;" alt="{\displaystyle \Omega =-p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}+\Omega _{\text{S}}.}"></span> </p><p>For sufficiently macroscopic and gently curved surfaces, the surface free energy must simply be proportional to the surface area:<sup id="cite_ref-gibbseq_15-1" class="reference"><a href="#cite_note-gibbseq-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-landaulifshitz_16-0" class="reference"><a href="#cite_note-landaulifshitz-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{\text{S}}=\gamma A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>S</mtext> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega _{\text{S}}=\gamma A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1d3ad9c6f1db80a2002cb474b1973df49f1bef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.575ex; height:2.676ex;" alt="{\displaystyle \Omega _{\text{S}}=\gamma A,}"></span> for surface tension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> and surface area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. </p><p>As stated above, this implies the mechanical work needed to increase a surface area <i>A</i> is <span class="texhtml"><i>dW</i> = <i>γ dA</i></span>, assuming the volumes on each side do not change. Thermodynamics requires that for systems held at constant chemical potential and temperature, all spontaneous changes of state are accompanied by a decrease in this free energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }"></span>, that is, an increase in total entropy taking into account the possible movement of energy and particles from the surface into the surrounding fluids. From this it is easy to understand why decreasing the surface area of a mass of liquid is always <a href="/wiki/Spontaneous_process" title="Spontaneous process">spontaneous</a>, provided it is not coupled to any other energy changes. It follows that in order to increase surface area, a certain amount of energy must be added. </p><p>Gibbs and other scientists have wrestled with the arbitrariness in the exact microscopic placement of the surface.<sup id="cite_ref-Rusanov2005_17-0" class="reference"><a href="#cite_note-Rusanov2005-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> For microscopic surfaces with very tight curvatures, it is not correct to assume the surface tension is independent of size, and topics like the <a href="/wiki/Tolman_length" title="Tolman length">Tolman length</a> come into play. For a macroscopic-sized surface (and planar surfaces), the surface placement does not have a significant effect on <span class="texhtml mvar" style="font-style:italic;">γ</span>; however, it does have a very strong effect on the values of the surface entropy, surface excess mass densities, and surface internal energy,<sup id="cite_ref-gibbseq_15-2" class="reference"><a href="#cite_note-gibbseq-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 237">: 237 </span></sup> which are the partial derivatives of the surface tension function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (T,\mu _{1},\mu _{2},\cdots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (T,\mu _{1},\mu _{2},\cdots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd7ade7f1bb8dca51ad6228c6c5fd16089f6c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.445ex; height:2.843ex;" alt="{\displaystyle \gamma (T,\mu _{1},\mu _{2},\cdots )}"></span>. </p><p>Gibbs emphasized that for solids, the surface free energy may be completely different from surface stress (what he called surface tension):<sup id="cite_ref-gibbseq_15-3" class="reference"><a href="#cite_note-gibbseq-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 315">: 315 </span></sup> the surface free energy is the work required to <i>form</i> the surface, while surface stress is the work required to <i>stretch</i> the surface. In the case of a two-fluid interface, there is no distinction between forming and stretching because the fluids and the surface completely replenish their nature when the surface is stretched. For a solid, stretching the surface, even elastically, results in a fundamentally changed surface. Further, the surface stress on a solid is a directional quantity (a <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">stress tensor</a>) while surface energy is scalar. </p><p>Fifteen years after Gibbs, <a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">J.D. van der Waals</a> developed the theory of capillarity effects based on the hypothesis of a continuous variation of density.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> He added to the energy density the term <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\nabla \rho )^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\nabla \rho )^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb59ff9b06980ea5b65036442d553015088afcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.655ex; height:3.176ex;" alt="{\displaystyle c(\nabla \rho )^{2},}"></span> where <i>c</i> is the capillarity coefficient and <i>ρ</i> is the density. For the multiphase <i>equilibria</i>, the results of the van der Waals approach practically coincide with the Gibbs formulae, but for modelling of the <i>dynamics</i> of phase transitions the van der Waals approach is much more convenient.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The van der Waals capillarity energy is now widely used in the <a href="/wiki/Phase_field_models" class="mw-redirect" title="Phase field models">phase field models</a> of multiphase flows. Such terms are also discovered in the dynamics of non-equilibrium gases.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Thermodynamics_of_bubbles">Thermodynamics of bubbles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=21" title="Edit section: Thermodynamics of bubbles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The pressure inside an ideal spherical bubble can be derived from thermodynamic free energy considerations.<sup id="cite_ref-landaulifshitz_16-1" class="reference"><a href="#cite_note-landaulifshitz-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The above free energy can be written as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =-\Delta PV_{\text{A}}-p_{\text{B}}V+\gamma A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>V</mi> <mo>+</mo> <mi>γ</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega =-\Delta PV_{\text{A}}-p_{\text{B}}V+\gamma A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4abc63f8f00ff7b32c66423aac23136a5dfb7d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.125ex; height:2.676ex;" alt="{\displaystyle \Omega =-\Delta PV_{\text{A}}-p_{\text{B}}V+\gamma A}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta P=p_{\text{A}}-p_{\text{B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta P=p_{\text{A}}-p_{\text{B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6282060230204c78102a95326386707d8dcbecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.82ex; height:2.509ex;" alt="{\displaystyle \Delta P=p_{\text{A}}-p_{\text{B}}}"></span> is the pressure difference between the inside (A) and outside (B) of the bubble, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a2e1fa8988238b508039e3eb3753d61e317c0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle V_{\text{A}}}"></span> is the bubble volume. In equilibrium, <span class="texhtml"><i>d</i>Ω = 0</span>, and so, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta P\,dV_{\text{A}}=\gamma \,dA.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta P\,dV_{\text{A}}=\gamma \,dA.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b39413a801c52426a49155c28ce6d1a866f8476" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.458ex; height:2.676ex;" alt="{\displaystyle \Delta P\,dV_{\text{A}}=\gamma \,dA.}"></span> </p><p>For a spherical bubble, the volume and surface area are given simply by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{A}}={\tfrac {4}{3}}\pi R^{3}\quad \rightarrow \quad dV_{\text{A}}=4\pi R^{2}\,dR,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="1em" /> <mo stretchy="false">→</mo> <mspace width="1em" /> <mi>d</mi> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{A}}={\tfrac {4}{3}}\pi R^{3}\quad \rightarrow \quad dV_{\text{A}}=4\pi R^{2}\,dR,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a61e28988785f48519b95b034a79303657b1e09d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:36.447ex; height:3.676ex;" alt="{\displaystyle V_{\text{A}}={\tfrac {4}{3}}\pi R^{3}\quad \rightarrow \quad dV_{\text{A}}=4\pi R^{2}\,dR,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=4\pi R^{2}\quad \rightarrow \quad dA=8\pi R\,dR.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mn>4</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="1em" /> <mo stretchy="false">→</mo> <mspace width="1em" /> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>8</mn> <mi>π</mi> <mi>R</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=4\pi R^{2}\quad \rightarrow \quad dA=8\pi R\,dR.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3294342b9cb4058fea48c47363622ccfad7c86a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:32.743ex; height:2.676ex;" alt="{\displaystyle A=4\pi R^{2}\quad \rightarrow \quad dA=8\pi R\,dR.}"></span> </p><p>Substituting these relations into the previous expression, we find <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta P={\frac {2}{R}}\gamma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>R</mi> </mfrac> </mrow> <mi>γ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta P={\frac {2}{R}}\gamma ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903fc25c98d0fba6ea47dc4f374d8ed7a08925d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.289ex; height:5.343ex;" alt="{\displaystyle \Delta P={\frac {2}{R}}\gamma ,}"></span> which is equivalent to the <a href="/wiki/Young%E2%80%93Laplace_equation" title="Young–Laplace equation">Young–Laplace equation</a> when <span class="texhtml"><i>R<sub>x</sub></i> = <i>R<sub>y</sub></i></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Influence_of_temperature">Influence of temperature</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=22" title="Edit section: Influence of temperature"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Temperature_dependence_surface_tension_of_water.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Temperature_dependence_surface_tension_of_water.svg/290px-Temperature_dependence_surface_tension_of_water.svg.png" decoding="async" width="290" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Temperature_dependence_surface_tension_of_water.svg/435px-Temperature_dependence_surface_tension_of_water.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Temperature_dependence_surface_tension_of_water.svg/580px-Temperature_dependence_surface_tension_of_water.svg.png 2x" data-file-width="589" data-file-height="421" /></a><figcaption>Temperature dependence of the surface tension between the liquid and vapor phases of pure water</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SFT-benzene.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4b/SFT-benzene.png/290px-SFT-benzene.png" decoding="async" width="290" height="255" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4b/SFT-benzene.png 1.5x" data-file-width="389" data-file-height="342" /></a><figcaption>Temperature dependency of the surface tension of <a href="/wiki/Benzene" title="Benzene">benzene</a></figcaption></figure> <p>Surface tension is dependent on temperature. For that reason, when a value is given for the surface tension of an interface, temperature must be explicitly stated. The general trend is that surface tension decreases with the increase of temperature, reaching a value of 0 at the <a href="/wiki/Critical_temperature" class="mw-redirect" title="Critical temperature">critical temperature</a>. For further details see <a href="/wiki/E%C3%B6tv%C3%B6s_rule" title="Eötvös rule">Eötvös rule</a>. There are only empirical equations to relate surface tension and temperature: </p> <ul><li>Eötvös:<sup id="cite_ref-phywe_22-0" class="reference"><a href="#cite_note-phywe-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-adam_23-0" class="reference"><a href="#cite_note-adam-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Physical_Properties_Sources_Index_(PPSI)_24-0" class="reference"><a href="#cite_note-Physical_Properties_Sources_Index_(PPSI)-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma V^{2/3}=k(T_{\mathrm {C} }-T).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>−</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma V^{2/3}=k(T_{\mathrm {C} }-T).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89da3a1078f08e11e89924dfff3d40f3515c0ba6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.896ex; height:3.343ex;" alt="{\displaystyle \gamma V^{2/3}=k(T_{\mathrm {C} }-T).}"></span> Here <span class="texhtml mvar" style="font-style:italic;">V</span> is the molar volume of a substance, <span class="texhtml"><i>T</i><sub>C</sub></span> is the <a href="/wiki/Critical_temperature" class="mw-redirect" title="Critical temperature">critical temperature</a> and <span class="texhtml mvar" style="font-style:italic;">k</span> is a constant valid for almost all substances.<sup id="cite_ref-phywe_22-1" class="reference"><a href="#cite_note-phywe-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> A typical value is <span class="texhtml mvar" style="font-style:italic;">k</span> = <span class="nowrap"><span data-sort-value="6993210000000000000♠"></span>2.1<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup> J K<sup>−1</sup> mol<sup>−<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">2</span>⁄<span class="den">3</span></span></sup></span>.<sup id="cite_ref-phywe_22-2" class="reference"><a href="#cite_note-phywe-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Physical_Properties_Sources_Index_(PPSI)_24-1" class="reference"><a href="#cite_note-Physical_Properties_Sources_Index_(PPSI)-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> For water one can further use <span class="texhtml mvar" style="font-style:italic;">V</span> = 18 ml/mol and <span class="texhtml"><i>T</i><sub>C</sub></span> = 647 K (374 °C).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> A variant on Eötvös is described by Ramay and Shields:<sup id="cite_ref-moore_26-0" class="reference"><a href="#cite_note-moore-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma V^{2/3}=k\left(T_{\mathrm {C} }-T-6\,\mathrm {K} \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>−</mo> <mi>T</mi> <mo>−</mo> <mn>6</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma V^{2/3}=k\left(T_{\mathrm {C} }-T-6\,\mathrm {K} \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a656db2d378dd7d2019c5fc48e481007ad5eb6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.835ex; height:3.343ex;" alt="{\displaystyle \gamma V^{2/3}=k\left(T_{\mathrm {C} }-T-6\,\mathrm {K} \right)}"></span> where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures.</li> <li>Guggenheim–Katayama:<sup id="cite_ref-adam_23-1" class="reference"><a href="#cite_note-adam-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma =\gamma ^{\circ }\left(1-{\frac {T}{T_{\mathrm {C} }}}\right)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>=</mo> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma =\gamma ^{\circ }\left(1-{\frac {T}{T_{\mathrm {C} }}}\right)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6477e7c50b61b4263985ca8c29a8b2245ead90a3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.95ex; height:6.176ex;" alt="{\displaystyle \gamma =\gamma ^{\circ }\left(1-{\frac {T}{T_{\mathrm {C} }}}\right)^{n}}"></span> <span class="texhtml"><i>γ</i>°</span> is a constant for each liquid and <span class="texhtml mvar" style="font-style:italic;">n</span> is an empirical factor, whose value is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">11</span><span class="sr-only">/</span><span class="den">9</span></span>⁠</span> for organic liquids. This equation was also proposed by <a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a>, who further proposed that <span class="texhtml"><i>γ</i>°</span> could be given by the expression <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{2}T_{\mathrm {C} }^{1/3}P_{\mathrm {C} }^{2/3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msubsup> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{2}T_{\mathrm {C} }^{1/3}P_{\mathrm {C} }^{2/3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a21cca42de52b5a26c509e313e9888c3c0959bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.612ex; height:3.676ex;" alt="{\displaystyle K_{2}T_{\mathrm {C} }^{1/3}P_{\mathrm {C} }^{2/3},}"></span> where <span class="texhtml"><i>K</i><sub>2</sub></span> is a universal constant for all liquids, and <span class="texhtml"><i>P</i><sub>C</sub></span> is the <a href="/wiki/Critical_pressure" class="mw-redirect" title="Critical pressure">critical pressure</a> of the liquid (although later experiments found <span class="texhtml"><i>K</i><sub>2</sub></span> to vary to some degree from one liquid to another).<sup id="cite_ref-adam_23-2" class="reference"><a href="#cite_note-adam-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup></li></ul> <p>Both Guggenheim–Katayama and Eötvös take into account the fact that surface tension reaches 0 at the critical temperature, whereas Ramay and Shields fails to match reality at this endpoint. </p> <div class="mw-heading mw-heading4"><h4 id="Influence_of_solute_concentration">Influence of solute concentration</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=23" title="Edit section: Influence of solute concentration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Solutes can have different effects on surface tension depending on the nature of the surface and the solute: </p> <ul><li>Little or no effect, for example <a href="/wiki/Sugar" title="Sugar">sugar</a> at water|air, most organic compounds at oil/air</li> <li>Increase surface tension, most <a href="/wiki/Inorganic_compounds" class="mw-redirect" title="Inorganic compounds">inorganic salts</a> at water|air</li> <li>Non-monotonic change, most inorganic acids at water|air</li> <li>Decrease surface tension progressively, as with most amphiphiles, e.g., <a href="/wiki/Alcohols" class="mw-redirect" title="Alcohols">alcohols</a> at water|air</li> <li>Decrease surface tension until certain critical concentration, and no effect afterwards: <a href="/wiki/Surfactants" class="mw-redirect" title="Surfactants">surfactants</a> that form micelles</li></ul> <p>What complicates the effect is that a solute can exist in a different concentration at the surface of a solvent than in its bulk. This difference varies from one solute–solvent combination to another. </p><p><a href="/wiki/Gibbs_isotherm" title="Gibbs isotherm">Gibbs isotherm</a> states that: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =-{\frac {1}{RT}}\left({\frac {\partial \gamma }{\partial \ln C}}\right)_{T,P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>γ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ln</mi> <mo>⁡</mo> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>,</mo> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma =-{\frac {1}{RT}}\left({\frac {\partial \gamma }{\partial \ln C}}\right)_{T,P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebde4ea34134b795127b67825e0df24999e06c36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.732ex; height:6.509ex;" alt="{\displaystyle \Gamma =-{\frac {1}{RT}}\left({\frac {\partial \gamma }{\partial \ln C}}\right)_{T,P}}"></span> </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">Γ</span> is known as surface concentration, it represents excess of solute per unit area of the surface over what would be present if the bulk concentration prevailed all the way to the surface. It has units of mol/m<sup>2</sup></li> <li><span class="texhtml mvar" style="font-style:italic;">C</span> is the concentration of the substance in the bulk solution.</li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a> and <span class="texhtml mvar" style="font-style:italic;">T</span> the <a href="/wiki/Temperature" title="Temperature">temperature</a></li></ul> <p>Certain assumptions are taken in its deduction, therefore Gibbs isotherm can only be applied to ideal (very dilute) solutions with two components. </p> <div class="mw-heading mw-heading4"><h4 id="Influence_of_particle_size_on_vapor_pressure">Influence of particle size on vapor pressure</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=24" title="Edit section: Influence of particle size on vapor pressure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Gibbs%E2%80%93Thomson_effect" class="mw-redirect" title="Gibbs–Thomson effect">Gibbs–Thomson effect</a></div> <p>The <a href="/wiki/Clausius%E2%80%93Clapeyron_relation" title="Clausius–Clapeyron relation">Clausius–Clapeyron relation</a> leads to another equation also attributed to Kelvin, as the <a href="/wiki/Kelvin_equation" title="Kelvin equation">Kelvin equation</a>. It explains why, because of surface tension, the <a href="/wiki/Vapor_pressure" title="Vapor pressure">vapor pressure</a> for small droplets of liquid in suspension is greater than standard vapor pressure of that same liquid when the interface is flat. That is to say that when a liquid is forming small droplets, the equilibrium concentration of its vapor in its surroundings is greater. This arises because the pressure inside the droplet is greater than outside.<sup id="cite_ref-moore_26-1" class="reference"><a href="#cite_note-moore-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\mathrm {v} }^{\mathrm {fog} }=P_{\mathrm {v} }^{\circ }e^{V2\gamma /(RTr_{\mathrm {k} })}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> <mn>2</mn> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>T</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">k</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\mathrm {v} }^{\mathrm {fog} }=P_{\mathrm {v} }^{\circ }e^{V2\gamma /(RTr_{\mathrm {k} })}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39e567174ae5db6002ace305b12c9d80ed0517eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.694ex; height:3.176ex;" alt="{\displaystyle P_{\mathrm {v} }^{\mathrm {fog} }=P_{\mathrm {v} }^{\circ }e^{V2\gamma /(RTr_{\mathrm {k} })}}"></span> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TinyDropletMolecules.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/TinyDropletMolecules.png/220px-TinyDropletMolecules.png" decoding="async" width="220" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/TinyDropletMolecules.png/330px-TinyDropletMolecules.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/TinyDropletMolecules.png/440px-TinyDropletMolecules.png 2x" data-file-width="1066" data-file-height="664" /></a><figcaption><a href="/wiki/Molecule" title="Molecule">Molecules</a> on the surface of a tiny droplet (left) have, on average, fewer neighbors than those on a flat surface (right). Hence they are bound more weakly to the droplet than are flat-surface molecules.</figcaption></figure> <ul><li><span class="texhtml"><i>P</i><sub>v</sub>°</span> is the standard vapor pressure for that liquid at that temperature and pressure.</li> <li><span class="texhtml mvar" style="font-style:italic;">V</span> is the molar volume.</li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a></li> <li><span class="texhtml"><i>r</i><sub>k</sub></span> is the Kelvin radius, the radius of the droplets.</li></ul> <p>The effect explains <a href="/wiki/Supersaturation" title="Supersaturation">supersaturation</a> of vapors. In the absence of <a href="/wiki/Nucleation" title="Nucleation">nucleation</a> sites, tiny droplets must form before they can evolve into larger droplets. This requires a vapor pressure many times the vapor pressure at the <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a> point.<sup id="cite_ref-moore_26-2" class="reference"><a href="#cite_note-moore-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p><p>This equation is also used in <a href="/wiki/Catalyst" class="mw-redirect" title="Catalyst">catalyst</a> chemistry to assess <a href="/wiki/Mesoporous_material" title="Mesoporous material">mesoporosity</a> for solids.<sup id="cite_ref-Handbook_27-0" class="reference"><a href="#cite_note-Handbook-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The effect can be viewed in terms of the average number of molecular neighbors of surface molecules (see diagram). </p><p>The table shows some calculated values of this effect for water at different drop sizes: </p> <table class="toccolours" border="1" style="float: center; margin: 0 0 1em 1em; border-collapse: collapse;"> <tbody><tr> <th style="text-align:center; background:#c0c0f0;" colspan="5"><span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>P</i></span><span class="sr-only">/</span><span class="den"><i>P</i><sub>0</sub></span></span>⁠</span></span> for water drops of different radii at <a href="/wiki/Standard_temperature_and_pressure" title="Standard temperature and pressure">STP</a><sup id="cite_ref-adam_23-3" class="reference"><a href="#cite_note-adam-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </th></tr> <tr style="text-align:center;"> <td style="width:120px;">Droplet radius (nm) </td> <td style="width:120px;">1000 </td> <td style="width:120px;">100 </td> <td style="width:120px;">10 </td> <td style="width:120px;">1 </td></tr> <tr style="text-align:center;"> <td><span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>P</i></span><span class="sr-only">/</span><span class="den"><i>P</i><sub>0</sub></span></span>⁠</span></span></td> <td style="text-align:center;">1.001</td> <td style="text-align:center;">1.011</td> <td style="text-align:center;">1.114</td> <td style="text-align:center;">2.95 </td></tr></tbody></table> <p>The effect becomes clear for very small drop sizes, as a drop of 1 nm radius has about 100 molecules inside, which is a quantity small enough to require a <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> analysis. </p> <div class="mw-heading mw-heading2"><h2 id="Methods_of_measurement">Methods of measurement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=25" title="Edit section: Methods of measurement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Because surface tension manifests itself in various effects, it offers a number of paths to its measurement. Which method is optimal depends upon the nature of the liquid being measured, the conditions under which its tension is to be measured, and the stability of its surface when it is deformed. An instrument that measures surface tension is called <a href="/wiki/Tensiometer_(surface_tension)" title="Tensiometer (surface tension)">tensiometer.</a> </p> <ul><li><a href="/wiki/Du_No%C3%BCy_ring_method" title="Du Noüy ring method">Du Noüy ring method</a>: The traditional method used to measure surface or interfacial tension. Wetting properties of the surface or interface have little influence on this measuring technique. Maximum pull exerted on the ring by the surface is measured.<sup id="cite_ref-phywe_22-3" class="reference"><a href="#cite_note-phywe-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Wilhelmy_plate" title="Wilhelmy plate">Wilhelmy plate method</a>: A universal method especially suited to check surface tension over long time intervals. A vertical plate of known perimeter is attached to a balance, and the force due to wetting is measured.</li> <li><a href="/wiki/Spinning_drop_method" title="Spinning drop method">Spinning drop method</a>: This technique is ideal for measuring low interfacial tensions. The diameter of a drop within a heavy phase is measured while both are rotated.</li> <li><a href="/wiki/Pendant_drop_method" class="mw-redirect" title="Pendant drop method">Pendant drop method</a>: Surface and interfacial tension can be measured by this technique, even at elevated temperatures and pressures. Geometry of a drop is analyzed optically. For pendant drops the maximum diameter and the ratio between this parameter and the diameter at the distance of the maximum diameter from the drop apex has been used to evaluate the size and shape parameters in order to determine surface tension.</li> <li><a href="/wiki/Bubble_pressure_method" class="mw-redirect" title="Bubble pressure method">Bubble pressure method</a> (Jaeger's method): A measurement technique for determining surface tension at short surface ages. Maximum pressure of each bubble is measured.</li> <li>Drop volume method: A method for determining interfacial tension as a function of interface age. Liquid of one density is pumped into a second liquid of a different density and time between drops produced is measured.</li> <li>Capillary rise method: The end of a capillary is immersed into the solution. The height at which the solution reaches inside the capillary is related to the surface tension by the equation <a href="#Liquid_in_a_vertical_tube">discussed above</a>.<sup id="cite_ref-calvert_28-0" class="reference"><a href="#cite_note-calvert-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Stalagmometric_method" title="Stalagmometric method">Stalagmometric method</a>: A method of weighting and reading a drop of liquid.</li> <li><a href="/wiki/Sessile_drop_method" class="mw-redirect" title="Sessile drop method">Sessile drop method</a>: A method for determining surface tension and <a href="/wiki/Density" title="Density">density</a> by placing a drop on a substrate and measuring the <a href="/wiki/Contact_angle" title="Contact angle">contact angle</a> (see <a href="/wiki/Sessile_drop_technique" title="Sessile drop technique">Sessile drop technique</a>).<sup id="cite_ref-dp_29-0" class="reference"><a href="#cite_note-dp-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Du_No%C3%BCy%E2%80%93Padday_method" title="Du Noüy–Padday method">Du Noüy–Padday method</a>: A minimized version of Du Noüy method uses a small diameter metal needle instead of a ring, in combination with a high sensitivity microbalance to record maximum pull. The advantage of this method is that very small sample volumes (down to few tens of microliters) can be measured with very high precision, without the need to correct for <a href="/wiki/Buoyancy" title="Buoyancy">buoyancy</a> (for a needle or rather, rod, with proper geometry). Further, the measurement can be performed very quickly, minimally in about 20 seconds.</li> <li>Vibrational frequency of levitated drops: The natural frequency of vibrational oscillations of magnetically levitated drops has been used to measure the surface tension of superfluid <sup>4</sup>He. This value is estimated to be 0.375 dyn/cm at <span class="texhtml mvar" style="font-style:italic;">T</span> = 0 K.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup></li> <li>Resonant oscillations of spherical and hemispherical liquid drop: The technique is based on measuring the resonant frequency of spherical and hemispherical pendant droplets driven in oscillations by a modulated electric field. The surface tension and viscosity can be evaluated from the obtained resonant curves.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></li> <li>Drop-bounce method: This method is based on aerodynamic levitation with a split-able nozzle design. After dropping a stably levitated droplet onto a platform, the sample deforms and bounces back, oscillating in mid-air as it tries to minimize its surface area. Through this oscillation behavior, the liquid's surface tension and viscosity can be measured.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Values">Values</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=26" title="Edit section: Values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Data_table">Data table</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=27" title="Edit section: Data table"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" style="margin:0; text-align: right;"> <caption style="background:#C0C0F0; border: 1px solid #AAA">Surface tension of various liquids in <a href="/wiki/Dyne" title="Dyne">dyn</a>/cm against air<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><br />Mixture compositions denoted "%" are by mass<br /> dyn/cm is equivalent to the <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a> of mN/m (millinewton per meter) </caption> <tbody><tr> <th>Liquid</th> <th>Temperature (°C)</th> <th>Surface tension, <span class="texhtml mvar" style="font-style:italic;">γ</span> </th></tr> <tr> <td><a href="/wiki/Acetic_acid_(data_page)" title="Acetic acid (data page)">Acetic acid</a></td> <td>20</td> <td>27.60 </td></tr> <tr> <td>Acetic acid (45.1%) + Water</td> <td>30</td> <td>40.68 </td></tr> <tr> <td>Acetic acid (10.0%) + Water</td> <td>30</td> <td>54.56 </td></tr> <tr> <td><a href="/wiki/Acetone_(data_page)" title="Acetone (data page)">Acetone</a></td> <td>20</td> <td>23.70 </td></tr> <tr> <td><a href="/wiki/Benzene" title="Benzene">Benzene</a></td> <td>20</td> <td>28.88 </td></tr> <tr> <td><a href="/wiki/Blood" title="Blood">Blood</a></td> <td>22</td> <td>55.89 </td></tr> <tr> <td><a href="/wiki/Butyl_acetate" title="Butyl acetate">Butyl acetate</a></td> <td>20</td> <td>25.09 </td></tr> <tr> <td><a href="/wiki/Butyric_acid" title="Butyric acid">Butyric acid</a></td> <td>20</td> <td>26.51 </td></tr> <tr> <td><a href="/wiki/Carbon_tetrachloride" title="Carbon tetrachloride">Carbon tetrachloride</a></td> <td>25</td> <td>26.43 </td></tr> <tr> <td><a href="/wiki/Chloroform" title="Chloroform">Chloroform</a></td> <td>25</td> <td>26.67 </td></tr> <tr> <td><a href="/wiki/Diethyl_ether_(data_page)" title="Diethyl ether (data page)">Diethyl ether</a></td> <td>20</td> <td>17.00 </td></tr> <tr> <td><a href="/wiki/Diethylene_glycol" title="Diethylene glycol">Diethylene glycol</a></td> <td>20</td> <td>30.09 </td></tr> <tr> <td><a href="/wiki/Dimethyl_sulfoxide" title="Dimethyl sulfoxide">Dimethyl sulfoxide</a></td> <td>20</td> <td>43.54 </td></tr> <tr> <td><a href="/wiki/Ethanol_(data_page)" title="Ethanol (data page)">Ethanol</a></td> <td>20</td> <td>22.27 </td></tr> <tr> <td>Ethanol (40%) + Water</td> <td>25</td> <td>29.63 </td></tr> <tr> <td>Ethanol (11.1%) + Water</td> <td>25</td> <td>46.03 </td></tr> <tr> <td><a href="/wiki/Ethylene_glycol" title="Ethylene glycol">Ethylene glycol</a></td> <td>25</td> <td>47.3 </td></tr> <tr> <td><a href="/wiki/Glycerol_(data_page)" title="Glycerol (data page)">Glycerol</a></td> <td>20</td> <td>63.00 </td></tr> <tr> <td><a href="/wiki/Heptane" title="Heptane">Heptane</a></td> <td>20</td> <td>20.14 </td></tr> <tr> <td><a href="/wiki/Hexane_(data_page)" title="Hexane (data page)"><i>n</i>-Hexane</a></td> <td>20</td> <td>18.40 </td></tr> <tr> <td><a href="/wiki/Hydrochloric_acid" title="Hydrochloric acid">Hydrochloric acid</a> 17.7 <a href="/wiki/Molar_solution" class="mw-redirect" title="Molar solution">M</a> aqueous solution</td> <td>20</td> <td>65.95 </td></tr> <tr> <td><a href="/wiki/Isopropanol_(data_page)" class="mw-redirect" title="Isopropanol (data page)">Isopropanol</a></td> <td>20</td> <td>21.70 </td></tr> <tr> <td><a href="/wiki/Superfluid_helium-4" title="Superfluid helium-4">Liquid helium II</a></td> <td>−273</td> <td>0.37<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Mercury_(element)" title="Mercury (element)">Mercury</a></td> <td>20</td> <td>486.5 </td></tr> <tr> <td><a href="/wiki/Nitrogen" title="Nitrogen">Liquid nitrogen</a></td> <td>−196</td> <td>8.85 </td></tr> <tr> <td><a href="/wiki/Nonane" title="Nonane">Nonane</a></td> <td>20</td> <td>22.85 </td></tr> <tr> <td><a href="/wiki/Oxygen" title="Oxygen">Liquid oxygen</a></td> <td>−182</td> <td>13.2 </td></tr> <tr> <td><a href="/wiki/Mercury_(element)" title="Mercury (element)">Mercury</a></td> <td>15</td> <td>487.00 </td></tr> <tr> <td><a href="/wiki/Methanol_(data_page)" title="Methanol (data page)">Methanol</a></td> <td>20</td> <td>22.60 </td></tr> <tr> <td><a href="/wiki/Methylene_iodide" class="mw-redirect" title="Methylene iodide">Methylene iodide</a></td> <td>20</td> <td>67.00 </td></tr> <tr> <td>Molten <a href="/wiki/Silver_chloride" title="Silver chloride">Silver chloride</a></td> <td>650</td> <td>163<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td>Molten <a href="/wiki/Sodium_chloride" title="Sodium chloride">Sodium chloride</a>/<a href="/wiki/Calcium_chloride" title="Calcium chloride">Calcium chloride</a> (47/53 mole %)</td> <td>650</td> <td>139<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </td></tr> <tr> <td><a href="/wiki/Octane" title="Octane"><i>n</i>-Octane</a></td> <td>20</td> <td>21.80 </td></tr> <tr> <td><a href="/wiki/Propionic_acid" title="Propionic acid">Propionic acid</a></td> <td>20</td> <td>26.69 </td></tr> <tr> <td><a href="/wiki/Propylene_carbonate" title="Propylene carbonate">Propylene carbonate</a></td> <td>20</td> <td>41.1 </td></tr> <tr> <td><a href="/wiki/Sodium_chloride" title="Sodium chloride">Sodium chloride</a> 6.0 <a href="/wiki/Molar_solution" class="mw-redirect" title="Molar solution">M</a> aqueous solution</td> <td>20</td> <td>82.55 </td></tr> <tr> <td><a href="/wiki/Sodium_chloride" title="Sodium chloride">Sodium chloride</a> (molten)</td> <td>1073</td> <td>115 </td></tr> <tr> <td><a href="/wiki/Sucrose" title="Sucrose">Sucrose</a> (55%) + water</td> <td>20</td> <td>76.45 </td></tr> <tr> <td><a href="/wiki/Water_(data_page)" title="Water (data page)">Water</a></td> <td>0</td> <td>75.64 </td></tr> <tr> <td>Water</td> <td>25</td> <td>71.97 </td></tr> <tr> <td>Water</td> <td>50</td> <td>67.91 </td></tr> <tr> <td>Water</td> <td>100</td> <td>58.85 </td></tr> <tr> <td><a href="/wiki/Toluene" title="Toluene">Toluene</a></td> <td>25</td> <td>27.73 </td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Surface_tension_of_water">Surface tension of water</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=28" title="Edit section: Surface tension of water"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The surface tension of pure liquid water in contact with its vapor has been given by IAPWS<sup id="cite_ref-IAPWS_40-0" class="reference"><a href="#cite_note-IAPWS-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> as </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\text{w}}=235.8\left(1-{\frac {T}{T_{\text{C}}}}\right)^{1.256}\left[1-0.625\left(1-{\frac {T}{T_{\text{C}}}}\right)\right]~{\text{mN/m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>w</mtext> </mrow> </msub> <mo>=</mo> <mn>235.8</mn> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1.256</mn> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>0.625</mn> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>C</mtext> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext>mN/m</mtext> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\text{w}}=235.8\left(1-{\frac {T}{T_{\text{C}}}}\right)^{1.256}\left[1-0.625\left(1-{\frac {T}{T_{\text{C}}}}\right)\right]~{\text{mN/m}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b65193820b2a29652f19d30a1adecdf0c13c8ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:57.989ex; height:6.509ex;" alt="{\displaystyle \gamma _{\text{w}}=235.8\left(1-{\frac {T}{T_{\text{C}}}}\right)^{1.256}\left[1-0.625\left(1-{\frac {T}{T_{\text{C}}}}\right)\right]~{\text{mN/m}},}"></span> </p><p>where both <span class="texhtml mvar" style="font-style:italic;">T</span> and the critical temperature <span class="texhtml"><i>T</i><sub>C</sub></span> = 647.096 K are expressed in <a href="/wiki/Kelvin" title="Kelvin">kelvins</a>. The region of validity the entire vapor–liquid saturation curve, from the triple point (0.01 °C) to the critical point. It also provides reasonable results when extrapolated to metastable (supercooled) conditions, down to at least −25 °C. This formulation was originally adopted by IAPWS in 1976 and was adjusted in 1994 to conform to the International Temperature Scale of 1990. </p><p>The uncertainty of this formulation is given over the full range of temperature by IAPWS.<sup id="cite_ref-IAPWS_40-1" class="reference"><a href="#cite_note-IAPWS-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> For temperatures below 100 °C, the uncertainty is ±0.5%. </p> <div class="mw-heading mw-heading3"><h3 id="Surface_tension_of_seawater">Surface tension of seawater</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=29" title="Edit section: Surface tension of seawater"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nayar et al.<sup id="cite_ref-Nayar_41-0" class="reference"><a href="#cite_note-Nayar-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> published reference data for the surface tension of seawater over the salinity range of <span class="nowrap">20 ≤ <span class="texhtml mvar" style="font-style:italic;">S</span> ≤ 131 g/kg</span> and a temperature range of <span class="nowrap">1 ≤ <span class="texhtml mvar" style="font-style:italic;">t</span> ≤ 92 °C</span> at atmospheric pressure. The range of temperature and salinity encompasses both the oceanographic range and the range of conditions encountered in thermal <a href="/wiki/Desalination" title="Desalination">desalination</a> technologies. The uncertainty of the measurements varied from 0.18 to 0.37 mN/m with the average uncertainty being 0.22 mN/m. </p><p>Nayar et al. correlated the data with the following equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{\mathrm {sw} }=\gamma _{\mathrm {w} }\left(1+3.766\times 10^{-4}S+2.347\times 10^{-6}St\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>3.766</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </msup> <mi>S</mi> <mo>+</mo> <mn>2.347</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>6</mn> </mrow> </msup> <mi>S</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{\mathrm {sw} }=\gamma _{\mathrm {w} }\left(1+3.766\times 10^{-4}S+2.347\times 10^{-6}St\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0ccbe090d3d6ea9f9e8ff2e80891579dab3fb3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:47.78ex; height:3.343ex;" alt="{\displaystyle \gamma _{\mathrm {sw} }=\gamma _{\mathrm {w} }\left(1+3.766\times 10^{-4}S+2.347\times 10^{-6}St\right)}"></span> where <span class="texhtml"><i>γ</i><sub>sw</sub></span> is the surface tension of seawater in mN/m, <span class="texhtml"><i>γ</i><sub>w</sub></span> is the surface tension of water in mN/m, <span class="texhtml mvar" style="font-style:italic;">S</span> is the reference salinity<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> in g/kg, and <span class="texhtml mvar" style="font-style:italic;">t</span> is temperature in degrees Celsius. The average absolute percentage deviation between measurements and the correlation was 0.19% while the maximum deviation is 0.60%. </p><p>The International Association for the Properties of Water and Steam (IAPWS) has adopted this correlation as an international standard guideline.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=30" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Agnes_Pockels" title="Agnes Pockels">Agnes Pockels</a>—early surface sciences researcher</li> <li><a href="/wiki/Anti-fog" title="Anti-fog">Anti-fog</a></li> <li><a href="/wiki/Capillary_wave" title="Capillary wave">Capillary wave</a>—short waves on a water surface, governed by surface tension and inertia</li> <li><a href="/wiki/Cheerio_effect" class="mw-redirect" title="Cheerio effect">Cheerio effect</a>—the tendency for small wettable floating objects to attract one another</li> <li><a href="/wiki/Cohesion_(chemistry)" title="Cohesion (chemistry)">Cohesion</a></li> <li>Dimensionless numbers <ul><li><a href="/wiki/E%C3%B6tv%C3%B6s_number" title="Eötvös number">Bond number or Eötvös number</a></li> <li><a href="/wiki/Capillary_number" title="Capillary number">Capillary number</a></li> <li><a href="/wiki/Marangoni_number" title="Marangoni number">Marangoni number</a></li> <li><a href="/wiki/Weber_number" title="Weber number">Weber number</a></li></ul></li> <li><a href="/wiki/Dortmund_Data_Bank" title="Dortmund Data Bank">Dortmund Data Bank</a>—contains experimental temperature-dependent surface tensions</li> <li><a href="/wiki/Electrodipping_force" title="Electrodipping force">Electrodipping force</a></li> <li><a href="/wiki/Electrowetting" title="Electrowetting">Electrowetting</a></li> <li><a href="/wiki/Electrocapillarity" title="Electrocapillarity">Electrocapillarity</a></li> <li><a href="/wiki/E%C3%B6tv%C3%B6s_rule" title="Eötvös rule">Eötvös rule</a>—a rule for predicting surface tension dependent on temperature</li> <li><a href="/wiki/Hydrostatic_equilibrium" title="Hydrostatic equilibrium">Hydrostatic equilibrium</a>—the effect of gravity pulling matter into a round shape</li> <li><a href="/wiki/Interface_(chemistry)" class="mw-redirect" title="Interface (chemistry)">Interface (chemistry)</a></li> <li><a href="/wiki/Meniscus_(liquid)" title="Meniscus (liquid)">Meniscus</a>—surface curvature formed by a liquid in a container</li> <li><a href="/wiki/Mercury_beating_heart" title="Mercury beating heart">Mercury beating heart</a>—a consequence of inhomogeneous surface tension</li> <li><a href="/wiki/Microfluidics" title="Microfluidics">Microfluidics</a></li> <li><a href="/wiki/Sessile_drop_technique" title="Sessile drop technique">Sessile drop technique</a></li> <li><a href="/wiki/Sow-Hsin_Chen" title="Sow-Hsin Chen">Sow-Hsin Chen</a></li> <li><a href="/wiki/Specific_surface_energy" class="mw-redirect" title="Specific surface energy">Specific surface energy</a>—same as surface tension in isotropic materials.</li> <li><a href="/wiki/Spinning_drop_method" title="Spinning drop method">Spinning drop method</a></li> <li><a href="/wiki/Stalagmometric_method" title="Stalagmometric method">Stalagmometric method</a></li> <li><a href="/wiki/Pressure#Surface_pressure_and_surface_tension" title="Pressure">Surface pressure</a></li> <li><a href="/wiki/Surface_science" title="Surface science">Surface science</a></li> <li><a href="/wiki/Surface_tension_biomimetics" title="Surface tension biomimetics">Surface tension biomimetics</a></li> <li><a href="/wiki/Surface_tension_values" class="mw-redirect" title="Surface tension values">Surface tension values</a></li> <li><a href="/wiki/Surfactant" title="Surfactant">Surfactants</a>—substances which reduce surface tension.</li> <li><a href="/wiki/Szyszkowski_equation" title="Szyszkowski equation">Szyszkowski equation</a>—calculating surface tension of aqueous solutions</li> <li><a href="/wiki/Tears_of_wine" title="Tears of wine">Tears of wine</a>—the surface tension induced phenomenon seen on the sides of glasses containing alcoholic beverages.</li> <li><a href="/wiki/Tolman_length" title="Tolman length">Tolman length</a>—leading term in correcting the surface tension for curved surfaces.</li> <li><a href="/wiki/Wetting" title="Wetting">Wetting</a> and <a href="/wiki/Dewetting" title="Dewetting">dewetting</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Explanatory_notes">Explanatory notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=31" title="Edit section: Explanatory notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">In a <a href="/wiki/Mercury_barometer" class="mw-redirect" title="Mercury barometer">mercury barometer</a>, the upper liquid surface is an interface between the liquid and a vacuum containing some molecules of evaporated liquid.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=32" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.sciencedirect.com/topics/earth-and-planetary-sciences/surface-tension#:~:text=surface%20tension%20results,adhesion">"Surface Tension - an overview | ScienceDirect 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H (November 2014). <a rel="nofollow" class="external text" href="http://dspace.mit.edu/bitstream/1721.1/96884/1/Surface_Tension_of_Seawater.pdf">"Surface tension of seawater"</a> <span class="cs1-format">(PDF)</span>. <i>J. Phys. Chem. Ref. Data</i>. <b>43</b> (4): 43103. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014JPCRD..43d3103N">2014JPCRD..43d3103N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.4899037">10.1063/1.4899037</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F96884">1721.1/96884</a></span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220922033641/http://dspace.mit.edu/bitstream/handle/1721.1/96884/Surface_Tension_of_Seawater.pdf;jsessionid=68E9601570E59C9B5BA69E912FF21A4A?sequence=1">Archived</a> from the original on 2022-09-22<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-04-20</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Phys.+Chem.+Ref.+Data&rft.atitle=Surface+tension+of+seawater&rft.volume=43&rft.issue=4&rft.pages=43103&rft.date=2014-11&rft_id=info%3Ahdl%2F1721.1%2F96884&rft_id=info%3Adoi%2F10.1063%2F1.4899037&rft_id=info%3Abibcode%2F2014JPCRD..43d3103N&rft.aulast=Nayar&rft.aufirst=K.+G&rft.au=Panchanathan%2C+D&rft.au=McKinley%2C+G.+H&rft.au=Lienhard%2C+J.+H&rft_id=http%3A%2F%2Fdspace.mit.edu%2Fbitstream%2F1721.1%2F96884%2F1%2FSurface_Tension_of_Seawater.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurface+tension" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilleroFeistelWrightMcDougall2008" class="citation journal cs1">Millero, Frank J; Feistel, Rainer; Wright, Daniel G; McDougall, Trevor J (January 2008). "The composition of Standard Seawater and the definition of the Reference-Composition Salinity Scale". <i>Deep-Sea Research Part I</i>. <b>55</b> (1): 50. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008DSRI...55...50M">2008DSRI...55...50M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.dsr.2007.10.001">10.1016/j.dsr.2007.10.001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Deep-Sea+Research+Part+I&rft.atitle=The+composition+of+Standard+Seawater+and+the+definition+of+the+Reference-Composition+Salinity+Scale&rft.volume=55&rft.issue=1&rft.pages=50&rft.date=2008-01&rft_id=info%3Adoi%2F10.1016%2Fj.dsr.2007.10.001&rft_id=info%3Abibcode%2F2008DSRI...55...50M&rft.aulast=Millero&rft.aufirst=Frank+J&rft.au=Feistel%2C+Rainer&rft.au=Wright%2C+Daniel+G&rft.au=McDougall%2C+Trevor+J&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurface+tension" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation techreport cs1"><a rel="nofollow" class="external text" href="http://www.iapws.org/relguide/Seawater-Surf.pdf"><i>Guideline on the Surface Tension of Seawater</i></a> <span class="cs1-format">(PDF)</span> (Technical report). International Association for the Properties of Water and Steam. October 2019. IAPWS G14-19. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200326232003/http://www.iapws.org/relguide/Seawater-Surf.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2020-03-26<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-03-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Guideline+on+the+Surface+Tension+of+Seawater&rft.pub=International+Association+for+the+Properties+of+Water+and+Steam&rft.date=2019-10&rft_id=http%3A%2F%2Fwww.iapws.org%2Frelguide%2FSeawater-Surf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurface+tension" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=33" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Michael_Berry_(physicist)" title="Michael Berry (physicist)">Berry, M V</a> (1971-03-01). "The molecular mechanism of surface tension". <i>Physics Education</i>. <b>6</b> (2): 79–84. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0031-9120%2F6%2F2%2F001">10.1088/0031-9120/6/2/001</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0031-9120">0031-9120</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarchandTerzi1971" class="citation journal cs1">Marchand, Antonin; Terzi, Mariana (March 1971). "Molten salts mixture surface tension". <i>The Journal of Chemical Thermodynamics</i>. <b>3</b> (2): 259–265. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0021-9614%2871%2980111-8">10.1016/S0021-9614(71)80111-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Chemical+Thermodynamics&rft.atitle=Molten+salts+mixture+surface+tension&rft.volume=3&rft.issue=2&rft.pages=259-265&rft.date=1971-03&rft_id=info%3Adoi%2F10.1016%2FS0021-9614%2871%2980111-8&rft.aulast=Marchand&rft.aufirst=Antonin&rft.au=Terzi%2C+Mariana&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASurface+tension" class="Z3988"></span></li> <li>Marchand, Antonin; Weijs, Joost H.; Snoeijer, Jacco H.; Andreotti, Bruno (2011-09-26). "Why is surface tension a force parallel to the interface?". <i>American Journal of Physics</i>. <b>79</b> (10): 999–1008. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3619866">10.1119/1.3619866</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://www.worldcat.org/search?fq=x0:jrnl&q=n2:0002-9505">0002-9505</a>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/abs/1211">abs/1211</a>.S. Sternberg</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Surface_tension&action=edit&section=34" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Surface_tension" class="extiw" title="commons:Surface tension"><span style="font-style:italic; font-weight:bold;">Surface tension</span></a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://physics.stackexchange.com/a/150853/81224">"Why is surface tension parallel to the interface?"</a>. <i>Physics Stack Exchange</i>. Retrieved 2021-03-19.3854</li> <li><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/surten.html">On surface tension and interesting real-world cases</a></li> <li><a rel="nofollow" class="external text" href="http://www.kayelaby.npl.co.uk/general_physics/2_2/2_2_5.html">Surface Tensions of Various Liquids</a></li> <li><a rel="nofollow" class="external text" href="http://ddbonline.ddbst.de/DIPPR106SFTCalculation/DIPPR106SFTCalculationCGI.exe">Calculation of temperature-dependent surface tensions for some common components</a></li> <li><a rel="nofollow" class="external text" href="http://www.aim.env.uea.ac.uk/aim/surftens/surftens.php">Surface tension calculator for aqueous solutions</a> containing the ions H<sup>+</sup>, <span class="chemf nowrap">NH<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">4</sub></span></span></span>, Na<sup>+</sup>, K<sup>+</sup>, Mg<sup>2+</sup>, Ca<sup>2+</sup>, <span class="chemf nowrap">SO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">4</sub></span></span></span>, <span class="chemf nowrap">NO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sub></span></span></span>, Cl<sup>−</sup>, <span class="chemf nowrap">CO<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">2−</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">3</sub></span></span></span>, Br<sup>−</sup> and OH<sup>−</sup>.</li> <li><a href="/wiki/T._Proctor_Hall" title="T. Proctor Hall">T. Proctor Hall</a> (1893) <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/122437#page/399/mode/1up">New methods of measuring surface tension in liquids</a>, <a href="/wiki/Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a> (series 5, 36: 385–415), link from <a href="/wiki/Biodiversity_Heritage_Library" title="Biodiversity Heritage Library">Biodiversity Heritage Library</a>.</li> <li><a rel="nofollow" class="external text" href="http://www.magnet.fsu.edu/education/community/slideshows/bubblewall/index.html">The Bubble Wall</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged March 2022">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup> (Audio slideshow from the National High Magnetic Field Laboratory explaining cohesion, surface tension and hydrogen bonds)</li> <li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/Interface_free_energy">C. Pfister: Interface Free Energy. Scholarpedia 2010</a> (from first principles of statistical mechanics)</li> <li><a rel="nofollow" class="external text" href="http://perminc.com/resources/fundamentals-of-fluid-flow-in-porous-media/chapter-2-the-porous-medium/multi-phase-saturated-rock-properties/surface-interfacial-tension/">Surface and Interfacial Tension</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output 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