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class="vector-toc-numb">1.3</span> <span>Stern</span> </div> </a> <ul id="toc-Stern-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grahame" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grahame"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Grahame</span> </div> </a> <ul id="toc-Grahame-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bockris/Devanathan/Müller_(BDM)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bockris/Devanathan/Müller_(BDM)"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Bockris/Devanathan/Müller (BDM)</span> </div> </a> <ul id="toc-Bockris/Devanathan/Müller_(BDM)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trasatti/Buzzanca" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trasatti/Buzzanca"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> 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id="toc-Mathematical_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrical_double_layers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electrical_double_layers"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Electrical double layers</span> </div> </a> <button aria-controls="toc-Electrical_double_layers-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 Electrical double layers subsection</span> </button> <ul id="toc-Electrical_double_layers-sublist" class="vector-toc-list"> <li id="toc-Differential_capacitance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Differential_capacitance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Differential capacitance</span> </div> </a> <ul id="toc-Differential_capacitance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electron_transfer_in_electrical_double_layer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electron_transfer_in_electrical_double_layer"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Electron transfer in electrical double layer</span> </div> </a> <ul id="toc-Electron_transfer_in_electrical_double_layer-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">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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">7</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 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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Double layer (surface science)</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 22 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-22" 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">22 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B7%D8%A8%D9%82%D8%A9_%D8%AB%D9%86%D8%A7%D8%A6%D9%8A%D8%A9" title="طبقة ثنائية – Arabic" lang="ar" hreflang="ar" data-title="طبقة ثنائية" 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/Doble_capa_el%C3%A8ctrica" title="Doble capa elèctrica – Catalan" lang="ca" hreflang="ca" data-title="Doble capa elèctrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elektrochemische_Doppelschicht" title="Elektrochemische Doppelschicht – German" lang="de" hreflang="de" data-title="Elektrochemische Doppelschicht" 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/Elektriline_kaksikkiht" title="Elektriline kaksikkiht – Estonian" lang="et" hreflang="et" data-title="Elektriline kaksikkiht" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Doble_capa_el%C3%A9ctrica" title="Doble capa eléctrica – Spanish" lang="es" hreflang="es" data-title="Doble capa eléctrica" 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%84%D8%A7%DB%8C%D9%87_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C_%D9%85%D8%B6%D8%A7%D8%B9%D9%81" title="لایه الکتریکی مضاعف – Persian" lang="fa" hreflang="fa" data-title="لایه الکتریکی مضاعف" 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/Double_couche_%C3%A9lectrique" title="Double couche électrique – French" lang="fr" hreflang="fr" data-title="Double couche électrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lapisan_ganda_(ilmu_permukaan)" title="Lapisan ganda (ilmu permukaan) – Indonesian" lang="id" hreflang="id" data-title="Lapisan ganda (ilmu permukaan)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Doppio_strato_elettrico" title="Doppio strato elettrico – Italian" lang="it" hreflang="it" data-title="Doppio strato elettrico" 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%A9%D7%9B%D7%91%D7%94_%D7%9B%D7%A4%D7%95%D7%9C%D7%94_%D7%97%D7%A9%D7%9E%D7%9C%D7%99%D7%AA" title="שכבה כפולה חשמלית – Hebrew" lang="he" hreflang="he" data-title="שכבה כפולה חשמלית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D2%9A%D0%BE%D1%81_%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BB%D1%96%D0%BA_%D2%9B%D0%B0%D0%B1%D0%B0%D1%82" title="Қос электрлік қабат – Kazakh" lang="kk" hreflang="kk" data-title="Қос электрлік қабат" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elektrochemische_dubbellaag" title="Elektrochemische dubbellaag – Dutch" lang="nl" hreflang="nl" data-title="Elektrochemische dubbellaag" 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/%E9%9B%BB%E6%B0%97%E4%BA%8C%E9%87%8D%E5%B1%A4" title="電気二重層 – Japanese" lang="ja" hreflang="ja" data-title="電気二重層" 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/Elektrisk_dobbeltlag" title="Elektrisk dobbeltlag – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elektrisk dobbeltlag" 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/Elektrisk_dobbeltlag" title="Elektrisk dobbeltlag – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elektrisk dobbeltlag" 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/Podw%C3%B3jna_warstwa_elektryczna" title="Podwójna warstwa elektryczna – Polish" lang="pl" hreflang="pl" data-title="Podwójna warstwa elektryczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Strat_dublu_electric" title="Strat dublu electric – Romanian" lang="ro" hreflang="ro" data-title="Strat dublu electric" 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%94%D0%B2%D0%BE%D0%B9%D0%BD%D0%BE%D0%B9_%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D1%81%D0%BB%D0%BE%D0%B9" title="Двойной электрический слой – Russian" lang="ru" hreflang="ru" data-title="Двойной электрический слой" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elektriska_dubbelskiktet" title="Elektriska dubbelskiktet – Swedish" lang="sv" hreflang="sv" data-title="Elektriska dubbelskiktet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elektriksel_%C3%A7ift_tabaka" title="Elektriksel çift tabaka – Turkish" lang="tr" hreflang="tr" data-title="Elektriksel çift tabaka" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%B2%D1%96%D0%B9%D0%BD%D0%B8%D0%B9_%D0%B5%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D1%88%D0%B0%D1%80" title="Подвійний електричний шар – Ukrainian" lang="uk" hreflang="uk" data-title="Подвійний електричний шар" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%8F%8C%E7%94%B5%E5%B1%82" title="双电层 – Chinese" lang="zh" hreflang="zh" data-title="双电层" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> 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data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Molecular interface between a surface and a fluid</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Double_Layer.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Double_Layer.png/300px-Double_Layer.png" decoding="async" width="300" height="261" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Double_Layer.png/450px-Double_Layer.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Double_Layer.png/600px-Double_Layer.png 2x" data-file-width="2224" data-file-height="1932" /></a><figcaption>Schematic of the electrical double layer (EDL) in <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solution</a> at the interface with a negatively-charged surface of a mineral solid. Blue + sphere: cations; red – spheres: anions. The number of cations is larger in the EDL close to the negatively-charged surface in order to neutralize these negative charges and to maintain electroneutrality. The drawing does not explicitly show the negative charges of the surface.</figcaption></figure> <p>In <a href="/wiki/Surface_science" title="Surface science">surface science</a>, a <b>double layer</b> (<b>DL</b>, also called an <b>electrical double layer</b>, <b>EDL</b>) is a structure that appears on the <a href="/wiki/Surface" title="Surface">surface</a> of an object when it is exposed to a <a href="/wiki/Fluid" title="Fluid">fluid</a>. The object might be a <a href="/wiki/Solid" title="Solid">solid</a> particle, a <a href="/wiki/Gas_bubble" class="mw-redirect" title="Gas bubble">gas bubble</a>, a liquid <a href="/wiki/Droplet" class="mw-redirect" title="Droplet">droplet</a>, or a <a href="/wiki/Porous_media" class="mw-redirect" title="Porous media">porous body</a>. The DL refers to two parallel layers of <a href="/wiki/Electric_charge" title="Electric charge">charge</a> surrounding the object. The first layer, the <a href="/wiki/Surface_charge" title="Surface charge">surface charge</a> (either positive or negative), consists of <a href="/wiki/Ion" title="Ion">ions</a> which are <a href="/wiki/Adsorbed" class="mw-redirect" title="Adsorbed">adsorbed</a> onto the object due to <a href="/wiki/Intermolecular_force" title="Intermolecular force">chemical interactions</a>. The second layer is composed of ions attracted to the surface charge via the <a href="/wiki/Coulomb_force" class="mw-redirect" title="Coulomb force">Coulomb force</a>, electrically <a href="/wiki/Electric-field_screening" title="Electric-field screening">screening</a> the first layer. This second layer is loosely associated with the object. It is made of free ions that move in the fluid under the influence of <a href="/wiki/Electromagnetism" title="Electromagnetism">electric attraction</a> and <a href="/wiki/Thermal_motion" class="mw-redirect" title="Thermal motion">thermal motion</a> rather than being firmly anchored. It is thus called the "diffuse layer". </p><p><a href="/wiki/Interface_(chemistry)" class="mw-redirect" title="Interface (chemistry)">Interfacial</a> DLs are most apparent in systems with a large <a href="/wiki/Surface-area-to-volume_ratio" title="Surface-area-to-volume ratio">surface-area-to-volume ratio</a>, such as a <a href="/wiki/Colloid" title="Colloid">colloid</a> or porous bodies with particles or pores (respectively) on the scale of <a href="/wiki/Micrometres" class="mw-redirect" title="Micrometres">micrometres</a> to <a href="/wiki/Nanometres" class="mw-redirect" title="Nanometres">nanometres</a>. However, DLs are important to other phenomena, such as the <a href="/wiki/Electrochemical" class="mw-redirect" title="Electrochemical">electrochemical</a> behaviour of <a href="/wiki/Electrode" title="Electrode">electrodes</a>. </p><p>DLs play a fundamental role in many everyday substances. For instance, <a href="/wiki/Homogenized_milk" class="mw-redirect" title="Homogenized milk">homogenized milk</a> exists only because <a href="/wiki/Fat" title="Fat">fat</a> droplets are covered with a DL that prevents their <a href="/wiki/Coagulation_(milk)" class="mw-redirect" title="Coagulation (milk)">coagulation</a> into <a href="/wiki/Butter" title="Butter">butter</a>. DLs exist in practically all <a href="/wiki/Heterogeneous" class="mw-redirect" title="Heterogeneous">heterogeneous</a> fluid-based systems, such as blood, paint, ink and ceramic and cement <a href="/wiki/Slurry" title="Slurry">slurry</a>. </p><p>The DL is closely related to <a href="/wiki/Electrokinetic_phenomena" title="Electrokinetic phenomena">electrokinetic phenomena</a> and <a href="/wiki/Electroacoustic_phenomena" title="Electroacoustic phenomena">electroacoustic phenomena</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Development_of_the_(interfacial)_double_layer"><span id="Development_of_the_.28interfacial.29_double_layer"></span>Development of the (interfacial) double layer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=1" title="Edit section: Development of the (interfacial) double layer"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Helmholtz">Helmholtz</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=2" title="Edit section: Helmholtz"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:EDLC-Potentialdistribution.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/EDLC-Potentialdistribution.png/200px-EDLC-Potentialdistribution.png" decoding="async" width="200" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/EDLC-Potentialdistribution.png/300px-EDLC-Potentialdistribution.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/EDLC-Potentialdistribution.png/400px-EDLC-Potentialdistribution.png 2x" data-file-width="632" data-file-height="698" /></a><figcaption>Simplified illustration of the potential development in the area and in the further course of a Helmholtz double layer.</figcaption></figure> <p>When an <i>electronic</i> conductor is brought in contact with a solid or liquid <i>ionic</i> conductor (electrolyte), a common boundary (<a href="/wiki/Interface_(chemistry)" class="mw-redirect" title="Interface (chemistry)">interface</a>) among the two <a href="/wiki/Phase_(matter)" title="Phase (matter)">phases</a> appears. <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> was the first to realize that <a href="/wiki/Electric_charge" title="Electric charge">charged</a> electrodes immersed in electrolyte solutions repel the co-ions of the charge while attracting counterions to their surfaces. Two layers of opposite <a href="/wiki/Electrical_polarity" class="mw-redirect" title="Electrical polarity">polarity</a> form at the interface between electrode and electrolyte. In 1853, he showed that an electrical double layer (DL) is essentially a <a href="/wiki/Molecule" title="Molecule">molecular</a> dielectric and stores charge electrostatically.<sup id="cite_ref-EDL_2-0" class="reference"><a href="#cite_note-EDL-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Below the electrolyte's decomposition voltage, the stored charge is linearly dependent on the voltage applied. </p><p>This early model predicted a constant <a href="/wiki/Differential_capacitance" title="Differential capacitance">differential capacitance</a> independent from the charge density depending on the <a href="/wiki/Dielectric_constant" class="mw-redirect" title="Dielectric constant">dielectric constant</a> of the electrolyte <a href="/wiki/Solvent" title="Solvent">solvent</a> and the thickness of the double-layer.<sup id="cite_ref-Namisnyk_3-0" class="reference"><a href="#cite_note-Namisnyk-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Srinivasan_4-0" class="reference"><a href="#cite_note-Srinivasan-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>This model, while a good foundation for the description of the interface, does not consider important factors including diffusion/mixing of ions in solution, the possibility of <a href="/wiki/Adsorption" title="Adsorption">adsorption</a> onto the surface, and the interaction between solvent <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">dipole moments</a> and the electrode. </p> <div class="mw-heading mw-heading3"><h3 id="Gouy–Chapman"><span id="Gouy.E2.80.93Chapman"></span>Gouy–Chapman</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=3" title="Edit section: Gouy–Chapman"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Louis_Georges_Gouy" title="Louis Georges Gouy">Louis Georges Gouy</a> in 1910 and <a href="/wiki/David_Chapman_(scientist)" class="mw-redirect" title="David Chapman (scientist)">David Leonard Chapman</a> in 1913 both observed that capacitance was not a constant and that it depended on the applied potential and the ionic concentration. The "Gouy–Chapman model" made significant improvements by introducing a <a href="/wiki/Diffusion" title="Diffusion">diffuse</a> model of the DL. In this model, the charge distribution of ions as a function of distance from the metal surface allows <a href="/wiki/Maxwell%E2%80%93Boltzmann_statistics" title="Maxwell–Boltzmann statistics">Maxwell–Boltzmann statistics</a> to be applied. Thus the <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a> <a href="/wiki/Exponential_decay" title="Exponential decay">decreases exponentially</a> away from the surface of the fluid bulk.<sup id="cite_ref-Namisnyk_3-1" class="reference"><a href="#cite_note-Namisnyk-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><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> </p><p>Gouy-Chapman layers may bear special relevance in bioelectrochemistry. The observation of long-distance inter-protein <a href="/wiki/Electron_transfer" title="Electron transfer">electron transfer</a> through the aqueous solution<sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> has been attributed to a diffuse region between redox partner proteins (<a href="/wiki/Cytochrome" title="Cytochrome">cytochromes</a> <i><a href="/wiki/Cytochrome_c" title="Cytochrome c">c</a></i> and <a href="/wiki/Coenzyme_Q_%E2%80%93_cytochrome_c_reductase" title="Coenzyme Q – cytochrome c reductase"><i>c</i><sub>1</sub></a>) that is depleted of cations in comparison to the solution bulk, thereby leading to reduced <a href="/wiki/Electric-field_screening" title="Electric-field screening">screening</a>, electric fields extending several nanometers, and currents decreasing quasi exponentially with the distance at rate ~1 nm<sup>−1</sup>. This region is termed "Gouy-Chapman conduit"<sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> and is strongly regulated by <a href="/wiki/Protein_phosphorylation" title="Protein phosphorylation">phosphorylation</a>, which adds one negative charge to the protein surface that disrupts cationic depletion and prevents long-distance charge transport.<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> Similar effects are observed at the redox active site of <a href="/wiki/Photosystem" title="Photosystem">photosynthetic complexes</a>.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Stern">Stern</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=4" title="Edit section: Stern"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Gouy-Chapman model fails for highly charged DLs. In 1924, <a href="/wiki/Otto_Stern" title="Otto Stern">Otto Stern</a> suggested combining the Helmholtz model with the Gouy-Chapman model: in Stern's model, some ions adhere to the electrode as suggested by Helmholtz, giving an internal Stern layer, while some form a Gouy-Chapman diffuse layer.<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>The Stern layer accounts for ions' finite size and consequently an ion's closest approach to the electrode is on the order of the ionic radius. The Stern model has its own limitations, namely that it effectively treats ions as point charges, assumes all significant interactions in the <a href="/wiki/Diffuse_layer" class="mw-redirect" title="Diffuse layer">diffuse layer</a> are <a href="/wiki/Coulomb" title="Coulomb">Coulombic</a>, assumes <a href="/wiki/Dielectric_permittivity" class="mw-redirect" title="Dielectric permittivity">dielectric permittivity</a> to be constant throughout the double layer, and that fluid viscosity is constant plane.<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> <div class="mw-heading mw-heading3"><h3 id="Grahame">Grahame</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=5" title="Edit section: Grahame"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Electric_double-layer_(BMD_model)_NT-int.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Electric_double-layer_%28BMD_model%29_NT-int.svg/200px-Electric_double-layer_%28BMD_model%29_NT-int.svg.png" decoding="async" width="200" height="224" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Electric_double-layer_%28BMD_model%29_NT-int.svg/300px-Electric_double-layer_%28BMD_model%29_NT-int.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Electric_double-layer_%28BMD_model%29_NT-int.svg/400px-Electric_double-layer_%28BMD_model%29_NT-int.svg.png 2x" data-file-width="733" data-file-height="822" /></a><figcaption>Schematic representation of a double layer on an electrode (BMD) model. 1. Inner Helmholtz plane, (IHP), 2. Outer Helmholtz plane (OHP), 3. Diffuse layer, 4. Solvated ions (cations) 5. Specifically adsorbed ions (redox ion, which contributes to the pseudocapacitance), 6. Molecules of the electrolyte solvent</figcaption></figure><p>D. C. Grahame modified the Stern model in 1947.<sup id="cite_ref-Grahame1947_12-0" class="reference"><a href="#cite_note-Grahame1947-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> He proposed that some ionic or uncharged species can penetrate the Stern layer, although the closest approach to the electrode is normally occupied by solvent molecules. This could occur if ions lose their <a href="/wiki/Solvation_shell" title="Solvation shell">solvation shell</a> as they approach the electrode. He called ions in direct contact with the electrode "specifically adsorbed ions". This model proposed the existence of three regions. The inner Helmholtz plane (IHP) passes through the centres of the specifically adsorbed ions. The outer Helmholtz plane (OHP) passes through the centres of solvated ions at the distance of their closest approach to the electrode.<sup id="cite_ref-NakamuraSato2011_13-0" class="reference"><a href="#cite_note-NakamuraSato2011-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Finally the diffuse layer is the region beyond the OHP. </p><div class="mw-heading mw-heading3"><h3 id="Bockris/Devanathan/Müller_(BDM)"><span id="Bockris.2FDevanathan.2FM.C3.BCller_.28BDM.29"></span>Bockris/Devanathan/Müller (BDM)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=6" title="Edit section: Bockris/Devanathan/Müller (BDM)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In 1963, <a href="/wiki/John_Bockris" title="John Bockris">J. O'M. Bockris</a>, M. A. V. Devanathan and <a href="/w/index.php?title=Klaus_M%C3%BCller_(chemist)&action=edit&redlink=1" class="new" title="Klaus Müller (chemist) (page does not exist)">Klaus Müller</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> proposed the BDM model of the double-layer that included the action of the solvent in the interface. They suggested that the attached molecules of the solvent, such as water, would have a fixed alignment to the electrode surface. This first layer of solvent molecules displays a strong orientation to the electric field depending on the charge. This orientation has great influence on the <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> of the solvent that varies with field strength. The IHP passes through the centers of these molecules. Specifically adsorbed, partially solvated ions appear in this layer. The solvated ions of the electrolyte are outside the IHP. Through the centers of these ions pass the OHP. The diffuse layer is the region beyond the OHP. </p> <div class="mw-heading mw-heading3"><h3 id="Trasatti/Buzzanca"><span id="Trasatti.2FBuzzanca"></span>Trasatti/Buzzanca</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=7" title="Edit section: Trasatti/Buzzanca"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Further research with double layers on ruthenium dioxide films in 1971 by Sergio Trasatti and Giovanni Buzzanca demonstrated that the electrochemical behavior of these electrodes at low voltages with specific adsorbed ions was like that of capacitors. The specific adsorption of the ions in this region of potential could also involve a partial charge transfer between the ion and the electrode. It was the first step towards understanding pseudocapacitance.<sup id="cite_ref-Srinivasan_4-1" class="reference"><a href="#cite_note-Srinivasan-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Conway">Conway</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=8" title="Edit section: Conway"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Between 1975 and 1980, <a href="/wiki/Brian_Evans_Conway" title="Brian Evans Conway">Brian Evans Conway</a> conducted extensive fundamental and development work on <a href="/wiki/Ruthenium_oxide" title="Ruthenium oxide">ruthenium oxide</a> electrochemical capacitors. In 1991, he described the difference between 'Supercapacitor' and 'Battery' behavior in electrochemical energy storage. In 1999, he coined the term supercapacitor to explain the increased capacitance by surface redox reactions with faradaic charge transfer between electrodes and ions.<sup id="cite_ref-Conway_Transition_15-0" class="reference"><a href="#cite_note-Conway_Transition-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>His "supercapacitor" stored electrical charge partially in the Helmholtz double-layer and partially as the result of faradaic reactions with "pseudocapacitance" charge transfer of electrons and protons between electrode and electrolyte. The working mechanisms of pseudocapacitors are redox reactions, intercalation and electrosorption. </p> <div class="mw-heading mw-heading3"><h3 id="Marcus">Marcus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=9" title="Edit section: Marcus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The physical and mathematical basics of electron charge transfer absent chemical bonds leading to pseudocapacitance was developed by <a href="/wiki/Rudolph_A._Marcus" title="Rudolph A. Marcus">Rudolph A. Marcus</a>. <a href="/wiki/Marcus_Theory" class="mw-redirect" title="Marcus Theory">Marcus Theory</a> explains the rates of electron transfer reactions—the rate at which an electron can move from one chemical species to another. It was originally formulated to address <a href="/wiki/Outer_sphere_electron_transfer" title="Outer sphere electron transfer">outer sphere electron transfer</a> reactions, in which two chemical species change only in their charge, with an electron jumping. For redox reactions without making or breaking bonds, Marcus theory takes the place of <a href="/wiki/Henry_Eyring_(chemist)" title="Henry Eyring (chemist)">Henry Eyring</a>'s <a href="/wiki/Transition_state_theory" title="Transition state theory">transition state theory</a> which was derived for reactions with structural changes. Marcus received the <a href="/wiki/List_of_Nobel_laureates_in_Chemistry" title="List of Nobel laureates in Chemistry">Nobel Prize in Chemistry</a> in 1992 for this theory.<sup id="cite_ref-nobel_17-0" class="reference"><a href="#cite_note-nobel-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mathematical_description">Mathematical description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=10" title="Edit section: Mathematical description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are detailed descriptions of the interfacial DL in many books on colloid and interface science<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><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><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> and microscale fluid transport.<sup id="cite_ref-Bruus_21-0" class="reference"><a href="#cite_note-Bruus-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kirby_22-0" class="reference"><a href="#cite_note-Kirby-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> There is also a recent IUPAC technical report<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> on the subject of interfacial double layer and related <a href="/wiki/Electrokinetic_phenomena" title="Electrokinetic phenomena">electrokinetic phenomena</a>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:DoubleLayer.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/c/c6/DoubleLayer.gif/360px-DoubleLayer.gif" decoding="async" width="360" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/c/c6/DoubleLayer.gif/540px-DoubleLayer.gif 1.5x, //upload.wikimedia.org/wikipedia/en/c/c6/DoubleLayer.gif 2x" data-file-width="630" data-file-height="525" /></a><figcaption>detailed illustration of interfacial DL</figcaption></figure> <p>As stated by Lyklema, "...the reason for the formation of a "relaxed" ("equilibrium") double layer is the non-electric affinity of charge-determining ions for a surface..."<sup id="cite_ref-Lyklema_24-0" class="reference"><a href="#cite_note-Lyklema-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> This process leads to the buildup of an <a href="/wiki/Electric_surface_charge" class="mw-redirect" title="Electric surface charge">electric surface charge</a>, expressed usually in C/m<sup>2</sup>. This surface charge creates an electrostatic field that then affects the ions in the bulk of the liquid. This electrostatic field, in combination with the thermal motion of the ions, creates a counter charge, and thus screens the electric surface charge. The net electric charge in this screening diffuse layer is equal in magnitude to the net surface charge, but has the opposite polarity. As a result, the complete structure is electrically neutral. </p><p>The diffuse layer, or at least part of it, can move under the influence of <a href="/wiki/Tangential" class="mw-redirect" title="Tangential">tangential</a> <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a>. There is a conventionally introduced slipping plane that separates mobile fluid from fluid that remains attached to the surface. Electric potential at this plane is called <a href="/wiki/Electrokinetic_potential" class="mw-redirect" title="Electrokinetic potential">electrokinetic potential</a> or <a href="/wiki/Zeta_potential" title="Zeta potential">zeta potential</a> (also denoted as ζ-potential).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The electric potential on the external boundary of the Stern layer versus the bulk electrolyte is referred to as <a href="/wiki/Stern_potential" class="mw-redirect" title="Stern potential">Stern potential</a>. Electric potential difference between the fluid bulk and the surface is called the electric surface potential. </p><p>Usually <a href="/wiki/Zeta_potential" title="Zeta potential">zeta potential</a> is used for estimating the degree of DL charge. A characteristic value of this electric potential in the DL is 25 mV with a maximum value around 100 mV (up to several volts on electrodes<sup id="cite_ref-Kirby_22-1" class="reference"><a href="#cite_note-Kirby-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>). The chemical composition of the sample at which the ζ-potential is 0 is called the <a href="/wiki/Point_of_zero_charge" title="Point of zero charge">point of zero charge</a> or the <a href="/wiki/Iso-electric_point" class="mw-redirect" title="Iso-electric point">iso-electric point</a>. It is usually determined by the solution pH value, since protons and hydroxyl ions are the charge-determining ions for most surfaces.<sup id="cite_ref-Kirby_22-2" class="reference"><a href="#cite_note-Kirby-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lyklema_24-1" class="reference"><a href="#cite_note-Lyklema-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>Zeta potential can be measured using <a href="/wiki/Electrophoresis" title="Electrophoresis">electrophoresis</a>, <a href="/wiki/Electroacoustic_phenomena" title="Electroacoustic phenomena">electroacoustic phenomena</a>, <a href="/wiki/Streaming_potential" class="mw-redirect" title="Streaming potential">streaming potential</a>, and <a href="/wiki/Electroosmotic_flow" class="mw-redirect" title="Electroosmotic flow">electroosmotic flow</a>. </p><p>The characteristic thickness of the DL is the <a href="/wiki/Debye_length" title="Debye length">Debye length</a>, κ<sup>−1</sup>. It is reciprocally proportional to the square root of the ion concentration <i>C</i>. In aqueous solutions it is typically on the scale of a few nanometers and the thickness decreases with increasing concentration of the electrolyte. </p><p>The electric field strength inside the DL can be anywhere from zero to over 10<sup>9</sup> V/m. These steep electric potential gradients are the reason for the importance of the DLs. </p><p>The theory for a flat surface and a symmetrical electrolyte<sup id="cite_ref-Lyklema_24-2" class="reference"><a href="#cite_note-Lyklema-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> is usually referred to as the Gouy-Chapman theory. It yields a simple relationship between electric charge in the diffuse layer σ<sup>d</sup> and the Stern potential Ψ<sup>d</sup>:<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{d}=-{\sqrt {{8\varepsilon _{0}}{\varepsilon _{m}}CRT}}\sinh {\frac {F\Psi ^{d}}{2RT}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mi>C</mi> <mi>R</mi> <mi>T</mi> </msqrt> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{d}=-{\sqrt {{8\varepsilon _{0}}{\varepsilon _{m}}CRT}}\sinh {\frac {F\Psi ^{d}}{2RT}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/511e41a10b5efe4c865700546a4d3eea94daf7f7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.278ex; height:5.843ex;" alt="{\displaystyle \sigma ^{d}=-{\sqrt {{8\varepsilon _{0}}{\varepsilon _{m}}CRT}}\sinh {\frac {F\Psi ^{d}}{2RT}}}"></span> </p><p>There is no general analytical solution for mixed electrolytes, curved surfaces or even spherical particles. There is an asymptotic solution for spherical particles with low charged DLs. In the case when electric potential over DL is less than 25 mV, the so-called Debye-Huckel approximation holds. It yields the following expression for electric potential<i> Ψ</i> in the spherical DL as a function of the distance <i>r</i> from the particle center: </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 {\Psi }(r)={\Psi ^{d}}{\frac {a}{r}}\exp({-\kappa }(r-a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ψ</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>r</mi> </mfrac> </mrow> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>κ</mi> </mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Psi }(r)={\Psi ^{d}}{\frac {a}{r}}\exp({-\kappa }(r-a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/444b1443f30e2d41914c33a6e4da927f92cb2728" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.555ex; height:4.676ex;" alt="{\displaystyle {\Psi }(r)={\Psi ^{d}}{\frac {a}{r}}\exp({-\kappa }(r-a))}"></span> </p><p>There are several asymptotic models which play important roles in theoretical developments associated with the interfacial DL. </p><p>The first one is "thin DL". This model assumes that DL is much thinner than the colloidal particle or capillary radius. This restricts the value of the Debye length and particle radius as following: <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 \kappa a\gg 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mi>a</mi> <mo>≫</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa a\gg 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca5ba4711473c97118eca3e2d7767a56c84bd43" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.345ex; height:2.176ex;" alt="{\displaystyle \kappa a\gg 1}"></span> </p><p>This model offers tremendous simplifications for many subsequent applications. Theory of <a href="/wiki/Electrophoresis" title="Electrophoresis">electrophoresis</a> is just one example.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The theory of <a href="/wiki/Electroacoustic_phenomena" title="Electroacoustic phenomena">electroacoustic phenomena</a> is another example.<sup id="cite_ref-Dukhin_30-0" class="reference"><a href="#cite_note-Dukhin-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>The thin DL model is valid for most aqueous systems because the Debye length is only a few nanometers in such cases. It breaks down only for nano-colloids in solution with ionic strengths close to water. </p><p>The opposing "thick DL" model assumes that the Debye length is larger than particle radius: </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 \kappa a<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> <mi>a</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa a<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/310ee60fc93a79deff38bafe676e8382fad6ccf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.83ex; height:2.176ex;" alt="{\displaystyle \kappa a<1}"></span></dd></dl> <p>This model can be useful for some nano-colloids and non-polar fluids, where the Debye length is much larger. </p><p>The last model introduces "overlapped DLs".<sup id="cite_ref-Dukhin_30-1" class="reference"><a href="#cite_note-Dukhin-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> This is important in concentrated dispersions and emulsions when distances between particles become comparable with the Debye length. </p> <div class="mw-heading mw-heading2"><h2 id="Electrical_double_layers">Electrical double layers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=11" title="Edit section: Electrical double layers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>electrical double layer</b> (<b>EDL</b>) is the result of the variation of <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a> near a surface, and has a significant influence on the behaviour of <a href="/wiki/Colloid" title="Colloid">colloids</a> and other surfaces in contact with <a href="/wiki/Solution_(chemistry)" title="Solution (chemistry)">solutions</a> or solid-state <a href="/wiki/Fast_ion_conductor" class="mw-redirect" title="Fast ion conductor">fast ion conductors</a>. </p><p>The primary difference between a double layer on an electrode and one on an interface is the mechanism of <a href="/wiki/Surface_charge" title="Surface charge">surface charge</a> formation. With an electrode, it is possible to regulate the surface charge by applying an external electric potential. This application, however, is impossible in colloidal and porous double layers, because for colloidal particles, one does not have access to the interior of the particle to apply a potential difference. </p><p>EDLs are analogous to the <a href="/wiki/Double_layer_(plasma)" class="mw-redirect" title="Double layer (plasma)">double layer</a> in <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Differential_capacitance">Differential capacitance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=12" title="Edit section: Differential capacitance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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">Main article: <a href="/wiki/Differential_capacitance" title="Differential capacitance">Differential capacitance</a></div> <p>EDLs have an additional parameter defining their characterization: <a href="/wiki/Differential_capacitance" title="Differential capacitance">differential capacitance</a>. Differential capacitance, denoted as <i>C</i>, is described by the equation below: </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 C={\frac {d\sigma }{d\Psi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>σ</mi> </mrow> <mrow> <mi>d</mi> <mi mathvariant="normal">Ψ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\frac {d\sigma }{d\Psi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcec6c171cc4c27ce3b3231fafabfa20b53f1fbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.725ex; height:5.509ex;" alt="{\displaystyle C={\frac {d\sigma }{d\Psi }}}"></span></dd></dl> <p>where σ is the <a href="/wiki/Surface_charge" title="Surface charge">surface charge</a> and ψ is the <a href="/wiki/Electric_surface_potential" class="mw-redirect" title="Electric surface potential">electric surface potential</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Electron_transfer_in_electrical_double_layer">Electron transfer in electrical double layer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=13" title="Edit section: Electron transfer in electrical double layer"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formation of electrical double layer (EDL) has been traditionally assumed to be entirely dominated by ion adsorption and redistribution. With considering the fact that the contact electrification between solid-solid is dominated by electron transfer, it is suggested by Wang that the EDL is formed by a two-step process.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> In the first step, when the molecules in the solution first approach a virgin surface that has no pre-existing surface charges, it may be possible that the atoms/molecules in the solution directly interact with the atoms on the solid surface to form strong overlap of electron clouds. Electron transfer occurs first to make the “neutral” atoms on solid surface become charged, i.e., the formation of ions. In the second step, if there are ions existing in the liquid, such as H<sup>+</sup> and OH<sup>–</sup>, the loosely distributed negative ions in the solution would be attracted to migrate toward the surface bonded ions due to electrostatic interactions, forming an EDL. Both electron transfer and ion transfer co-exist at liquid-solid interface.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Two-step_model_for_EDL.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Two-step_model_for_EDL.jpg/220px-Two-step_model_for_EDL.jpg" decoding="async" width="220" height="191" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Two-step_model_for_EDL.jpg/330px-Two-step_model_for_EDL.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Two-step_model_for_EDL.jpg/440px-Two-step_model_for_EDL.jpg 2x" data-file-width="799" data-file-height="693" /></a><figcaption>The "two-step" model (Wang model) for the formation of electric double-layer (EDL) at a liquid-solid interface, in which the electron transfer plays a dominant role in the first step.</figcaption></figure> <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=Double_layer_(surface_science)&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Electrical_double_layer" class="extiw" title="commons:Category:Electrical double layer">Electrical double layer</a></span>.</div></div> </div> <ul><li><a href="/wiki/Depletion_region" title="Depletion region">Depletion region</a> (structure of semiconductor junction)</li> <li><a href="/wiki/DLVO_theory" title="DLVO theory">DLVO theory</a></li> <li><a href="/wiki/Electroosmotic_pump" title="Electroosmotic pump">Electroosmotic pump</a></li> <li><a href="/wiki/Interface_and_colloid_science" title="Interface and colloid science">Interface and colloid science</a></li> <li><a href="/wiki/Nanofluidics" title="Nanofluidics">Nanofluidics</a></li> <li><a href="/wiki/Poisson-Boltzmann_equation" class="mw-redirect" title="Poisson-Boltzmann equation">Poisson-Boltzmann equation</a></li> <li><a href="/wiki/Supercapacitor" title="Supercapacitor">Supercapacitor</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHelmholtz1853" class="citation cs2 cs1-prop-foreign-lang-source">Helmholtz, H. 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Marcus: The Nobel Prize in Chemistry 1992</a></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Dukhin, S.S. & Derjaguin, B.V. "Electrokinetic Phenomena", J.Willey and Sons, 1974</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Russel, W.B., Saville, D.A. and Schowalter, W.R. "Colloidal Dispersions", Cambridge University Press,1989</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">Kruyt, H.R. "Colloid Science", Elsevier: Volume 1, Irreversible systems, (1952)</span> </li> <li id="cite_note-Bruus-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bruus_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBruus,_H.2007" class="citation book cs1">Bruus, H. (2007). <i>Theoretical Microfluidics</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+Microfluidics&rft.date=2007&rft.au=Bruus%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></span> </li> <li id="cite_note-Kirby-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kirby_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kirby_22-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Kirby_22-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirby,_B.J.2010" class="citation book cs1">Kirby, B.J. (2010). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190428234717/http://www.kirbyresearch.com/textbook/"><i>Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-11903-0" title="Special:BookSources/978-0-521-11903-0"><bdi>978-0-521-11903-0</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://www.kirbyresearch.com/textbook">the original</a> on 2019-04-28<span class="reference-accessdate">. Retrieved <span class="nowrap">2010-01-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Micro-+and+Nanoscale+Fluid+Mechanics%3A+Transport+in+Microfluidic+Devices.&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-11903-0&rft.au=Kirby%2C+B.J.&rft_id=http%3A%2F%2Fwww.kirbyresearch.com%2Ftextbook&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">"Measurement and Interpretation of Electrokinetic Phenomena", International Union of Pure and Applied Chemistry, Technical Report, published in Pure Appl.Chem., vol 77, 10, pp.1753-1805, 2005 <a rel="nofollow" class="external text" href="http://old.iupac.org/publications/pac/2005/pdf/7710x1753.pdf">(pdf)</a></span> </li> <li id="cite_note-Lyklema-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lyklema_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lyklema_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lyklema_24-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Lyklema, J. "Fundamentals of Interface and Colloid Science", vol.2, page.3.208, 1995</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMorrisonRoss2002" class="citation book cs1">Morrison, Ian D.; Ross, Sydney (2002). <i>Colloidal dispersions : suspensions, emulsions, and foams</i> (2nd ed.). 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Bogotsky, Fundamentals of Electrochemistry, Wiley-Interscience, 2006.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHanaorGhadiriChrzanowskiGan2014" class="citation journal cs1">Hanaor, D.A.H.; Ghadiri, M.; Chrzanowski, W.; Gan, Y. (2014). <a rel="nofollow" class="external text" href="http://drgan.org/wp-content/uploads/2015/05/025_Langmuir_2014.pdf">"Scalable Surface Area Characterization by Electrokinetic Analysis of Complex Anion Adsorption"</a> <span class="cs1-format">(PDF)</span>. <i>Langmuir</i>. <b>30</b> (50): 15143–15152. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2106.03411">2106.03411</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fla503581e">10.1021/la503581e</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/25495551">25495551</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:4697498">4697498</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Langmuir&rft.atitle=Scalable+Surface+Area+Characterization+by+Electrokinetic+Analysis+of+Complex+Anion+Adsorption&rft.volume=30&rft.issue=50&rft.pages=15143-15152&rft.date=2014&rft_id=info%3Aarxiv%2F2106.03411&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4697498%23id-name%3DS2CID&rft_id=info%3Apmid%2F25495551&rft_id=info%3Adoi%2F10.1021%2Fla503581e&rft.aulast=Hanaor&rft.aufirst=D.A.H.&rft.au=Ghadiri%2C+M.&rft.au=Chrzanowski%2C+W.&rft.au=Gan%2C+Y.&rft_id=http%3A%2F%2Fdrgan.org%2Fwp-content%2Fuploads%2F2015%2F05%2F025_Langmuir_2014.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Hunter, R.J. "Foundations of Colloid Science", Oxford University Press, 1989</span> </li> <li id="cite_note-Dukhin-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dukhin_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dukhin_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Dukhin, A. S. and Goetz, P. J. <i>Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound</i>, Elsevier, 2017 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-63908-0" title="Special:BookSources/978-0-444-63908-0">978-0-444-63908-0</a></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangWang2019" class="citation journal cs1">Wang, Z.L.; Wang, A.C. (2019). "On the origin of contact electrification". <i>Materials Today</i>. <b>30</b>: 34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.mattod.2019.05.016">10.1016/j.mattod.2019.05.016</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:189987682">189987682</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Materials+Today&rft.atitle=On+the+origin+of+contact+electrification&rft.volume=30&rft.pages=34&rft.date=2019&rft_id=info%3Adoi%2F10.1016%2Fj.mattod.2019.05.016&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A189987682%23id-name%3DS2CID&rft.aulast=Wang&rft.aufirst=Z.L.&rft.au=Wang%2C+A.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLinXuWangWang2020" class="citation journal cs1">Lin, S.Q.; Xu, L.; Wang, A.C.; Wang, Z.L. (2020). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6972942">"Quantifying electron-transfer and ion-transfer in liquid-solid contact electrification and the formation mechanism of electric double-layer"</a>. <i>Nature Communications</i>. <b>11</b> (1): 399. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs41467-019-14278-9">10.1038/s41467-019-14278-9</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6972942">6972942</a></span>. <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/31964882">31964882</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature+Communications&rft.atitle=Quantifying+electron-transfer+and+ion-transfer+in+liquid-solid+contact+electrification+and+the+formation+mechanism+of+electric+double-layer&rft.volume=11&rft.issue=1&rft.pages=399&rft.date=2020&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6972942%23id-name%3DPMC&rft_id=info%3Apmid%2F31964882&rft_id=info%3Adoi%2F10.1038%2Fs41467-019-14278-9&rft.aulast=Lin&rft.aufirst=S.Q.&rft.au=Xu%2C+L.&rft.au=Wang%2C+A.C.&rft.au=Wang%2C+Z.L.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6972942&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStillingerKirkwood1960" class="citation journal cs1">Stillinger, Frank H.; Kirkwood, John G. (1960). "Theory of the Diffuse Double Layer". <i>The Journal of Chemical Physics</i>. <b>33</b> (5): 1282–1290. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1960JChPh..33.1282S">1960JChPh..33.1282S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1731401">10.1063/1.1731401</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-9606">0021-9606</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Chemical+Physics&rft.atitle=Theory+of+the+Diffuse+Double+Layer&rft.volume=33&rft.issue=5&rft.pages=1282-1290&rft.date=1960&rft.issn=0021-9606&rft_id=info%3Adoi%2F10.1063%2F1.1731401&rft_id=info%3Abibcode%2F1960JChPh..33.1282S&rft.aulast=Stillinger&rft.aufirst=Frank+H.&rft.au=Kirkwood%2C+John+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_C._Hiemenz1986" class="citation book cs1">Paul C. Hiemenz (1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TEFRAAAAMAAJ"><i>Principles of Colloid and Surface Chemistry</i></a>. M. Dekker. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-7476-9" title="Special:BookSources/978-0-8247-7476-9"><bdi>978-0-8247-7476-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Colloid+and+Surface+Chemistry&rft.pub=M.+Dekker&rft.date=1986&rft.isbn=978-0-8247-7476-9&rft.au=Paul+C.+Hiemenz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTEFRAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_C._HiemenzRaj_Rajagopalan1997" class="citation book cs1">Paul C. Hiemenz; Raj Rajagopalan (18 March 1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CBvrS8rfPlYC"><i>Principles of Colloid and Surface Chemistry, Third Edition, Revised and Expanded</i></a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-9397-5" title="Special:BookSources/978-0-8247-9397-5"><bdi>978-0-8247-9397-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Colloid+and+Surface+Chemistry%2C+Third+Edition%2C+Revised+and+Expanded&rft.pub=CRC+Press&rft.date=1997-03-18&rft.isbn=978-0-8247-9397-5&rft.au=Paul+C.+Hiemenz&rft.au=Raj+Rajagopalan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCBvrS8rfPlYC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADouble+layer+%28surface+science%29" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Double_layer_(surface_science)&action=edit&section=17" title="Edit section: External 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