Fick's laws of diffusion

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<span>Fick's first law</span> </div> </a> <button aria-controls="toc-Fick's_first_law-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Fick's first law subsection</span> </button> <ul id="toc-Fick's_first_law-sublist" class="vector-toc-list"> <li id="toc-Variations_of_the_first_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variations_of_the_first_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Variations of the first law</span> </div> </a> <ul id="toc-Variations_of_the_first_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_of_Fick's_first_law_for_gases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_Fick's_first_law_for_gases"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Derivation of Fick's first law for gases</span> </div> </a> <ul id="toc-Derivation_of_Fick's_first_law_for_gases-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Fick's_second_law" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fick's_second_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Fick's second law</span> </div> </a> <button aria-controls="toc-Fick's_second_law-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Fick's second law subsection</span> </button> <ul id="toc-Fick's_second_law-sublist" class="vector-toc-list"> <li id="toc-Derivation_of_Fick's_second_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_Fick's_second_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Derivation of Fick's second law</span> </div> </a> <ul id="toc-Derivation_of_Fick's_second_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Example_solutions_and_generalization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example_solutions_and_generalization"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Example solutions and generalization</span> </div> </a> <button aria-controls="toc-Example_solutions_and_generalization-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Example solutions and generalization subsection</span> </button> <ul id="toc-Example_solutions_and_generalization-sublist" class="vector-toc-list"> <li id="toc-Example_solution_1:_constant_concentration_source_and_diffusion_length" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_solution_1:_constant_concentration_source_and_diffusion_length"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Example solution 1: constant concentration source and diffusion length</span> </div> </a> <ul id="toc-Example_solution_1:_constant_concentration_source_and_diffusion_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_solution_2:_Brownian_particle_and_mean_squared_displacement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_solution_2:_Brownian_particle_and_mean_squared_displacement"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Example solution 2: Brownian particle and mean squared displacement</span> </div> </a> <ul id="toc-Example_solution_2:_Brownian_particle_and_mean_squared_displacement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Generalizations</span> </div> </a> <ul id="toc-Generalizations-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"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Fick's_flow_in_liquids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fick's_flow_in_liquids"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Fick's flow in liquids</span> </div> </a> <ul id="toc-Fick's_flow_in_liquids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sorption_rate_and_collision_frequency_of_diluted_solute" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sorption_rate_and_collision_frequency_of_diluted_solute"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Sorption rate and collision frequency of diluted solute</span> </div> </a> <ul id="toc-Sorption_rate_and_collision_frequency_of_diluted_solute-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biological_perspective" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biological_perspective"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Biological perspective</span> </div> </a> <ul id="toc-Biological_perspective-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semiconductor_fabrication_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semiconductor_fabrication_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Semiconductor fabrication applications</span> </div> </a> <ul id="toc-Semiconductor_fabrication_applications-sublist" class="vector-toc-list"> <li id="toc-CVD_method_of_fabricate_semiconductor" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#CVD_method_of_fabricate_semiconductor"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4.1</span> <span>CVD method of fabricate semiconductor</span> </div> </a> <ul id="toc-CVD_method_of_fabricate_semiconductor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Invalidity_of_Fickian_diffusion" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Invalidity_of_Fickian_diffusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4.2</span> <span>Invalidity of Fickian diffusion</span> </div> </a> <ul id="toc-Invalidity_of_Fickian_diffusion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Food_production_and_cooking" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Food_production_and_cooking"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Food production and cooking</span> </div> </a> <ul id="toc-Food_production_and_cooking-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"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Fick's laws of diffusion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 28 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-28" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">28 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D8%A7%D9%86%D9%88%D9%86%D9%8A_%D9%81%D9%8A%D9%83_%D9%84%D9%84%D8%A7%D9%86%D8%AA%D8%B4%D8%A7%D8%B1" title="قانوني فيك للانتشار – Arabic" lang="ar" hreflang="ar" data-title="قانوني فيك للانتشار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Llei_de_Fick" title="Llei de Fick – Catalan" lang="ca" hreflang="ca" data-title="Llei de Fick" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Prvn%C3%AD_Fick%C5%AFv_z%C3%A1kon" title="První Fickův zákon – Czech" lang="cs" hreflang="cs" data-title="První Fickův zákon" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ficks_love" title="Ficks love – Danish" lang="da" hreflang="da" data-title="Ficks love" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Diffusion#Erstes_Fick'sches_Gesetz" title="Diffusion – German" lang="de" hreflang="de" data-title="Diffusion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ficki_difusiooniseadused" title="Ficki difusiooniseadused – Estonian" lang="et" hreflang="et" data-title="Ficki difusiooniseadused" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9D%CF%8C%CE%BC%CE%BF%CE%B9_%CF%84%CE%BF%CF%85_%CE%A6%CE%B9%CE%BA_%CE%B3%CE%B9%CE%B1_%CF%84%CE%B7_%CE%B4%CE%B9%CE%AC%CF%87%CF%85%CF%83%CE%B7" title="Νόμοι του Φικ για τη διάχυση – Greek" lang="el" hreflang="el" data-title="Νόμοι του Φικ για τη διάχυση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Leyes_de_Fick" title="Leyes de Fick – Spanish" lang="es" hreflang="es" data-title="Leyes de Fick" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D9%88%D8%A7%D9%86%DB%8C%D9%86_%D9%86%D9%81%D9%88%D8%B0_%D9%81%DB%8C%DA%A9" title="قوانین نفوذ فیک – Persian" lang="fa" hreflang="fa" data-title="قوانین نفوذ فیک" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Lois_de_Fick" title="Lois de Fick – French" lang="fr" hreflang="fr" data-title="Lois de Fick" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Dl%C3%ADthe_Fick" title="Dlíthe Fick – Irish" lang="ga" hreflang="ga" data-title="Dlíthe Fick" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%94%BC%ED%81%AC%EC%9D%98_%ED%99%95%EC%82%B0_%EB%B2%95%EC%B9%99" title="피크의 확산 법칙 – Korean" lang="ko" hreflang="ko" data-title="피크의 확산 법칙" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AB%E0%A4%BF%E0%A4%95_%E0%A4%95%E0%A5%87_%E0%A4%B5%E0%A4%BF%E0%A4%B8%E0%A4%B0%E0%A4%A3_%E0%A4%95%E0%A5%87_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="फिक के विसरण के नियम – Hindi" lang="hi" hreflang="hi" data-title="फिक के विसरण के नियम" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Leggi_di_Fick" title="Leggi di Fick – Italian" lang="it" hreflang="it" data-title="Leggi di Fick" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7%D7%99_%D7%94%D7%93%D7%99%D7%A4%D7%95%D7%96%D7%99%D7%94_%D7%A9%D7%9C_%D7%A4%D7%99%D7%A7" title="חוקי הדיפוזיה של פיק – Hebrew" lang="he" hreflang="he" data-title="חוקי הדיפוזיה של פיק" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wetten_van_Fick" title="Wetten van Fick – Dutch" lang="nl" hreflang="nl" data-title="Wetten van Fick" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A3%E3%83%83%E3%82%AF%E3%81%AE%E6%B3%95%E5%89%87" title="フィックの法則 – Japanese" lang="ja" hreflang="ja" data-title="フィックの法則" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ficks_diffusjonslover" title="Ficks diffusjonslover – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ficks diffusjonslover" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ficks_diffusjonslover" title="Ficks diffusjonslover – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ficks diffusjonslover" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Prawa_Ficka" title="Prawa Ficka – Polish" lang="pl" hreflang="pl" data-title="Prawa Ficka" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Lei_de_Fick" title="Lei de Fick – Portuguese" lang="pt" hreflang="pt" data-title="Lei de Fick" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Legile_lui_Fick" title="Legile lui Fick – Romanian" lang="ro" hreflang="ro" data-title="Legile lui Fick" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%A4%D0%B8%D0%BA%D0%B0" title="Закон Фика – Russian" lang="ru" hreflang="ru" data-title="Закон Фика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Fickove_z%C3%A1kony" title="Fickove zákony – Slovak" lang="sk" hreflang="sk" data-title="Fickove zákony" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Difuzijski_zakon" title="Difuzijski zakon – Slovenian" lang="sl" hreflang="sl" data-title="Difuzijski zakon" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ficks_lag" title="Ficks lag – Swedish" lang="sv" hreflang="sv" data-title="Ficks lag" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the technique of measuring <a href="/wiki/Cardiac_output" title="Cardiac output">cardiac output</a>, see <a href="/wiki/Fick_principle" title="Fick principle">Fick principle</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DiffusionMicroMacro.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/DiffusionMicroMacro.gif/250px-DiffusionMicroMacro.gif" decoding="async" width="250" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4d/DiffusionMicroMacro.gif 1.5x" data-file-width="360" data-file-height="300" /></a><figcaption><a href="/wiki/Molecular_diffusion" title="Molecular diffusion">Molecular diffusion</a> from a microscopic and macroscopic point of view. Initially, there are <a href="/wiki/Solute" class="mw-redirect" title="Solute">solute</a> molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. <b>Top</b>: A single molecule moves around randomly. <b>Middle</b>: With more molecules, there is a clear trend where the solute fills the container more and more uniformly. <b>Bottom</b>: With an enormous number of solute molecules, randomness becomes undetectable: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. This smooth flow is described by Fick's laws.</figcaption></figure> <p><b>Fick's laws of diffusion</b> describe <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> and were first posited by <a href="/wiki/Adolf_Fick" class="mw-redirect" title="Adolf Fick">Adolf Fick</a> in 1855 on the basis of largely experimental results. They can be used to solve for the <a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusion coefficient</a>, <span class="texhtml mvar" style="font-style:italic;">D</span>. Fick's first law can be used to derive his second law which in turn is identical to the <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a>. </p><p><i>Fick's first law</i>: Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p><i>Fick's second law</i>: Prediction of change in concentration gradient with time due to diffusion. </p><p>A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called <a href="/wiki/Anomalous_diffusion" title="Anomalous diffusion">anomalous diffusion</a> or non-Fickian diffusion. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1855, physiologist Adolf Fick first reported<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> his now well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of <a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Thomas Graham</a>, which fell short of proposing the fundamental laws for which Fick would become famous. Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists: <a href="/wiki/Darcy%27s_law" title="Darcy's law">Darcy's law</a> (hydraulic flow), <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> (charge transport), and <a href="/wiki/Fourier%27s_law" class="mw-redirect" title="Fourier's law">Fourier's law</a> (heat transport). </p><p>Fick's experiments (modeled on Graham's) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Fick's work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally possible.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Today, Fick's laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does <i>not</i> follow Fick's laws (which happens in cases of diffusion through porous media and diffusion of swelling penetrants, among others),<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-GorbanMMNP2011_5-0" class="reference"><a href="#cite_note-GorbanMMNP2011-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> it is referred to as <i>non-Fickian</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Fick's_first_law"><span id="Fick.27s_first_law"></span>Fick's first law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=2" title="Edit section: Fick's first law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Fick's first law</b> relates the diffusive <a href="/wiki/Flux" title="Flux">flux</a> to the <span class="anchor" id="concentration_gradient"></span>gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative), or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low concentration across a concentration gradient. In one (spatial) dimension, the law can be written in various forms, where the most common form (see<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>) is in a molar basis: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D{\frac {d\varphi }{dx}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D{\frac {d\varphi }{dx}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8e2ee62d173a34b6764d24e0875770a20feb95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.521ex; height:5.509ex;" alt="{\displaystyle J=-D{\frac {d\varphi }{dx}},}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">J</span> is the <b>diffusion flux</b>, of which the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimension</a> is the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> per unit area per unit time. <span class="texhtml mvar" style="font-style:italic;">J</span> measures the amount of substance that will flow through a unit area during a unit time interval,</li> <li><span class="texhtml mvar" style="font-style:italic;">D</span> is the <b>diffusion coefficient</b> or <b><a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusivity</a></b>. Its dimension is area per unit time,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\varphi }{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>φ</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\varphi }{dx}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4281cce886d55cc828c886d02a8bcc4c74363f3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.572ex; height:5.509ex;" alt="{\displaystyle {\frac {d\varphi }{dx}}}"></span> is the <b>concentration gradient</b>,</li> <li><span class="texhtml mvar" style="font-style:italic;">φ</span> (for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume,</li> <li><span class="texhtml mvar" style="font-style:italic;">x</span> is position, the dimension of which is length.</li></ul> <p><span class="texhtml mvar" style="font-style:italic;">D</span> is proportional to the squared velocity of the diffusing particles, which depends on the temperature, <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> of the fluid and the size of the particles according to the <a href="/wiki/Einstein_relation_(kinetic_theory)#Stokes-Einstein_equation" title="Einstein relation (kinetic theory)">Stokes–Einstein relation</a>. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of <span class="nowrap"><span data-sort-value="6990600000000000000♠"></span>(0.6–2)<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−9</sup> m<sup>2</sup>/s</span>. For biological molecules the diffusion coefficients normally range from 10<sup>−10</sup> to 10<sup>−11</sup> m<sup>2</sup>/s. </p><p>In two or more dimensions we must use <span class="texhtml">∇</span>, the <a href="/wiki/Del" title="Del">del</a> or <a href="/wiki/Gradient" title="Gradient">gradient</a> operator, which generalises the first derivative, obtaining </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} =-D\nabla \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =-D\nabla \varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af80dcb0343386859c329b66219c3a8fbba226b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.314ex; height:2.676ex;" alt="{\displaystyle \mathbf {J} =-D\nabla \varphi ,}"></span></dd></dl> <p>where <span class="texhtml"><b>J</b></span> denotes the diffusion flux vector. </p><p>The driving force for the one-dimensional diffusion is the quantity <span class="texhtml">−<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">∂<i>φ</i></span><span class="sr-only">/</span><span class="den">∂<i>x</i></span></span>⁠</span></span>, which for ideal mixtures is the concentration gradient. </p> <div class="mw-heading mw-heading3"><h3 id="Variations_of_the_first_law">Variations of the first law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=3" title="Edit section: Variations of the first law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another form for the first law is to write it with the primary variable as <a href="/wiki/Mass_fraction_(chemistry)" title="Mass fraction (chemistry)">mass fraction</a> (<span class="texhtml mvar" style="font-style:italic;">y<sub>i</sub></span>, given for example in kg/kg), then the equation changes to: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} _{i}=-{\frac {\rho D}{M_{i}}}\nabla y_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>D</mi> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=-{\frac {\rho D}{M_{i}}}\nabla y_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2305fe867345c68b9334469822cc3706cafb39d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.571ex; height:5.843ex;" alt="{\displaystyle \mathbf {J} _{i}=-{\frac {\rho D}{M_{i}}}\nabla y_{i},}"></span></dd></dl> <p>where </p> <ul><li>the index <span class="texhtml mvar" style="font-style:italic;">i</span> denotes the <span class="texhtml mvar" style="font-style:italic;">i</span>th species,</li> <li><span class="texhtml"><b>J</b><sub><i>i</i></sub></span> is the <b>diffusion flux vector</b> of the <span class="texhtml mvar" style="font-style:italic;">i</span>th species (for example in mol/m<sup>2</sup>-s),</li> <li><span class="texhtml"><i>M</i><sub><i>i</i></sub></span> is the <a href="/wiki/Molar_mass" title="Molar mass">molar mass</a> of the <span class="texhtml mvar" style="font-style:italic;">i</span>th species,</li> <li><span class="texhtml mvar" style="font-style:italic;">ρ</span> is the mixture <a href="/wiki/Density" title="Density">density</a> (for example in kg/m<sup>3</sup>).</li></ul> <p>The <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> is outside the <a href="/wiki/Gradient" title="Gradient">gradient</a> operator. This is because: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>i</mi> </mrow> </msub> <mi>ρ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3efc8925e5f288a7e203fe8bb1f2e5aefe998e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.293ex; height:5.343ex;" alt="{\displaystyle y_{i}={\frac {\rho _{si}}{\rho }},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">ρ<sub>si</sub></span> is the partial density of the <span class="texhtml mvar" style="font-style:italic;">i</span>th species. </p><p>Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> of this species. Then Fick's first law (one-dimensional case) can be written </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046628db58834bc00120ecfa79dacb7ad1cf5e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.566ex; height:5.676ex;" alt="{\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}},}"></span></dd></dl> <p>where </p> <ul><li>the index <span class="texhtml mvar" style="font-style:italic;">i</span> denotes the <span class="texhtml mvar" style="font-style:italic;">i</span>th species,</li> <li><span class="texhtml mvar" style="font-style:italic;">c</span> is the concentration (mol/m<sup>3</sup>),</li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is the <a href="/wiki/Universal_gas_constant" class="mw-redirect" title="Universal gas constant">universal gas constant</a> (J/K/mol),</li> <li><span class="texhtml mvar" style="font-style:italic;">T</span> is the absolute temperature (K),</li> <li><span class="texhtml mvar" style="font-style:italic;">μ</span> is the chemical potential (J/mol).</li></ul> <p>The driving force of Fick's law can be expressed as a <a href="/wiki/Fugacity" title="Fugacity">fugacity</a> difference: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a378d99f3a158a17f1a8c6e214be6b848d4d522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.973ex; height:5.676ex;" alt="{\displaystyle J_{i}=-{\frac {D}{RT}}{\frac {\partial f_{i}}{\partial x}},}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span> is the fugacity in Pa. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span> is a partial pressure of component <span class="texhtml"><i>i</i></span> in a vapor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}^{\text{G}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}^{\text{G}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6c0c327ffe43312b7ebc1ccd341ef01ea887e3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.843ex; height:3.176ex;" alt="{\displaystyle f_{i}^{\text{G}}}"></span> or liquid <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}^{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}^{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2582d244407ea329c5b0a251de15ffee032cfa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.58ex; height:3.176ex;" alt="{\displaystyle f_{i}^{\text{L}}}"></span> phase. At vapor liquid equilibrium the evaporation flux is zero because <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}^{\text{G}}=f_{i}^{\text{L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>G</mtext> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>L</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}^{\text{G}}=f_{i}^{\text{L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648825d559be9c9386004c8eb141f6d5c51a41ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.521ex; height:3.176ex;" alt="{\displaystyle f_{i}^{\text{G}}=f_{i}^{\text{L}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_Fick's_first_law_for_gases"><span id="Derivation_of_Fick.27s_first_law_for_gases"></span>Derivation of Fick's first law for gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=4" title="Edit section: Derivation of Fick's first law for gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Four versions of Fick's law for binary gas mixtures are given below. These assume: thermal diffusion is negligible; the body force per unit mass is the same on both species; and either pressure is constant or both species have the same molar mass. Under these conditions, Ref.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> shows in detail how the diffusion equation from the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory of gases</a> reduces to this version of Fick's law: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {V_{i}} =-D\nabla \ln {\left(y_{i}\right)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {V_{i}} =-D\nabla \ln {\left(y_{i}\right)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44edb989f92516d653d949b8af9df2ca3f25f72" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.652ex; height:2.843ex;" alt="{\displaystyle \mathbf {V_{i}} =-D\nabla \ln {\left(y_{i}\right)},}"></span> where <span class="texhtml"><b>V<sub>i</sub></b></span> is the diffusion velocity of species <span class="texhtml mvar" style="font-style:italic;">i</span>. In terms of species flux this is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>D</mi> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799f1efe59a0f3ef88d89e710aa3e04f1faac460" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.528ex; height:5.843ex;" alt="{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla y_{i}.}"></span> </p><p>If, additionally, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \rho =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba91be1a60a5ba0501790fe73419516979d466a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.399ex; height:2.676ex;" alt="{\displaystyle \nabla \rho =0}"></span>, this reduces to the most common form of Fick's law, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J_{i}} =-D\nabla \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J_{i}} =-D\nabla \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984b38bbf47679f6557d1660aae5a056301d8af0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.071ex; height:2.676ex;" alt="{\displaystyle \mathbf {J_{i}} =-D\nabla \varphi .}"></span> </p><p>If (instead of or in addition to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \rho =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \rho =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba91be1a60a5ba0501790fe73419516979d466a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.399ex; height:2.676ex;" alt="{\displaystyle \nabla \rho =0}"></span>) both species have the same molar mass, Fick's law becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla x_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </msub> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>D</mi> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla x_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/944358e1a4bf17e566ad28bb03223aeaf71f7d50" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.719ex; height:5.843ex;" alt="{\displaystyle \mathbf {J_{i}} =-{\frac {\rho D}{M_{i}}}\nabla x_{i},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span> is the mole fraction of species <span class="texhtml mvar" style="font-style:italic;">i</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Fick's_second_law"><span id="Fick.27s_second_law"></span>Fick's second law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=5" title="Edit section: Fick's second law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Fick's second law</b> predicts how diffusion causes the concentration to change with respect to time. It is a <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equation</a> which in one dimension reads: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7da69b4dc49c94991e3b3ce0b570314df0626d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.485ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}},}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">φ</span> is the concentration in dimensions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\mathsf {N}}{\mathsf {L}}^{-3}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\mathsf {N}}{\mathsf {L}}^{-3}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bed483987f740ca97896b4d6bfaa6c5fab02a7f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.532ex; height:3.176ex;" alt="{\displaystyle [{\mathsf {N}}{\mathsf {L}}^{-3}]}"></span>, example mol/m<sup>3</sup>; <span class="texhtml"><i>φ</i> = <i>φ</i>(<i>x</i>,<i>t</i>)</span> is a function that depends on location <span class="texhtml mvar" style="font-style:italic;">x</span> and time <span class="texhtml mvar" style="font-style:italic;">t</span>,</li> <li><span class="texhtml mvar" style="font-style:italic;">t</span> is time, example s,</li> <li><span class="texhtml mvar" style="font-style:italic;">D</span> is the diffusion coefficient in dimensions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [{\mathsf {L}}^{2}{\mathsf {T}}^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\mathsf {L}}^{2}{\mathsf {T}}^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3323477a35e6803d96a50f9e7a2a3ecd4ce1aaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.524ex; height:3.176ex;" alt="{\displaystyle [{\mathsf {L}}^{2}{\mathsf {T}}^{-1}]}"></span>, example m<sup>2</sup>/s,</li> <li><span class="texhtml mvar" style="font-style:italic;">x</span> is the position, example m.</li></ul> <p>In two or more dimensions we must use the <a href="/wiki/Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a> <span class="texhtml">Δ = ∇<sup>2</sup></span>, which generalises the second derivative, obtaining the equation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\Delta \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\Delta \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ff64878e413fd0c632caa975bd170832387081" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.8ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\Delta \varphi .}"></span></dd></dl> <p>Fick's second law has the same mathematical form as the <a href="/wiki/Heat_equation" title="Heat equation">Heat equation</a> and its <a href="/wiki/Fundamental_solution" title="Fundamental solution">fundamental solution</a> is the same as the <a href="/wiki/Heat_kernel" title="Heat kernel">Heat kernel</a>, except switching thermal conductivity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> with diffusion coefficient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {x^{2}}{4Dt}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>4</mn> <mi>π</mi> <mi>D</mi> <mi>t</mi> </msqrt> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {x^{2}}{4Dt}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14729f6947322e53e285227852d22daff9822bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.24ex; height:6.676ex;" alt="{\displaystyle \varphi (x,t)={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {x^{2}}{4Dt}}\right).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_Fick's_second_law"><span id="Derivation_of_Fick.27s_second_law"></span>Derivation of Fick's second law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=6" title="Edit section: Derivation of Fick's second law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fick's second law can be derived from Fick's first law and the <a href="/wiki/Mass_conservation" class="mw-redirect" title="Mass conservation">mass conservation</a> in absence of any chemical reactions: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}+{\frac {\partial }{\partial x}}J=0\Rightarrow {\frac {\partial \varphi }{\partial t}}-{\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)\,=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>J</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}+{\frac {\partial }{\partial x}}J=0\Rightarrow {\frac {\partial \varphi }{\partial t}}-{\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)\,=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d4e99e2ca79e7f09ea94c80a2383a48e96d3d5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.375ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}+{\frac {\partial }{\partial x}}J=0\Rightarrow {\frac {\partial \varphi }{\partial t}}-{\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)\,=0.}"></span></dd></dl> <p>Assuming the diffusion coefficient <span class="texhtml mvar" style="font-style:italic;">D</span> to be a constant, one can exchange the orders of the differentiation and multiply by the constant: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)=D{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\varphi =D{\frac {\partial ^{2}\varphi }{\partial x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>φ</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>φ</mi> <mo>=</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)=D{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\varphi =D{\frac {\partial ^{2}\varphi }{\partial x^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2e88f9c72d436b40073feda453f5850d386dbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.154ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)=D{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\varphi =D{\frac {\partial ^{2}\varphi }{\partial x^{2}}},}"></span></dd></dl> <p>and, thus, receive the form of the Fick's equations as was stated above. </p><p>For the case of diffusion in two or more dimensions Fick's second law becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\nabla ^{2}\varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\nabla ^{2}\varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353dea076952c13655ecded73655218e24d510c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.241ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\nabla ^{2}\varphi ,}"></span></dd></dl> <p>which is analogous to the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a>. </p><p>If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\,\nabla \varphi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\,\nabla \varphi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a7a05753a541d1f12b7c2a55718d74779617ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.611ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\,\nabla \varphi ).}"></span></dd></dl> <p>An important example is the case where <span class="texhtml"><b>φ</b></span> is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant <span class="texhtml mvar" style="font-style:italic;">D</span>, the solution for the concentration will be a linear change of concentrations along <span class="texhtml mvar" style="font-style:italic;">x</span>. In two or more dimensions we obtain </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\varphi =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\varphi =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f49b830ecb3c09f6d18dd2cbb877b8b3a5ff82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.418ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}\varphi =0,}"></span></dd></dl> <p>which is <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a>, the solutions to which are referred to by mathematicians as <a href="/wiki/Harmonic_functions" class="mw-redirect" title="Harmonic functions">harmonic functions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Example_solutions_and_generalization">Example solutions and generalization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=7" title="Edit section: Example solutions and generalization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Fick's second law is a special case of the <a href="/wiki/Convection%E2%80%93diffusion_equation" title="Convection–diffusion equation">convection–diffusion equation</a> in which there is no <a href="/wiki/Advection" title="Advection">advective flux</a> and no net volumetric source. It can be derived from the <a href="/wiki/Continuity_equation#Differential_form" title="Continuity equation">continuity equation</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {j} =R,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mi>R</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {j} =R,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d5772f9dd5f0f74e7dbac7752ce8bf404af3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.455ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot \mathbf {j} =R,}"></span></dd></dl> <p>where <span class="texhtml"><b>j</b></span> is the total <a href="/wiki/Flux" title="Flux">flux</a> and <span class="texhtml mvar" style="font-style:italic;">R</span> is a net volumetric source for <span class="texhtml"><b>φ</b></span>. The only source of flux in this situation is assumed to be <i>diffusive flux</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} _{\text{diffusion}}=-D\nabla \varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>diffusion</mtext> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} _{\text{diffusion}}=-D\nabla \varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560477a504cd355cb1db2af7003f2f9e8de69e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.164ex; width:18.279ex; height:2.676ex;" alt="{\displaystyle \mathbf {j} _{\text{diffusion}}=-D\nabla \varphi .}"></span></dd></dl> <p>Plugging the definition of diffusive flux to the continuity equation and assuming there is no source (<span class="texhtml"><i>R</i> = 0</span>), we arrive at Fick's second law: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}=D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a5bc1d17575752b20812c76a803734711d930a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:14.098ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial t}}=D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}.}"></span></dd></dl> <p>If flux were the result of both diffusive flux and <a href="/wiki/Advection" title="Advection">advective flux</a>, the <a href="/wiki/Convection%E2%80%93diffusion_equation" title="Convection–diffusion equation">convection–diffusion equation</a> is the result. </p> <div class="mw-heading mw-heading3"><h3 id="Example_solution_1:_constant_concentration_source_and_diffusion_length">Example solution 1: constant concentration source and diffusion length</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=8" title="Edit section: Example solution 1: constant concentration source and diffusion length"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple case of diffusion with time <span class="texhtml mvar" style="font-style:italic;">t</span> in one dimension (taken as the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis) from a boundary located at position <span class="texhtml"><i>x</i> = 0</span>, where the concentration is maintained at a value <span class="texhtml"><i>n</i><sub>0</sub></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left(x,t\right)=n_{0}\operatorname {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>erfc</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>D</mi> <mi>t</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\left(x,t\right)=n_{0}\operatorname {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa97948933254f7150273ec34c4f304caace7b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.183ex; height:6.509ex;" alt="{\displaystyle n\left(x,t\right)=n_{0}\operatorname {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right),}"></span></dd></dl> <p>where <span class="texhtml">erfc</span> is the complementary <a href="/wiki/Error_function" title="Error function">error function</a>. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space – that is, the corrosion product layer – is <i>semi-infinite</i>, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is <i>infinite</i> (lasting both through the layer with <span class="texhtml"><i>n</i>(<i>x</i>, 0) = 0</span>, <span class="texhtml"><i>x</i> > 0</span> and that with <span class="texhtml"><i>n</i>(<i>x</i>, 0) = <i>n</i><sub>0</sub></span>, <span class="texhtml"><i>x</i> ≤ 0</span>), then the solution is amended only with coefficient <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> in front of <span class="texhtml"><i>n</i><sub>0</sub></span> (as the diffusion now occurs in both directions). This case is valid when some solution with concentration <span class="texhtml"><i>n</i><sub>0</sub></span> is put in contact with a layer of pure solvent. (Bokstein, 2005) The length <span class="texhtml">2<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>Dt</i></span></span></span> is called the <i>diffusion length</i> and provides a measure of how far the concentration has propagated in the <span class="texhtml mvar" style="font-style:italic;">x</span>-direction by diffusion in time <span class="texhtml mvar" style="font-style:italic;">t</span> (Bird, 1976). </p><p>As a quick approximation of the error function, the first two terms of the <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> can be used: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(x,t)=n_{0}\left[1-2\left({\frac {x}{2{\sqrt {Dt\pi }}}}\right)\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>D</mi> <mi>t</mi> <mi>π</mi> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(x,t)=n_{0}\left[1-2\left({\frac {x}{2{\sqrt {Dt\pi }}}}\right)\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b412f149f106b8c7e581c17144e82dfd24ecfa53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.835ex; height:6.509ex;" alt="{\displaystyle n(x,t)=n_{0}\left[1-2\left({\frac {x}{2{\sqrt {Dt\pi }}}}\right)\right].}"></span></dd></dl> <p>If <span class="texhtml mvar" style="font-style:italic;">D</span> is time-dependent, the diffusion length becomes </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2{\sqrt {\int _{0}^{t}D(\tau )\,d\tau }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>D</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {\int _{0}^{t}D(\tau )\,d\tau }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea31ead6d1aefc18751d006e75864af305885ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.551ex; height:7.509ex;" alt="{\displaystyle 2{\sqrt {\int _{0}^{t}D(\tau )\,d\tau }}.}"></span></dd></dl> <p>This idea is useful for estimating a diffusion length over a heating and cooling cycle, where <span class="texhtml mvar" style="font-style:italic;">D</span> varies with temperature. </p> <div class="mw-heading mw-heading3"><h3 id="Example_solution_2:_Brownian_particle_and_mean_squared_displacement">Example solution 2: Brownian particle and mean squared displacement</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=9" title="Edit section: Example solution 2: Brownian particle and mean squared displacement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Another simple case of diffusion is the <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> of one particle. The particle's <a href="/wiki/Mean_squared_displacement" title="Mean squared displacement">Mean squared displacement</a> from its original position is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{MSD}}\equiv \left\langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\right\rangle =2nDt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>MSD</mtext> </mrow> <mo>≡</mo> <mrow> <mo>⟨</mo> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mi>D</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{MSD}}\equiv \left\langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\right\rangle =2nDt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9167d4195ef1b0af11015b6d32022f3f4ee15f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.262ex; height:3.343ex;" alt="{\displaystyle {\text{MSD}}\equiv \left\langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\right\rangle =2nDt,}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the particle's Brownian motion. For example, the diffusion of a molecule across a <a href="/wiki/Cell_membrane" title="Cell membrane">cell membrane</a> 8 nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a <a href="/wiki/Eukaryotic_Cell" class="mw-redirect" title="Eukaryotic Cell">eukaryotic cell</a> is a 3-D diffusion. For a cylindrical <a href="/wiki/Cactus" title="Cactus">cactus</a>, the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion. </p><p>The square root of MSD, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2nDt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>n</mi> <mi>D</mi> <mi>t</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2nDt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8f794e1c49c674621d2d215146303cb7d5c113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.257ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2nDt}}}"></span>, is often used as a characterization of how far the particle has moved after time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> has elapsed. The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is Gaussian and 3D is a Maxwell-Boltzmann distribution. </p> <div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=10" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>In <i>non-homogeneous media</i>, the diffusion coefficient varies in space, <span class="texhtml"><i>D</i> = <i>D</i>(<i>x</i>)</span>. This dependence does not affect Fick's first law but the second law changes: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>D</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d39de19579202245ea61069e70a2dc4923f732d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:70.826ex; height:7.176ex;" alt="{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}.}"></span></li> <li>In <i><a href="/wiki/Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a> media</i>, the diffusion coefficient depends on the direction. It is a symmetric <a href="/wiki/Tensor" title="Tensor">tensor</a> <span class="texhtml"><i>D<sub>ji</sub></i> = <i>D<sub>ij</sub></i></span>. Fick's first law changes to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D\nabla \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D\nabla \varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/533c9e55794ca1efb7d494e2ff2c7e4e6c06a9bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.405ex; height:2.676ex;" alt="{\displaystyle J=-D\nabla \varphi ,}"></span> it is the product of a tensor and a vector: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4891c4e1301c923df7920788a39d91cea18d6c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.567ex; height:7.509ex;" alt="{\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.}"></span> For the diffusion equation this formula gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d739916c79c046dddb0ea5ad6cd117c9699c41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:52.32ex; height:7.509ex;" alt="{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.}"></span> The symmetric matrix of diffusion coefficients <span class="texhtml"><i>D<sub>ij</sub></i></span> should be <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a>. It is needed to make the right-hand side operator <a href="/wiki/Elliptic_operator" title="Elliptic operator">elliptic</a>.</li> <li>For <i>inhomogeneous anisotropic media</i> these two forms of the diffusion equation should be combined in <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>D</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f38a403a10ce0842c346adb8a6e0b4beefbfee51" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:79.594ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).}"></span></li> <li>The approach based on <a href="/wiki/Diffusion#Einstein's_mobility_and_Teorell_formula" title="Diffusion">Einstein's mobility and Teorell formula</a> gives the following generalization of Fick's equation for the <i>multicomponent diffusion</i> of the perfect components: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mspace width="thinmathspace" /> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8495d26d21997398fee250941ae7f9208c47e41a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.805ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right),}"></span> where <span class="texhtml mvar" style="font-style:italic;">φ<sub>i</sub></span> are concentrations of the components and <span class="texhtml mvar" style="font-style:italic;">D<sub>ij</sub></span> is the matrix of coefficients. Here, indices <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml mvar" style="font-style:italic;">j</span> are related to the various components and not to the space coordinates.</li></ul> <p>The <a href="/wiki/Diffusion#The_theory_of_diffusion_in_gases_based_on_Boltzmann's_equation" title="Diffusion">Chapman–Enskog formulae for diffusion in gases</a> include exactly the same terms. These physical models of diffusion are different from the test models <span class="texhtml">∂<sub><i>t</i></sub><i>φ<sub>i</sub></i> = Σ<sub><i>j</i></sub> <i>D<sub>ij</sub></i> Δ<i>φ<sub>j</sub></i></span> which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the <a href="/wiki/Maxwell%E2%80%93Stefan_diffusion" title="Maxwell–Stefan diffusion">Maxwell–Stefan diffusion</a> equation. </p><p>For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example <span class="texhtml"><i>D</i><sub><i>ij</i>,<i>αβ</i></sub></span>, where <span class="texhtml"><i>i</i>, <i>j</i></span> refer to the components and <span class="texhtml"><i>α</i>, <i>β</i> = 1, 2, 3</span> correspond to the space coordinates. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=11" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Equations based on Fick's law have been commonly used to model <a href="/wiki/Passive_transport" title="Passive transport">transport processes</a> in foods, <a href="/wiki/Neuron" title="Neuron">neurons</a>, <a href="/wiki/Biopolymer" title="Biopolymer">biopolymers</a>, <a href="/wiki/Pharmacology" title="Pharmacology">pharmaceuticals</a>, <a href="/wiki/Porous" class="mw-redirect" title="Porous">porous</a> <a href="/wiki/Soil" title="Soil">soils</a>, <a href="/wiki/Population_dynamics" title="Population dynamics">population dynamics</a>, nuclear materials, <a href="/wiki/Plasma_physics" class="mw-redirect" title="Plasma physics">plasma physics</a>, and <a href="/wiki/Doping_(semiconductor)" title="Doping (semiconductor)">semiconductor doping</a> processes. The theory of <a href="/wiki/Voltammetry" title="Voltammetry">voltammetric</a> methods is based on solutions of Fick's equation. On the other hand, in some cases a "Fickian (another common approximation of the transport equation is that of the diffusion theory)" description is inadequate. For example, in <a href="/wiki/Polymer" title="Polymer">polymer</a> science and food science a more general approach is required to describe transport of components in materials undergoing a <a href="/wiki/Glass_transition" title="Glass transition">glass transition</a>. One more general framework is the <a href="/wiki/Maxwell%E2%80%93Stefan_diffusion" title="Maxwell–Stefan diffusion">Maxwell–Stefan diffusion</a> equations<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> of multi-component <a href="/wiki/Mass_transfer" title="Mass transfer">mass transfer</a>, from which Fick's law can be obtained as a limiting case, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell–Stefan equations are used. See also non-diagonal coupled transport processes (<a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager</a> relationship). </p> <div class="mw-heading mw-heading3"><h3 id="Fick's_flow_in_liquids"><span id="Fick.27s_flow_in_liquids"></span>Fick's flow in liquids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=12" title="Edit section: Fick's flow in liquids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When two <a href="/wiki/Miscibility" title="Miscibility">miscible</a> liquids are brought into contact, and diffusion takes place, the macroscopic (or average) concentration evolves following Fick's law. On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and molecular scale, where molecular <a href="/wiki/Random_walk" title="Random walk">random walks</a> take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a <a href="/wiki/Tautology_(logic)" title="Tautology (logic)">tautology</a>, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by <a href="/wiki/Renormalization" title="Renormalization">renormalizing</a> the fluctuating hydrodynamics equations. </p> <div class="mw-heading mw-heading3"><h3 id="Sorption_rate_and_collision_frequency_of_diluted_solute">Sorption rate and collision frequency of diluted solute</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=13" title="Edit section: Sorption rate and collision frequency of diluted solute"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Adsorption, absorption, and collision of molecules, particles, and surfaces are important problems in many fields. These fundamental processes regulate chemical, biological, and environmental reactions. Their rate can be calculated using the diffusion constant and Fick's laws of diffusion especially when these interactions happen in diluted solutions. </p><p>Typically, the diffusion constant of molecules and particles defined by Fick's equation can be calculated using the <a href="/wiki/Stokes%E2%80%93Einstein_equation" class="mw-redirect" title="Stokes–Einstein equation">Stokes–Einstein equation</a>. In the ultrashort time limit, in the order of the diffusion time <i>a</i><sup>2</sup>/<i>D</i>, where <i>a</i> is the particle radius, the diffusion is described by the <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a>. At a longer time, the <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> merges into the <a href="/wiki/Stokes%E2%80%93Einstein_equation" class="mw-redirect" title="Stokes–Einstein equation">Stokes–Einstein equation</a>. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the <a href="/wiki/Fluctuation-dissipation_theorem" class="mw-redirect" title="Fluctuation-dissipation theorem">fluctuation-dissipation theorem</a> based on the <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is:<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(t)=\mu \,k_{\rm {B}}T\left(1-e^{-t/(m\mu )}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(t)=\mu \,k_{\rm {B}}T\left(1-e^{-t/(m\mu )}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704b409f226d6dc79f3957d7dc61b2f4e2e91dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.626ex; height:4.843ex;" alt="{\displaystyle D(t)=\mu \,k_{\rm {B}}T\left(1-e^{-t/(m\mu )}\right),}"></span></dd></dl> <p>where (all in SI units) </p> <ul><li><i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>,</li> <li><i>T</i> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a>,</li> <li><i>μ</i> is the mobility of the particle in the fluid or gas, which can be calculated using the <a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a>,</li> <li><i>m</i> is the mass of the particle,</li> <li><i>t</i> is time.</li></ul> <p>For a single molecule such as organic molecules or <a href="/wiki/Biomolecule" title="Biomolecule">biomolecules</a> (e.g. proteins) in water, the exponential term is negligible due to the small product of <i>mμ</i> in the ultrafast picosecond region, thus irrelevant to the relatively slower adsorption of diluted solute. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diffusive_sorption_probability.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/dc/Diffusive_sorption_probability.png" decoding="async" width="181" height="216" class="mw-file-element" data-file-width="181" data-file-height="216" /></a><figcaption>Scheme of molecular diffusion in the solution. Orange dots are solute molecules, solvent molecules are not drawn, black arrow is an example random walk trajectory, and the red curve is the diffusive Gaussian broadening probability function from the Fick's law of diffusion.<sup id="cite_ref-Pyle-BJNano_12-0" class="reference"><a href="#cite_note-Pyle-BJNano-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup><sup>:Fig. 9</sup></figcaption></figure> <p>The <a href="/wiki/Adsorption" title="Adsorption">adsorption</a> or <a href="/wiki/Absorption_(chemistry)" title="Absorption (chemistry)">absorption</a> rate of a dilute solute to a surface or interface in a (gas or liquid) solution can be calculated using Fick's laws of diffusion. The accumulated number of molecules adsorbed on the surface is expressed by the Langmuir-Schaefer equation by integrating the diffusion flux equation over time as shown in the simulated molecular diffusion in the first section of this page:<sup id="cite_ref-LangmuirSchaefer1937JACS_13-0" class="reference"><a href="#cite_note-LangmuirSchaefer1937JACS-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =2AC_{b}{\sqrt {\frac {Dt}{\pi }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mn>2</mn> <mi>A</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>D</mi> <mi>t</mi> </mrow> <mi>π</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma =2AC_{b}{\sqrt {\frac {Dt}{\pi }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/534baae7659fc929941869b51eee913412aa5ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.627ex; height:6.176ex;" alt="{\displaystyle \Gamma =2AC_{b}{\sqrt {\frac {Dt}{\pi }}}.}"></span></dd></dl> <ul><li><span class="texhtml mvar" style="font-style:italic;">A</span> is the surface area (m<sup>2</sup>).</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf365bf59c72d2c4836aaadcda126759f293318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.599ex; height:2.509ex;" alt="{\displaystyle C_{b}}"></span> is the number concentration of the adsorber molecules (solute) in the bulk solution (#/m<sup>3</sup>).</li> <li><span class="texhtml mvar" style="font-style:italic;">D</span> is diffusion coefficient of the adsorber (m<sup>2</sup>/s).</li> <li><span class="texhtml mvar" style="font-style:italic;">t</span> is elapsed time (s).</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> is the accumulated number of molecules in unit # molecules adsorbed during the time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>.</li></ul> <p>The equation is named after American chemists <a href="/wiki/Irving_Langmuir" title="Irving Langmuir">Irving Langmuir</a> and <a href="/wiki/Vincent_Schaefer" title="Vincent Schaefer">Vincent Schaefer</a>. </p><p>Briefly as explained in,<sup id="cite_ref-WardTordai1946_14-0" class="reference"><a href="#cite_note-WardTordai1946-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> the concentration gradient profile near a newly created (from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}"></span>) absorptive surface (placed at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span>) in a once uniform bulk solution is solved in the above sections from Fick's equation, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial C}{\partial x}}={\frac {C_{b}}{\sqrt {\pi Dt}}}{\text{exp}}\left(-{\frac {x^{2}}{4Dt}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msqrt> <mi>π</mi> <mi>D</mi> <mi>t</mi> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>exp</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial C}{\partial x}}={\frac {C_{b}}{\sqrt {\pi Dt}}}{\text{exp}}\left(-{\frac {x^{2}}{4Dt}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb1fea329f3feece75fdad34ded836a71e267b78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.852ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial C}{\partial x}}={\frac {C_{b}}{\sqrt {\pi Dt}}}{\text{exp}}\left(-{\frac {x^{2}}{4Dt}}\right),}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">C</span> is the number concentration of adsorber molecules at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef10a0c06f8a5238d439b9a7bde431605db5190" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.203ex; height:2.343ex;" alt="{\displaystyle x,t}"></span> (#/m<sup>3</sup>). </p><p>The concentration gradient at the subsurface at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> is simplified to the pre-exponential factor of the distribution </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial C}{\partial x}}\right)_{x=0}={\frac {C_{b}}{\sqrt {\pi Dt}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>C</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <msqrt> <mi>π</mi> <mi>D</mi> <mi>t</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial C}{\partial x}}\right)_{x=0}={\frac {C_{b}}{\sqrt {\pi Dt}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42efb3d639d3d6349ccedb52781b4cf0fd389cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.228ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial C}{\partial x}}\right)_{x=0}={\frac {C_{b}}{\sqrt {\pi Dt}}}.}"></span></dd></dl> <p>And the rate of diffusion (flux) across area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a71bf21ad35b8fe05555041d54d1e17eeb0f490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.39ex; height:2.176ex;" alt="{\displaystyle A.}"></span> of the plane is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\partial \Gamma }{\partial t}}\right)_{x=0}=-{\frac {DAC_{b}}{\sqrt {\pi Dt}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mi>A</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <msqrt> <mi>π</mi> <mi>D</mi> <mi>t</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \Gamma }{\partial t}}\right)_{x=0}=-{\frac {DAC_{b}}{\sqrt {\pi Dt}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa74fd918e70f75a5fce173cd651471b23f67064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.958ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial \Gamma }{\partial t}}\right)_{x=0}=-{\frac {DAC_{b}}{\sqrt {\pi Dt}}}.}"></span></dd></dl> <p>Integrating over time, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =\int _{0}^{t}\left({\frac {\partial \Gamma }{\partial t}}\right)_{x=0}=2AC_{b}{\sqrt {\frac {Dt}{\pi }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Γ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>A</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>D</mi> <mi>t</mi> </mrow> <mi>π</mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma =\int _{0}^{t}\left({\frac {\partial \Gamma }{\partial t}}\right)_{x=0}=2AC_{b}{\sqrt {\frac {Dt}{\pi }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c521487942a2b7d35739f2628c8a71520d42ccc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.704ex; height:6.343ex;" alt="{\displaystyle \Gamma =\int _{0}^{t}\left({\frac {\partial \Gamma }{\partial t}}\right)_{x=0}=2AC_{b}{\sqrt {\frac {Dt}{\pi }}}.}"></span></dd></dl> <p>The Langmuir–Schaefer equation can be extended to the Ward–Tordai Equation to account for the "back-diffusion" of rejected molecules from the surface:<sup id="cite_ref-WardTordai1946_14-1" class="reference"><a href="#cite_note-WardTordai1946-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =2A{C_{\text{b}}}{\sqrt {\frac {Dt}{\pi }}}-A{\sqrt {\frac {D}{\pi }}}\int _{0}^{\sqrt {t}}{\frac {C(\tau )}{\sqrt {t-\tau }}}\,d\tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mn>2</mn> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>b</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>D</mi> <mi>t</mi> </mrow> <mi>π</mi> </mfrac> </msqrt> </mrow> <mo>−</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>D</mi> <mi>π</mi> </mfrac> </msqrt> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>t</mi> </msqrt> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <msqrt> <mi>t</mi> <mo>−</mo> <mi>τ</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma =2A{C_{\text{b}}}{\sqrt {\frac {Dt}{\pi }}}-A{\sqrt {\frac {D}{\pi }}}\int _{0}^{\sqrt {t}}{\frac {C(\tau )}{\sqrt {t-\tau }}}\,d\tau ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e903dd121b5fd8d13ea679497941b14f6bb4a6b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:42.395ex; height:7.009ex;" alt="{\displaystyle \Gamma =2A{C_{\text{b}}}{\sqrt {\frac {Dt}{\pi }}}-A{\sqrt {\frac {D}{\pi }}}\int _{0}^{\sqrt {t}}{\frac {C(\tau )}{\sqrt {t-\tau }}}\,d\tau ,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf365bf59c72d2c4836aaadcda126759f293318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.599ex; height:2.509ex;" alt="{\displaystyle C_{b}}"></span> is the bulk concentration, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> is the sub-surface concentration (which is a function of time depending on the reaction model of the adsorption), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> is a dummy variable. </p><p>Monte Carlo simulations show that these two equations work to predict the adsorption rate of systems that form predictable concentration gradients near the surface but have troubles for systems without or with unpredictable concentration gradients, such as typical biosensing systems or when flow and convection are significant.<sup id="cite_ref-JixinMCSimuAdsorption_15-0" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:DiffusiveAdsorptionHistory.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/DiffusiveAdsorptionHistory.jpg/220px-DiffusiveAdsorptionHistory.jpg" decoding="async" width="220" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/DiffusiveAdsorptionHistory.jpg/330px-DiffusiveAdsorptionHistory.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/DiffusiveAdsorptionHistory.jpg/440px-DiffusiveAdsorptionHistory.jpg 2x" data-file-width="5446" data-file-height="4216" /></a><figcaption>A brief history of the theories on diffusive adsorption.<sup id="cite_ref-JixinMCSimuAdsorption_15-1" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>A brief history of diffusive adsorption is shown in the right figure.<sup id="cite_ref-JixinMCSimuAdsorption_15-2" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> A noticeable challenge of understanding the diffusive adsorption at the single-molecule level is the <a href="/wiki/Fractal" title="Fractal">fractal</a> nature of diffusion. Most computer simulations pick a time step for diffusion which ignores the fact that there are self-similar finer diffusion events (fractal) within each step. Simulating the fractal diffusion shows that a factor of two corrections should be introduced for the result of a fixed time-step adsorption simulation, bringing it to be consistent with the above two equations.<sup id="cite_ref-JixinMCSimuAdsorption_15-3" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>A more problematic result of the above equations is they predict the lower limit of adsorption under ideal situations but is very difficult to predict the actual adsorption rates. The equations are derived at the long-time-limit condition when a stable concentration gradient has been formed near the surface. But real adsorption is often done much faster than this infinite time limit i.e. the concentration gradient, decay of concentration at the sub-surface, is only partially formed before the surface has been saturated or flow is on to maintain a certain gradient, thus the adsorption rate measured is almost always faster than the equations have predicted for low or none energy barrier adsorption (unless there is a significant adsorption energy barrier that slows down the absorption significantly), for example, thousands to millions time faster in the self-assembly of monolayers at the water-air or water-substrate interfaces.<sup id="cite_ref-LangmuirSchaefer1937JACS_13-1" class="reference"><a href="#cite_note-LangmuirSchaefer1937JACS-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> As such, it is necessary to calculate the evolution of the concentration gradient near the surface and find out a proper time to stop the imagined infinite evolution for practical applications. While it is hard to predict when to stop but it is reasonably easy to calculate the shortest time that matters, the critical time when the first nearest neighbor from the substrate surface feels the building-up of the concentration gradient. This yields the upper limit of the adsorption rate under an ideal situation when there are no other factors than diffusion that affect the absorber dynamics:<sup id="cite_ref-JixinMCSimuAdsorption_15-4" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r\rangle ={\frac {4}{\pi }}AC_{b}^{4/3}D,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>r</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>π</mi> </mfrac> </mrow> <mi>A</mi> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msubsup> <mi>D</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle r\rangle ={\frac {4}{\pi }}AC_{b}^{4/3}D,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3310d2c1b237d518d62459b8e1218ec21963f16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.935ex; height:5.176ex;" alt="{\displaystyle \langle r\rangle ={\frac {4}{\pi }}AC_{b}^{4/3}D,}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>r</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle r\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f356a9911a7960c689308b20eec4bd62203445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.858ex; height:2.843ex;" alt="{\displaystyle \langle r\rangle }"></span> is the adsorption rate assuming under adsorption energy barrier-free situation, in unit #/s,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is the area of the surface of interest on an "infinite and flat" substrate (m<sup>2</sup>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf365bf59c72d2c4836aaadcda126759f293318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.599ex; height:2.509ex;" alt="{\displaystyle C_{b}}"></span> is the concentration of the absorber molecule in the bulk solution (#/m<sup>3</sup>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the diffusion constant of the absorber (solute) in the solution (m<sup>2</sup>/s) defined with Fick's law.</li></ul> <p>This equation can be used to predict the initial adsorption rate of any system; It can be used to predict the steady-state adsorption rate of a typical biosensing system when the binding site is just a very small fraction of the substrate surface and a near-surface concentration gradient is never formed; It can also be used to predict the adsorption rate of molecules on the surface when there is a significant flow to push the concentration gradient very shallowly in the sub-surface. </p><p>This critical time is significantly different from the first passenger arriving time or the mean free-path time. Using the average first-passenger time and Fick's law of diffusion to estimate the average binding rate will significantly over-estimate the concentration gradient because the first passenger usually comes from many layers of neighbors away from the target, thus its arriving time is significantly longer than the nearest neighbor diffusion time. Using the mean free path time plus the Langmuir equation will cause an artificial concentration gradient between the initial location of the first passenger and the target surface because the other neighbor layers have no change yet, thus significantly lower estimate the actual binding time, i.e., the actual first passenger arriving time itself, the inverse of the above rate, is difficult to calculate. If the system can be simplified to 1D diffusion, then the average first passenger time can be calculated using the same nearest neighbor critical diffusion time for the first neighbor distance to be the MSD,<sup id="cite_ref-Pandey-JPCB2024_16-0" class="reference"><a href="#cite_note-Pandey-JPCB2024-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\sqrt {2Dt}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>D</mi> <mi>t</mi> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\sqrt {2Dt}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7c89908e0e0658ae661e37283da7d2455b06b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.19ex; height:3.009ex;" alt="{\displaystyle L={\sqrt {2Dt}},}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L~=C_{b}^{-1/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mtext> </mtext> <mo>=</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L~=C_{b}^{-1/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33da94e30902a5b4a81a3434db2b5784c114416c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.036ex; height:3.676ex;" alt="{\displaystyle L~=C_{b}^{-1/3}}"></span> (unit m) is the average nearest neighbor distance approximated as cubic packing, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{b}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{b}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf365bf59c72d2c4836aaadcda126759f293318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.599ex; height:2.509ex;" alt="{\displaystyle C_{b}}"></span> is the solute concentration in the bulk solution (unit # molecule / m<sup>3</sup>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the diffusion coefficient defined by Fick's equation (unit m<sup>2</sup>/s),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is the critical time (unit s).</li></ul> <p>In this critical time, it is unlikely the first passenger has arrived and adsorbed. But it sets the speed of the layers of neighbors to arrive. At this speed with a concentration gradient that stops around the first neighbor layer, the gradient does not project virtually in the longer time when the actual first passenger arrives. Thus, the average first passenger coming rate (unit # molecule/s) for this 3D diffusion simplified in 1D problem, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <r>={\frac {a}{t}}=2aC_{b}^{2/3}D,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo><</mo> <mi>r</mi> <mo>>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>a</mi> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msubsup> <mi>D</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle <r>={\frac {a}{t}}=2aC_{b}^{2/3}D,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad8d111721309c02b163b9123b60b152fa8cffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.032ex; height:4.676ex;" alt="{\displaystyle <r>={\frac {a}{t}}=2aC_{b}^{2/3}D,}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is a factor of converting the 3D diffusive adsorption problem into a 1D diffusion problem whose value depends on the system, e.g., a fraction of adsorption area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> over solute nearest neighbor sphere surface area <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\pi L^{2}/4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\pi L^{2}/4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b005641d17af81b6532a13cd1417c8b2c00437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.456ex; height:3.176ex;" alt="{\displaystyle 4\pi L^{2}/4}"></span> assuming cubic packing each unit has 8 neighbors shared with other units. This example fraction converges the result to the 3D diffusive adsorption solution shown above with a slight difference in pre-factor due to different packing assumptions and ignoring other neighbors. </p><p>When the area of interest is the size of a molecule (specifically, a <i>long cylindrical molecule</i> such as DNA), the adsorption rate equation represents the collision frequency of two molecules in a diluted solution, with one molecule a specific side and the other no steric dependence, i.e., a molecule (random orientation) hit one side of the other. The diffusion constant need to be updated to the relative diffusion constant between two diffusing molecules. This estimation is especially useful in studying the interaction between a small molecule and a larger molecule such as a protein. The effective diffusion constant is dominated by the smaller one whose diffusion constant can be used instead. </p><p>The above hitting rate equation is also useful to predict the kinetics of molecular <a href="/wiki/Self-assembly" title="Self-assembly">self-assembly</a> on a surface. Molecules are randomly oriented in the bulk solution. Assuming 1/6 of the molecules has the right orientation to the surface binding sites, i.e. 1/2 of the z-direction in x, y, z three dimensions, thus the concentration of interest is just 1/6 of the bulk concentration. Put this value into the equation one should be able to calculate the theoretical adsorption kinetic curve using the <a href="/wiki/Langmuir_adsorption_model" title="Langmuir adsorption model">Langmuir adsorption model</a>. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:JChen2022JPCA.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/JChen2022JPCA.png/220px-JChen2022JPCA.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/JChen2022JPCA.png/330px-JChen2022JPCA.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/JChen2022JPCA.png/440px-JChen2022JPCA.png 2x" data-file-width="1834" data-file-height="1394" /></a><figcaption>Comparing collision theory and diffusive collision theory.<sup id="cite_ref-JChen2022JPCA_17-0" class="reference"><a href="#cite_note-JChen2022JPCA-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>The bimolecular collision frequency related to many reactions including protein coagulation/aggregation is initially described by <a href="/wiki/Smoluchowski_coagulation_equation" title="Smoluchowski coagulation equation">Smoluchowski coagulation equation</a> proposed by <a href="/wiki/Marian_Smoluchowski" title="Marian Smoluchowski">Marian Smoluchowski</a> in a seminal 1916 publication,<sup id="cite_ref-Smoluchowski1916_18-0" class="reference"><a href="#cite_note-Smoluchowski1916-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> derived from <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> and Fick's laws of diffusion. Under an idealized reaction condition for A + B → product in a diluted solution, Smoluchovski suggested that the molecular flux at the infinite time limit can be calculated from Fick's laws of diffusion yielding a fixed/stable concentration gradient from the target molecule, e.g. B is the target molecule holding fixed relatively, and A is the moving molecule that creates a concentration gradient near the target molecule B due to the coagulation reaction between A and B. Smoluchowski calculated the collision frequency between A and B in the solution with unit #/s/m<sup>3</sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{AB}=4{\pi }RD_{r}C_{A}C_{B},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> <mi>R</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{AB}=4{\pi }RD_{r}C_{A}C_{B},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c278bc2b41e0cb00591971a1bc1282b0141f0d10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.469ex; height:2.509ex;" alt="{\displaystyle Z_{AB}=4{\pi }RD_{r}C_{A}C_{B},}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"></span> is the radius of the collision,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{r}=D_{A}+D_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{r}=D_{A}+D_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/955de844fb0e9e28c342b2a258c0cb841b6eb2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.63ex; height:2.509ex;" alt="{\displaystyle D_{r}=D_{A}+D_{B}}"></span> is the relative diffusion constant between A and B (m<sup>2</sup>/s),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c5f6fde0832b5aa24cc8d90aca913e132e9929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle C_{A}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e66cdda0ceb560bfab11747176e76650cf35d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.141ex; height:2.509ex;" alt="{\displaystyle C_{B}}"></span> are number concentrations of A and B respectively (#/m<sup>3</sup>).</li></ul> <p>The reaction order of this bimolecular reaction is 2 which is the analogy to the result from <a href="/wiki/Collision_theory" title="Collision theory">collision theory</a> by replacing the moving speed of the molecule with diffusive flux. In the collision theory, the traveling time between A and B is proportional to the distance which is a similar relationship for the diffusion case if the flux is fixed. </p><p>However, under a practical condition, the concentration gradient near the target molecule is evolving over time with the molecular flux evolving as well,<sup id="cite_ref-JixinMCSimuAdsorption_15-5" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and on average the flux is much bigger than the infinite time limit flux Smoluchowski has proposed. Before the first passenger arrival time, Fick's equation predicts a concentration gradient over time which does not build up yet in reality. Thus, this Smoluchowski frequency represents the lower limit of the real collision frequency. </p><p>In 2022, Chen calculates the upper limit of the collision frequency between A and B in a solution assuming the bulk concentration of the moving molecule is fixed after the first nearest neighbor of the target molecule.<sup id="cite_ref-JChen2022JPCA_17-1" class="reference"><a href="#cite_note-JChen2022JPCA-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Thus the concentration gradient evolution stops at the first nearest neighbor layer given a stop-time to calculate the actual flux. He named this the critical time and derived the diffusive collision frequency in unit #/s/m<sup>3</sup>:<sup id="cite_ref-JChen2022JPCA_17-2" class="reference"><a href="#cite_note-JChen2022JPCA-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{AB}={\frac {8}{\pi }}{\sigma }D_{r}C_{A}C_{B}{\sqrt[{3}]{C_{A}+C_{B}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mi>π</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{AB}={\frac {8}{\pi }}{\sigma }D_{r}C_{A}C_{B}{\sqrt[{3}]{C_{A}+C_{B}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41903d0226d67be96d7e01b2a9ba545a79a020a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.141ex; height:5.176ex;" alt="{\displaystyle Z_{AB}={\frac {8}{\pi }}{\sigma }D_{r}C_{A}C_{B}{\sqrt[{3}]{C_{A}+C_{B}}},}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314ac5cfd38ebefe34513c781f56fa6510fa4fb6" 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 }}"></span> is the area of the cross-section of the collision (m<sup>2</sup>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{r}=D_{A}+D_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{r}=D_{A}+D_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/955de844fb0e9e28c342b2a258c0cb841b6eb2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.63ex; height:2.509ex;" alt="{\displaystyle D_{r}=D_{A}+D_{B}}"></span> is the relative diffusion constant between A and B (m<sup>2</sup>/s),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c5f6fde0832b5aa24cc8d90aca913e132e9929" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.127ex; height:2.509ex;" alt="{\displaystyle C_{A}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{B}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{B}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e66cdda0ceb560bfab11747176e76650cf35d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.141ex; height:2.509ex;" alt="{\displaystyle C_{B}}"></span> are number concentrations of A and B respectively (#/m<sup>3</sup>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{3}]{C_{A}+C_{B}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{3}]{C_{A}+C_{B}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34a686f0e8e4588a7ffa30f7912c08596bf62177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.432ex; height:3.343ex;" alt="{\displaystyle {\sqrt[{3}]{C_{A}+C_{B}}}}"></span> represents 1/<d>, where d is the average distance between two molecules.</li></ul> <p>This equation assumes the upper limit of a diffusive collision frequency between A and B is when the first neighbor layer starts to feel the evolution of the concentration gradient, whose reaction order is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠2<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span> instead of 2. Both the Smoluchowski equation and the JChen equation satisfy dimensional checks with SI units. But the former is dependent on the radius and the latter is on the area of the collision sphere. From dimensional analysis, there will be an equation dependent on the volume of the collision sphere but eventually, all equations should converge to the same numerical rate of the collision that can be measured experimentally. The actual reaction order for a bimolecular unit reaction could be between 2 and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠2<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, which makes sense because the diffusive collision time is squarely dependent on the distance between the two molecules. </p><p>These new equations also avoid the singularity on the adsorption rate at time zero for the Langmuir-Schaefer equation. The infinity rate is justifiable under ideal conditions because when you introduce target molecules magically in a solution of probe molecule or vice versa, there always be a probability of them overlapping at time zero, thus the rate of that two molecules association is infinity. It does not matter that other millions of molecules have to wait for their first mate to diffuse and arrive. The average rate is thus infinity. But statistically this argument is meaningless. The maximum rate of a molecule in a period of time larger than zero is 1, either meet or not, thus the infinite rate at time zero for that molecule pair really should just be one, making the average rate 1/millions or more and statistically negligible. This does not even count in reality no two molecules can magically meet at time zero. </p> <div class="mw-heading mw-heading3"><h3 id="Biological_perspective">Biological perspective</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=14" title="Edit section: Biological perspective"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first law gives rise to the following formula:<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>flux</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e863e91acc8f39d0954b2d3441a50cf057857b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.337ex; height:2.843ex;" alt="{\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)},}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">P</span> is the permeability, an experimentally determined membrane "<a href="/wiki/Electrical_conductance" class="mw-redirect" title="Electrical conductance">conductance</a>" for a given gas at a given temperature,</li> <li><span class="texhtml"><i>c</i><sub>2</sub> − <i>c</i><sub>1</sub></span> is the difference in <a href="/wiki/Concentration" title="Concentration">concentration</a> of the gas across the <a href="/wiki/Artificial_membrane" class="mw-redirect" title="Artificial membrane">membrane</a> for the direction of flow (from <span class="texhtml"><i>c</i><sub>1</sub></span> to <span class="texhtml"><i>c</i><sub>2</sub></span>).</li></ul> <p>Fick's first law is also important in radiation transfer equations. However, in this context, it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a <a href="/wiki/Flux_limiter" title="Flux limiter">flux limiter</a>. </p><p>The exchange rate of a gas across a fluid membrane can be determined by using this law together with <a href="/wiki/Graham%27s_law" title="Graham's law">Graham's law</a>. </p><p>Under the condition of a diluted solution when diffusion takes control, the membrane permeability mentioned in the above section can be theoretically calculated for the solute using the equation mentioned in the last section (use with particular care because the equation is derived for dense solutes, while biological molecules are not denser than water. Also, this equation assumes ideal concentration gradient forms near the membrane and evolves):<sup id="cite_ref-Pyle-BJNano_12-1" class="reference"><a href="#cite_note-Pyle-BJNano-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=2A_{p}\eta _{tm}{\sqrt {\frac {D}{\pi t}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>D</mi> <mrow> <mi>π</mi> <mi>t</mi> </mrow> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=2A_{p}\eta _{tm}{\sqrt {\frac {D}{\pi t}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/034599bf10556e0eae50ac193f8c93ee84c5616f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.211ex; height:6.176ex;" alt="{\displaystyle P=2A_{p}\eta _{tm}{\sqrt {\frac {D}{\pi t}}},}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{P}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{P}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8449430a7248eecf6b4bfff33f4098664e31c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.21ex; height:2.509ex;" alt="{\displaystyle A_{P}}"></span> is the total area of the pores on the membrane (unit m<sup>2</sup>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{tm}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{tm}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200c6057ad3972d6e08c51adccea55d423a7743e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.424ex; height:2.176ex;" alt="{\displaystyle \eta _{tm}}"></span> transmembrane efficiency (unitless), which can be calculated from the stochastic theory of <a href="/wiki/Chromatography" title="Chromatography">chromatography</a>,</li> <li><i>D</i> is the diffusion constant of the solute unit m<sup>2</sup>⋅s<sup>−1</sup>,</li> <li><i>t</i> is time unit s,</li> <li><i>c</i><sub>2</sub>, <i>c</i><sub>1</sub> concentration should use unit mol m<sup>−3</sup>, so flux unit becomes mol s<sup>−1</sup>.</li></ul> <p>The flux is decay over the square root of time because a concentration gradient builds up near the membrane over time under ideal conditions. When there is flow and convection, the flux can be significantly different than the equation predicts and show an effective time t with a fixed value,<sup id="cite_ref-JixinMCSimuAdsorption_15-6" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> which makes the flux stable instead of decay over time. A critical time has been estimated under idealized flow conditions when there is no gradient formed.<sup id="cite_ref-JixinMCSimuAdsorption_15-7" class="reference"><a href="#cite_note-JixinMCSimuAdsorption-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-JChen2022JPCA_17-3" class="reference"><a href="#cite_note-JChen2022JPCA-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This strategy is adopted in biology such as blood circulation. </p> <div class="mw-heading mw-heading3"><h3 id="Semiconductor_fabrication_applications">Semiconductor fabrication applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=15" title="Edit section: Semiconductor fabrication applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Semiconductor" title="Semiconductor">semiconductor</a> is a collective term for a series of devices. It mainly includes three categories：two-terminal devices, three-terminal devices, and four-terminal devices. The combination of the semiconductors is called an integrated circuit. </p><p>The relationship between Fick's law and semiconductors: the principle of the semiconductor is transferring chemicals or dopants from a layer to a layer. Fick's law can be used to control and predict the diffusion by knowing how much the concentration of the dopants or chemicals move per meter and second through mathematics. </p><p>Therefore, different types and levels of semiconductors can be fabricated. </p><p><a href="/wiki/Integrated_circuit" title="Integrated circuit">Integrated circuit</a> fabrication technologies, model processes like CVD, thermal oxidation, wet oxidation, doping, etc. use diffusion equations obtained from Fick's law. </p> <div class="mw-heading mw-heading4"><h4 id="CVD_method_of_fabricate_semiconductor">CVD method of fabricate semiconductor</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=16" title="Edit section: CVD method of fabricate semiconductor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The wafer is a kind of semiconductor whose silicon substrate is coated with a layer of CVD-created polymer chain and films. This film contains n-type and p-type dopants and takes responsibility for dopant conductions. The principle of CVD relies on the gas phase and gas-solid chemical reaction to create thin films. </p><p>The viscous flow regime of CVD is driven by a pressure gradient. CVD also includes a diffusion component distinct from the surface diffusion of adatoms. In CVD, reactants and products must also diffuse through a boundary layer of stagnant gas that exists next to the substrate. The total number of steps required for CVD film growth are gas phase diffusion of reactants through the boundary layer, adsorption and surface diffusion of adatoms, reactions on the substrate, and gas phase diffusion of products away through the boundary layer. </p><p>The velocity profile for gas flow is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)=\left({\frac {5x}{\mathrm {Re} ^{1/2}}}\right)\mathrm {Re} ={\frac {v\rho L}{\eta }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mi>x</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mi>ρ</mi> <mi>L</mi> </mrow> <mi>η</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (x)=\left({\frac {5x}{\mathrm {Re} ^{1/2}}}\right)\mathrm {Re} ={\frac {v\rho L}{\eta }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cad9a1f90555b34fabf52943b85ef5aa512600f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.608ex; height:6.176ex;" alt="{\displaystyle \delta (x)=\left({\frac {5x}{\mathrm {Re} ^{1/2}}}\right)\mathrm {Re} ={\frac {v\rho L}{\eta }},}"></span> where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> is the thickness,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9a978f729af1a34028e60d45f484f0908e4be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }"></span> is the Reynolds number,</li> <li><span class="texhtml mvar" style="font-style:italic;">x</span> is the length of the substrate,</li> <li><span class="texhtml"><i>v</i> = 0</span> at any surface,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is viscosity,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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></span><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 }"></span> is density.</li></ul> <p>Integrated the <span class="texhtml mvar" style="font-style:italic;">x</span> from <span class="texhtml">0</span> to <span class="texhtml mvar" style="font-style:italic;">L</span>, it gives the average thickness: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\frac {10L}{3\mathrm {Re} ^{1/2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>10</mn> <mi>L</mi> </mrow> <mrow> <mn>3</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {10L}{3\mathrm {Re} ^{1/2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5c235e080751920103643c0c14f1f687d7318f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.234ex; height:5.843ex;" alt="{\displaystyle \delta ={\frac {10L}{3\mathrm {Re} ^{1/2}}}.}"></span> </p><p>To keep the reaction balanced, reactants must diffuse through the stagnant boundary layer to reach the substrate. So a thin boundary layer is desirable. According to the equations, increasing vo would result in more wasted reactants. The reactants will not reach the substrate uniformly if the flow becomes turbulent. Another option is to switch to a new carrier gas with lower viscosity or density. </p><p>The Fick's first law describes diffusion through the boundary layer. As a function of pressure (<i>p</i>) and temperature (<i>T</i>) in a gas, diffusion is determined. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=D_{0}\left({\frac {p_{0}}{p}}\right)\left({\frac {T}{T_{0}}}\right)^{3/2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=D_{0}\left({\frac {p_{0}}{p}}\right)\left({\frac {T}{T_{0}}}\right)^{3/2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a44e8235a161cba42477214a3666c446b00d212" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.271ex; height:6.676ex;" alt="{\displaystyle D=D_{0}\left({\frac {p_{0}}{p}}\right)\left({\frac {T}{T_{0}}}\right)^{3/2},}"></span> where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" 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:2.009ex;" alt="{\displaystyle p_{0}}"></span> is the standard pressure,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"></span> is the standard temperature,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b943acf3ade50743482c4ee4570ad3de335d90d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.979ex; height:2.509ex;" alt="{\displaystyle D_{0}}"></span> is the standard diffusitivity.</li></ul> <p>The equation tells that increasing the temperature or decreasing the pressure can increase the diffusivity. </p><p>Fick's first law predicts the flux of the reactants to the substrate and product away from the substrate: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D_{i}\left({\frac {dc_{i}}{dx}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D_{i}\left({\frac {dc_{i}}{dx}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898a51d7f2a64a95f7b5c573fb187711b4305826" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.803ex; height:6.176ex;" alt="{\displaystyle J=-D_{i}\left({\frac {dc_{i}}{dx}}\right),}"></span> where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> is the thickness <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span>,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dc_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dc_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33066cd92f987850afc61913fd3abb992208d14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.022ex; height:2.509ex;" alt="{\displaystyle dc_{i}}"></span> is the first reactant's concentration.</li></ul> <p>In ideal gas law <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pV=nRT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>V</mi> <mo>=</mo> <mi>n</mi> <mi>R</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pV=nRT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c364f7f0b47178cfd35c3bbaf6dcb22a98cf44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.94ex; height:2.509ex;" alt="{\displaystyle pV=nRT}"></span>, the concentration of the gas is expressed by partial pressure. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D_{i}\left({\frac {p_{i}-p_{0}}{\delta RT}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>δ</mi> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D_{i}\left({\frac {p_{i}-p_{0}}{\delta RT}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6905d4535e9060a23aa4169321a3d92d0563ca85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.813ex; height:6.176ex;" alt="{\displaystyle J=-D_{i}\left({\frac {p_{i}-p_{0}}{\delta RT}}\right),}"></span> where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"></span> is the gas constant,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p_{i}-p_{0}}{\delta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>δ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{i}-p_{0}}{\delta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7228c366e4f175ca2c05c12bafe601a4af06684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.869ex; height:5.343ex;" alt="{\displaystyle {\frac {p_{i}-p_{0}}{\delta }}}"></span> is the partial pressure gradient.</li></ul> <p>As a result, Fick's first law tells us we can use a partial pressure gradient to control the diffusivity and control the growth of thin films of semiconductors. </p><p>In many realistic situations, the simple Fick's law is not an adequate formulation for the semiconductor problem. It only applies to certain conditions, for example, given the semiconductor boundary conditions: constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate). </p> <div class="mw-heading mw-heading4"><h4 id="Invalidity_of_Fickian_diffusion">Invalidity of Fickian diffusion</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=17" title="Edit section: Invalidity of Fickian diffusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Even though Fickian diffusion has been used to model diffusion processes in semiconductor manufacturing (including CVD reactors) in early days, it often fails to validate the diffusion in advanced semiconductor nodes (< 90 nm). This mostly stems from the inability of Fickian diffusion to model diffusion processes accurately at molecular level and smaller. In advanced semiconductor manufacturing, it is important to understand the movement at atomic scales, which is failed by continuum diffusion. Today, most semiconductor manufacturers use <a href="/wiki/Random_walk" title="Random walk">random walk</a> to study and model diffusion processes. This allows us to study the effects of diffusion in a discrete manner to understand the movement of individual atoms, molecules, plasma etc. </p><p>In such a process, the movements of diffusing species (atoms, molecules, plasma etc.) are treated as a discrete entity, following a random walk through the CVD reactor, boundary layer, material structures etc. Sometimes, the movements might follow a biased-random walk depending on the processing conditions. Statistical analysis is done to understand variation/stochasticity arising from the random walk of the species, which in-turn affects the overall process and electrical variations. </p> <div class="mw-heading mw-heading3"><h3 id="Food_production_and_cooking">Food production and cooking</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=18" title="Edit section: Food production and cooking"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formulation of Fick's first law can explain a variety of complex phenomena in the context of food and cooking: Diffusion of molecules such as ethylene promotes plant growth and ripening, salt and sugar molecules promotes meat brining and marinating, and water molecules promote dehydration. Fick's first law can also be used to predict the changing moisture profiles across a spaghetti noodle as it hydrates during cooking. These phenomena are all about the spontaneous movement of particles of solutes driven by the concentration gradient. In different situations, there is different diffusivity which is a constant.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>By controlling the concentration gradient, the cooking time, shape of the food, and salting can be controlled. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Advection" title="Advection">Advection</a></li> <li><a href="/wiki/Churchill%E2%80%93Bernstein_equation" title="Churchill–Bernstein equation">Churchill–Bernstein equation</a></li> <li><a href="/wiki/Diffusion" title="Diffusion">Diffusion</a></li> <li><a href="/wiki/False_diffusion" title="False diffusion">False diffusion</a></li> <li><a href="/wiki/Gas_exchange" title="Gas exchange">Gas exchange</a></li> <li><a href="/wiki/Mass_flux" title="Mass flux">Mass flux</a></li> <li><a href="/wiki/Maxwell%E2%80%93Stefan_diffusion" title="Maxwell–Stefan diffusion">Maxwell–Stefan diffusion</a></li> <li><a href="/wiki/Nernst%E2%80%93Planck_equation" title="Nernst–Planck equation">Nernst–Planck equation</a></li> <li><a href="/wiki/Osmosis" title="Osmosis">Osmosis</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=20" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output 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Archived from <a rel="nofollow" class="external text" href="http://humanphysiology.tuars.com/program/section3/3ch9/s3ch9_2.htm">the original</a> on 24 March 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+3%2F3ch9%2Fs3ch9_2&rft.btitle=Essentials+of+Human+Physiology&rft.aulast=Nosek&rft.aufirst=TM&rft_id=http%3A%2F%2Fhumanphysiology.tuars.com%2Fprogram%2Fsection3%2F3ch9%2Fs3ch9_2.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZhouNybergRowat2015" class="citation journal cs1">Zhou L, Nyberg K, Rowat AC (September 2015). <a rel="nofollow" class="external text" href="http://www.escholarship.org/uc/item/1t87565r">"Understanding diffusion theory and Fick's law through food and cooking"</a>. <i>Advances in Physiology Education</i>. <b>39</b> (3): 192–197. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1152%2Fadvan.00133.2014">10.1152/advan.00133.2014</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26330037">26330037</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3921833">3921833</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Physiology+Education&rft.atitle=Understanding+diffusion+theory+and+Fick%27s+law+through+food+and+cooking&rft.volume=39&rft.issue=3&rft.pages=192-197&rft.date=2015-09&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A3921833%23id-name%3DS2CID&rft_id=info%3Apmid%2F26330037&rft_id=info%3Adoi%2F10.1152%2Fadvan.00133.2014&rft.aulast=Zhou&rft.aufirst=L&rft.au=Nyberg%2C+K&rft.au=Rowat%2C+AC&rft_id=http%3A%2F%2Fwww.escholarship.org%2Fuc%2Fitem%2F1t87565r&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=21" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerg1977" class="citation book cs1">Berg HC (1977). <i>Random Walks in Biology</i>. Princeton.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Random+Walks+in+Biology&rft.pub=Princeton&rft.date=1977&rft.aulast=Berg&rft.aufirst=HC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirdStewartLightfoot1976" class="citation book cs1">Bird RB, Stewart WE, Lightfoot EN (1976). <i>Transport Phenomena</i>. John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transport+Phenomena&rft.pub=John+Wiley+%26+Sons&rft.date=1976&rft.aulast=Bird&rft.aufirst=RB&rft.au=Stewart%2C+WE&rft.au=Lightfoot%2C+EN&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBokshteinMendelevSrolovitz2005" class="citation book cs1">Bokshtein BS, Mendelev MI, Srolovitz DJ, eds. (2005). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/thermodynamicski00boks"><i>Thermodynamics and Kinetics in Materials Science: A Short Course</i></a></span>. Oxford: Oxford University Press. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/thermodynamicski00boks/page/n181">167</a>–171.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+and+Kinetics+in+Materials+Science%3A+A+Short+Course&rft.place=Oxford&rft.pages=167-171&rft.pub=Oxford+University+Press&rft.date=2005&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fthermodynamicski00boks&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrank1980" class="citation book cs1">Crank J (1980). <i>The Mathematics of Diffusion</i>. Oxford University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematics+of+Diffusion&rft.pub=Oxford+University+Press&rft.date=1980&rft.aulast=Crank&rft.aufirst=J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFick1855" class="citation journal cs1">Fick A (1855). "On liquid diffusion". <i>Annalen der Physik und Chemie</i>. <b>94</b>: 59.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik+und+Chemie&rft.atitle=On+liquid+diffusion&rft.volume=94&rft.pages=59&rft.date=1855&rft.aulast=Fick&rft.aufirst=A&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span> – reprinted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFick1995" class="citation journal cs1">Fick, Adolph (1995). "On liquid diffusion". <i>Journal of Membrane Science</i>. <b>100</b>: 33–38. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0376-7388%2894%2900230-v">10.1016/0376-7388(94)00230-v</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Membrane+Science&rft.atitle=On+liquid+diffusion&rft.volume=100&rft.pages=33-38&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0376-7388%2894%2900230-v&rft.aulast=Fick&rft.aufirst=Adolph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2004" class="citation book cs1">Smith WF (2004). <i>Foundations of Materials Science and Engineering</i> (3rd ed.). McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Materials+Science+and+Engineering&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=2004&rft.aulast=Smith&rft.aufirst=WF&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFick%27s+laws+of+diffusion" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fick%27s_laws_of_diffusion&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://dragon.unideb.hu/~zerdelyi/Diffusion-on-the-nanoscale/node2.html">Fick's equations, Boltzmann's transformation, etc.</a> (with figures and animations)</li> <li><a rel="nofollow" class="external text" href="http://cnx.org/content/m1036/2.11/">Fick's Second Law</a> on <a href="/wiki/OpenStax" title="OpenStax">OpenStax</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Fick%27s_laws_of_diffusion&oldid=1251815959">https://en.wikipedia.org/w/index.php?title=Fick%27s_laws_of_diffusion&oldid=1251815959</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Diffusion" title="Category:Diffusion">Diffusion</a></li><li><a href="/wiki/Category:Eponymous_laws_of_physics" title="Category:Eponymous laws of physics">Eponymous laws of physics</a></li><li><a href="/wiki/Category:Mathematics_in_medicine" title="Category:Mathematics in medicine">Mathematics in medicine</a></li><li><a href="/wiki/Category:Physical_chemistry" title="Category:Physical chemistry">Physical chemistry</a></li><li><a href="/wiki/Category:Statistical_mechanics" title="Category:Statistical mechanics">Statistical mechanics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Wikipedia_articles_needing_page_number_citations_from_September_2024" title="Category:Wikipedia articles needing page number citations from September 2024">Wikipedia articles needing page number citations from September 2024</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Use_dmy_dates_from_July_2020" title="Category:Use dmy dates from July 2020">Use dmy dates from July 2020</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 October 2024, at 06:53<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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Fick's laws of diffusion - Wikipedia