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class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Ostwald ripening</span></h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected mw-list-item"> <a class="minerva__tab-text" href="/wiki/Ostwald_ripening" rel="" data-event-name="tabs.main">Article</a> </li> <li class="minerva__tab mw-list-item"> <a class="minerva__tab-text" href="/wiki/Talk:Ostwald_ripening" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large 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cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <p><b>Ostwald ripening</b> is a phenomenon observed in <a href="/wiki/Solid_solution" title="Solid solution">solid solutions</a> and <a href="/wiki/Sol_(colloid)" title="Sol (colloid)">liquid sols</a> that involves the change of an inhomogeneous structure over time, in that small crystals or sol particles first dissolve and then redeposit onto larger crystals or sol particles.<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> </p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg/220px-TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg" decoding="async" width="220" height="234" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg/330px-TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg/440px-TEM_of_Ostwald_ripening_in_Pd_nanoparticles.jpg 2x" data-file-width="939" data-file-height="997"></a><figcaption>Ostwald ripening in <a href="/wiki/Palladium" title="Palladium">palladium</a> <a href="/wiki/Nanoparticle" title="Nanoparticle">nanoparticles</a> dissolved in <a href="/wiki/Formaldehyde" title="Formaldehyde">formaldehyde</a> at 6 (a), 24 (b), 48 (c) and 72 hours (d). The small palladium particles are being consumed as the larger ones grow bigger.<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></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/220px--A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="47" data-mwtitle="A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.480p.vp9.webm" type='video/webm; codecs="vp9, opus"' data-transcodekey="480p.vp9.webm" data-width="854" data-height="480"></source><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.720p.vp9.webm" type='video/webm; codecs="vp9, opus"' data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720"></source><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.1080p.vp9.webm" type='video/webm; codecs="vp9, opus"' data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080"></source><source src="//upload.wikimedia.org/wikipedia/commons/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv" type='video/ogg; codecs="theora, vorbis"' data-width="1920" data-height="1080"></source><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144"></source><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.240p.vp9.webm" type='video/webm; codecs="vp9, opus"' data-transcodekey="240p.vp9.webm" data-width="426" data-height="240"></source><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.360p.vp9.webm" type='video/webm; codecs="vp9, opus"' data-transcodekey="360p.vp9.webm" data-width="640" data-height="360"></source><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/b/bc/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv/A-general-patterning-approach-by-manipulating-the-evolution-of-two-dimensional-liquid-foams-ncomms14110-s2.ogv.360p.webm" type='video/webm; codecs="vp8, vorbis"' data-transcodekey="360p.webm" data-width="640" data-height="360"></source></video></span><figcaption>Growth of bubbles in a liquid foam via Ostwald ripening.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Dissolution of small crystals or sol particles and the redeposition of the dissolved species on the surfaces of larger crystals or sol particles was first described by <a href="/wiki/Wilhelm_Ostwald" title="Wilhelm Ostwald">Wilhelm Ostwald</a> in 1896.<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-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For <a href="/wiki/Colloid" title="Colloid">colloidal</a> systems, Ostwald ripening is also found in water-in-oil <a href="/wiki/Emulsion" title="Emulsion">emulsions</a>, while <a href="/wiki/Flocculation" title="Flocculation">flocculation</a> is found in oil-in-water emulsions.<sup id="cite_ref-Hubbard_2004_4230_6-0" class="reference"><a href="#cite_note-Hubbard_2004_4230-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Mechanism"><span class="tocnumber">1</span> <span class="toctext">Mechanism</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Controlled_Ostwald_Ripening"><span class="tocnumber">2</span> <span class="toctext">Controlled Ostwald Ripening</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Specific_examples"><span class="tocnumber">3</span> <span class="toctext">Specific examples</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#See_also"><span class="tocnumber">4</span> <span class="toctext">See also</span></a></li> <li class="toclevel-1 tocsection-5"><a href="#References"><span class="tocnumber">5</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-6"><a href="#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Mechanism">Mechanism</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Ostwald_ripening&action=edit&section=1" title="Edit section: Mechanism" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"><p> This <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamically</a>-driven spontaneous process occurs because larger particles are more energetically favored than smaller particles.<sup id="cite_ref-GandC_7-0" class="reference"><a href="#cite_note-GandC-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This stems from the fact that molecules on the surface of a particle are energetically less stable than the ones in the interior.</p><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sodium-chloride-3D-ionic.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Sodium-chloride-3D-ionic.png/220px-Sodium-chloride-3D-ionic.png" decoding="async" width="220" height="209" class="mw-file-element" data-file-width="1000" data-file-height="948"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 209px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Sodium-chloride-3D-ionic.png/220px-Sodium-chloride-3D-ionic.png" data-width="220" data-height="209" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Sodium-chloride-3D-ionic.png/330px-Sodium-chloride-3D-ionic.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/Sodium-chloride-3D-ionic.png/440px-Sodium-chloride-3D-ionic.png 2x" data-class="mw-file-element"> </span></a><figcaption>Cubic crystal structure (sodium chloride)</figcaption></figure> <p>Consider a cubic crystal of atoms: all the atoms inside are bonded to 6 neighbours and are quite stable, but atoms on the surface are only bonded to 5 neighbors or fewer, which makes these surface atoms less stable. Large particles are more energetically favorable since, continuing with this example, more atoms are bonded to 6 neighbors and fewer atoms are at the unfavorable surface. As the system tries to lower its overall energy, molecules on the surface of a small particle (energetically unfavorable, with only 3 or 4 or 5 bonded neighbors) will tend to detach from the particle and diffuse into the solution. </p><p><a href="/wiki/Kelvin_equation" title="Kelvin equation">Kelvin's equation</a> describes the relationship between the radius of curvature and the chemical potential between the surface and the inner volume: </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 \Delta \mu ={\frac {2\sigma \nu _{\mathrm {at} }}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>μ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>σ</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mu ={\frac {2\sigma \nu _{\mathrm {at} }}{r}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f3ad23c3e011a30b1c8416c74d450347949a17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.607ex; height:5.176ex;" alt="{\displaystyle \Delta \mu ={\frac {2\sigma \nu _{\mathrm {at} }}{r}}}"></noscript><span class="lazy-image-placeholder" style="width: 12.607ex;height: 5.176ex;vertical-align: -1.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f3ad23c3e011a30b1c8416c74d450347949a17" data-alt="{\displaystyle \Delta \mu ={\frac {2\sigma \nu _{\mathrm {at} }}{r}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></noscript><span class="lazy-image-placeholder" style="width: 1.402ex;height: 2.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" data-alt="{\displaystyle \mu }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> corresponds to the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a>, <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"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></noscript><span class="lazy-image-placeholder" style="width: 1.33ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" data-alt="{\displaystyle \sigma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to the <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a>, <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 \nu _{\mathrm {at} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu _{\mathrm {at} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093bd1361d285b3d310300c55d94afb4b09a4256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.842ex; height:2.009ex;" alt="{\displaystyle \nu _{\mathrm {at} }}"></noscript><span class="lazy-image-placeholder" style="width: 2.842ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093bd1361d285b3d310300c55d94afb4b09a4256" data-alt="{\displaystyle \nu _{\mathrm {at} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to the <a href="/wiki/Molar_volume" title="Molar volume">atomic volume</a> 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 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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to the radius of the particle. The chemical potential of an ideal solution can also be expressed as a function of the solute’s concentration if liquid and solid phases are in equilibrium. </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 \mu =k_{\mathrm {B} }T\log(C_{\mathrm {eq} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <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> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =k_{\mathrm {B} }T\log(C_{\mathrm {eq} })}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825e29b5c48a58ac0157f3927ac3ba0e3bd24f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.415ex; height:3.009ex;" alt="{\displaystyle \mu =k_{\mathrm {B} }T\log(C_{\mathrm {eq} })}"></noscript><span class="lazy-image-placeholder" style="width: 17.415ex;height: 3.009ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825e29b5c48a58ac0157f3927ac3ba0e3bd24f82" data-alt="{\displaystyle \mu =k_{\mathrm {B} }T\log(C_{\mathrm {eq} })}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></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 k_{\mathrm {B} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}"></noscript><span class="lazy-image-placeholder" style="width: 2.607ex;height: 2.509ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d" data-alt="{\displaystyle k_{\mathrm {B} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> corresponds to the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></noscript><span class="lazy-image-placeholder" style="width: 1.636ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" data-alt="{\displaystyle T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to the temperature 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_{\mathrm {eq} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\mathrm {eq} }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849f273fa5740685b43e9b65658ce0a8ef415c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.504ex; height:2.843ex;" alt="{\displaystyle C_{\mathrm {eq} }}"></noscript><span class="lazy-image-placeholder" style="width: 3.504ex;height: 2.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/849f273fa5740685b43e9b65658ce0a8ef415c57" data-alt="{\displaystyle C_{\mathrm {eq} }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to the solute concentration in a solution in which the solid and the liquid phase are in equilibrium. </p><p>Combining both expressions the following equation is obtained: </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 k_{\mathrm {B} }T\log \left({\frac {C_{\mathrm {eq} }}{C_{\infty }}}\right)={\frac {2\sigma \nu _{\mathrm {at} }}{r}}\rightarrow C_{\mathrm {eq} }(r)=C_{\mathrm {eq} }(\infty )\mathrm {e} ^{\frac {2\sigma \nu _{\mathrm {at} }}{rk_{\mathrm {B} }T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>σ</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> <mo stretchy="false">→</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>σ</mi> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mi>r</mi> <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> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\mathrm {B} }T\log \left({\frac {C_{\mathrm {eq} }}{C_{\infty }}}\right)={\frac {2\sigma \nu _{\mathrm {at} }}{r}}\rightarrow C_{\mathrm {eq} }(r)=C_{\mathrm {eq} }(\infty )\mathrm {e} ^{\frac {2\sigma \nu _{\mathrm {at} }}{rk_{\mathrm {B} }T}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc894ee1905d623490ef31548290d92fc647b5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.887ex; height:6.509ex;" alt="{\displaystyle k_{\mathrm {B} }T\log \left({\frac {C_{\mathrm {eq} }}{C_{\infty }}}\right)={\frac {2\sigma \nu _{\mathrm {at} }}{r}}\rightarrow C_{\mathrm {eq} }(r)=C_{\mathrm {eq} }(\infty )\mathrm {e} ^{\frac {2\sigma \nu _{\mathrm {at} }}{rk_{\mathrm {B} }T}}}"></noscript><span class="lazy-image-placeholder" style="width: 50.887ex;height: 6.509ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc894ee1905d623490ef31548290d92fc647b5e" data-alt="{\displaystyle k_{\mathrm {B} }T\log \left({\frac {C_{\mathrm {eq} }}{C_{\infty }}}\right)={\frac {2\sigma \nu _{\mathrm {at} }}{r}}\rightarrow C_{\mathrm {eq} }(r)=C_{\mathrm {eq} }(\infty )\mathrm {e} ^{\frac {2\sigma \nu _{\mathrm {at} }}{rk_{\mathrm {B} }T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Thus, the equilibrium 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_{eq}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{eq}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec222311d1378908939616164180e0a02feb420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.417ex; height:2.843ex;" alt="{\displaystyle C_{eq}}"></noscript><span class="lazy-image-placeholder" style="width: 3.417ex;height: 2.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec222311d1378908939616164180e0a02feb420" data-alt="{\displaystyle C_{eq}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, is lower around bigger particles than it is around smaller particles. </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_{\mathrm {eq} }(r)>C_{\mathrm {eq} }(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>></mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">q</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{\mathrm {eq} }(r)>C_{\mathrm {eq} }(R)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500a241fa2324f0c09d65ce1e96011b9c4956b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.537ex; height:3.009ex;" alt="{\displaystyle C_{\mathrm {eq} }(r)>C_{\mathrm {eq} }(R)}"></noscript><span class="lazy-image-placeholder" style="width: 16.537ex;height: 3.009ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500a241fa2324f0c09d65ce1e96011b9c4956b74" data-alt="{\displaystyle C_{\mathrm {eq} }(r)>C_{\mathrm {eq} }(R)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></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 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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></noscript><span class="lazy-image-placeholder" style="width: 1.049ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" data-alt="{\displaystyle r}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></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 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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.764ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" data-alt="{\displaystyle R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are the particles radius, 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 r<R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo><</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r<R}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda3f5af44b388094e37b9ddeb9524a3433ef30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle r<R}"></noscript><span class="lazy-image-placeholder" style="width: 5.911ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bda3f5af44b388094e37b9ddeb9524a3433ef30c" data-alt="{\displaystyle r<R}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. Inferring from <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick’s first law of diffusion</a>, the particles will move from big concentrations, corresponding to areas surrounding small particles, to small concentrations, corresponding to areas surrounding large nanoparticles. Thus, the small particles will tend to shrink while the big particles will grow. As a result, the average size of the nanoparticles in the solution will grow, and the dispersion of sizes will decrease. Therefore, if a solution is left for a long time, in the extreme case 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 t\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\rightarrow \infty }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b543f76f961ec3f52d78fa3d72c3d87a521dd3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\rightarrow \infty }"></noscript><span class="lazy-image-placeholder" style="width: 6.777ex;height: 2.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b543f76f961ec3f52d78fa3d72c3d87a521dd3a7" data-alt="{\displaystyle t\rightarrow \infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, its particles would evolve until they would finally form a single huge spherical particle to minimize the total surface area. </p><p>The history of research progress in quantitatively modeling Ostwald ripening is long, with many derivations.<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> In 1958, Lifshitz and Slyozov<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> performed a mathematical investigation of Ostwald ripening in the case where <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> of material is the slowest process. They began by stating how a single particle grows in a solution. This equation describes where the boundary is between small, shrinking particles and large, growing particles. They finally conclude that the average radius of the particles ⟨R⟩, grows as follows: </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 ^{3}-\langle R\rangle _{0}^{3}={\frac {8\gamma c_{\infty }v^{2}D}{9R_{g}T}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>R</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>R</mi> <msubsup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>γ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>D</mi> </mrow> <mrow> <mn>9</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle R\rangle ^{3}-\langle R\rangle _{0}^{3}={\frac {8\gamma c_{\infty }v^{2}D}{9R_{g}T}}t}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebcd62fab0931af02467220d58e10ddad8f0103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.283ex; height:6.343ex;" alt="{\displaystyle \langle R\rangle ^{3}-\langle R\rangle _{0}^{3}={\frac {8\gamma c_{\infty }v^{2}D}{9R_{g}T}}t}"></noscript><span class="lazy-image-placeholder" style="width: 26.283ex;height: 6.343ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebcd62fab0931af02467220d58e10ddad8f0103" data-alt="{\displaystyle \langle R\rangle ^{3}-\langle R\rangle _{0}^{3}={\frac {8\gamma c_{\infty }v^{2}D}{9R_{g}T}}t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where </p> <table> <tbody><tr> <td align="right"><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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb4c97fd87c5588653dc40086b739df1d4d04bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.573ex; height:2.843ex;" alt="{\displaystyle \langle R\rangle }"></noscript><span class="lazy-image-placeholder" style="width: 3.573ex;height: 2.843ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb4c97fd87c5588653dc40086b739df1d4d04bf" data-alt="{\displaystyle \langle R\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>average radius of all the particles </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></noscript><span class="lazy-image-placeholder" style="width: 1.262ex;height: 2.176ex;vertical-align: -0.838ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" data-alt="{\displaystyle \gamma }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>particle <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a> or <a href="/wiki/Surface_energy" title="Surface energy">surface energy</a> </td></tr> <tr> <td align="right"><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_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\infty }}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bd8353b8a00366a48746b014abb63a266e471e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.882ex; height:2.009ex;" alt="{\displaystyle c_{\infty }}"></noscript><span class="lazy-image-placeholder" style="width: 2.882ex;height: 2.009ex;vertical-align: -0.671ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bd8353b8a00366a48746b014abb63a266e471e" data-alt="{\displaystyle c_{\infty }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td><a href="/wiki/Solubility" title="Solubility">solubility</a> of the particle material </td></tr> <tr> <td align="right"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></noscript><span class="lazy-image-placeholder" style="width: 1.128ex;height: 1.676ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" data-alt="{\displaystyle v}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td><a href="/wiki/Molar_volume" title="Molar volume">molar volume</a> of the particle material </td></tr> <tr> <td align="right"><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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.924ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" data-alt="{\displaystyle D}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td><a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusion coefficient</a> of the particle material </td></tr> <tr> <td align="right"><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_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{g}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b87b78f0b90b93107b1f2e15c7a8961363a648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.785ex; height:2.843ex;" alt="{\displaystyle R_{g}}"></noscript><span class="lazy-image-placeholder" style="width: 2.785ex;height: 2.843ex;vertical-align: -1.005ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b87b78f0b90b93107b1f2e15c7a8961363a648" data-alt="{\displaystyle R_{g}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td><a href="/wiki/Gas_constant" title="Gas constant">ideal gas constant</a> </td></tr> <tr> <td align="right"><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><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></noscript><span class="lazy-image-placeholder" style="width: 1.636ex;height: 2.176ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" data-alt="{\displaystyle T}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">absolute temperature</a> and </td></tr> <tr> <td align="right"><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><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 0.84ex;height: 2.009ex;vertical-align: -0.338ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" data-alt="{\displaystyle t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></td> <td>time. </td></tr></tbody></table> <p>Note that the quantity <span class="texhtml">⟨R⟩<sup>3</sup></span> is different from <span class="texhtml">⟨R<sup>3</sup>⟩</span>, and that the statement that ⟨R⟩ goes as <span class="texhtml">t<sup>1/3</sup></span> relies on <span class="texhtml">⟨R⟩<sub>0</sub></span> being zero; but because <a href="/wiki/Nucleation" title="Nucleation">nucleation</a> is a separate process from growth, this places <span class="texhtml">⟨R⟩<sub>0</sub></span> outside the bounds of validity of the equation. In contexts where the actual value of <span class="texhtml">⟨R⟩<sub>0</sub></span> is irrelevant, an approach that respects the meanings of all terms is to take the time derivative of the equation to eliminate <span class="texhtml">⟨R⟩<sub>0</sub></span> and <span class="texhtml">t</span>. Another such approach is to change the <span class="texhtml">⟨R⟩<sub>0</sub></span> to <span class="texhtml">⟨R⟩<sub>i</sub></span> with the initial time <span class="texhtml">i</span> having a positive value.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2015)">citation needed</span></a></i>]</sup> </p><p>Also contained in the Lifshitz and Slyozov derivation is an equation for the size <a href="/wiki/Distribution_function_(physics)" title="Distribution function (physics)">distribution function</a> <span class="texhtml">f(R, t)</span> of particles. For convenience, the radius of particles is divided by the average radius to form a new variable, ρ = <span class="texhtml">R(⟨R⟩)<sup>−1</sup></span>. </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 f(R,t)={\frac {4}{9}}\rho ^{2}\left({\frac {3}{3+\rho }}\right)^{\frac {7}{3}}\left({\frac {1.5}{1.5-\rho }}\right)^{\frac {11}{3}}\exp \left(-{\frac {1.5}{1.5-\rho }}\right),\rho <1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>9</mn> </mfrac> </mrow> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>3</mn> <mo>+</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>3</mn> </mfrac> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.5</mn> <mrow> <mn>1.5</mn> <mo>−</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>3</mn> </mfrac> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.5</mn> <mrow> <mn>1.5</mn> <mo>−</mo> <mi>ρ</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mi>ρ</mi> <mo><</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(R,t)={\frac {4}{9}}\rho ^{2}\left({\frac {3}{3+\rho }}\right)^{\frac {7}{3}}\left({\frac {1.5}{1.5-\rho }}\right)^{\frac {11}{3}}\exp \left(-{\frac {1.5}{1.5-\rho }}\right),\rho <1.5}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2a18c9c7ce979237b19841f0b0ed2d33607c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:64.276ex; height:7.343ex;" alt="{\displaystyle f(R,t)={\frac {4}{9}}\rho ^{2}\left({\frac {3}{3+\rho }}\right)^{\frac {7}{3}}\left({\frac {1.5}{1.5-\rho }}\right)^{\frac {11}{3}}\exp \left(-{\frac {1.5}{1.5-\rho }}\right),\rho <1.5}"></noscript><span class="lazy-image-placeholder" style="width: 64.276ex;height: 7.343ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2a18c9c7ce979237b19841f0b0ed2d33607c57" data-alt="{\displaystyle f(R,t)={\frac {4}{9}}\rho ^{2}\left({\frac {3}{3+\rho }}\right)^{\frac {7}{3}}\left({\frac {1.5}{1.5-\rho }}\right)^{\frac {11}{3}}\exp \left(-{\frac {1.5}{1.5-\rho }}\right),\rho <1.5}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Three years after that Lifshitz and Slyozov published their findings (in Russian, 1958), <a href="/wiki/Carl_Wagner" title="Carl Wagner">Carl Wagner</a> performed his own mathematical investigation of Ostwald ripening,<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> examining both systems where <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> was slow and also where attachment and detachment at the particle surface was slow. Although his calculations and approach were different, Wagner came to the same conclusions as Lifshitz and Slyozov for slow-diffusion systems. This duplicate derivation went unnoticed for years because the two scientific papers were published on opposite sides of the <a href="/wiki/Iron_Curtain" title="Iron Curtain">Iron Curtain</a> in 1961.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2011)">citation needed</span></a></i>]</sup> It was not until 1975 that Kahlweit addressed the fact that the theories were identical<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> and combined them into the Lifshitz-Slyozov-Wagner or LSW theory of Ostwald ripening. Many experiments and <a href="/wiki/Simulations" class="mw-redirect" title="Simulations">simulations</a> have shown LSW theory to be robust and accurate. Even some systems that undergo <a href="/wiki/Spinodal_decomposition" title="Spinodal decomposition">spinodal decomposition</a> have been shown to <a href="/wiki/Quantitatively" class="mw-redirect" title="Quantitatively">quantitatively</a> obey LSW theory after initial stages of growth.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Wagner derived that when attachment and detachment of molecules is slower than diffusion, then the growth rate 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 \langle R\rangle ^{2}={\frac {64\gamma c_{\infty }v^{2}k_{s}}{81R_{g}T}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>R</mi> <msup> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>64</mn> <mi>γ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mn>81</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle R\rangle ^{2}={\frac {64\gamma c_{\infty }v^{2}k_{s}}{81R_{g}T}}t}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e309c1ee85743cb580422d140a614d5a233966be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.268ex; height:6.343ex;" alt="{\displaystyle \langle R\rangle ^{2}={\frac {64\gamma c_{\infty }v^{2}k_{s}}{81R_{g}T}}t}"></noscript><span class="lazy-image-placeholder" style="width: 20.268ex;height: 6.343ex;vertical-align: -2.505ex;" data-mw-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e309c1ee85743cb580422d140a614d5a233966be" data-alt="{\displaystyle \langle R\rangle ^{2}={\frac {64\gamma c_{\infty }v^{2}k_{s}}{81R_{g}T}}t}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>where <span class="texhtml">k<sub>s</sub></span> is the <a href="/wiki/Reaction_rate_constant" title="Reaction rate constant">reaction rate constant</a> of attachment with <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units</a> of length per time. Since the average radius is usually something that can be measured in experiments, it is fairly easy to tell if a system is obeying the slow-diffusion equation or the slow-attachment equation. If the experimental data obeys neither equation, then it is likely that another mechanism is taking place and Ostwald ripening is not occurring. </p><p>Although LSW theory and Ostwald ripening were intended for solids ripening in a fluid, Ostwald ripening is also observed in liquid-liquid systems, for example, in an oil-in-water <a href="/wiki/Emulsion_polymerization" title="Emulsion polymerization">emulsion polymerization</a>.<sup id="cite_ref-Hubbard_2004_4230_6-1" class="reference"><a href="#cite_note-Hubbard_2004_4230-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> In this case, Ostwald ripening causes the <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> of <a href="/wiki/Monomer" title="Monomer">monomers</a> (i.e. individual molecules or atoms) from smaller droplets to larger droplets due to greater solubility of the single monomer molecules in the larger monomer droplets. The rate of this diffusion process is linked to the solubility of the monomer in the continuous (water) phase of the emulsion. This can lead to the destabilization of emulsions (for example, by creaming and sedimentation).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Controlled_Ostwald_Ripening">Controlled Ostwald Ripening</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Ostwald_ripening&action=edit&section=2" title="Edit section: Controlled Ostwald Ripening" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <p>Inhibition of sulfathiazole crystal growth by polyvinylpyrrolidone. The polymer forms a noncondensed netlike film over the sulfathiazole crystal, allowing the crystal to grow out only through the openings of the net. The growth is thus controlled by the pore size of the polymer network at the crystal surface. The smaller the pore size, the higher is the supersaturation of the solution required for the crystals to grow.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Specific_examples">Specific examples</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Ostwald_ripening&action=edit&section=3" title="Edit section: Specific examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pastis1.jpg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Pastis1.jpg/170px-Pastis1.jpg" decoding="async" width="170" height="227" class="mw-file-element" data-file-width="480" data-file-height="640"></noscript><span class="lazy-image-placeholder" style="width: 170px;height: 227px;" data-mw-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Pastis1.jpg/170px-Pastis1.jpg" data-width="170" data-height="227" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Pastis1.jpg/255px-Pastis1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Pastis1.jpg/340px-Pastis1.jpg 2x" data-class="mw-file-element"> </span></a><figcaption>Oil droplets in <a href="/wiki/Pastis" title="Pastis">pastis</a> mixed with water grow by Ostwald ripening.</figcaption></figure> <p>One example of Ostwald ripening is the re-crystallization of water within <a href="/wiki/Ice_cream" title="Ice cream">ice cream</a> which gives old ice cream a gritty, crunchy texture. Larger ice crystals grow at the expense of smaller ones within the ice cream, creating a coarser texture.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>Another gastronomical example is the <a href="/wiki/Ouzo_effect" title="Ouzo effect">ouzo effect</a>, where the droplets in the cloudy microemulsion grow by Ostwald ripening. </p><p>In <a href="/wiki/Geology" title="Geology">geology</a>, it is the textural coarsening, aging or growth of <a href="/wiki/Phenocrysts" class="mw-redirect" title="Phenocrysts">phenocrysts</a> and crystals in solid rock which is below the <a href="/wiki/Solidus_(chemistry)" class="mw-redirect" title="Solidus (chemistry)">solidus</a> temperature. It is often ascribed as a process in the formation of <a href="/wiki/Orthoclase" title="Orthoclase">orthoclase</a> <a href="/wiki/Megacryst" title="Megacryst">megacrysts</a>,<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> as an alternative to the physical processes governing <a href="/wiki/Crystal_growth" title="Crystal growth">crystal growth</a> from <a href="/wiki/Nucleation" title="Nucleation">nucleation</a> and growth rate <a href="/wiki/Thermochemistry" title="Thermochemistry">thermochemical</a> limitations. </p><p>In <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solution</a> chemistry and <a href="/wiki/Precipitation_(chemistry)" title="Precipitation (chemistry)">precipitates</a> ageing, the term refers to the growth of larger crystals from those of smaller size which have a higher solubility than the larger ones. In the process, many small crystals formed initially (<i>nuclei</i>) slowly disappear, except for a few that grow larger, at the expense of the small crystals (<i>crystal growth</i>). The smaller crystals act as fuel for the growth of bigger crystals. Limiting Ostwald ripening is fundamental in modern technology for the solution synthesis of <a href="/wiki/Quantum_dot" title="Quantum dot">quantum dots</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> Ostwald ripening is also the key process in the <a href="/wiki/Precipitation_(chemistry)#Digestion_and_precipitates_ageing" title="Precipitation (chemistry)">digestion</a> and aging of precipitates, an important step in <a href="/wiki/Gravimetric_analysis" title="Gravimetric analysis">gravimetric analysis</a>. The digested precipitate is generally purer, and easier to wash and filter. </p><p>Ostwald ripening can also occur in <a href="/wiki/Emulsion" title="Emulsion">emulsion</a> systems, with molecules diffusing from small droplets to large ones through the continuous phase. When a <a href="/wiki/Miniemulsion" title="Miniemulsion">miniemulsion</a> is desired, an extremely <a href="/wiki/Hydrophobic" class="mw-redirect" title="Hydrophobic">hydrophobic</a> compound is added to stop this process from taking place.<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> </p><p>Diffusional growth of larger drops in liquid water <a href="/wiki/Cloud" title="Cloud">clouds</a> in the atmosphere at the expense of smaller drops is also characterized as Ostwald ripening.<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> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="See_also">See also</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Ostwald_ripening&action=edit&section=4" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <ul><li><a href="/wiki/Particle_aggregation" title="Particle aggregation">Aggregation</a></li> <li><a href="/wiki/Coalescence_(chemistry)" title="Coalescence (chemistry)">Coalescence (chemistry)</a></li> <li><a href="/wiki/Coalescence_(physics)" title="Coalescence (physics)">Coalescence (physics)</a></li> <li><a href="/wiki/Critical_radius" title="Critical radius">Critical radius</a></li> <li><a href="/wiki/Kirkendall_effect" title="Kirkendall effect">Kirkendall effect</a></li> <li><a href="/wiki/Rock_microstructure" class="mw-redirect" title="Rock microstructure">Rock microstructure</a></li> <li><a href="/wiki/Solubility_equilibrium#Particle_size_effect" title="Solubility equilibrium">Solubility equilibrium § Particle size effect</a></li> <li><a href="/wiki/Viedma_ripening" title="Viedma ripening">Viedma ripening</a></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Ostwald_ripening&action=edit&section=5" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <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 .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="CITEREFZhangWangHeWang2015" class="citation journal cs1">Zhang, Zhaorui; Wang, Zhenni; He, Shengnan; Wang, Chaoqi; Jin, Mingshang; Yin, Yadong (2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5669216">"Redox reaction induced Ostwald ripening for size- and shape-focusing of palladium nanocrystals"</a>. <i>Chem. 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Michael; Decker, Eric Andrew; Choi, Seung Jun (2012). "Inhibition of Ostwald Ripening in Model Beverage Emulsions by Addition of Poorly Water Soluble Triglyceride Oils". <i>Journal of Food Science</i>. <b>77</b> (1): <span class="nowrap">C33 –</span> <span class="nowrap">C38</span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1750-3841.2011.02484.x">10.1111/j.1750-3841.2011.02484.x</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/1750-3841">1750-3841</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/22133014">22133014</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+Food+Science&rft.atitle=Inhibition+of+Ostwald+Ripening+in+Model+Beverage+Emulsions+by+Addition+of+Poorly+Water+Soluble+Triglyceride+Oils&rft.volume=77&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3EC33+-%3C%2Fspan%3E+%3Cspan+class%3D%22nowrap%22%3EC38%3C%2Fspan%3E&rft.date=2012&rft.issn=1750-3841&rft_id=info%3Apmid%2F22133014&rft_id=info%3Adoi%2F10.1111%2Fj.1750-3841.2011.02484.x&rft.aulast=McClements&rft.aufirst=David+Julian&rft.au=Henson%2C+Lulu&rft.au=Popplewell%2C+L.+Michael&rft.au=Decker%2C+Eric+Andrew&rft.au=Choi%2C+Seung+Jun&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOstwald+ripening" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoodIronsJonas2002" class="citation journal cs1">Wood, R.; Irons, S.; Jonas, P. 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Studies using simple trajectory analysis"</a>. <i>Journal of the Atmospheric Sciences</i>. <b>59</b> (18): <span class="nowrap">2681–</span>2693. <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/2002JAtS...59.2681W">2002JAtS...59.2681W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0469%282002%29059%3C2681%3AHIITSR%3E2.0.CO%3B2">10.1175/1520-0469(2002)059<2681:HIITSR>2.0.CO;2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Atmospheric+Sciences&rft.atitle=How+important+is+the+spectral+ripening+effect+in+stratiform+boundary+layer+clouds%3F+Studies+using+simple+trajectory+analysis&rft.volume=59&rft.issue=18&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2681-%3C%2Fspan%3E2693&rft.date=2002&rft_id=info%3Adoi%2F10.1175%2F1520-0469%282002%29059%3C2681%3AHIITSR%3E2.0.CO%3B2&rft_id=info%3Abibcode%2F2002JAtS...59.2681W&rft.aulast=Wood&rft.aufirst=R.&rft.au=Irons%2C+S.&rft.au=Jonas%2C+P.+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1175%252F1520-0469%25282002%2529059%253C2681%253AHIITSR%253E2.0.CO%253B2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AOstwald+ripening" class="Z3988"></span></span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External 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