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subsection </button> <ul id="toc-Extensions_and_modifications-sublist" class="vector-toc-list"> <li id="toc-Lennard-Jones_truncated_&_shifted_(LJTS)_potential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lennard-Jones_truncated_&_shifted_(LJTS)_potential"> <div class="vector-toc-text"> 5.1 Lennard-Jones truncated & shifted (LJTS) potential </div> </a> <ul id="toc-Lennard-Jones_truncated_&_shifted_(LJTS)_potential-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 6 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Lennard-Jones_substance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lennard-Jones_substance"> <div class="vector-toc-text"> 6.1 Lennard-Jones substance </div> </a> <ul id="toc-Lennard-Jones_substance-sublist" class="vector-toc-list"> <li id="toc-Thermophysical_properties_of_the_Lennard-Jones_substance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Thermophysical_properties_of_the_Lennard-Jones_substance"> <div class="vector-toc-text"> 6.1.1 Thermophysical properties of the Lennard-Jones substance </div> </a> <ul id="toc-Thermophysical_properties_of_the_Lennard-Jones_substance-sublist" class="vector-toc-list"> <li id="toc-Characteristic_points_and_curves" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Characteristic_points_and_curves"> <div class="vector-toc-text"> 6.1.1.1 Characteristic points and curves </div> </a> <ul id="toc-Characteristic_points_and_curves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_the_Lennard-Jones_fluid" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Properties_of_the_Lennard-Jones_fluid"> <div class="vector-toc-text"> 6.1.1.2 Properties of the Lennard-Jones fluid </div> </a> <ul id="toc-Properties_of_the_Lennard-Jones_fluid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_the_Lennard-Jones_solid" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Properties_of_the_Lennard-Jones_solid"> <div class="vector-toc-text"> 6.1.1.3 Properties of the Lennard-Jones solid </div> </a> <ul id="toc-Properties_of_the_Lennard-Jones_solid-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mixtures_of_Lennard-Jones_substances" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Mixtures_of_Lennard-Jones_substances"> <div class="vector-toc-text"> 6.1.2 Mixtures of Lennard-Jones substances </div> </a> <ul id="toc-Mixtures_of_Lennard-Jones_substances-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equations_of_state" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Equations_of_state"> <div class="vector-toc-text"> 6.1.3 Equations of state </div> </a> <ul id="toc-Equations_of_state-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lennard-Jones_potential_as_building_block_for_force_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lennard-Jones_potential_as_building_block_for_force_fields"> <div class="vector-toc-text"> 6.2 Lennard-Jones potential as building block for force fields </div> </a> <ul id="toc-Lennard-Jones_potential_as_building_block_for_force_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 7 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </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"> 8 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 9 External links </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 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data-title="Potencial de Lennard-Jones" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Lennard-Jones-potentialet" title="Lennard-Jones-potentialet – Danish" lang="da" hreflang="da" data-title="Lennard-Jones-potentialet" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lennard-Jones-Potential" title="Lennard-Jones-Potential – German" lang="de" hreflang="de" data-title="Lennard-Jones-Potential" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lennard-Jonesi_potentsiaal" title="Lennard-Jonesi potentsiaal – Estonian" lang="et" hreflang="et" data-title="Lennard-Jonesi potentsiaal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Potencial_de_Lennard-Jones" title="Potencial de Lennard-Jones – Spanish" lang="es" hreflang="es" data-title="Potencial de Lennard-Jones" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%AA%D8%A7%D9%86%D8%B3%DB%8C%D9%84_%D9%84%D9%86%D8%A7%D8%B1%D8%AF-%D8%AC%D9%88%D9%86%D8%B2" title="پتانسیل لنارد-جونز – Persian" lang="fa" hreflang="fa" data-title="پتانسیل لنارد-جونز" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Potentiel_de_Lennard-Jones" title="Potentiel de Lennard-Jones – French" lang="fr" hreflang="fr" data-title="Potentiel de Lennard-Jones" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Potenziale_di_Lennard-Jones" title="Potenziale di Lennard-Jones – Italian" lang="it" hreflang="it" data-title="Potenziale di Lennard-Jones" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C_%D7%9C%D7%A0%D7%90%D7%A8%D7%93-%D7%92%27%D7%95%D7%A0%D7%A1" title="פוטנציאל לנארד-ג'ונס – Hebrew" lang="he" hreflang="he" data-title="פוטנציאל לנארד-ג'ונס" data-language-autonym="עברית" 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class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Lennard-Jonesov_potencijal" title="Lennard-Jonesov potencijal – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Lennard-Jonesov potencijal" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lennard-Jonesin_potentiaali" title="Lennard-Jonesin potentiaali – Finnish" lang="fi" hreflang="fi" data-title="Lennard-Jonesin potentiaali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Lennard-Jones_potential" title="Lennard-Jones potential – Swedish" lang="sv" hreflang="sv" data-title="Lennard-Jones potential" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Lennard-Jones_potansiyeli" title="Lennard-Jones potansiyeli – Turkish" lang="tr" hreflang="tr" data-title="Lennard-Jones potansiyeli" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB_%D0%9B%D0%B5%D0%BD%D0%BD%D0%B0%D1%80%D0%B4-%D0%94%D0%B6%D0%BE%D0%BD%D1%81%D0%B0" title="Потенціал Леннард-Джонса – Ukrainian" lang="uk" hreflang="uk" data-title="Потенціал Леннард-Джонса" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li 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src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Graph_of_Lennard-Jones_potential.png/320px-Graph_of_Lennard-Jones_potential.png" decoding="async" width="320" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Graph_of_Lennard-Jones_potential.png/480px-Graph_of_Lennard-Jones_potential.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Graph_of_Lennard-Jones_potential.png/640px-Graph_of_Lennard-Jones_potential.png 2x" data-file-width="5726" data-file-height="3567" /></a><figcaption>Graph of the Lennard-Jones potential function: Intermolecular potential energy VLJ as a function of the distance of a pair of particles. The potential minimum is at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{\rm {min}}=2^{1/6}\sigma .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <mi>σ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{\rm {min}}=2^{1/6}\sigma .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da11761197546a01c686681361952cc08d57f7a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.104ex; 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Richtmyer">Richtmyer</a></li></ul></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:Computational_physics" title="Template:Computational physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computational_physics" title="Template talk:Computational physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computational_physics" title="Special:EditPage/Template:Computational physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Computational_chemistry" title="Computational chemistry">computational chemistry</a>, <a href="/wiki/Molecular_physics" title="Molecular physics">molecular physics</a>, and <a href="/wiki/Physical_chemistry" title="Physical chemistry">physical chemistry</a>, the Lennard-Jones potential (also termed the LJ potential or 12-6 potential; named for <a href="/wiki/John_Lennard-Jones" title="John Lennard-Jones">John Lennard-Jones</a>) is an intermolecular <a href="/wiki/Pair_potential" title="Pair potential">pair potential</a>. Out of all the <a href="/wiki/Interatomic_potential" title="Interatomic potential">intermolecular potentials</a>, the Lennard-Jones potential is probably the one that has been the most extensively studied.<a href="#cite_note-FischerWendland-1">[1]</a><a href="#cite_note-:17-2">[2]</a> It is considered an archetype model for simple yet realistic <a href="/wiki/Intermolecular_interaction" class="mw-redirect" title="Intermolecular interaction">intermolecular interactions</a>. The Lennard-Jones potential is often used as a building block in <a href="/wiki/Molecular_modelling" title="Molecular modelling">molecular models</a> (a.k.a. <a href="/wiki/Force_field_(chemistry)" title="Force field (chemistry)">force fields</a>) for more complex substances.<a href="#cite_note-SchwerdtfegerWales-3">[3]</a> Many studies of the idealized "Lennard-Jones substance" use the potential to understand the physical nature of matter. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=1" title="Edit section: Overview">edit</a>]</div> The Lennard-Jones potential is a simple model that still manages to describe the essential features of interactions between simple atoms and molecules: Two interacting particles repel each other at very close distance, attract each other at moderate distance, and eventually stop interacting at infinite distance, as shown in the Figure. The Lennard-Jones potential is a pair potential, i.e. no three- or multi-body interactions are covered by the potential.<a href="#cite_note-SchwerdtfegerWales-3">[3]</a><a href="#cite_note-4">[4]</a> The general Lennard-Jones potential combines a repulsive potential, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/r^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18ba457515b8ce99f32c1baed3278abca54df3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle 1/r^{n}}">, with an attractive potential, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1/r^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1/r^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7665e2fc3ce37a41f486bdd8c217a9df463df17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.857ex; height:2.843ex;" alt="{\displaystyle -1/r^{m}}">, using empirically determined coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a456541580ea1bf26b1e331ff3c2d7a71425c069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.439ex; height:2.509ex;" alt="{\displaystyle B_{m}}">:<a href="#cite_note-LJ_1931-5">[5]</a><a href="#cite_note-:32-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{LJ}}(r)={\frac {A_{n}}{r^{n}}}-{\frac {B_{m}}{r^{m}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>LJ</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{LJ}}(r)={\frac {A_{n}}{r^{n}}}-{\frac {B_{m}}{r^{m}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303e03a9fa8b4a304c399d8cf7c030cd54dd4f02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.976ex; height:5.343ex;" alt="{\displaystyle V_{\text{LJ}}(r)={\frac {A_{n}}{r^{n}}}-{\frac {B_{m}}{r^{m}}}.}"> In his 1931 review<a href="#cite_note-LJ_1931-5">[5]</a> Lennard-Jones suggested using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23d2adac58ea4d0848d7fe553d7f3da00f9a45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=6}"> to match the <a href="/wiki/London_dispersion_force" title="London dispersion force">London dispersion force</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38fb7dfd38b5db65d78bb6f2c1bf9e6e6cd07c14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=12}"> based matching experimental data.<a href="#cite_note-FischerWendland-1">[1]</a> Setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}=4\varepsilon \sigma ^{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>ε</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}=4\varepsilon \sigma ^{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4b7300e0b3b9bd8ec3b06d420999f9d7e192cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.513ex; height:3.009ex;" alt="{\displaystyle A_{n}=4\varepsilon \sigma ^{12}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{m}=4\varepsilon \sigma ^{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mi>ε</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{m}=4\varepsilon \sigma ^{6}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bbe9ac111e8c7c4f345694f3d861b8c1abcee26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.168ex; height:3.009ex;" alt="{\displaystyle B_{m}=4\varepsilon \sigma ^{6}}"> gives the widely used Lennard-Jones 12-6 potential:<a href="#cite_note-WhenToUse-7">[7]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>LJ</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>ε</mi> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82d16f77cae964a5c4c52fb89165dd5d596ee03f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.192ex; height:6.176ex;" alt="{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right],}"> where r is the distance between two interacting particles, ε is the depth of the <a href="/wiki/Potential_well" title="Potential well">potential well</a>, and σ is the distance at which the particle-particle potential energy V is zero. The Lennard-Jones 12-6 potential has its minimum at a distance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r_{\rm {min}}=2^{1/6}\sigma ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> </mrow> </msup> <mi>σ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r_{\rm {min}}=2^{1/6}\sigma ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb226b8a7c821063a18eaa90413f86eb90ae410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.104ex; height:3.176ex;" alt="{\displaystyle r=r_{\rm {min}}=2^{1/6}\sigma ,}"> where the potential energy has the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=-\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=-\varepsilon .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd4f5f5e84d49724da899b3a3c3e8c9f85d67c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.424ex; height:2.343ex;" alt="{\displaystyle V=-\varepsilon .}"> The Lennard-Jones potential is usually the standard choice for the development of theories for <a href="/wiki/Phase_(matter)" title="Phase (matter)">matter</a> (especially soft-matter) as well as for the development and testing of computational methods and algorithms. Numerous <a href="/wiki/Interatomic_potential" title="Interatomic potential">intermolecular potentials</a> have been proposed in the past for the modeling of simple soft repulsive and attractive interactions between spherically symmetric particles, i.e. the general shape shown in the Figure. Examples for other potentials are the <a href="/wiki/Morse_potential" title="Morse potential">Morse potential</a>, the <a href="/wiki/Mie_potential" title="Mie potential">Mie potential</a>,<a href="#cite_note-:30-8">[8]</a> the <a href="/wiki/Buckingham_potential" title="Buckingham potential">Buckingham potential</a> and the Tang-Tönnies potential.<a href="#cite_note-9">[9]</a> While some of these may be more suited to modelling <a href="/wiki/Fluid" title="Fluid">real fluids</a>,<a href="#cite_note-10">[10]</a> the simplicity of the Lennard-Jones potential, as well as its often surprising ability to accurately capture real fluid behavior, has historically made it the <a href="/wiki/Pair_potential" title="Pair potential">pair-potential</a> of greatest general importance.<a href="#cite_note-:2-11">[11]</a> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=2" title="Edit section: History">edit</a>]</div> In 1924, the year that Lennard-Jones received his PhD from <a href="/wiki/Cambridge_University" class="mw-redirect" title="Cambridge University">Cambridge University</a>, he published<a href="#cite_note-:32-6">[6]</a><a href="#cite_note-12">[12]</a> a series of landmark papers on the pair potentials that would ultimately be named for him.<a href="#cite_note-:17-2">[2]</a><a href="#cite_note-SchwerdtfegerWales-3">[3]</a><a href="#cite_note-Lennard-Jonesium-13">[13]</a><a href="#cite_note-FischerWendland-1">[1]</a> In these papers he adjusted the parameters of the potential then using the result in a model of gas viscosity, seeking a set of values consistent with experiment. His initial results suggested a repulsive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=13.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>13.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=13.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20840031cd7695870729300261d1762878cf6779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.627ex; height:2.176ex;" alt="{\displaystyle n=13.5}"> and an attractive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918bb386a7ca6891255b62ef91ccc022883f3809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=3}">. Before Lennard-Jones, back in 1903, <a href="/wiki/Gustav_Mie" title="Gustav Mie">Gustav Mie</a> had worked on effective field theories; <a href="/wiki/Eduard_Gr%C3%BCneisen" title="Eduard Grüneisen">Eduard Grüneisen</a> built on Mie work for solids, showing that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e64e2a4a5b6cb58f1553c6a65551b4898bb82403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle n>m}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m>3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>></mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m>3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9e7bab471b7136372ab52804f1769982f49306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m>3}"> is required for solids. As a result of this work the Lennard-Jones potential is sometimes called the Mie− Grüneisen potential in <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>.<a href="#cite_note-SchwerdtfegerWales-3">[3]</a> In 1930, after the discovery of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="/wiki/Fritz_London" title="Fritz London">Fritz London</a> showed that theory predicts the long-range attractive force should have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c23d2adac58ea4d0848d7fe553d7f3da00f9a45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=6}">. In 1931, Lennard-Jones applied this form of the potential to describe many properties of fluids setting the stage for many subsequent studies.<a href="#cite_note-FischerWendland-1">[1]</a> <div class="mw-heading mw-heading2"><h2 id="Dimensionless_(reduced_units)">Dimensionless (reduced units)</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=3" title="Edit section: Dimensionless (reduced units)">edit</a>]</div> <table class="wikitable" style="float:right"> <caption>dimensionless (reduced) units </caption> <tbody><tr> <th>Property</th> <th>Symbol</th> <th>Reduced form </th></tr> <tr> <td>Length</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc488e611bcc916d2da5dec54181e4909297e088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.103ex; height:2.343ex;" alt="{\displaystyle r^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {r}{\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>σ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {r}{\sigma }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b897a0d216d74e1e8c12e02943ac5e036896b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.166ex; height:4.676ex;" alt="{\displaystyle {\frac {r}{\sigma }}}"> </td></tr> <tr> <td>Time</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c73524c328c099fbfcff931451272a61e74fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t{\sqrt {\frac {\varepsilon }{m\sigma ^{2}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>ε</mi> <mrow> <mi>m</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t{\sqrt {\frac {\varepsilon }{m\sigma ^{2}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7787a57d84b5c6d2909932e10350c2c9db7e59f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.425ex; height:6.176ex;" alt="{\displaystyle t{\sqrt {\frac {\varepsilon }{m\sigma ^{2}}}}}"> </td></tr> <tr> <td>Temperature</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5426ed7abdea7d2ff995ad0e01bc4ca62a273855" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.343ex;" alt="{\displaystyle T^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {k_{B}T}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k_{B}T}{\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc268724a0550ba3a3e3462804409e057e704e68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.163ex; height:5.343ex;" alt="{\displaystyle {\frac {k_{B}T}{\varepsilon }}}"> </td></tr> <tr> <td>Force</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ab6d9b2b48fcfd2f5c741de789391d4cdd0f13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.869ex; height:2.343ex;" alt="{\displaystyle F^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {F\sigma }{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mi>σ</mi> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {F\sigma }{\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e72fec50782249daa620e26c5ce390d7f0aeb884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.907ex; height:5.176ex;" alt="{\displaystyle {\frac {F\sigma }{\varepsilon }}}"> </td></tr> <tr> <td>Energy</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8def9b6d2f0acfca452861c11dc2a08419765efe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.895ex; height:2.343ex;" alt="{\displaystyle U^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {U}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {U}{\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6eedde5f371d391014bffb268736aa153ec6032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.619ex; height:5.176ex;" alt="{\displaystyle {\frac {U}{\varepsilon }}}"> </td></tr> <tr> <td>Pressure</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180f741aa3ec0ac9e97e4777842b446c75bd4fa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.676ex;" alt="{\displaystyle p^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p\sigma ^{3}}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p\sigma ^{3}}{\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfff73d56d55a2760bec95dc7c416d0b2c4a5c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.39ex; height:5.676ex;" alt="{\displaystyle {\frac {p\sigma ^{3}}{\varepsilon }}}"> </td></tr> <tr> <td>Density</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b9afc5b13d9b11ab7cc83f865daa1f7fbee033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.843ex;" alt="{\displaystyle \rho ^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \sigma ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \sigma ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/870d3925b5d7b1920dafce47d66af0e7cbf42773" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.587ex; height:3.176ex;" alt="{\displaystyle \rho \sigma ^{3}}"> </td></tr> <tr> <td>Surface tension</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b94d9c54dd2c77e4571ed51bd635158b22c204a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.334ex; height:2.843ex;" alt="{\displaystyle \gamma ^{*}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\gamma \sigma ^{2}}{\varepsilon }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>γ</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>ε</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\gamma \sigma ^{2}}{\varepsilon }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80e2580ad85b9ecc04191c6e3311feb21f229c4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.483ex; height:5.676ex;" alt="{\displaystyle {\frac {\gamma \sigma ^{2}}{\varepsilon }}}"> </td></tr></tbody></table> Dimensionless reduced units can be defined based on the Lennard-Jones potential parameters, which is convenient for molecular simulations. From a numerical point of view, the advantages of this unit system include computing values which are closer to unity, using simplified equations and being able to easily scale the results.<a href="#cite_note-:28-14">[14]</a><a href="#cite_note-Rapaport2004-15">[15]</a> This reduced units system requires the specification of the size parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> and the energy parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> of the Lennard-Jones potential and the mass of the particle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">. All physical properties can be converted straightforwardly taking the respective dimension into account, see table. The reduced units are often abbreviated and indicated by an asterisk. In general, reduced units can also be built up on other molecular interaction potentials that consist of a length parameter and an energy parameter. <div class="mw-heading mw-heading2"><h2 id="Long-range_interactions">Long-range interactions</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=4" title="Edit section: Long-range interactions">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:ExpemplaricObservable.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/ExpemplaricObservable.png/318px-ExpemplaricObservable.png" decoding="async" width="318" height="204" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/ExpemplaricObservable.png/477px-ExpemplaricObservable.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/ExpemplaricObservable.png/636px-ExpemplaricObservable.png 2x" data-file-width="5637" data-file-height="3624" /></a><figcaption>Illustrative example of the convergence of a correction scheme to account for the long-range interactions of the Lennard-Jones potential. Therein, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> indicates an exemplaric observable and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cf28a6f5f2d25c1c9b419e01571e9f4e8953f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {c} }}"> the applied cut-off radius. The long-range corrected value is indicated as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathrm {corr} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathrm {corr} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b368a280808a05d16f38bde13aa6e4f3131977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.998ex; height:2.509ex;" alt="{\displaystyle X_{\mathrm {corr} }}"> (symbols and line as a guide for the eye); the hypothetical 'true' value as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathrm {true} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathrm {true} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776347661e512f3255eea60593e15203316c8464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.085ex; height:2.509ex;" alt="{\displaystyle X_{\mathrm {true} }}"> (dashed line).</figcaption></figure> The Lennard-Jones potential, cf. Eq. (1) and Figure on the top, has an infinite range. Only under its consideration, the 'true' and 'full' Lennard-Jones potential is examined. For the evaluation of an <a href="/wiki/Observable" title="Observable">observable</a> of an ensemble of particles interacting by the Lennard-Jones potential using molecular simulations, the interactions can only be evaluated explicitly up to a certain distance – simply due to the fact that the number of particles will always be finite. The maximum distance applied in a simulation is usually referred to as 'cut-off' radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8cf28a6f5f2d25c1c9b419e01571e9f4e8953f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {c} }}"> (because the Lennard-Jones potential is radially symmetric). To obtain thermophysical properties (both macroscopic or microscopic) of the 'true' and 'full' Lennard-Jones (LJ) potential, the contribution of the potential beyond the cut-off radius has to be accounted for. Different correction schemes have been developed to account for the influence of the long-range interactions in simulations and to sustain a sufficiently good approximation of the 'full' potential.<a href="#cite_note-:0-16">[16]</a><a href="#cite_note-:28-14">[14]</a> They are based on simplifying assumptions regarding the structure of the fluid. For simple cases, such as in studies of the equilibrium of homogeneous fluids, simple correction terms yield excellent results. In other cases, such as in studies of inhomogeneous systems with different phases, accounting for the long-range interactions is more tedious. These corrections are usually referred to as 'long-range corrections'. For most properties, simple analytical expressions are known and well established. For a given observable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}">, the 'corrected' simulation result <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathrm {corr} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathrm {corr} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b368a280808a05d16f38bde13aa6e4f3131977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.998ex; height:2.509ex;" alt="{\displaystyle X_{\mathrm {corr} }}"> is then simply computed from the actually sampled value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathrm {sampled} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathrm {sampled} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36f922135f131ba5537cc7b7cea07c80f3dd1ef0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.011ex; height:2.843ex;" alt="{\displaystyle X_{\mathrm {sampled} }}"> and the long-range correction value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathrm {lrc} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathrm {lrc} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a5a5dd8e6cb79cbed83bf97922dd3e11c4e0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.989ex; height:2.509ex;" alt="{\displaystyle X_{\mathrm {lrc} }}">, e.g. for the internal energy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\mathrm {corr} }=U_{\mathrm {sampled} }+U_{\mathrm {lrc} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {corr} }=U_{\mathrm {sampled} }+U_{\mathrm {lrc} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0016163aeb770484d070c2776c0504f4a58e558" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.925ex; height:2.843ex;" alt="{\displaystyle U_{\mathrm {corr} }=U_{\mathrm {sampled} }+U_{\mathrm {lrc} }}">.<a href="#cite_note-:28-14">[14]</a> The hypothetical true value of the observable of the Lennard-Jones potential at truly infinite cut-off distance (thermodynamic limit) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathrm {true} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathrm {true} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/776347661e512f3255eea60593e15203316c8464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.085ex; height:2.509ex;" alt="{\displaystyle X_{\mathrm {true} }}"> can in general only be estimated. Furthermore, the quality of the long-range correction scheme depends on the cut-off radius. The assumptions made with the correction schemes are usually not justified at (very) short cut-off radii. This is illustrated in the example shown in Figure on the right. The long-range correction scheme is said to be converged, if the remaining error of the correction scheme is sufficiently small at a given cut-off distance, cf. Figure. <div class="mw-heading mw-heading2"><h2 id="Extensions_and_modifications">Extensions and modifications</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=5" title="Edit section: Extensions and modifications">edit</a>]</div> The Lennard-Jones potential – as an archetype for <a href="/wiki/Interatomic_potential" title="Interatomic potential">intermolecular potentials</a> – has been used numerous times as starting point for the development of more elaborate or more generalized intermolecular potentials. Various extensions and modifications of the Lennard-Jones potential have been proposed in the literature; a more extensive list is given in the '<a href="/wiki/Interatomic_potential" title="Interatomic potential">interatomic potential</a>' article. The following list refers only to several example potentials that are directly related to the Lennard-Jones potential and are of both historic importance and still relevant for present research. <ul><li><a href="/wiki/Mie_potential" title="Mie potential">Mie potential</a> The Mie potential is the generalized version of the Lennard-Jones potential, i.e. the exponents 12 and 6 are introduced as parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {rep} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {rep} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8c52ea9086501d1dbc13735db54a1138543a809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle \lambda _{\mathrm {rep} }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\mathrm {attr} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{\mathrm {attr} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9dbea3b09aac3169de7ab48c79fb70699a39ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.333ex; height:2.509ex;" alt="{\displaystyle \lambda _{\mathrm {attr} }}">. Especially thermodynamic derivative properties, e.g. the <a href="/wiki/Compressibility" title="Compressibility">compressibility</a> and the <a href="/wiki/Speed_of_sound" title="Speed of sound">speed of sound</a>, are known to be very sensitive to the steepness of the repulsive part of the intermolecular potential, which can therefore be modeled more sophisticated by the Mie potential.<a href="#cite_note-:31-17">[17]</a> The first explicit formulation of the Mie potential is attributed to <a href="/wiki/Eduard_Gr%C3%BCneisen" title="Eduard Grüneisen">Eduard Grüneisen</a>.<a href="#cite_note-18">[18]</a><a href="#cite_note-19">[19]</a> Hence, the Mie potential was actually proposed before the Lennard-Jones potential. The Mie potential is named after <a href="/wiki/Gustav_Mie" title="Gustav Mie">Gustav Mie</a>.<a href="#cite_note-:30-8">[8]</a></li> <li><a href="/wiki/Buckingham_potential" title="Buckingham potential">Buckingham potential</a> The Buckingham potential was proposed by <a href="/wiki/Richard_Buckingham" title="Richard Buckingham">Richard Buckingham</a>. The repulsive part of the Lennard-Jones potential is therein replaced by an exponential function and it incorporates an additional parameter.</li> <li><a href="/wiki/Stockmayer_potential" title="Stockmayer potential">Stockmayer potential</a> The Stockmayer potential is named after W.H. Stockmayer.<a href="#cite_note-20">[20]</a> The Stockmayer potential is a combination of a Lennard-Jones potential superimposed by a dipole. Hence, Stockmayer particles are not spherically symmetric, but rather have an important orientational structure.</li> <li>Two center Lennard-Jones potential The two center Lennard-Jones potential consists of two identical Lennard-Jones interaction sites (same <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">) that are bonded as a rigid body. It is often abbreviated as 2CLJ. Usually, the elongation (distance between the Lennard-Jones sites) is significantly smaller than the size parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">. Hence, the two interaction sites are significantly fused.</li> <li>Lennard-Jones truncated & splined potential The Lennard-Jones truncated & splined potential is a rarely used yet useful potential. Similar to the more popular LJTS potential, it is sturdily truncated at a certain 'end' distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a42708fb3b10b44e596a034264a4ab8470e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {end} }}"> and no long-range interactions are considered beyond. Opposite to the LJTS potential, which is shifted such that the potential is continuous, the Lennard-Jones truncated & splined potential is made continuous by using an arbitrary but favorable spline function.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Lennard-Jones_truncated_&_shifted_(LJTS)_potential">Lennard-Jones truncated & shifted (LJTS) potential</h3>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=6" title="Edit section: Lennard-Jones truncated & shifted (LJTS) potential">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png/317px-Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png" decoding="async" width="317" height="770" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png/475px-Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png/633px-Vapor_liquid_equilibrium_properties_of_LJ_and_LJTS_potential.png 2x" data-file-width="6374" data-file-height="15492" /></a><figcaption>Comparison of the vapor–liquid equilibrium of the 'full' Lennard-Jones potential (black) and the 'Lennard-Jones truncated & shifted' potential (blue). The symbols indicate molecular simulation results;<a href="#cite_note-:20-21">[21]</a><a href="#cite_note-22">[22]</a> the lines indicate results from equations of state.<a href="#cite_note-:2-11">[11]</a><a href="#cite_note-23">[23]</a></figcaption></figure> The Lennard-Jones truncated & shifted (LJTS) potential is an often used alternative to the 'full' Lennard-Jones potential (see Eq. (1)). The 'full' and the 'truncated & shifted' Lennard-Jones potential have to be kept strictly separate. They are simply two different intermolecular potentials yielding different thermophysical properties. The Lennard-Jones truncated & shifted potential is defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{LJTS}}(r)={\begin{cases}V_{\text{LJ}}(r)-V_{\text{LJ}}(r_{\text{end}})&~~~~r\leq r_{\text{end}}\\0&~~~~r>r_{\text{end}},\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>LJTS</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>LJ</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>LJ</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>end</mtext> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>r</mi> <mo>≤</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>end</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>r</mi> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>end</mtext> </mrow> </msub> <mo>,</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{LJTS}}(r)={\begin{cases}V_{\text{LJ}}(r)-V_{\text{LJ}}(r_{\text{end}})&~~~~r\leq r_{\text{end}}\\0&~~~~r>r_{\text{end}},\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/644da5a4224b56ee1e5d36152b97e566eeea84de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.555ex; height:6.176ex;" alt="{\displaystyle V_{\text{LJTS}}(r)={\begin{cases}V_{\text{LJ}}(r)-V_{\text{LJ}}(r_{\text{end}})&~~~~r\leq r_{\text{end}}\\0&~~~~r>r_{\text{end}},\end{cases}}}"> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>LJ</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>ε</mi> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b6904bd4b6457ae0608cd387781bbee6f9a126" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.192ex; height:6.176ex;" alt="{\displaystyle V_{\text{LJ}}(r)=4\varepsilon \left[\left({\frac {\sigma }{r}}\right)^{12}-\left({\frac {\sigma }{r}}\right)^{6}\right].}"> Hence, the LJTS potential is truncated at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a42708fb3b10b44e596a034264a4ab8470e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {end} }}"> and shifted by the corresponding energy value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {LJ} }(r_{\mathrm {end} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">J</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {LJ} }(r_{\mathrm {end} })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f5be6a4e1c6d01971805abbd8b132b0db091f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.108ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {LJ} }(r_{\mathrm {end} })}">. The latter is applied to avoid a discontinuity jump of the potential at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a42708fb3b10b44e596a034264a4ab8470e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {end} }}">. For the LJTS potential, no long-range interactions beyond <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a42708fb3b10b44e596a034264a4ab8470e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {end} }}"> are required – neither explicitly nor implicitly. The most frequently used version of the Lennard-Jones truncated & shifted potential is the one with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }=2.5\,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2.5</mn> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }=2.5\,\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63198178adc8d6e735dc2659507966e80acbd186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.626ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {end} }=2.5\,\sigma }">.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Nevertheless, different <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a42708fb3b10b44e596a034264a4ab8470e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {end} }}"> values have been used in the literature.<a href="#cite_note-24">[24]</a><a href="#cite_note-25">[25]</a><a href="#cite_note-:29-26">[26]</a><a href="#cite_note-27">[27]</a> Each LJTS potential with a given truncation radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d6a42708fb3b10b44e596a034264a4ab8470e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.839ex; height:2.009ex;" alt="{\displaystyle r_{\mathrm {end} }}"> has to be considered as a potential and accordingly a substance of its own. The LJTS potential is computationally significantly cheaper than the 'full' Lennard-Jones potential, but still covers the essential physical features of matter (the presence of a critical and a triple point, soft repulsive and attractive interactions, phase equilibria etc.). Therefore, the LJTS potential is used for the testing of new algorithms, simulation methods, and new physical theories.<a href="#cite_note-28">[28]</a><a href="#cite_note-29">[29]</a><a href="#cite_note-30">[30]</a> Interestingly, for homogeneous systems, the intermolecular forces that are calculated from the LJ and the LJTS potential at a given distance are the same (since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{d}}V/{\text{d}}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}V/{\text{d}}r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a698c2d23b427cfa84b3232073efd1a73430c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.583ex; height:2.843ex;" alt="{\displaystyle {\text{d}}V/{\text{d}}r}"> is the same), whereas the potential energy and the pressure are affected by the shifting. Also, the properties of the LJTS substance may furthermore be affected by the chosen simulation algorithm, i.e. MD or MC sampling (this is in general not the case for the 'full' Lennard-Jones potential). For the LJTS potential with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{\mathrm {end} }=2.5\,\sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2.5</mn> <mspace width="thinmathspace" /> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{\mathrm {end} }=2.5\,\sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63198178adc8d6e735dc2659507966e80acbd186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.626ex; height:2.509ex;" alt="{\displaystyle r_{\mathrm {end} }=2.5\,\sigma }">, the potential energy shift is approximately 1/60 of the dispersion energy at the potential well: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {LJ} }(r_{\mathrm {end} }=2.5\,\sigma )=-0.0163\,\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">J</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>2.5</mn> <mspace width="thinmathspace" /> <mi>σ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>0.0163</mn> <mspace width="thinmathspace" /> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {LJ} }(r_{\mathrm {end} }=2.5\,\sigma )=-0.0163\,\varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38b29dfcad8b780bf9d1c7afd3cd62abde23afc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.731ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {LJ} }(r_{\mathrm {end} }=2.5\,\sigma )=-0.0163\,\varepsilon }">. The Figure on the right shows the comparison of the <a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">vapor–liquid equilibrium</a> of the 'full' Lennard-Jones potential and the 'Lennard-Jones truncated & shifted' potential. The 'full' Lennard-Jones potential results prevail a significantly higher <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical temperature</a> and pressure compared to the LJTS potential results, but the critical density is very similar.<a href="#cite_note-Stephan_e1699185-31">[31]</a><a href="#cite_note-:6-32">[32]</a><a href="#cite_note-:29-26">[26]</a> The vapor pressure and the enthalpy of vaporization are influenced more strongly by the long-range interactions than the saturated densities. This is due to the fact that the potential is manipulated mainly energetically by the truncation and shifting. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=7" title="Edit section: Applications">edit</a>]</div> The Lennard-Jones potential is not only of fundamental importance in <a href="/wiki/Computational_chemistry" title="Computational chemistry">computational chemistry</a> and <a href="/wiki/Soft_matter" title="Soft matter">soft-matter physics</a>, but also for the modeling of real substances. The Lennard-Jones potential is used for fundamental studies on the behavior of matter and for elucidating atomistic phenomena. It is also often used for somewhat special use cases, e.g. for studying thermophysical properties of two- or four-dimensional substances<a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a><a href="#cite_note-35">[35]</a> (instead of the classical three spatial directions of our universe). There are two main applications of the Lennard-Jones potentials: (i) for studying the hypothetical Lennard-Jones substance<a href="#cite_note-Lennard-Jonesium-13">[13]</a> and (ii) for modeling interactions in real substance models.<a href="#cite_note-SchwerdtfegerWales-3">[3]</a><a href="#cite_note-:17-2">[2]</a> These two applications are discussed in the following. <div class="mw-heading mw-heading3"><h3 id="Lennard-Jones_substance">Lennard-Jones substance</h3>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=8" title="Edit section: Lennard-Jones substance">edit</a>]</div> A Lennard-Jones substance or "Lennard-Jonesium" is the name given to an idealized substance which would result from atoms or molecules interacting exclusively through the Lennard-Jones potential.<a href="#cite_note-Lennard-Jonesium-13">[13]</a> <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a><a href="#cite_note-:16-36">[36]</a> and <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">computer simulations</a><a href="#cite_note-Rapaport2004-15">[15]</a><a href="#cite_note-:0-16">[16]</a> can be used to study the Lennard-Jones potential and to obtain thermophysical properties of the 'Lennard-Jones substance'. The Lennard-Jones substance is often referred to as 'Lennard-Jonesium,'<a href="#cite_note-Lennard-Jonesium-13">[13]</a> suggesting that it is viewed as a (fictive) <a href="/wiki/Chemical_element" title="Chemical element">chemical element</a>.<a href="#cite_note-:20-21">[21]</a> Moreover, its energy and length parameters can be adjusted to fit many different real substances. Both the Lennard-Jones potential and, accordingly, the Lennard-Jones substance are simplified yet realistic models, such as they accurately capture essential physical principles like the presence of a <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical</a> and a <a href="/wiki/Triple_point" title="Triple point">triple point</a>, <a href="/wiki/Condensation" title="Condensation">condensation</a> and <a href="/wiki/Freezing" title="Freezing">freezing</a>. Due in part to its mathematical simplicity, the Lennard-Jones potential has been extensively used in studies on matter since the early days of computer simulation.<a href="#cite_note-:1-37">[37]</a><a href="#cite_note-:11-38">[38]</a><a href="#cite_note-39">[39]</a><a href="#cite_note-40">[40]</a> <div class="mw-heading mw-heading4"><h4 id="Thermophysical_properties_of_the_Lennard-Jones_substance">Thermophysical properties of the Lennard-Jones substance</h4>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=9" title="Edit section: Thermophysical properties of the Lennard-Jones substance">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:LJ_PhaseDiagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/LJ_PhaseDiagram.png/347px-LJ_PhaseDiagram.png" decoding="async" width="347" height="313" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/LJ_PhaseDiagram.png/521px-LJ_PhaseDiagram.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/LJ_PhaseDiagram.png/694px-LJ_PhaseDiagram.png 2x" data-file-width="6566" data-file-height="5916" /></a><figcaption>Phase diagram of the Lennard-Jones substance. Correlations and numeric values for the critical point and triple point(s) are taken from Refs.<a href="#cite_note-:20-21">[21]</a><a href="#cite_note-:18-41">[41]</a><a href="#cite_note-:2-11">[11]</a> The star indicates the critical point.<a href="#cite_note-:20-21">[21]</a> The circle indicates the vapor–liquid–solid triple point and the triangle indicates the vapor–solid (fcc)–solid (hcp) triple point.<a href="#cite_note-:18-41">[41]</a><a href="#cite_note-:10-42">[42]</a> The solid lines indicate coexistence lines of two phases.<a href="#cite_note-:20-21">[21]</a><a href="#cite_note-:18-41">[41]</a> The dashed lines indicate the vapor–liquid spinodal.<a href="#cite_note-:2-11">[11]</a> </figcaption></figure> Thermophysical properties of the Lennard-Jones substance,<a href="#cite_note-Lennard-Jonesium-13">[13]</a> i.e. particles interacting with the Lennard-Jones potential can be obtained using statistical mechanics. Some properties can be computed analytically, i.e. with machine precision, whereas most properties can only be obtained by performing molecular simulations.<a href="#cite_note-Rapaport2004-15">[15]</a> The latter will in general be superimposed by both statistical and systematic uncertainties.<a href="#cite_note-:13-43">[43]</a><a href="#cite_note-:20-21">[21]</a><a href="#cite_note-44">[44]</a><a href="#cite_note-:15-45">[45]</a> The virial coefficients can for example be computed directly from the Lennard-potential using algebraic expressions<a href="#cite_note-:16-36">[36]</a> and reported data has therefore no uncertainty. Molecular simulation results, e.g. the pressure at a given temperature and density has both statistical and systematic uncertainties.<a href="#cite_note-:13-43">[43]</a><a href="#cite_note-:15-45">[45]</a> Molecular simulations of the Lennard-Jones potential can in general be performed using either <a href="/wiki/Molecular_dynamics" title="Molecular dynamics">molecular dynamics</a> (MD) simulations or <a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo</a> (MC) simulation. For MC simulations, the Lennard-Jones potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {LJ} }(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">J</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {LJ} }(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/506bd61e001d152de2b61d98428abac6a3779aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.318ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {LJ} }(r)}"> is directly used, whereas MD simulations are always based on the derivative of the potential, i.e. the force <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\mathrm {d} V/\mathrm {d} r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\mathrm {d} V/\mathrm {d} r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643427af3c630c2bf21d1be1dcd0446d95d8ecb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.423ex; height:2.843ex;" alt="{\displaystyle F=\mathrm {d} V/\mathrm {d} r}">. These differences in combination with differences in the treatment of the long-range interactions (see below) can influence computed thermophysical properties.<a href="#cite_note-46">[46]</a><a href="#cite_note-:6-32">[32]</a> Since the Lennard-Jonesium is the archetype for the modeling of simple yet realistic intermolecular interactions, a large number of thermophysical properties were studied and reported in the literature.<a href="#cite_note-:20-21">[21]</a> Computer experiment data of the Lennard-Jones potential is presently considered the most accurately known data in classical mechanics computational chemistry. Hence, such data is also mostly used as a benchmark for validating and testing new algorithms and theories. The Lennard-Jones potential has been constantly used since the early days of molecular simulations. The first results from computer experiments for the Lennard-Jones potential were reported by Rosenbluth and Rosenbluth<a href="#cite_note-:11-38">[38]</a> and Wood and Parker<a href="#cite_note-:1-37">[37]</a> after molecular simulations on "<a href="/wiki/Equation_of_State_Calculations_by_Fast_Computing_Machines" title="Equation of State Calculations by Fast Computing Machines">fast computing machines</a>" became available in 1953.<a href="#cite_note-:7-47">[47]</a> Since then many studies reported data of the Lennard-Jones substance;<a href="#cite_note-:20-21">[21]</a> approximately 50,000 data points are publicly available. The current state of research on the thermophysical properties of the Lennard-Jones substance is summarized by Stephan et al.<a href="#cite_note-:20-21">[21]</a> (which did not cover transport and mixture properties). The US <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">National Institute of Standards and Technology</a> (NIST) provides examples of molecular dynamics and Monte Carlo codes along with results obtained from them.<a href="#cite_note-48">[48]</a> Transport property data of Lennard-Jones fluids have been compiled by Bell et al.<a href="#cite_note-49">[49]</a> and Lautenschaeger and Hasse.<a href="#cite_note-50">[50]</a> Figure on the right shows the phase diagram of the Lennard-Jones fluid. Phase equilibria of the Lennard-Jones potential have been studied numerous times and are accordingly known today with good precision.<a href="#cite_note-:18-41">[41]</a><a href="#cite_note-:20-21">[21]</a><a href="#cite_note-:19-51">[51]</a> The Figure shows results correlations derived from computer experiment results (hence, lines instead of data points are shown). The mean intermolecular interaction of a Lennard-Jones particle strongly depends on the thermodynamic state, i.e., temperature and pressure (or density). For solid states, the attractive Lennard-Jones interaction plays a dominant role – especially at low temperatures. For liquid states, no ordered structure is present compared to solid states. The mean potential energy per particle is negative. For gaseous states, attractive interactions of the Lennard-Jones potential play a minor role – since they are far distanced. The main part of the internal energy is stored as kinetic energy for gaseous states. At supercritical states, the attractive Lennard-Jones interaction plays a minor role. With increasing temperature, the mean kinetic energy of the particles increases and exceeds the energy well of the Lennard-Jones potential. Hence, the particles mainly interact by the potentials' soft repulsive interactions and the mean potential energy per particle is accordingly positive. Overall, due to the large timespan the Lennard-Jones potential has been studied and thermophysical property data has been reported in the literature and computational resources were insufficient for accurate simulations (to modern standards), a noticeable amount of data is known to be dubious.<a href="#cite_note-:20-21">[21]</a> Nevertheless, in many studies such data is used as reference. The lack of data repositories and data assessment is a crucial element for future work in the long-going field of Lennard-Jones potential research. <div class="mw-heading mw-heading5"><h5 id="Characteristic_points_and_curves">Characteristic points and curves</h5>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=10" title="Edit section: Characteristic points and curves">edit</a>]</div> The most important characteristic points of the Lennard-Jones potential are the <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical point</a> and the vapor–liquid–solid <a href="/wiki/Triple_point" title="Triple point">triple point</a>. They were studied numerous times in the literature and compiled in Ref.<a href="#cite_note-:20-21">[21]</a> The critical point was thereby assessed to be located at <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {c} }=1.321\pm 0.007\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1.321</mn> <mo>±</mo> <mn>0.007</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {c} }=1.321\pm 0.007\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6791f69e21ba903242c78c56eeff204fda62692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.867ex; height:3.343ex;" alt="{\displaystyle T_{\mathrm {c} }=1.321\pm 0.007\,\varepsilon k_{\mathrm {B} }^{-1}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {c} }=0.316\pm 0.005\,\sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.316</mn> <mo>±</mo> <mn>0.005</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {c} }=0.316\pm 0.005\,\sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24f8f7e27f6d3192c6a991ec07b54d3a8bbb29c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.747ex; height:3.176ex;" alt="{\displaystyle \rho _{\mathrm {c} }=0.316\pm 0.005\,\sigma ^{-3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathrm {c} }=0.129\pm 0.005\,\varepsilon \sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.129</mn> <mo>±</mo> <mn>0.005</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\mathrm {c} }=0.129\pm 0.005\,\varepsilon \sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dda9d45f7c332f83452039c261307b5650d675b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:23.887ex; height:3.009ex;" alt="{\displaystyle p_{\mathrm {c} }=0.129\pm 0.005\,\varepsilon \sigma ^{-3}}"></li></ul> The given uncertainties were calculated from the standard deviation of the critical parameters derived from the most reliable available <a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">vapor–liquid equilibrium</a> data sets.<a href="#cite_note-:20-21">[21]</a> These uncertainties can be assumed as a lower limit to the accuracy with which the critical point of fluid can be obtained from molecular simulation results. <figure typeof="mw:File/Thumb"><a href="/wiki/File:CharacteristicCurves.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/CharacteristicCurves.png/315px-CharacteristicCurves.png" decoding="async" width="315" height="317" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/CharacteristicCurves.png/474px-CharacteristicCurves.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/CharacteristicCurves.png/631px-CharacteristicCurves.png 2x" data-file-width="5727" data-file-height="5757" /></a><figcaption>Characteristic curves of the Lennard-Jones substance. The thick black line indicates the vapor–liquid equilibrium; the star indicates the critical point. The brown line indicates the solid–fluid equilibrium. Other black solid lines and symbols indicate Brown's characteristic curves (see text for details) of the Lennard-Jones substance: lines are results from an <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a>, symbols from molecular simulations and triangles exact data in the ideal gas limit obtained from the virial coefficients. Data taken from.<a href="#cite_note-:3-52">[52]</a><a href="#cite_note-Deiters_2720–2728-53">[53]</a><a href="#cite_note-:22-54">[54]</a></figcaption></figure> The triple point is presently assumed to be located at <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {tr} }=0.69\pm 0.005\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.69</mn> <mo>±</mo> <mn>0.005</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {tr} }=0.69\pm 0.005\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2dd3f79f14d76f93a62c992b91fa9e9ad6ea4df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.258ex; height:3.343ex;" alt="{\displaystyle T_{\mathrm {tr} }=0.69\pm 0.005\,\varepsilon k_{\mathrm {B} }^{-1}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {tr,gas} }=0.0017\pm 0.004\,\sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mo>,</mo> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.0017</mn> <mo>±</mo> <mn>0.004</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {tr,gas} }=0.0017\pm 0.004\,\sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d8672fc5279b125023412f7b7dadd146a39353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.213ex; height:3.343ex;" alt="{\displaystyle \rho _{\mathrm {tr,gas} }=0.0017\pm 0.004\,\sigma ^{-3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {tr,liq} }=0.845\pm 0.009\,\sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mo>,</mo> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.845</mn> <mo>±</mo> <mn>0.009</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {tr,liq} }=0.845\pm 0.009\,\sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ad51eb2a3d30923ea94fb880b31e16564ddaed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.553ex; height:3.343ex;" alt="{\displaystyle \rho _{\mathrm {tr,liq} }=0.845\pm 0.009\,\sigma ^{-3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {tr,sol} }=0.961\pm 0.007\,\sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mo>,</mo> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">l</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.961</mn> <mo>±</mo> <mn>0.007</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {tr,sol} }=0.961\pm 0.007\,\sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2215fbf149d714e806dbd79b7e4c735dee24ec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.686ex; height:3.343ex;" alt="{\displaystyle \rho _{\mathrm {tr,sol} }=0.961\pm 0.007\,\sigma ^{-3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathrm {tr} }=0.0012\pm 0.0007\,\varepsilon \sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.0012</mn> <mo>±</mo> <mn>0.0007</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\mathrm {tr} }=0.0012\pm 0.0007\,\varepsilon \sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f08f59da2fb624cfca8d68caa009eee60f9cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:26.766ex; height:3.009ex;" alt="{\displaystyle p_{\mathrm {tr} }=0.0012\pm 0.0007\,\varepsilon \sigma ^{-3}}"></li></ul> The uncertainties represent the scattering of data from different authors.<a href="#cite_note-:18-41">[41]</a> The critical point of the Lennard-Jones substance has been studied far more often than the triple point. For both the critical point and the vapor–liquid–solid triple point, several studies reported results out of the above stated ranges. The above stated data is the presently assumed correct and reliable data. Nevertheless, the determinateness of the critical temperature and the triple point temperature is still unsatisfactory. Evidently, the phase coexistence curves (cf. figures) are of fundamental importance to characterize the Lennard-Jones potential. Furthermore, Brown's characteristic curves<a href="#cite_note-55">[55]</a> yield an illustrative description of essential features of the Lennard-Jones potential. Brown's characteristic curves are defined as curves on which a certain thermodynamic property of the substance matches that of an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a>. For a real fluid, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"> and its derivatives can match the values of the ideal gas for special <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><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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"> combinations only as a result of Gibbs' phase rule. The resulting points collectively constitute a characteristic curve. Four main characteristic curves are defined: One 0th-order (named Zeno curve) and three 1st-order curves (named Amagat, Boyle, and Charles curve). The characteristic curve are required to have a negative or zero curvature throughout and a single maximum in a double-logarithmic pressure-temperature diagram. Furthermore, Brown's characteristic curves and the virial coefficients are directly linked in the limit of the ideal gas and are therefore known exactly at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \rightarrow 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \rightarrow 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d7b253c2e58fd5dc99a9fc736692f3f63263a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.978ex; height:2.676ex;" alt="{\displaystyle \rho \rightarrow 0}">. Both computer simulation results and equation of state results have been reported in the literature for the Lennard-Jones potential.<a href="#cite_note-Deiters_2720–2728-53">[53]</a><a href="#cite_note-:20-21">[21]</a><a href="#cite_note-:3-52">[52]</a><a href="#cite_note-56">[56]</a><a href="#cite_note-57">[57]</a> Points on the Zeno curve Z have a <a href="/wiki/Compressibility_factor" title="Compressibility factor">compressibility factor</a> of unity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=p/(\rho T)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>ρ</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=p/(\rho T)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/001c5e6de4e00bc61d6e0fbe5e462694ff941ce0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.019ex; height:2.843ex;" alt="{\displaystyle Z=p/(\rho T)=1}">. The Zeno curve originates at the <a href="/wiki/Boyle_temperature" title="Boyle temperature">Boyle temperature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {B} }=3.417927982\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>3.417927982</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {B} }=3.417927982\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ed7f693823e75c42a973a134bec61da611b063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.138ex; height:3.343ex;" alt="{\displaystyle T_{\mathrm {B} }=3.417927982\,\varepsilon k_{\mathrm {B} }^{-1}}">, surrounds the critical point, and has a slope of unity in the low temperature limit.<a href="#cite_note-:3-52">[52]</a> Points on the Boyle curve B have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} (1/\rho )}}\right|_{T}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>Z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} (1/\rho )}}\right|_{T}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13f2b9df7a06cc87a460f46677dac966de33f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.762ex; height:6.509ex;" alt="{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} (1/\rho )}}\right|_{T}=0}">. The Boyle curve originates with the Zeno curve at the Boyle temperature, faintly surrounds the critical point, and ends on the vapor pressure curve. Points on the Charles curve (a.k.a. <a href="/wiki/Joule%E2%80%93Thomson_effect" title="Joule–Thomson effect">Joule-Thomson inversion curve</a>) have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} T}}\right|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>Z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} T}}\right|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e8f8e2817ce30c3b0456ffd28c2913b0401436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.776ex; height:6.176ex;" alt="{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} T}}\right|_{p}=0}"> and more importantly <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {\mathrm {d} T}{\mathrm {d} p}}\right|_{h}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>p</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\mathrm {d} T}{\mathrm {d} p}}\right|_{h}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76568779daab1430301c1f58ad3bdca53a4cbf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.852ex; height:6.009ex;" alt="{\displaystyle \left.{\frac {\mathrm {d} T}{\mathrm {d} p}}\right|_{h}=0}">, i.e. no temperature change upon isenthalpic throttling. It originates at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=6.430798418\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>6.430798418</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=6.430798418\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560231f5464af1b4219e705d945efcf70e21a3d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.021ex; height:3.343ex;" alt="{\displaystyle T=6.430798418\,\varepsilon k_{\mathrm {B} }^{-1}}"> in the ideal gas limit, crosses the Zeno curve, and terminates on the vapor pressure curve. Points on the Amagat curve A have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} T}}\right|_{\rho }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>Z</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} T}}\right|_{\rho }=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a2c6fc524217723a790ac4fde20544cd6d727d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:9.799ex; height:6.176ex;" alt="{\displaystyle \left.{\frac {\mathrm {d} Z}{\mathrm {d} T}}\right|_{\rho }=0}">. It also starts in the ideal gas limit at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=25.15242837\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>25.15242837</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=25.15242837\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02dea1de4df7d32bce262956a579b913b308a103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.021ex; height:3.343ex;" alt="{\displaystyle T=25.15242837\,\varepsilon k_{\mathrm {B} }^{-1}}">, surrounds the critical point and the other three characteristic curves and passes into the solid phase region. A comprehensive discussion of the characteristic curves of the Lennard-Jones potential is given by Stephan and Deiters.<a href="#cite_note-:3-52">[52]</a> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:VirialCoeff.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/VirialCoeff.png/329px-VirialCoeff.png" decoding="async" width="329" height="388" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/VirialCoeff.png/494px-VirialCoeff.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/VirialCoeff.png/658px-VirialCoeff.png 2x" data-file-width="5852" data-file-height="6900" /></a><figcaption>Virial coefficients from the Lennard-Jones potential as a function of the temperature: Second virial coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> (top) and third virial coefficient <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> (bottom). The circle indicates the Boyle temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {B} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1331259c82febdd8f099573f1b77e0a54c181bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.753ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {B} }}">. Results taken from.<a href="#cite_note-:3-52">[52]</a></figcaption></figure> <div class="mw-heading mw-heading5"><h5 id="Properties_of_the_Lennard-Jones_fluid">Properties of the Lennard-Jones fluid</h5>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=11" title="Edit section: Properties of the Lennard-Jones fluid">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Vapor_liquid_equilibrium_properties_of_LJ_potential.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Vapor_liquid_equilibrium_properties_of_LJ_potential.png/291px-Vapor_liquid_equilibrium_properties_of_LJ_potential.png" decoding="async" width="291" height="726" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Vapor_liquid_equilibrium_properties_of_LJ_potential.png/437px-Vapor_liquid_equilibrium_properties_of_LJ_potential.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Vapor_liquid_equilibrium_properties_of_LJ_potential.png/582px-Vapor_liquid_equilibrium_properties_of_LJ_potential.png 2x" data-file-width="6224" data-file-height="15516" /></a><figcaption>Vapor–liquid equilibrium of the Lennard-Jones substance: Vapor pressure (top), saturated densities (middle) and interfacial tension (bottom). Symbols indicate molecular simulation results.<a href="#cite_note-Stephan_e1699185-31">[31]</a><a href="#cite_note-:20-21">[21]</a> Lines indicate results from equation of state (and square gradient theory for the interfacial tension).<a href="#cite_note-Stephan_e1699185-31">[31]</a><a href="#cite_note-:2-11">[11]</a></figcaption></figure> Properties of the Lennard-Jones fluid have been studied extensively in the literature due to the outstanding importance of the Lennard-Jones potential in soft-matter physics and related fields.<a href="#cite_note-Lennard-Jonesium-13">[13]</a> About 50 datasets of computer experiment data for the <a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">vapor–liquid equilibrium</a> have been published to date.<a href="#cite_note-:20-21">[21]</a> Furthermore, more than 35,000 data points at homogeneous fluid states have been published over the years and recently been compiled and assessed for outliers in an open access database.<a href="#cite_note-:20-21">[21]</a> The vapor–liquid equilibrium of the Lennard-Jones substance is presently known with a precision, i.e. mutual agreement of thermodynamically consistent data, of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1\%}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d68d1106982e14eff7d5a826d0a2af69aa8274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.906ex; height:2.343ex;" alt="{\displaystyle \pm 1\%}"> for the vapor pressure, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 0.2\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>0.2</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 0.2\%}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b0efe068799da9078401b434112a8bae0219c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.716ex; height:2.343ex;" alt="{\displaystyle \pm 0.2\%}"> for the saturated liquid density, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 1\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1\%}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d68d1106982e14eff7d5a826d0a2af69aa8274" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.906ex; height:2.343ex;" alt="{\displaystyle \pm 1\%}"> for the saturated vapor density, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 0.75\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>0.75</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 0.75\%}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d84f32715c7217e2b4df5d76cc98cbf6a7a78c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.878ex; height:2.343ex;" alt="{\displaystyle \pm 0.75\%}"> for the enthalpy of vaporization, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm 4\%}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>4</mn> <mi mathvariant="normal">%</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 4\%}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92d2a7f4e187f6ed08019aa6745a75bae88f09a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.906ex; height:2.343ex;" alt="{\displaystyle \pm 4\%}"> for the surface tension.<a href="#cite_note-:20-21">[21]</a> This status quo can not be considered satisfactory considering the fact that statistical uncertainties usually reported for single data sets are significantly below the above stated values (even for far more complex molecular force fields). Both phase equilibrium properties and homogeneous state properties at arbitrary density can in general only be obtained from molecular simulations, whereas virial coefficients can be computed directly from the Lennard-Jones potential.<a href="#cite_note-:16-36">[36]</a> Numerical data for the second and third virial coefficient is available in a wide temperature range.<a href="#cite_note-58">[58]</a><a href="#cite_note-:3-52">[52]</a><a href="#cite_note-:20-21">[21]</a> For higher virial coefficients (up to the sixteenth), the number of available data points decreases with increasing number of the virial coefficient.<a href="#cite_note-59">[59]</a><a href="#cite_note-60">[60]</a> Also transport properties (viscosity, heat conductivity, and self diffusion coefficient) of the Lennard-Jones fluid have been studied,<a href="#cite_note-61">[61]</a><a href="#cite_note-62">[62]</a> but the database is significantly less dense than for homogeneous equilibrium properties like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pvT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>v</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pvT}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66c2ad7e5c97bf6379fc0e59a7085da859b864e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.023ex; height:2.509ex;" alt="{\displaystyle pvT}"> – or internal energy data. Moreover, a large number of analytical models (<a href="/wiki/Equation_of_state" title="Equation of state">equations of state</a>) have been developed for the description of the Lennard-Jones fluid (see below for details). <div class="mw-heading mw-heading5"><h5 id="Properties_of_the_Lennard-Jones_solid">Properties of the Lennard-Jones solid</h5>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=12" title="Edit section: Properties of the Lennard-Jones solid">edit</a>]</div> The database and knowledge for the Lennard-Jones solid is significantly poorer than for the fluid phases. It was realized early that the interactions in solid phases should not be approximated to be pair-wise additive – especially for metals.<a href="#cite_note-:8-63">[63]</a><a href="#cite_note-:9-64">[64]</a> Nevertheless, the Lennard-Jones potential is used in solid-state physics due to its simplicity and computational efficiency. Hence, the basic properties of the solid phases and the solid–fluid phase equilibria have been investigated several times, e.g. Refs.<a href="#cite_note-:19-51">[51]</a><a href="#cite_note-:18-41">[41]</a><a href="#cite_note-:10-42">[42]</a><a href="#cite_note-:23-65">[65]</a><a href="#cite_note-66">[66]</a><a href="#cite_note-:22-54">[54]</a> The Lennard-Jones substance form fcc (face centered cubic), hcp (hexagonal close-packed) and other close-packed polytype <a href="/wiki/Crystal_structure" title="Crystal structure">lattices</a> – depending on temperature and pressure, cf. figure above with phase diagram. At low temperature and up to moderate pressure, the hcp lattice is energetically favored and therefore the equilibrium structure. The fcc lattice structure is energetically favored at both high temperature and high pressure and therefore overall the equilibrium structure in a wider state range. The coexistence line between the fcc and hcp phase starts at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6a5b6d0370b358a8d5f3df6d17eeca08d3629b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle T=0}"> at approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=878.5\,\varepsilon \sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>878.5</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=878.5\,\varepsilon \sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cacb903794456ec68cc1d72ac04a92afd6846b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.788ex; height:3.009ex;" alt="{\displaystyle p=878.5\,\varepsilon \sigma ^{-3}}">, passes through a temperature maximum at approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=0.4\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>0.4</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=0.4\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f0e3729cd79eeb19b74deb728745c3ef7cb1053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.721ex; height:3.343ex;" alt="{\displaystyle T=0.4\,\varepsilon k_{\mathrm {B} }^{-1}}">, and then ends on the vapor–solid phase boundary at approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=0.32\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>0.32</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=0.32\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d727b5841a4c00fd55f7fd5eef667a7d1e09ebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.884ex; height:3.343ex;" alt="{\displaystyle T=0.32\,\varepsilon k_{\mathrm {B} }^{-1}}">, which thereby forms a triple point.<a href="#cite_note-:23-65">[65]</a><a href="#cite_note-:18-41">[41]</a> Hence, only the fcc solid phase exhibits phase equilibria with the liquid and supercritical phase, cf. figure above with phase diagram. The triple point of the two solid phases (fcc and hcp) and the vapor phase is reported to be located at:<a href="#cite_note-:23-65">[65]</a><a href="#cite_note-:18-41">[41]</a> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {tr} }=0.32\pm 0.001\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.32</mn> <mo>±</mo> <mn>0.001</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {tr} }=0.32\pm 0.001\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32013227f1f966401145f0a178060b5c3e67507" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.258ex; height:3.343ex;" alt="{\displaystyle T_{\mathrm {tr} }=0.32\pm 0.001\,\varepsilon k_{\mathrm {B} }^{-1}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {tr,gas} }=..}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mo>,</mo> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">s</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {tr,gas} }=..}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee6450ede24c105326da66ef74f0beb3d06d6812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.602ex; height:2.343ex;" alt="{\displaystyle \rho _{\mathrm {tr,gas} }=..}"> not reported yet</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {tr,fcc} }=1.03859\pm 0.0008\,\sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mo>,</mo> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1.03859</mn> <mo>±</mo> <mn>0.0008</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {tr,fcc} }=1.03859\pm 0.0008\,\sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6435f635c721aee8d4900dcc350214f5f6940d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.318ex; height:3.343ex;" alt="{\displaystyle \rho _{\mathrm {tr,fcc} }=1.03859\pm 0.0008\,\sigma ^{-3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{\mathrm {tr,hcp} }=1.03861\pm 0.0007\,\sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> <mo>,</mo> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1.03861</mn> <mo>±</mo> <mn>0.0007</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{\mathrm {tr,hcp} }=1.03861\pm 0.0007\,\sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8139e7c5777fdeeed770c0e46b2fcf94401fecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.804ex; height:3.343ex;" alt="{\displaystyle \rho _{\mathrm {tr,hcp} }=1.03861\pm 0.0007\,\sigma ^{-3}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\mathrm {tr} }=0.96\cdot 10^{-9}\,\varepsilon \sigma ^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.96</mn> <mo>⋅</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>ε</mi> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\mathrm {tr} }=0.96\cdot 10^{-9}\,\varepsilon \sigma ^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0254877001751e35ef817df832b937d94597ad0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:21.479ex; height:3.009ex;" alt="{\displaystyle p_{\mathrm {tr} }=0.96\cdot 10^{-9}\,\varepsilon \sigma ^{-3}}"></li></ul> Note, that other and significantly differing values have also been reported in the literature. Hence, the database for the fcc-hcp–vapor triple point should be further solidified in the future. <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:LJ_mixtures.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/LJ_mixtures.png/440px-LJ_mixtures.png" decoding="async" width="440" height="366" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/LJ_mixtures.png/660px-LJ_mixtures.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/LJ_mixtures.png/880px-LJ_mixtures.png 2x" data-file-width="14412" data-file-height="11984" /></a><figcaption>Vapor–liquid equilibria of binary Lennard-Jones mixtures. In all shown cases, component 2 is the more volatile component (enriching in the vapor phase). The units are given in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> of component 1, which is the same in all four shown mixtures. The temperature is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=0.92\,\varepsilon k_{\mathrm {B} }^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mn>0.92</mn> <mspace width="thinmathspace" /> <mi>ε</mi> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=0.92\,\varepsilon k_{\mathrm {B} }^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df04838f42a15a3e8e9794cfeeb57cf0fe08c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.884ex; height:3.343ex;" alt="{\displaystyle T=0.92\,\varepsilon k_{\mathrm {B} }^{-1}}">. Symbols are molecular simulation results and lines are results from an <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a>. Data taken from Ref.<a href="#cite_note-Stephan_e1699185-31">[31]</a></figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Mixtures_of_Lennard-Jones_substances">Mixtures of Lennard-Jones substances</h4>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=13" title="Edit section: Mixtures of Lennard-Jones substances">edit</a>]</div> <a href="/wiki/Mixture" title="Mixture">Mixtures</a> of Lennard-Jones particles are mostly used as a prototype for the development of theories and methods of solutions, but also to study properties of solutions in general. This dates back to the fundamental work of conformal solution theory of <a href="/wiki/Christopher_Longuet-Higgins" title="Christopher Longuet-Higgins">Longuet-Higgins</a><a href="#cite_note-67">[67]</a> and Leland and <a href="/wiki/John_Shipley_Rowlinson" title="John Shipley Rowlinson">Rowlinson</a> and co-workers.<a href="#cite_note-68">[68]</a><a href="#cite_note-69">[69]</a> Those are today the basis of most theories for mixtures.<a href="#cite_note-70">[70]</a><a href="#cite_note-:21-71">[71]</a> Mixtures of two or more Lennard-Jones components are set up by changing at least one potential interaction parameter (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }">) of one of the components with respect to the other. For a binary mixture, this yields three types of pair interactions that are all modeled by the Lennard-Jones potential: 1-1, 2-2, and 1-2 interactions. For the cross interactions 1–2, additional assumptions are required for the specification of parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\mathrm {12} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\mathrm {12} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5247e06904ee40682cf4aa847bf0009b8d8a2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\mathrm {12} }}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {12} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {12} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9679afeb7b34557b195d04ca7ef0308fedb6642" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.204ex; height:2.009ex;" alt="{\displaystyle \sigma _{\mathrm {12} }}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\mathrm {11} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\mathrm {11} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88fe6e2d0813a2376f2b1b51e0274b789f7990e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\mathrm {11} }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {11} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {11} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a19c3f5a101fbf60f3c62f6a1d28a70b8dc02f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.204ex; height:2.009ex;" alt="{\displaystyle \sigma _{\mathrm {11} }}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\mathrm {22} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\mathrm {22} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75662035dd1dcb316e048364051a3b9a9d2d3bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.96ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{\mathrm {22} }}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {22} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {22} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/596532e0bb7c9efa2667e6bcf0b76cfe80f097a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.204ex; height:2.009ex;" alt="{\displaystyle \sigma _{\mathrm {22} }}">. Various choices (all more or less empirical and not rigorously based on physical arguments) can be used for these so-called combination rules.<a href="#cite_note-Schnabel2007-72">[72]</a> The most widely used<a href="#cite_note-Schnabel2007-72">[72]</a> combination rule is the one of <a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a> and Berthelot<a href="#cite_note-73">[73]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{12}=\eta _{12}{\frac {\sigma _{11}+\sigma _{22}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{12}=\eta _{12}{\frac {\sigma _{11}+\sigma _{22}}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd9857b7c93a878e24032b39298a589b3ebe3590" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.417ex; height:5.009ex;" alt="{\displaystyle \sigma _{12}=\eta _{12}{\frac {\sigma _{11}+\sigma _{22}}{2}}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{12}=\xi _{12}{\sqrt {\varepsilon _{11}\varepsilon _{22}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{12}=\xi _{12}{\sqrt {\varepsilon _{11}\varepsilon _{22}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07007ca083aee55cbacd5e3f4f1fbc2da96ca3aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.808ex; height:3.009ex;" alt="{\displaystyle \varepsilon _{12}=\xi _{12}{\sqrt {\varepsilon _{11}\varepsilon _{22}}}}"> The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17a576d6a549ec856098d3ce209fc778661ebb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.895ex; height:2.509ex;" alt="{\displaystyle \xi _{12}}"> is an additional state-independent interaction parameter for the mixture. The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f005f643ee6eca87cea08b5262a0f9211c367a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.032ex; height:2.176ex;" alt="{\displaystyle \eta _{12}}"> is usually set to unity since the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a> can be considered physically plausible for the cross-interaction size parameter. The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17a576d6a549ec856098d3ce209fc778661ebb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.895ex; height:2.509ex;" alt="{\displaystyle \xi _{12}}"> on the other hand is often used to adjust the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a> so as to reproduce the phase behavior of the model mixture. For analytical models, e.g. <a href="/wiki/Equation_of_state" title="Equation of state">equations of state</a>, the deviation parameter is usually written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{12}=1-\xi _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{12}=1-\xi _{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57eb0d4a899a1dbd3b0e234693f7ccdaaf8994e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.083ex; height:2.509ex;" alt="{\displaystyle k_{12}=1-\xi _{12}}">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d050b0d2237e61f28a0b3a98cbad1a89c874b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.156ex; height:2.509ex;" alt="{\displaystyle \xi _{12}>1}">, the cross-interaction dispersion energy and accordingly the attractive force between unlike particles is intensified, and the attractive forces between unlike particles are diminished for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee294eeac420821c096be0b43c35c7728a3afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.156ex; height:2.509ex;" alt="{\displaystyle \xi _{12}<1}">. For Lennard-Jones mixtures, both fluid and solid <a href="/wiki/Phase_rule" title="Phase rule">phase equilibria</a> can be studied, i.e. <a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">vapor–liquid</a>, <a href="/wiki/Liquid%E2%80%93liquid_extraction" title="Liquid–liquid extraction">liquid–liquid</a>, gas–gas, solid–vapor, <a href="/wiki/Crystallization" title="Crystallization">solid–liquid</a>, and solid–solid. Accordingly, different types of <a href="/wiki/Triple_point" title="Triple point">triple points</a> (three-phase equilibria) and <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical points</a> can exist as well as different <a href="/wiki/Eutectic_system" title="Eutectic system">eutectic</a> and <a href="/wiki/Azeotrope" title="Azeotrope">azeotropic points</a>.<a href="#cite_note-74">[74]</a><a href="#cite_note-:21-71">[71]</a> Binary Lennard-Jones mixtures in the fluid region (various types of equilibria of liquid and gas phases)<a href="#cite_note-Stephan_e1699185-31">[31]</a><a href="#cite_note-75">[75]</a><a href="#cite_note-76">[76]</a><a href="#cite_note-77">[77]</a><a href="#cite_note-78">[78]</a> have been studied more comprehensively then phase equilibria comprising solid phases.<a href="#cite_note-:24-79">[79]</a><a href="#cite_note-:25-80">[80]</a><a href="#cite_note-:26-81">[81]</a><a href="#cite_note-82">[82]</a><a href="#cite_note-:27-83">[83]</a> A large number of different Lennard-Jones mixtures have been studied in the literature. To date, no standard for such has been established. Usually, the binary interaction parameters and the two component parameters are chosen such that a mixture with properties convenient for a given task are obtained. Yet, this often makes comparisons tricky. For the fluid phase behavior, mixtures exhibit practically ideal behavior (in the sense of <a href="/wiki/Raoult%27s_law" title="Raoult's law">Raoult's law</a>) for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547f966af5420e2c8b8ca2b4f2a0605dbf09926c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.156ex; height:2.509ex;" alt="{\displaystyle \xi _{12}=1}">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d050b0d2237e61f28a0b3a98cbad1a89c874b46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.156ex; height:2.509ex;" alt="{\displaystyle \xi _{12}>1}"> attractive interactions prevail and the mixtures tend to form high-boiling azeotropes, i.e. a lower pressure than pure components' vapor pressures is required to stabilize the vapor–liquid equilibrium. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}<1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ee294eeac420821c096be0b43c35c7728a3afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.156ex; height:2.509ex;" alt="{\displaystyle \xi _{12}<1}"> repulsive interactions prevail and mixtures tend to form low-boiling azeotropes, i.e. a higher pressure than pure components' vapor pressures is required to stabilize the vapor–liquid equilibrium since the mean dispersive forces are decreased. Particularly low values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17a576d6a549ec856098d3ce209fc778661ebb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.895ex; height:2.509ex;" alt="{\displaystyle \xi _{12}}"> furthermore will result in liquid–liquid miscibility gaps. Also various types of phase equilibria comprising solid phases have been studied in the literature, e.g. by <a href="/wiki/Carol_K._Hall" title="Carol K. Hall">Carol</a> and co-workers.<a href="#cite_note-:26-81">[81]</a><a href="#cite_note-:27-83">[83]</a><a href="#cite_note-:25-80">[80]</a><a href="#cite_note-:24-79">[79]</a> Also, cases exist where the solid phase boundaries interrupt fluid phase equilibria. However, for phase equilibria that comprise solid phases, the amount of published data is sparse. <div class="mw-heading mw-heading4"><h4 id="Equations_of_state">Equations of state</h4>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=14" title="Edit section: Equations of state">edit</a>]</div> A large number of <a href="/wiki/Equation_of_state" title="Equation of state">equations of state</a> (EOS) for the Lennard-Jones potential/ substance have been proposed since its characterization and evaluation became available with the first computer simulations.<a href="#cite_note-:7-47">[47]</a> Due to the fundamental importance of the Lennard-Jones potential, most currently available molecular-based EOS are built around the Lennard-Jones fluid. They have been comprehensively reviewed by Stephan et al.<a href="#cite_note-:2-11">[11]</a><a href="#cite_note-:3-52">[52]</a> Equations of state for the Lennard-Jones fluid are of particular importance in <a href="/wiki/Soft_matter" title="Soft matter">soft-matter physics</a> and <a href="/wiki/Physical_chemistry" title="Physical chemistry">physical chemistry</a>, used as starting point for the development of EOS for complex fluids, e.g. <a href="/wiki/Polymer" title="Polymer">polymers</a> and associating fluids. The monomer units of these models are usually directly adapted from Lennard-Jones EOS as a building block, e.g. the PHC EOS,<a href="#cite_note-84">[84]</a> the BACKONE EOS,<a href="#cite_note-85">[85]</a><a href="#cite_note-86">[86]</a> and <a href="/wiki/Statistical_associating_fluid_theory" title="Statistical associating fluid theory">SAFT</a> type EOS.<a href="#cite_note-:31-17">[17]</a><a href="#cite_note-87">[87]</a><a href="#cite_note-88">[88]</a><a href="#cite_note-89">[89]</a> More than 30 Lennard-Jones EOS have been proposed in the literature. A comprehensive evaluation<a href="#cite_note-:2-11">[11]</a><a href="#cite_note-:3-52">[52]</a> of such EOS showed that several EOS<a href="#cite_note-90">[90]</a><a href="#cite_note-:4-91">[91]</a><a href="#cite_note-92">[92]</a><a href="#cite_note-93">[93]</a> describe the Lennard-Jones potential with good and similar accuracy, but none of them is outstanding. Three of those EOS show an unacceptable unphysical behavior in some fluid region, e.g. multiple van der Waals loops, while being elsewise reasonably precise. Only the Lennard-Jones EOS of Kolafa and Nezbeda<a href="#cite_note-:4-91">[91]</a> was found to be robust and precise for most thermodynamic properties of the Lennard-Jones fluid.<a href="#cite_note-:3-52">[52]</a><a href="#cite_note-:2-11">[11]</a> Furthermore, the Lennard-Jones EOS of Johnson et al.<a href="#cite_note-:5-94">[94]</a> was found to be less precise for practically all available reference data<a href="#cite_note-:20-21">[21]</a><a href="#cite_note-:2-11">[11]</a> than the Kolafa and Nezbeda EOS.<a href="#cite_note-:4-91">[91]</a> <div class="mw-heading mw-heading3"><h3 id="Lennard-Jones_potential_as_building_block_for_force_fields">Lennard-Jones potential as building block for force fields</h3>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=15" title="Edit section: Lennard-Jones potential as building block for force fields">edit</a>]</div> The Lennard-Jones potential is extensively used for molecular modeling of real substances. There are essentially two ways the Lennard-Jones potential can be used for molecular modeling: (1) A real substance atom or molecule is modeled directly by the Lennard-Jones potential, which yields very good results for <a href="/wiki/Noble_gas" title="Noble gas">noble gases</a> and <a href="/wiki/Methane" title="Methane">methane</a>, i.e. dispersively interacting spherical particles. In the case of methane, the molecule is assumed to be spherically symmetric and the hydrogen atoms are fused with the carbon atom to a common unit. This simplification can in general also be applied to more complex molecules, but yields usually poor results. (2) A real substance molecule is built of multiple Lennard-Jones interactions sites, which can be connected either by rigid bonds or flexible additional potentials (and eventually also consists of other potential types, e.g. partial charges). <a href="/wiki/Molecular_modelling" title="Molecular modelling">Molecular models</a> (often referred to as '<a href="/wiki/Force_field_(chemistry)" title="Force field (chemistry)">force fields</a>') for practically all molecular and ionic particles can be constructed using this scheme for example for <a href="/wiki/Alkane" title="Alkane">alkanes</a>. Upon using the first outlined approach, the molecular model has only the two parameters of the Lennard-Jones potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> that can be used for the fitting, e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon /k_{\mathrm {B} }=120\,\mathrm {K} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>120</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">K</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon /k_{\mathrm {B} }=120\,\mathrm {K} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b19aa9b562ccb369d07a1b156410e5b76e9db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.634ex; height:2.843ex;" alt="{\displaystyle \varepsilon /k_{\mathrm {B} }=120\,\mathrm {K} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =0.34\,\mathrm {nm} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mn>0.34</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =0.34\,\mathrm {nm} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c96623eed442916b7e4a0f71cfb727d3179f5a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.178ex; height:2.176ex;" alt="{\displaystyle \sigma =0.34\,\mathrm {nm} }"> can be used for <a href="/wiki/Argon" title="Argon">argon</a>. Upon adjusting the model parameters ε and σ to real substance properties, the Lennard-Jones potential can be used to describe simple substance (like <a href="/wiki/Noble_gas" title="Noble gas">noble gases</a>) with good accuracy. Evidently, this approach is only a good approximation for spherical and simply dispersively interacting molecules and atoms. The direct use of the Lennard-Jones potential has the great advantage that simulation results and theories for the Lennard-Jones potential can be used directly. Hence, available results for the Lennard-Jones potential and substance can be directly scaled using the appropriate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> (see reduced units). The Lennard-Jones potential parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> can in general be fitted to any desired real substance property. In soft-matter physics, usually experimental data for the vapor–liquid phase equilibrium or the critical point are used for the parametrization; in solid-state physics, rather the compressibility, heat capacity or lattice constants are employed.<a href="#cite_note-:8-63">[63]</a><a href="#cite_note-:9-64">[64]</a> The second outlined approach of using the Lennard-Jones potential as a building block of elongated and complex molecules is far more sophisticated. <a href="/wiki/Molecular_modelling" title="Molecular modelling">Molecular models</a> are thereby tailor-made in a sense that simulation results are only applicable for that particular model. This development approach for molecular force fields is today mainly performed in <a href="/wiki/Soft_matter" title="Soft matter">soft-matter physics</a> and associated fields such as <a href="/wiki/Chemical_engineering" title="Chemical engineering">chemical engineering</a>, chemistry, and computational biology. A large number of <a href="/wiki/Force_field_(chemistry)" title="Force field (chemistry)">force fields</a> are based on the Lennard-Jones potential, e.g. the <a rel="nofollow" class="external text" href="http://trappe.oit.umn.edu">TraPPE force field</a>,<a href="#cite_note-:14-95">[95]</a> the OPLS force field,<a href="#cite_note-96">[96]</a> and the <a rel="nofollow" class="external text" href="https://molmod.boltzmann-zuse.de/">MolMod force field</a><a href="#cite_note-:12-97">[97]</a> (an overview of <a href="/wiki/Force_field_(chemistry)" title="Force field (chemistry)">molecular force fields</a> is out of the scope of the present article). For the state-of-the-art modeling of solid-state materials, more elaborate multi-body potentials (e.g. <a href="/wiki/Embedded_atom_model" title="Embedded atom model">EAM potentials</a><a href="#cite_note-98">[98]</a>) are used. The Lennard-Jones potential yields a good approximation of intermolecular interactions for many applications: The macroscopic properties computed using the Lennard-Jones potential are in good agreement with experimental data for simple substances like argon on one side and the potential function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {LJ} }(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> <mi mathvariant="normal">J</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {LJ} }(r)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/506bd61e001d152de2b61d98428abac6a3779aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.318ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {LJ} }(r)}"> is in fair agreement with results from <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a> on the other side. The Lennard-Jones potential gives a good description of molecular interactions in <a href="/wiki/Fluid" title="Fluid">fluid</a> phases, whereas molecular interactions in solid phases are only roughly well described. This is mainly due to the fact that multi-body interactions play a significant role in solid phases, which are not comprised in the Lennard-Jones potential. Therefore, the Lennard-Jones potential is extensively used in <a href="/wiki/Soft_matter" title="Soft matter">soft-matter physics</a> and associated fields, whereas it is less frequently used in <a href="/wiki/Solid-state_physics" title="Solid-state physics">solid-state physics</a>. Due to its simplicity, the Lennard-Jones potential is often used to describe the properties of gases and simple fluids and to model dispersive and repulsive interactions in <a href="/wiki/Molecular_modelling" title="Molecular modelling">molecular models</a>. It is especially accurate for <a href="/wiki/Noble_gas" title="Noble gas">noble gas</a> atoms and <a href="/wiki/Methane" title="Methane">methane</a>. It is furthermore a good approximation for molecular interactions at long and short distances for neutral atoms and molecules. Therefore, the Lennard-Jones potential is very often used as a building block of <a href="/wiki/Molecular_modelling" title="Molecular modelling">molecular models</a> of complex molecules, e.g. <a href="/wiki/Alkane" title="Alkane">alkanes</a> or <a href="/wiki/Water" title="Water">water</a>.<a href="#cite_note-:14-95">[95]</a><a href="#cite_note-99">[99]</a><a href="#cite_note-:12-97">[97]</a> The Lennard-Jones potential can also be used to model the <a href="/wiki/Adsorption" title="Adsorption">adsorption</a> interactions at solid–fluid interfaces, i.e. <a href="/wiki/Physisorption" title="Physisorption">physisorption</a> or <a href="/wiki/Chemisorption" title="Chemisorption">chemisorption</a>. It is well accepted, that the main limitations of the Lennard-Jones potential lie in the fact the potential is a <a href="/wiki/Pair_potential" title="Pair potential">pair potential</a> (does not cover multi-body interactions) and that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r^{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/r^{12}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518cedd2f7f6619bcd1bc5df3b8a6dafe4f2caf5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.25ex; height:3.176ex;" alt="{\displaystyle 1/r^{12}}"> exponent term is used for the repulsion. Results from quantum chemistry suggest that a higher exponent than 12 has to be used, i.e. a steeper potential. Furthermore, the Lennard-Jones potential has a limited flexibility, i.e. only the two model parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> can be used for the fitting to describe a real substance. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=16" title="Edit section: See also">edit</a>]</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"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><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" /></a></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Lennard-Jones_potentials" class="extiw" title="commons:Category:Lennard-Jones potentials">Lennard-Jones potentials</a>.</div></div> </div> <ul><li><a href="/wiki/Comparison_of_force-field_implementations" title="Comparison of force-field implementations">Comparison of force-field implementations</a></li> <li><a href="/wiki/Embedded_atom_model" title="Embedded atom model">Embedded atom model</a></li> <li><a href="/wiki/Force_field_(chemistry)" title="Force field (chemistry)">Force field (chemistry)</a></li> <li><a href="/wiki/Molecular_mechanics" title="Molecular mechanics">Molecular mechanics</a></li> <li><a href="/wiki/Morse_potential" title="Morse potential">Morse potential</a> and <a href="/wiki/Morse/Long-range_potential" title="Morse/Long-range potential">Morse/Long-range potential</a></li> <li><a href="/wiki/Virial_expansion" title="Virial expansion">Virial expansion</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Lennard-Jones_potential&action=edit&section=17" title="Edit section: References">edit</a>]</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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FischerWendland-1">^ <a href="#cite_ref-FischerWendland_1-0">a</a> <a href="#cite_ref-FischerWendland_1-1">b</a> <a href="#cite_ref-FischerWendland_1-2">c</a> <a href="#cite_ref-FischerWendland_1-3">d</a> <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 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