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id="toc-Characteristics-sublist" class="vector-toc-list"> <li id="toc-Heterogeneous_nucleation_often_dominates_homogeneous_nucleation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heterogeneous_nucleation_often_dominates_homogeneous_nucleation"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Heterogeneous nucleation often dominates homogeneous nucleation</span> </div> </a> <ul id="toc-Heterogeneous_nucleation_often_dominates_homogeneous_nucleation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computer_simulation_studies_of_simple_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Computer_simulation_studies_of_simple_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Computer simulation studies of simple models</span> </div> </a> <ul id="toc-Computer_simulation_studies_of_simple_models-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_spinodal_region" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_spinodal_region"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>The spinodal region</span> </div> </a> <ul id="toc-The_spinodal_region-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_nucleation_of_crystals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_nucleation_of_crystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>The nucleation of crystals</span> </div> </a> <button aria-controls="toc-The_nucleation_of_crystals-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 The nucleation of crystals subsection</span> </button> <ul id="toc-The_nucleation_of_crystals-sublist" class="vector-toc-list"> <li id="toc-Primary_and_secondary_nucleation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primary_and_secondary_nucleation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Primary and secondary nucleation</span> </div> </a> <ul id="toc-Primary_and_secondary_nucleation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experimental_observations_on_the_nucleation_times_for_the_crystallisation_of_small_volumes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Experimental_observations_on_the_nucleation_times_for_the_crystallisation_of_small_volumes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Experimental observations on the nucleation times for the crystallisation of small volumes</span> </div> </a> <ul id="toc-Experimental_observations_on_the_nucleation_times_for_the_crystallisation_of_small_volumes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ice"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Ice</span> </div> </a> <ul id="toc-Ice-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-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 Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Nucleation_of_fluids_(gases_and_liquids)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nucleation_of_fluids_(gases_and_liquids)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Nucleation of fluids (gases and liquids)</span> </div> </a> <ul id="toc-Nucleation_of_fluids_(gases_and_liquids)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nucleation_of_crystals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nucleation_of_crystals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Nucleation of crystals</span> </div> </a> <ul id="toc-Nucleation_of_crystals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nucleation_in_solids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nucleation_in_solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Nucleation in solids</span> </div> </a> <ul id="toc-Nucleation_in_solids-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>References</span> </div> </a> <ul id="toc-References-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 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title="Nucleació – Catalan" lang="ca" hreflang="ca" data-title="Nucleació" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Keimbildung" title="Keimbildung – German" lang="de" hreflang="de" data-title="Keimbildung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Nucleaci%C3%B3n" title="Nucleación – Spanish" lang="es" hreflang="es" data-title="Nucleación" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D8%B3%D8%AA%D9%87%E2%80%8C%D8%B2%D8%A7%DB%8C%DB%8C_(%D8%B4%DB%8C%D9%85%DB%8C)" title="هسته‌زایی (شیمی) – Persian" lang="fa" hreflang="fa" data-title="هسته‌زایی (شیمی)" 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/Nucl%C3%A9ation" title="Nucléation – French" lang="fr" hreflang="fr" data-title="Nucléation" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Nucleaci%C3%B3n" title="Nucleación – Galician" lang="gl" hreflang="gl" data-title="Nucleación" 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%95%B5_%EC%83%9D%EC%84%B1" title="핵 생성 – Korean" lang="ko" hreflang="ko" data-title="핵 생성" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Nukleasi" title="Nukleasi – Indonesian" lang="id" hreflang="id" data-title="Nukleasi" 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/Nucleazione" title="Nucleazione – Italian" lang="it" hreflang="it" data-title="Nucleazione" 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%94%D7%AA%D7%92%D7%A8%D7%A2%D7%A0%D7%95%D7%AA" title="התגרענות – Hebrew" lang="he" hreflang="he" data-title="התגרענות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Nucleatie" title="Nucleatie – Dutch" lang="nl" hreflang="nl" data-title="Nucleatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%A0%B8%E7%94%9F%E6%88%90" title="核生成 – Japanese" lang="ja" hreflang="ja" data-title="核生成" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Nukleacja" title="Nukleacja – Polish" lang="pl" hreflang="pl" data-title="Nukleacja" 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/Nuclea%C3%A7%C3%A3o" title="Nucleação – Portuguese" lang="pt" hreflang="pt" data-title="Nucleação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Nuclea%C8%9Bie" title="Nucleație – Romanian" lang="ro" hreflang="ro" data-title="Nucleație" 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%93%D0%B5%D1%82%D0%B5%D1%80%D0%BE%D0%B3%D0%B5%D0%BD%D0%BD%D0%B0%D1%8F_%D0%BD%D1%83%D0%BA%D0%BB%D0%B5%D0%B0%D1%86%D0%B8%D1%8F" title="Гетерогенная нуклеация – Russian" lang="ru" hreflang="ru" data-title="Гетерогенная нуклеация" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%82%E0%B7%8A%E0%B6%A7%E0%B7%93%E0%B6%9A%E0%B6%BB%E0%B6%AB" title="න්‍යෂ්ටීකරණ – Sinhala" lang="si" hreflang="si" data-title="න්‍යෂ්ටීකරණ" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Nukleaatio" title="Nukleaatio – Finnish" lang="fi" hreflang="fi" data-title="Nukleaatio" 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thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</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:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</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:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><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=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><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 \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><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 \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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></td><td style="border-top:solid 1px black;"><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 \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><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 \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><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(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><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 H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><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(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><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 G(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</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:#ddf;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/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Other</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a class="mw-selflink selflink">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></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:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <b>nucleation</b> is the first step in the formation of either a new <a href="/wiki/Phase_(matter)" title="Phase (matter)">thermodynamic phase</a> or <a href="/wiki/Crystal_structure" title="Crystal structure">structure</a> via <a href="/wiki/Self-assembly" title="Self-assembly">self-assembly</a> or <a href="/wiki/Self-organization" title="Self-organization">self-organization</a> within a substance or <a href="/wiki/Mixture" title="Mixture">mixture</a>. Nucleation is typically defined to be the process that determines how long an observer has to wait before the new phase or self-organized structure appears. For example, if a volume of <a href="/wiki/Water" title="Water">water</a> is cooled (at <a href="/wiki/Atmospheric_pressure" title="Atmospheric pressure">atmospheric pressure</a>) significantly below 0<span class="nowrap"> </span>°C, it will tend to <a href="/wiki/Freezing" title="Freezing">freeze</a> into <a href="/wiki/Ice" title="Ice">ice</a>, but volumes of water cooled only a few degrees below 0<span class="nowrap"> </span>°C often stay completely free of ice for long periods (<a href="/wiki/Supercooling" title="Supercooling">supercooling</a>). At these conditions, nucleation of ice is either slow or does not occur at all. However, at lower temperatures nucleation is fast, and ice <a href="/wiki/Crystal" title="Crystal">crystals</a> appear after little or no delay.<sup id="cite_ref-pruppacher_book_1-0" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sear_review_2-0" class="reference"><a href="#cite_note-sear_review-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Nucleation is a common mechanism which generates first-order <a href="/wiki/Phase_transition" title="Phase transition">phase transitions</a>, and it is the start of the process of forming a new thermodynamic phase. In contrast, new phases at continuous phase transitions start to form immediately. </p><p>Nucleation is often very sensitive to <a href="/wiki/Impurity" class="mw-redirect" title="Impurity">impurities</a> in the system. These impurities may be too small to be seen by the naked eye, but still can control the rate of nucleation. Because of this, it is often important to distinguish between heterogeneous nucleation and homogeneous nucleation. Heterogeneous nucleation occurs at <i>nucleation sites</i> on surfaces in the system.<sup id="cite_ref-pruppacher_book_1-1" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Homogeneous nucleation occurs away from a surface. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Characteristics">Characteristics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=1" title="Edit section: Characteristics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv/220px--Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="165" data-durationhint="12" data-mwtitle="Nucleation_of_the_equilibrium_phase_(red)_from_a_metastable_phase_(white)_in_the_Ising_model.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Nucleation_of_the_equilibrium_phase_(red)_from_a_metastable_phase_(white)_in_the_Ising_model.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.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/9/92/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/9/92/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv" type="video/ogg; codecs="theora"" data-width="320" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/92/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv/Nucleation_of_the_equilibrium_phase_%28red%29_from_a_metastable_phase_%28white%29_in_the_Ising_model.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="192" data-height="144" /></video></span><figcaption>Nucleation at a surface (black) in the 2D <a href="/wiki/Ising_model" title="Ising model">Ising model</a>.<sup id="cite_ref-sear_review14_3-0" class="reference"><a href="#cite_note-sear_review14-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Up spins (particles in lattice-gas terminology) shown in red, down spins shown in white.</figcaption></figure> <p>Nucleation is usually a <a href="/wiki/Stochastic" title="Stochastic">stochastic</a> (random) process, so even in two identical systems nucleation will occur at different times.<sup id="cite_ref-pruppacher_book_1-2" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sear_review_2-1" class="reference"><a href="#cite_note-sear_review-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sear_review14_3-1" class="reference"><a href="#cite_note-sear_review14-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kreer_beckerdöring_4-0" class="reference"><a href="#cite_note-kreer_beckerdöring-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A common mechanism is illustrated in the animation to the right. This shows nucleation of a new phase (shown in red) in an existing phase (white). In the existing phase microscopic fluctuations of the red phase appear and decay continuously, until an unusually large fluctuation of the new red phase is so large it is more favourable for it to grow than to shrink back to nothing. This nucleus of the red phase then grows and converts the system to this phase. The standard theory that describes this behaviour for the nucleation of a new thermodynamic phase is called <a href="/wiki/Classical_nucleation_theory" title="Classical nucleation theory">classical nucleation theory</a>. However, the CNT fails in describing experimental results of vapour to liquid nucleation even for model substances like argon by several orders of magnitude.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>For nucleation of a new thermodynamic phase, such as the formation of ice in water below 0<span class="nowrap"> </span>°C, if the system is not evolving with time and nucleation occurs in one step, then the probability that nucleation has <i>not</i> occurred should undergo <a href="/wiki/Exponential_decay" title="Exponential decay">exponential decay</a>. This is seen for example in the nucleation of ice in <a href="/wiki/Supercooled" class="mw-redirect" title="Supercooled">supercooled</a> small water droplets.<sup id="cite_ref-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The decay rate of the exponential gives the nucleation rate. <a href="/wiki/Classical_nucleation_theory" title="Classical nucleation theory">Classical nucleation theory</a> is a widely used approximate theory for estimating these rates, and how they vary with variables such as temperature. It correctly predicts that the time you have to wait for nucleation decreases extremely rapidly when <a href="/wiki/Supersaturation" title="Supersaturation">supersaturated</a>.<sup id="cite_ref-pruppacher_book_1-3" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sear_review_2-2" class="reference"><a href="#cite_note-sear_review-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-kreer_beckerdöring_4-1" class="reference"><a href="#cite_note-kreer_beckerdöring-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>It is not just new phases such as liquids and crystals that form via nucleation followed by growth. The self-assembly process that forms objects like the <a href="/wiki/Amyloid" title="Amyloid">amyloid</a> aggregates associated with <a href="/wiki/Alzheimer%27s_disease" title="Alzheimer's disease">Alzheimer's disease</a> also starts with nucleation.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Energy consuming self-organising systems such as the <a href="/wiki/Microtubules" class="mw-redirect" title="Microtubules">microtubules</a> in cells also show <a href="/wiki/Microtubule_nucleation" title="Microtubule nucleation">nucleation</a> and growth. </p> <div class="mw-heading mw-heading3"><h3 id="Heterogeneous_nucleation_often_dominates_homogeneous_nucleation">Heterogeneous nucleation often dominates homogeneous nucleation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=2" title="Edit section: Heterogeneous nucleation often dominates homogeneous nucleation"><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_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>Three nuclei on a surface, illustrating decreasing contact angles. The <a href="/wiki/Contact_angle" title="Contact angle">contact angle</a> the nucleus surface makes with the solid horizontal surface decreases from left to right. The surface area of the nucleus decreases as the contact angle decreases. This geometrical effect reduces the barrier in <a href="/wiki/Classical_nucleation_theory" title="Classical nucleation theory">classical nucleation theory</a> and hence results in faster nucleation on surfaces with smaller contact angles. Also, if instead of the surface being flat it curves towards fluid, then this also reduces the interfacial area and so the nucleation barrier.</figcaption></figure> <p>Heterogeneous nucleation, nucleation with the nucleus at a surface, is much more common than homogeneous nucleation.<sup id="cite_ref-pruppacher_book_1-4" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sear_review14_3-2" class="reference"><a href="#cite_note-sear_review14-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> For example, in the nucleation of ice from supercooled water droplets, purifying the water to remove all or almost all impurities results in water droplets that freeze below around −35<span class="nowrap"> </span>°C,<sup id="cite_ref-pruppacher_book_1-5" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-sear_review14_3-3" class="reference"><a href="#cite_note-sear_review14-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_6-1" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> whereas water that contains impurities may freeze at −5<span class="nowrap"> </span>°C or warmer.<sup id="cite_ref-pruppacher_book_1-6" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>This observation that heterogeneous nucleation can occur when the rate of homogeneous nucleation is essentially zero, is often understood using <a href="/wiki/Classical_nucleation_theory" title="Classical nucleation theory">classical nucleation theory</a>. This predicts that the nucleation slows exponentially with the height of a <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">free energy</a> <a href="/wiki/Activation_energy#Relationship_with_Gibbs_energy" title="Activation energy">barrier</a> ΔG*. This barrier comes from the free energy penalty of forming the surface of the growing nucleus. For homogeneous nucleation the nucleus is approximated by a sphere, but as we can see in the schematic of macroscopic droplets to the right, droplets on surfaces are not complete spheres and so the area of the interface between the droplet and the surrounding fluid is less than a sphere's <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 4\pi r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b81fcce302776a01dc66fc186a1ce0a616b4d772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.597ex; height:2.676ex;" alt="{\displaystyle 4\pi r^{2}}"></span>. This reduction in surface area of the nucleus reduces the height of the barrier to nucleation and so speeds nucleation up exponentially.<sup id="cite_ref-sear_review_2-3" class="reference"><a href="#cite_note-sear_review-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Nucleation can also start at the surface of a liquid. For example, computer simulations of <a href="/wiki/Gold_nanoparticle" class="mw-redirect" title="Gold nanoparticle">gold nanoparticles</a> show that the crystal phase sometimes nucleates at the liquid-gold surface.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Computer_simulation_studies_of_simple_models">Computer simulation studies of simple models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=3" title="Edit section: Computer simulation studies of simple models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Classical_nucleation_theory" title="Classical nucleation theory">Classical nucleation theory</a> makes a number of assumptions, for example it treats a microscopic nucleus as if it is a macroscopic droplet with a well-defined surface whose free energy is estimated using an equilibrium property: the interfacial tension σ. For a nucleus that may be only of order ten molecules across it is not always clear that we can treat something so small as a volume plus a surface. Also nucleation is an inherently out of <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> phenomenon so it is not always obvious that its rate can be estimated using equilibrium properties. </p><p>However, modern computers are powerful enough to calculate essentially exact nucleation rates for simple models. These have been compared with the classical theory, for example for the case of nucleation of the crystal phase in the model of hard spheres. This is a model of perfectly hard spheres in thermal motion, and is a simple model of some <a href="/wiki/Colloids" class="mw-redirect" title="Colloids">colloids</a>. For the crystallization of hard spheres the classical theory is a very reasonable approximate theory.<sup id="cite_ref-auer04_9-0" class="reference"><a href="#cite_note-auer04-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> So for the simple models we can study, classical nucleation theory works quite well, but we do not know if it works equally well for (say) complex molecules crystallising out of solution. </p> <div class="mw-heading mw-heading3"><h3 id="The_spinodal_region">The spinodal region</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=4" title="Edit section: The spinodal region"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phase-transition processes can also be explained in terms of <a href="/wiki/Spinodal_decomposition" title="Spinodal decomposition">spinodal decomposition</a>, where phase separation is delayed until the system enters the unstable region where a small perturbation in composition leads to a decrease in energy and, thus, spontaneous growth of the perturbation.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This region of a phase diagram is known as the spinodal region and the phase separation process is known as spinodal decomposition and may be governed by the <a href="/wiki/Cahn%E2%80%93Hilliard_equation" title="Cahn–Hilliard equation">Cahn–Hilliard equation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="The_nucleation_of_crystals">The nucleation of crystals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=5" title="Edit section: The nucleation of crystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In many cases, liquids and solutions can be cooled down or concentrated up to conditions where the liquid or solution is significantly less thermodynamically stable than the crystal, but where no crystals will form for minutes, hours, weeks or longer; this process is called <a href="/wiki/Supercooling" title="Supercooling">supercooling</a>. Nucleation of the crystal is then being prevented by a substantial barrier. This has consequences, for example cold high altitude clouds may contain large numbers of small liquid water droplets that are far below 0<span class="nowrap"> </span>°C. </p><p> In small volumes, such as in small droplets, only one nucleation event may be needed for crystallisation. In these small volumes, the time until the first crystal appears is usually defined to be the nucleation time. Calcium carbonate crystal nucleation depends not only on degree of supersaturation but also the ratio of calcium to carbonate ions in aqueous solutions.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In larger volumes many nucleation events will occur. A simple model for crystallisation in that case, that combines nucleation and growth is the <a href="/wiki/Avrami_equation" title="Avrami equation">KJMA or Avrami model</a>. </p><figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rock-Candy-Closeup.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Rock-Candy-Closeup.jpg/170px-Rock-Candy-Closeup.jpg" decoding="async" width="170" height="262" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Rock-Candy-Closeup.jpg/255px-Rock-Candy-Closeup.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Rock-Candy-Closeup.jpg/340px-Rock-Candy-Closeup.jpg 2x" data-file-width="1790" data-file-height="2760" /></a><figcaption>When sugar is <a href="/wiki/Supersaturation" title="Supersaturation">supersaturated</a> in water, nucleation will occur, allowing sugar molecules to stick together and form large crystal structures.</figcaption></figure><p>Although the existing theories including the classical nucleation theory explain well the steady nucleation state when the crystal nucleation rate is not time dependent, the initial non-steady state transient nucleation,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> and even more mysterious incubation period, require more attention of the scientific community. Chemical ordering of the undercooling liquid prior to crystal nucleation was suggested to be responsible for that<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> feature by reducing the energy barrier for nucleation.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading3"><h3 id="Primary_and_secondary_nucleation">Primary and secondary nucleation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=6" title="Edit section: Primary and secondary nucleation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The time until the appearance of the first crystal is also called primary nucleation time, to distinguish it from secondary nucleation times. Primary here refers to the first nucleus to form, while secondary nuclei are crystal nuclei produced from a preexisting crystal. Primary nucleation describes the transition to a new phase that does not rely on the new phase already being present, either because it is the very first nucleus of that phase to form, or because the nucleus forms far from any pre-existing piece of the new phase. Particularly in the study of crystallisation, secondary nucleation can be important. This is the formation of nuclei of a new crystal directly caused by pre-existing crystals.<sup id="cite_ref-botsaris1976_15-0" class="reference"><a href="#cite_note-botsaris1976-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>For example, if the crystals are in a solution and the system is subject to shearing forces, small crystal nuclei could be sheared off a growing crystal, thus increasing the number of crystals in the system. So both primary and secondary nucleation increase the number of crystals in the system but their mechanisms are very different, and secondary nucleation relies on crystals already being present. </p> <div class="mw-heading mw-heading3"><h3 id="Experimental_observations_on_the_nucleation_times_for_the_crystallisation_of_small_volumes">Experimental observations on the nucleation times for the crystallisation of small volumes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=7" title="Edit section: Experimental observations on the nucleation times for the crystallisation of small volumes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is typically difficult to experimentally study the nucleation of crystals. The nucleus is microscopic, and thus too small to be directly observed. In large liquid volumes there are typically multiple nucleation events, and it is difficult to disentangle the effects of nucleation from those of growth of the nucleated phase. These problems can be overcome by working with small droplets. As nucleation is <a href="/wiki/Stochastic" title="Stochastic">stochastic</a>, many droplets are needed so that statistics for the nucleation events can be obtained. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen,_as_a_function_of_time(png).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen%2C_as_a_function_of_time%28png%29.png/220px-Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen%2C_as_a_function_of_time%28png%29.png" decoding="async" width="220" height="151" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen%2C_as_a_function_of_time%28png%29.png/330px-Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen%2C_as_a_function_of_time%28png%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen%2C_as_a_function_of_time%28png%29.png/440px-Fraction_of_a_set_of_supercooled_liquid_tin_droplets_that_have_not_frozen%2C_as_a_function_of_time%28png%29.png 2x" data-file-width="872" data-file-height="599" /></a><figcaption>The black triangles are the fraction of a large set of small supercooled liquid tin droplets that are still liquid, i.e., where the crystal state has not nucleated, as a function of time. The data are from Pound and La Mer (1952). The red curve is a fit of a function of the Gompertz form to these data.</figcaption></figure><p> To the right is shown an example set of nucleation data. It is for the nucleation at constant temperature and hence supersaturation of the crystal phase in small droplets of supercooled liquid tin; this is the work of Pound and La Mer.<sup id="cite_ref-pound52_16-0" class="reference"><a href="#cite_note-pound52-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>Nucleation occurs in different droplets at different times, hence the fraction is not a simple step function that drops sharply from one to zero at one particular time. The red curve is a fit of a <a href="/wiki/Gompertz_function" title="Gompertz function">Gompertz function</a> to the data. This is a simplified version of the model Pound and La Mer used to model their data.<sup id="cite_ref-pound52_16-1" class="reference"><a href="#cite_note-pound52-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> The model assumes that nucleation occurs due to impurity particles in the liquid tin droplets, and it makes the simplifying assumption that all impurity particles produce nucleation at the same rate. It also assumes that these particles are <a href="/wiki/Poisson_distribution" title="Poisson distribution">Poisson distributed</a> among the liquid tin droplets. The fit values are that the nucleation rate due to a single impurity particle is 0.02/s, and the average number of impurity particles per droplet is 1.2. Note that about 30% of the tin droplets never freeze; the data plateaus at a fraction of about 0.3. Within the model this is assumed to be because, by chance, these droplets do not have even one impurity particle and so there is no heterogeneous nucleation. Homogeneous nucleation is assumed to be negligible on the timescale of this experiment. The remaining droplets freeze in a stochastic way, at rates 0.02/s if they have one impurity particle, 0.04/s if they have two, and so on. </p><p>These data are just one example, but they illustrate common features of the nucleation of crystals in that there is clear evidence for heterogeneous nucleation, and that nucleation is clearly stochastic. </p> <div class="mw-heading mw-heading3"><h3 id="Ice">Ice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=8" title="Edit section: Ice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> The freezing of small water droplets to ice is an important process, particularly in the formation and dynamics of clouds.<sup id="cite_ref-pruppacher_book_1-7" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Water (at atmospheric pressure) does not freeze at 0<span class="nowrap"> </span>°C, but rather at temperatures that tend to decrease as the volume of the water decreases and as the concentration of dissolved chemicals in the water increases.<sup id="cite_ref-pruppacher_book_1-8" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png/220px-Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png/330px-Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png/440px-Survival_curve_34.5_micrometre_water_droplets_1950_NACA_Dorsch_%26_Hacker_ice_nucleation.png 2x" data-file-width="4000" data-file-height="3000" /></a><figcaption>Survival curve for water droplets 34.5 μm in diameter. Blue circles are data, and the red curve is a fit of a <a href="/wiki/Gumbel_distribution" title="Gumbel distribution">Gumbel distribution</a>.</figcaption></figure><p> Thus small droplets of water, as found in clouds, may remain liquid far below 0<span class="nowrap"> </span>°C. </p><p>An example of experimental data on the freezing of small water droplets is shown at the right. The plot shows the fraction of a large set of water droplets, that are still liquid water, i.e., have not yet frozen, as a function of temperature. Note that the highest temperature at which any of the droplets freezes is close to -19<span class="nowrap"> </span>°C, while the last droplet to freeze does so at almost -35<span class="nowrap"> </span>°C.<sup id="cite_ref-dorsch50_17-0" class="reference"><a href="#cite_note-dorsch50-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=9" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Nucleation_of_fluids_(gases_and_liquids)"><span id="Nucleation_of_fluids_.28gases_and_liquids.29"></span>Nucleation of fluids (gases and liquids)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=10" title="Edit section: Nucleation of fluids (gases and liquids)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Nucleation_finger.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Nucleation_finger.jpg/250px-Nucleation_finger.jpg" decoding="async" width="250" height="186" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Nucleation_finger.jpg/375px-Nucleation_finger.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Nucleation_finger.jpg/500px-Nucleation_finger.jpg 2x" data-file-width="1030" data-file-height="768" /></a><figcaption>Nucleation of carbon dioxide bubbles around a finger</figcaption></figure> <ul><li><a href="/wiki/Clouds" class="mw-redirect" title="Clouds">Clouds</a> form when wet air cools (often <a href="/wiki/Lapse_rate" title="Lapse rate">because the air rises</a>) and many small water droplets nucleate from the supersaturated air.<sup id="cite_ref-pruppacher_book_1-9" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The amount of water vapour that air can carry <a href="/wiki/Relative_humidity" class="mw-redirect" title="Relative humidity">decreases with lower temperatures</a>. The excess vapor begins to nucleate and to form small water droplets which form a cloud. Nucleation of the droplets of liquid water is heterogeneous, occurring on particles referred to as <a href="/wiki/Cloud_condensation_nuclei" title="Cloud condensation nuclei">cloud condensation nuclei</a>. <a href="/wiki/Cloud_seeding" title="Cloud seeding">Cloud seeding</a> is the process of adding artificial condensation nuclei to quicken the formation of clouds.</li> <li>Bubbles of <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a> nucleate shortly after the pressure is released from a container of <a href="/wiki/Carbonation" title="Carbonation">carbonated</a> liquid.</li> <li>Nucleation in <a href="/wiki/Boiling" title="Boiling">boiling</a> can occur in the bulk <a href="/wiki/Liquid" title="Liquid">liquid</a> if the pressure is reduced so that the liquid becomes <a href="/wiki/Superheating" title="Superheating">superheated</a> with respect to the pressure-dependent boiling point. More often, nucleation occurs on the heating surface, at nucleation sites. Typically, nucleation sites are tiny crevices where free gas-liquid surface is maintained or spots on the heating surface with lower <a href="/wiki/Wetting" title="Wetting">wetting</a> properties. Substantial superheating of a liquid can be achieved after the liquid is de-gassed and if the heating surfaces are clean, smooth and made of materials well wetted by the liquid.</li> <li>Some <a href="/wiki/Champagne_stirrer" class="mw-redirect" title="Champagne stirrer">champagne stirrers</a> operate by providing many nucleation sites via high surface-area and sharp corners, speeding the release of bubbles and removing carbonation from the wine.</li> <li>The <a href="/wiki/Diet_Coke_and_Mentos_eruption" class="mw-redirect" title="Diet Coke and Mentos eruption">Diet Coke and Mentos eruption</a> offers another example. The surface of Mentos candy provides nucleation sites for the formation of carbon-dioxide bubbles from carbonated soda.</li> <li>Both the <a href="/wiki/Bubble_chamber" title="Bubble chamber">bubble chamber</a> and the <a href="/wiki/Cloud_chamber" title="Cloud chamber">cloud chamber</a> rely on nucleation, of bubbles and droplets, respectively.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Nucleation_of_crystals">Nucleation of crystals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=11" title="Edit section: Nucleation of crystals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The most common <a href="/wiki/Crystallisation" class="mw-redirect" title="Crystallisation">crystallisation</a> process on Earth is the formation of ice. Liquid water does not freeze at 0<span class="nowrap"> </span>°C unless there is ice already present; cooling significantly below 0<span class="nowrap"> </span>°C is required to nucleate ice and for the water to freeze. For example, small droplets of very pure water can remain liquid down to below -30 °C although ice is the stable state below 0<span class="nowrap"> </span>°C.<sup id="cite_ref-pruppacher_book_1-10" class="reference"><a href="#cite_note-pruppacher_book-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>Many of the materials we make and use are crystalline, but are made from liquids, e.g. crystalline iron made from liquid iron cast into a mold, so the nucleation of crystalline materials is widely studied in industry.<sup id="cite_ref-Kelton2010_18-0" class="reference"><a href="#cite_note-Kelton2010-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> It is used heavily in the chemical industry for cases such as in the preparation of metallic ultradispersed powders that can serve as catalysts. For example, platinum deposited onto TiO<sub>2</sub> <a href="/wiki/Nanoparticle" title="Nanoparticle">nanoparticles</a> catalyses the decomposition of water.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> It is an important factor in the semiconductor industry, as the band gap energy in semiconductors is influenced by the size of nanoclusters.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Nucleation_in_solids">Nucleation in solids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Nucleation&action=edit&section=12" title="Edit section: Nucleation in solids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to the nucleation and growth of crystals e.g. in non-crystalline glasses, the nucleation and growth of impurity precipitates in crystals at, and between, grain boundaries is quite important industrially. For example in metals solid-state nucleation and precipitate growth plays an important role e.g. in modifying mechanical properties like ductility, while in semiconductors it plays an important role e.g. in trapping impurities during integrated circuit manufacture.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <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=Nucleation&action=edit&section=13" 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-pruppacher_book-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-pruppacher_book_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-pruppacher_book_1-10"><sup><i><b>k</b></i></sup></a></span> <span class="reference-text">H. R. Pruppacher and J. D. 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