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searchaux" style="display:none">Major stage of a crystallization process</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output 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(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox hlist" style="width:18em; text-align:center; font-size:95%;"><tbody><tr><th colspan="2" class="infobox-above" style="padding-bottom:0.3em; background:transparent; line-height:1.1em; font-size:125%; font-weight:bold;"><a href="/wiki/Crystallization" title="Crystallization">Crystallization</a></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Process-of-Crystallization-200px.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/ce/Process-of-Crystallization-200px.png" decoding="async" width="200" height="74" class="mw-file-element" data-file-width="200" data-file-height="74" /></a></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:#E7C6A5;">Fundamentals</th></tr><tr><td colspan="2" class="infobox-full-data" style="line-height:1.4em;"> <ul><li><a href="/wiki/Crystal" title="Crystal">Crystal</a></li> <li><a href="/wiki/Crystal_structure" title="Crystal structure">Crystal structure</a></li> <li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li></ul> </td></tr><tr><th colspan="2" class="infobox-header" style="background:#E7C6A5;">Concepts</th></tr><tr><td colspan="2" class="infobox-full-data" style="line-height:1.4em;"> <ul><li><a href="/wiki/Crystallization" title="Crystallization">Crystallization</a></li> <li><a class="mw-selflink selflink">Crystal growth</a></li> <li><a href="/wiki/Recrystallization_(chemistry)" title="Recrystallization (chemistry)">Recrystallization</a></li> <li><a href="/wiki/Seed_crystal" title="Seed crystal">Seed crystal</a></li> <li><a href="/wiki/Protocrystalline" title="Protocrystalline">Protocrystalline</a></li> <li><a href="/wiki/Single_crystal" title="Single crystal">Single crystal</a></li></ul> </td></tr><tr><th colspan="2" class="infobox-header" style="background:#E7C6A5;">Methods and technology</th></tr><tr><td colspan="2" class="infobox-full-data" style="line-height:1.4em;"> <ul><li><a href="/wiki/Boule_(crystal)" title="Boule (crystal)">Boules</a></li> <li><a href="/wiki/Bridgman%E2%80%93Stockbarger_method" title="Bridgman–Stockbarger method">Bridgman–Stockbarger method</a></li> <li><a href="/wiki/Van_Arkel%E2%80%93de_Boer_process" title="Van Arkel–de Boer process">Van Arkel–de Boer process</a></li> <li><a href="/wiki/Czochralski_method" title="Czochralski method">Czochralski method</a></li> <li><a href="/wiki/Epitaxy" title="Epitaxy">Epitaxy</a></li> <li><a href="/wiki/Flux_method" title="Flux method">Flux method</a></li> <li><a href="/wiki/Fractional_crystallization_(chemistry)" title="Fractional crystallization (chemistry)">Fractional crystallization</a></li> <li><a href="/wiki/Fractional_freezing" title="Fractional freezing">Fractional freezing</a></li> <li><a href="/wiki/Hydrothermal_synthesis" title="Hydrothermal synthesis">Hydrothermal synthesis</a></li> <li><a href="/wiki/Kyropoulos_method" title="Kyropoulos method">Kyropoulos method</a></li> <li><a href="/wiki/Laser-heated_pedestal_growth" title="Laser-heated pedestal growth">Laser-heated pedestal growth</a></li> <li><a href="/wiki/Micro-pulling-down" title="Micro-pulling-down">Micro-pulling-down</a></li> <li><a href="/wiki/Shaping_processes_in_crystal_growth" title="Shaping processes in crystal growth">Shaping processes in crystal growth</a></li> <li><a href="/wiki/Skull_crucible" title="Skull crucible">Skull crucible</a></li> <li><a href="/wiki/Verneuil_method" title="Verneuil method">Verneuil method</a></li> <li><a href="/wiki/Zone_melting" title="Zone melting">Zone melting</a></li></ul> </td></tr><tr><td colspan="2" class="infobox-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Crystallization" title="Template:Crystallization"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Crystallization" title="Template talk:Crystallization"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Crystallization" title="Special:EditPage/Template:Crystallization"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Schematic_of_crystal_growth_of_simple_cubic_lattice,_showing_additional_molecule_adding_in_corner.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Schematic_of_crystal_growth_of_simple_cubic_lattice%2C_showing_additional_molecule_adding_in_corner.png/220px-Schematic_of_crystal_growth_of_simple_cubic_lattice%2C_showing_additional_molecule_adding_in_corner.png" decoding="async" width="220" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Schematic_of_crystal_growth_of_simple_cubic_lattice%2C_showing_additional_molecule_adding_in_corner.png/330px-Schematic_of_crystal_growth_of_simple_cubic_lattice%2C_showing_additional_molecule_adding_in_corner.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Schematic_of_crystal_growth_of_simple_cubic_lattice%2C_showing_additional_molecule_adding_in_corner.png/440px-Schematic_of_crystal_growth_of_simple_cubic_lattice%2C_showing_additional_molecule_adding_in_corner.png 2x" data-file-width="10447" data-file-height="8273" /></a><figcaption>Schematic of a small part of a growing crystal. The crystal is of (blue) cubic particles on a simple cubic lattice. The top layer is incomplete, only ten of the sixteen lattice positions are occupied by particles. A particle in the fluid (shown with red edges) is joining the crystal, growing the crystal by one particle. It is joining the lattice at the point where its energy will be a minimum, which is in the corner of the incomplete top layer (on top of the particle shown with yellow edges). Its energy will be a minimum because in that position it has three neighbors (one below, one to its left and one above right) which it will interact with. All other positions on an incomplete crystal layer have only one or two neighbours.</figcaption></figure> <p>A <a href="/wiki/Crystal" title="Crystal">crystal</a> is a <a href="/wiki/Solid" title="Solid">solid</a> <a href="/wiki/Material" title="Material">material</a> whose constituent <a href="/wiki/Atom" title="Atom">atoms</a>, <a href="/wiki/Molecule" title="Molecule">molecules</a>, or <a href="/wiki/Ion" title="Ion">ions</a> are arranged in an orderly repeating pattern extending in all three spatial dimensions. <b>Crystal growth</b> is a major stage of a <a href="/wiki/Crystallization" title="Crystallization">crystallization process</a>, and consists of the addition of new atoms, ions, or <a href="/wiki/Polymer" title="Polymer">polymer</a> strings into the characteristic arrangement of the crystalline lattice.<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><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> The growth typically follows an initial stage of either homogeneous or heterogeneous (surface catalyzed) <a href="/wiki/Nucleation" title="Nucleation">nucleation</a>, unless a "seed" crystal, purposely added to start the growth, was already present. </p><p>The action of crystal growth yields a crystalline solid whose atoms or molecules are close packed, with fixed positions in <a href="/wiki/Space" title="Space">space</a> relative to each other. The crystalline <a href="/wiki/States_of_matter" class="mw-redirect" title="States of matter">state of matter</a> is characterized by a distinct <a href="/wiki/Structural_rigidity" title="Structural rigidity">structural rigidity</a> and very high resistance to <a href="/wiki/Plastic_deformation_in_solids" class="mw-redirect" title="Plastic deformation in solids">deformation</a> (i.e. changes of shape and/or volume). Most crystalline solids have high values both of <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a> and of the <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a> of <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>. This contrasts with most <a href="/wiki/Liquid" title="Liquid">liquids</a> or <a href="/wiki/Fluid" title="Fluid">fluids</a>, which have a low shear modulus, and typically exhibit the capacity for macroscopic <a href="/wiki/Viscous_flow" class="mw-redirect" title="Viscous flow">viscous flow</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>After successful formation of a stable nucleus, a growth stage ensues in which free particles (atoms or molecules) adsorb onto the nucleus and propagate its crystalline structure outwards from the nucleating site. This process is significantly faster than nucleation. The reason for such rapid growth is that real crystals contain <a href="/wiki/Dislocations#Screw_dislocations" class="mw-redirect" title="Dislocations">dislocations</a> and other defects, which act as a catalyst for the addition of particles to the existing crystalline structure. By contrast, perfect crystals (lacking defects) would grow exceedingly slowly.<sup id="cite_ref-frank49_3-0" class="reference"><a href="#cite_note-frank49-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> On the other hand, <b>impurities</b> can act as crystal growth inhibitors and can also modify <a href="/wiki/Crystal_habit" title="Crystal habit">crystal habit</a>.<sup id="cite_ref-nguyen17_4-0" class="reference"><a href="#cite_note-nguyen17-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Nucleation">Nucleation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=2" title="Edit section: Nucleation"><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/Nucleation" title="Nucleation">Nucleation</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Silver_surface_crystal_growth_SEM.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Silver_surface_crystal_growth_SEM.png/300px-Silver_surface_crystal_growth_SEM.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Silver_surface_crystal_growth_SEM.png/450px-Silver_surface_crystal_growth_SEM.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Silver_surface_crystal_growth_SEM.png/600px-Silver_surface_crystal_growth_SEM.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>Silver crystal growing on a ceramic substrate.</figcaption></figure> <p>Nucleation can be either <a href="https://en.wiktionary.org/wiki/Homogeneous" class="extiw" title="wiktionary:Homogeneous">homogeneous</a>, without the influence of foreign particles, or <a href="/wiki/Mixture#Distinguishing_between_mixture_types" title="Mixture">heterogeneous</a>, with the influence of foreign particles. Generally, heterogeneous nucleation takes place more quickly since the foreign particles act as a <a href="/wiki/Scaffold" class="mw-redirect" title="Scaffold">scaffold</a> for the crystal to grow on, thus eliminating the necessity of creating a new surface and the incipient surface energy requirements. </p><p>Heterogeneous nucleation can take place by several methods. Some of the most typical are small inclusions, or cuts, in the container the crystal is being grown on. This includes scratches on the sides and bottom of glassware. A common practice in crystal growing is to add a foreign substance, such as a string or a rock, to the solution, thereby providing nucleation sites for facilitating crystal growth and reducing the time to fully crystallize. </p><p>The number of nucleating sites can also be controlled in this manner. If a brand-new piece of glassware or a plastic container is used, crystals may not form because the container surface is too smooth to allow heterogeneous nucleation. On the other hand, a badly scratched container will result in many lines of small crystals. To achieve a moderate number of medium-sized crystals, a container which has a few scratches works best. Likewise, adding small previously made crystals, or seed crystals, to a crystal growing project will provide nucleating sites to the solution. The addition of only one seed crystal should result in a larger single crystal. </p> <div class="mw-heading mw-heading2"><h2 id="Mechanisms_of_growth">Mechanisms of growth</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=3" title="Edit section: Mechanisms of growth"><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:ImgSalt.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/ImgSalt.jpg/300px-ImgSalt.jpg" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/b9/ImgSalt.jpg 1.5x" data-file-width="360" data-file-height="270" /></a><figcaption>An example of the <a href="/wiki/Cubic_crystal" class="mw-redirect" title="Cubic crystal">cubic crystals</a> typical of the <a href="#Rock-salt_structure">rock-salt structure</a><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/MOS:BROKENSECTIONLINKS" class="mw-redirect" title="MOS:BROKENSECTIONLINKS"><span title=" (2024-06-05)">broken anchor</span></a></i>]</sup>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/CitricAcid_Crystalisation_Timelapse.ogv/250px--CitricAcid_Crystalisation_Timelapse.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="250" height="190" data-durationhint="4" data-mwtitle="CitricAcid_Crystalisation_Timelapse.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:CitricAcid_Crystalisation_Timelapse.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/77/CitricAcid_Crystalisation_Timelapse.ogv/CitricAcid_Crystalisation_Timelapse.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="630" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/77/CitricAcid_Crystalisation_Timelapse.ogv/CitricAcid_Crystalisation_Timelapse.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="946" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/7/77/CitricAcid_Crystalisation_Timelapse.ogv" type="video/ogg; codecs="theora"" data-width="1344" data-height="1024" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/77/CitricAcid_Crystalisation_Timelapse.ogv/CitricAcid_Crystalisation_Timelapse.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="190" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/77/CitricAcid_Crystalisation_Timelapse.ogv/CitricAcid_Crystalisation_Timelapse.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="316" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/77/CitricAcid_Crystalisation_Timelapse.ogv/CitricAcid_Crystalisation_Timelapse.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="472" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/7/77/CitricAcid_Crystalisation_Timelapse.ogv/CitricAcid_Crystalisation_Timelapse.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="472" data-height="360" /></video></span><figcaption><a href="/wiki/Time-lapse" class="mw-redirect" title="Time-lapse">Time-lapse</a> of growth of a <a href="/wiki/Citric_acid" title="Citric acid">citric acid</a> <a href="/wiki/Crystal" title="Crystal">crystal</a>. The video covers an area of 2.0 by 1.5 mm and was captured over 7.2 <a href="/wiki/Minute" title="Minute">min</a>.</figcaption></figure> <p>The interface between a crystal and its vapor can be molecularly sharp at temperatures well below the melting point. An ideal crystalline surface grows by the spreading of single layers, or equivalently, by the lateral advance of the growth steps bounding the layers. For perceptible growth rates, this mechanism requires a finite driving force (or degree of supercooling) in order to lower the nucleation barrier sufficiently for nucleation to occur by means of thermal fluctuations.<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> In the theory of crystal growth from the melt, Burton and Cabrera have distinguished between two major mechanisms:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Non-uniform_lateral_growth">Non-uniform lateral growth</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=4" title="Edit section: Non-uniform lateral growth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The surface advances by the lateral motion of steps which are one interplanar spacing in height (or some integral multiple thereof). An element of surface undergoes no change and does not advance normal to itself except during the passage of a step, and then it advances by the step height. It is useful to consider the step as the transition between two adjacent regions of a surface which are parallel to each other and thus identical in configuration—displaced from each other by an integral number of lattice planes. Note here the distinct possibility of a step in a diffuse surface, even though the step height would be much smaller than the thickness of the diffuse surface. </p> <div class="mw-heading mw-heading3"><h3 id="Uniform_normal_growth">Uniform normal growth</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=5" title="Edit section: Uniform normal growth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The surface advances normal to itself without the necessity of a stepwise growth mechanism. This means that in the presence of a sufficient thermodynamic driving force, every element of surface is capable of a continuous change contributing to the advancement of the interface. For a sharp or discontinuous surface, this continuous change may be more or less uniform over large areas for each successive new layer. For a more diffuse surface, a continuous growth mechanism may require changes over several successive layers simultaneously. </p><p>Non-uniform lateral growth is a geometrical motion of steps—as opposed to motion of the entire surface normal to itself. Alternatively, uniform normal growth is based on the time sequence of an element of surface. In this mode, there is no motion or change except when a step passes via a continual change. The prediction of which mechanism will be operative under any set of given conditions is fundamental to the understanding of crystal growth. Two criteria have been used to make this prediction: </p><p>Whether or not the surface is <i>diffuse</i>: a diffuse surface is one in which the change from one phase to another is continuous, occurring over several atomic planes. This is in contrast to a sharp surface for which the major change in property (e.g. density or composition) is discontinuous, and is generally confined to a depth of one interplanar distance.<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><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>Whether or not the surface is <i>singular</i>: a singular surface is one in which the surface tension as a function of orientation has a pointed minimum. Growth of singular surfaces is known to requires steps, whereas it is generally held that non-singular surfaces can continuously advance normal to themselves.<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="Driving_force">Driving force</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=6" title="Edit section: Driving force"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider next the necessary requirements for the appearance of lateral growth. It is evident that the lateral growth mechanism will be found when any area in the surface can reach a metastable equilibrium in the presence of a driving force. It will then tend to remain in such an equilibrium configuration until the passage of a step. Afterward, the configuration will be identical except that each part of the step will have advanced by the step height. If the surface cannot reach equilibrium in the presence of a driving force, then it will continue to advance without waiting for the lateral motion of steps. </p><p>Thus, Cahn concluded that the distinguishing feature is the ability of the surface to reach an equilibrium state in the presence of the driving force. He also concluded that for every surface or interface in a crystalline medium, there exists a critical driving force, which, if exceeded, will enable the surface or interface to advance normal to itself, and, if not exceeded, will require the lateral growth mechanism. </p><p>Thus, for sufficiently large driving forces, the interface can move uniformly without the benefit of either a heterogeneous nucleation or screw dislocation mechanism. What constitutes a sufficiently large driving force depends upon the diffuseness of the interface, so that for extremely diffuse interfaces, this critical driving force will be so small that any measurable driving force will exceed it. Alternatively, for sharp interfaces, the critical driving force will be very large, and most growth will occur by the lateral step mechanism. </p><p>Note that in a typical <a href="/wiki/Solidification" class="mw-redirect" title="Solidification">solidification</a> or <a href="/wiki/Crystallization" title="Crystallization">crystallization</a> process, the thermodynamic driving force is dictated by the degree of <a href="/wiki/Supercooling" title="Supercooling">supercooling</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Morphology">Morphology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=7" title="Edit section: Morphology"><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:SilverSulfideWhiskers1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/SilverSulfideWhiskers1.jpg/300px-SilverSulfideWhiskers1.jpg" decoding="async" width="300" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/SilverSulfideWhiskers1.jpg/450px-SilverSulfideWhiskers1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/SilverSulfideWhiskers1.jpg/600px-SilverSulfideWhiskers1.jpg 2x" data-file-width="1099" data-file-height="752" /></a><figcaption>Silver sulfide <a href="/wiki/Whisker_(metallurgy)" title="Whisker (metallurgy)">whiskers</a> growing out of surface-mount resistors.</figcaption></figure> <p>It is generally believed that the mechanical and other properties of the crystal are also pertinent to the subject matter, and that crystal <a href="/w/index.php?title=Morphology_(chemistry)&action=edit&redlink=1" class="new" title="Morphology (chemistry) (page does not exist)">morphology</a> provides the missing link between growth kinetics and physical properties. The necessary thermodynamic apparatus was provided by <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Josiah Willard Gibbs</a>' study of heterogeneous equilibrium. He provided a clear definition of surface energy, by which the concept of surface tension is made applicable to solids as well as liquids. He also appreciated that <i>an anisotropic surface free energy implied a non-spherical equilibrium shape</i>, which should be thermodynamically defined as <i>the shape which minimizes the total surface free energy</i>.<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>It may be instructional to note that <a href="/wiki/Monocrystalline_whisker" title="Monocrystalline whisker">whisker</a> growth provides the link between the mechanical phenomenon of high strength in whiskers and the various growth mechanisms which are responsible for their fibrous morphologies. (Prior to the discovery of carbon nanotubes, <a href="/wiki/Monocrystalline_whisker" title="Monocrystalline whisker">single-crystal whiskers</a> had the highest tensile strength of any materials known). Some mechanisms produce defect-free whiskers, while others may have single screw dislocations along the main axis of growth—producing high strength whiskers. </p><p>The mechanism behind whisker growth is not well understood, but seems to be encouraged by compressive mechanical <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stresses</a> including mechanically induced stresses, stresses induced by <a href="/wiki/Diffusion" title="Diffusion">diffusion</a> of different elements, and thermally induced stresses. Metal whiskers differ from metallic <a href="/wiki/Dendrite_(crystal)" title="Dendrite (crystal)">dendrites</a> in several respects. Dendrites are <a href="/wiki/Fern" title="Fern">fern</a>-shaped like the branches of a tree, and grow across the surface of the metal. In contrast, whiskers are fibrous and project at a right angle to the surface of growth, or substrate. </p> <div class="mw-heading mw-heading3"><h3 id="Diffusion-control">Diffusion-control</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=8" title="Edit section: Diffusion-control"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diffusioncontrollgrowthmodel2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Diffusioncontrollgrowthmodel2.jpg/220px-Diffusioncontrollgrowthmodel2.jpg" decoding="async" width="220" height="183" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Diffusioncontrollgrowthmodel2.jpg/330px-Diffusioncontrollgrowthmodel2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Diffusioncontrollgrowthmodel2.jpg/440px-Diffusioncontrollgrowthmodel2.jpg 2x" data-file-width="1244" data-file-height="1036" /></a><figcaption>Concentration profile in a diffusion-controlled system for a spherical nucleus with radius <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75b83e050da28fa0d83e2aa786963805742ab756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.01ex; height:2.009ex;" alt="{\displaystyle c_{s}}"></span> is the concentration of atoms in the solid nucleus, <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_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/161bd00c9b87a82436365401180e618862a74b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.729ex; height:2.009ex;" alt="{\displaystyle c_{l}}"></span> is the concentration in the liquid right at the surface if the nucleus, <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1882ba8f1dc60f0c68a642abb5af093c73910921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}"></span> is the equilibrium concentration in the liquid phase 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 \textstyle r+\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle r+\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c0feb13c9d0709e9f2150cd220a31bfe27bb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.938ex; height:2.509ex;" alt="{\displaystyle \textstyle r+\delta }"></span> is the distance from the nucleus where 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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1882ba8f1dc60f0c68a642abb5af093c73910921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}"></span> is recovered.</figcaption></figure> <p>Very commonly when the supersaturation (or degree of supercooling) is high, and sometimes even when it is not high, growth kinetics may be diffusion-controlled, which means the transport of atoms or molecules to the growing nucleus is limiting the velocity of crystal growth. Assuming the nucleus in such a diffusion-controlled system is a perfect sphere, the growth velocity, corresponding to the change of the radius with time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle {\frac {\partial r}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial r}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/618dcb6312f410c4060a797be881366cd537e9ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.51ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {\partial r}{\partial t}}}"></span>, can be determined with Fick’s Laws. </p><p>1. Fick' s Law: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=-D\nabla c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mi mathvariant="normal">∇</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D\nabla c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04f53470dc4e5ea6c44e1518db9f93e4153db4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.245ex; height:2.343ex;" alt="{\displaystyle J=-D\nabla c}"></span>, </p><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 \textstyle J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>J</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/231aa2cfda77bb5eb91c8daa0d5ba4b9494269bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle \textstyle J}"></span> is the flux of atoms in the dimension 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 \textstyle {\frac {[quantity]}{[time]\cdot [area]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>n</mi> <mi>t</mi> <mi>i</mi> <mi>t</mi> <mi>y</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mi>i</mi> <mi>m</mi> <mi>e</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mi>r</mi> <mi>e</mi> <mi>a</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {[quantity]}{[time]\cdot [area]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96843b18d87bb53174426365fccdd3081aef28b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.74ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {[quantity]}{[time]\cdot [area]}}}"></span>, <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 \textstyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>D</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac920029391c3024f45400d1da8285b1c9300d8" 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 \textstyle D}"></span> is the diffusion coefficient 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 \textstyle \nabla c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>c</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \nabla c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ad235178c9721e908c303a1a8ce1202152b688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.943ex; height:2.176ex;" alt="{\displaystyle \textstyle \nabla c}"></span> is the concentration gradient. </p><p>2. Fick' s Law: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a38626fb165e191e356cf6ffa0000db1a2caa265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.181ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}c}"></span>, </p><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 \textstyle {\frac {\partial c}{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial c}{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7087ba39b7069d4fd10d8b264ecfd611a2dd2055" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.48ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {\partial c}{\partial t}}}"></span> is the change of the concentration with time. The first Law can be adjusted to the flux of matter onto a specific surface, in this case the surface of the spherical nucleus: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J_{matter}=D4\pi \cdot r^{2}{\frac {\partial c}{\partial r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mi>D</mi> <mn>4</mn> <mi>π</mi> <mo>⋅</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{matter}=D4\pi \cdot r^{2}{\frac {\partial c}{\partial r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48021544c731a61975760c863608335372280d4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.032ex; height:5.509ex;" alt="{\displaystyle J_{matter}=D4\pi \cdot r^{2}{\frac {\partial c}{\partial r}}}"></span>, </p><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 \textstyle J_{matter}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle J_{matter}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f312a989734cb9df6bbb495145635cfa99a3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.53ex; height:2.509ex;" alt="{\displaystyle \textstyle J_{matter}}"></span> now is the flux onto the spherical surface in the dimension 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 \textstyle {\frac {[quantity]}{[time]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">[</mo> <mi>q</mi> <mi>u</mi> <mi>a</mi> <mi>n</mi> <mi>t</mi> <mi>i</mi> <mi>t</mi> <mi>y</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mi>i</mi> <mi>m</mi> <mi>e</mi> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {[quantity]}{[time]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c8bb90389963cb35c68f99b895246b4b5f00583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.875ex; height:4.843ex;" alt="{\displaystyle \textstyle {\frac {[quantity]}{[time]}}}"></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 \textstyle 4\pi \cdot r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mn>4</mn> <mi>π</mi> <mo>⋅</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle 4\pi \cdot r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/408f1d3854ce25b7d7175f889d9f8f56562e812e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:2.509ex;" alt="{\displaystyle \textstyle 4\pi \cdot r^{2}}"></span> being the area of the spherical nucleus. <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 \textstyle J_{matter}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle J_{matter}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f312a989734cb9df6bbb495145635cfa99a3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.53ex; height:2.509ex;" alt="{\displaystyle \textstyle J_{matter}}"></span> can also be expressed as the change of number of atoms in the nucleus over time, with the number of atoms in the nucleus being: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(t)={\frac {{\frac {4}{3}}\pi \cdot r(t)^{3}}{V_{at}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <mo>⋅</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(t)={\frac {{\frac {4}{3}}\pi \cdot r(t)^{3}}{V_{at}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110ea0b836bad0faeede56481dff205ac688628a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.068ex; height:7.009ex;" alt="{\displaystyle N(t)={\frac {{\frac {4}{3}}\pi \cdot r(t)^{3}}{V_{at}}}}"></span> , </p><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 \textstyle {\frac {4}{3}}\pi r^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {4}{3}}\pi r^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d685ddd147c84e123fe4c01d6a7ee9813f7b04d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.093ex; height:3.676ex;" alt="{\displaystyle \textstyle {\frac {4}{3}}\pi r^{3}}"></span> is the volume of the spherical nucleus 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 \textstyle V_{at}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle V_{at}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/378cd0f44ab89bbe406d936e69996d81f124b42e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.051ex; height:2.509ex;" alt="{\displaystyle \textstyle V_{at}}"></span> is the atomic volume. Therefore, the change if number of atoms in the nucleus over time will be: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial N(t)}{\partial t}}={\frac {4\pi \cdot r(t)^{2}}{V_{a}t}}{\frac {\partial r}{\partial t}}=J_{matter}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> <mo>⋅</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial N(t)}{\partial t}}={\frac {4\pi \cdot r(t)^{2}}{V_{a}t}}{\frac {\partial r}{\partial t}}=J_{matter}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29cd70675f74f0b56ecf3b2ffd67468beec45f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.558ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial N(t)}{\partial t}}={\frac {4\pi \cdot r(t)^{2}}{V_{a}t}}{\frac {\partial r}{\partial t}}=J_{matter}}"></span> </p><p>Combining both equations for <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 \textstyle J_{matter}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>a</mi> <mi>t</mi> <mi>t</mi> <mi>e</mi> <mi>r</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle J_{matter}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f312a989734cb9df6bbb495145635cfa99a3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.53ex; height:2.509ex;" alt="{\displaystyle \textstyle J_{matter}}"></span> the following expression for the growth velocity is obtained: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial r}{\partial t}}=V_{at}D{\frac {\partial c}{\partial r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> </mrow> </msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial r}{\partial t}}=V_{at}D{\frac {\partial c}{\partial r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f344fe400a02d96a14375a7c0c312afd9b39acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.479ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial r}{\partial t}}=V_{at}D{\frac {\partial c}{\partial r}}}"></span> </p><p>From second Fick’s Law for spheres the equation below can be obtained: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial c}{\partial t}}=D{\frac {\partial }{\partial t}}(r^{2}{\frac {\partial c}{\partial r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c}{\partial t}}=D{\frac {\partial }{\partial t}}(r^{2}{\frac {\partial c}{\partial r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d1c533278919dd0809105ffa1f32bdbf6fe66a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.293ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial c}{\partial t}}=D{\frac {\partial }{\partial t}}(r^{2}{\frac {\partial c}{\partial r}})}"></span> </p><p>Assuming that the diffusion profile does not change over time but is only shifted with the growing radius it can be said that <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 \textstyle {\frac {\partial c}{\partial t}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial c}{\partial t}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63dde302337854ba33ed69074aab6d243f4971cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.741ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {\partial c}{\partial t}}=0}"></span>, which leads to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r^{2}{\frac {\partial c}{\partial r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle r^{2}{\frac {\partial c}{\partial r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66461d83dab89774bde8e0ed893a87039300a760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.613ex; height:3.843ex;" alt="{\displaystyle \textstyle r^{2}{\frac {\partial c}{\partial r}}}"></span> being constant. This constant can be indicated with the letter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> and integrating will result in the following equation: </p><p><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^{2}{\frac {\partial c}{\partial r}}=A\Rightarrow {\frac {A}{r^{2}}}dr=dc\Rightarrow \int _{r}^{r+\delta }{\frac {A}{r^{2}}}dr=\int _{c_{0}}^{c_{l}}dc\Rightarrow c_{0}-c_{l}=A[{\frac {1}{r}}-{\frac {1}{r+\delta }}]\Rightarrow A={\frac {c_{0}-c_{l}}{[{\frac {1}{r}}-{\frac {1}{r+\delta }}]}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>d</mi> <mi>r</mi> <mo>=</mo> <mi>d</mi> <mi>c</mi> <mo stretchy="false">⇒</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>d</mi> <mi>r</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>c</mi> <mo stretchy="false">⇒</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">]</mo> <mo stretchy="false">⇒</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">]</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}{\frac {\partial c}{\partial r}}=A\Rightarrow {\frac {A}{r^{2}}}dr=dc\Rightarrow \int _{r}^{r+\delta }{\frac {A}{r^{2}}}dr=\int _{c_{0}}^{c_{l}}dc\Rightarrow c_{0}-c_{l}=A[{\frac {1}{r}}-{\frac {1}{r+\delta }}]\Rightarrow A={\frac {c_{0}-c_{l}}{[{\frac {1}{r}}-{\frac {1}{r+\delta }}]}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77999188243bf22bce2dddd9e3edf09707ec1b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:94.302ex; height:7.676ex;" alt="{\displaystyle r^{2}{\frac {\partial c}{\partial r}}=A\Rightarrow {\frac {A}{r^{2}}}dr=dc\Rightarrow \int _{r}^{r+\delta }{\frac {A}{r^{2}}}dr=\int _{c_{0}}^{c_{l}}dc\Rightarrow c_{0}-c_{l}=A[{\frac {1}{r}}-{\frac {1}{r+\delta }}]\Rightarrow A={\frac {c_{0}-c_{l}}{[{\frac {1}{r}}-{\frac {1}{r+\delta }}]}}}"></span>, </p><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 \textstyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99cb7e76c55f4354b1462b7dd2909ebb4ffe80f0" 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 \textstyle r}"></span> is the radius of the nucleus, <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 \textstyle r+\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle r+\delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c0feb13c9d0709e9f2150cd220a31bfe27bb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.938ex; height:2.509ex;" alt="{\displaystyle \textstyle r+\delta }"></span> is the distance from the nucleus where 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 \textstyle c_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle c_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cb75560b086f23a6ef2fa19d50bd6813720bf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle \textstyle c_{0}}"></span> is recovered 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 \textstyle c_{l}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle c_{l}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c85b012739aa98c75b2b5bb07e70d2ea95fe4f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.729ex; height:2.009ex;" alt="{\displaystyle \textstyle c_{l}}"></span> is the concentration right at the surface of the nucleus. Now the expression for <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 \textstyle {\frac {\partial c}{\partial r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {\partial c}{\partial r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f21845c9b87ddce347c88fd0b922566c50d42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.51ex; height:3.843ex;" alt="{\displaystyle \textstyle {\frac {\partial c}{\partial r}}}"></span> can be found by: </p><p><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^{2}{\frac {\partial c}{\partial r}}=A\Rightarrow {\frac {\partial c}{\partial r}}={\frac {A}{r^{2}}}={\frac {c_{0}-c_{l}}{[{\frac {1}{r}}-{\frac {1}{r+\delta }}]r^{2}}}=(c_{0}-c_{l})\cdot ({\frac {1}{r}}+{\frac {1}{\delta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <mo stretchy="false">⇒</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>c</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mo>+</mo> <mi>δ</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">]</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>δ</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}{\frac {\partial c}{\partial r}}=A\Rightarrow {\frac {\partial c}{\partial r}}={\frac {A}{r^{2}}}={\frac {c_{0}-c_{l}}{[{\frac {1}{r}}-{\frac {1}{r+\delta }}]r^{2}}}=(c_{0}-c_{l})\cdot ({\frac {1}{r}}+{\frac {1}{\delta }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058fc3b3fa621f0675fc09213b59bcee71971d38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:60.294ex; height:7.176ex;" alt="{\displaystyle r^{2}{\frac {\partial c}{\partial r}}=A\Rightarrow {\frac {\partial c}{\partial r}}={\frac {A}{r^{2}}}={\frac {c_{0}-c_{l}}{[{\frac {1}{r}}-{\frac {1}{r+\delta }}]r^{2}}}=(c_{0}-c_{l})\cdot ({\frac {1}{r}}+{\frac {1}{\delta }})}"></span> </p><p>Therefore, the growth velocity for a diffusion-controlled system can be described as: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial r}{\partial t}}=V_{at}D(c_{0}-c_{l})\cdot ({\frac {1}{r}}+{\frac {1}{\delta }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>t</mi> </mrow> </msub> <mi>D</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>δ</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial r}{\partial t}}=V_{at}D(c_{0}-c_{l})\cdot ({\frac {1}{r}}+{\frac {1}{\delta }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff29271125ee2f8103b5d6fb422765a313eeda1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.042ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial r}{\partial t}}=V_{at}D(c_{0}-c_{l})\cdot ({\frac {1}{r}}+{\frac {1}{\delta }})}"></span> </p><p><br /> </p> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Dendrite_formation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/4/4d/Dendrite_formation.gif" decoding="async" width="130" height="98" class="mw-file-element" data-file-width="130" data-file-height="98" /></a><figcaption>NASA animation of dendrite formation in microgravity.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dendrites01.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Dendrites01.jpg/350px-Dendrites01.jpg" decoding="async" width="350" height="237" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Dendrites01.jpg/525px-Dendrites01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Dendrites01.jpg/700px-Dendrites01.jpg 2x" data-file-width="1000" data-file-height="676" /></a><figcaption><a href="/wiki/Pyrolusite" title="Pyrolusite">Pyrolusite</a> (<a href="/wiki/Manganese(IV)_oxide" class="mw-redirect" title="Manganese(IV) oxide">manganese(IV) oxides</a>) <a href="/wiki/Dendrite_(crystal)" title="Dendrite (crystal)">dendrites</a> on a <a href="/wiki/Limestone" title="Limestone">limestone</a> <a href="/wiki/Bedding_plane" class="mw-redirect" title="Bedding plane">bedding plane</a> from <a href="/wiki/Solnhofen" title="Solnhofen">Solnhofen</a>, Germany. Scale in mm.</figcaption></figure> <p>Under such diffusion controlled conditions, the polyhedral crystal form will be unstable, it will sprout protrusions at its corners and edges where the degree of supersaturation is at its highest level. The tips of these protrusions will clearly be the points of highest supersaturation. It is generally believed that the protrusion will become longer (and thinner at the tip) until the effect of interfacial free energy in raising the chemical potential slows the tip growth and maintains a constant value for the tip thickness. <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><p>In the subsequent tip-thickening process, there should be a corresponding instability of shape. Minor bumps or "bulges" should be exaggerated—and develop into rapidly growing side branches. In such an unstable (or metastable) situation, minor degrees of anisotropy should be sufficient to determine directions of significant branching and growth. The most appealing aspect of this argument, of course, is that it yields the primary morphological features of <a href="/wiki/Dendrite_(crystal)" title="Dendrite (crystal)">dendritic</a> growth. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Abnormal_grain_growth" title="Abnormal grain growth">Abnormal grain growth</a></li> <li><a href="/wiki/Chvorinov%27s_rule" title="Chvorinov's rule">Chvorinov's rule</a></li> <li><a href="/wiki/Cloud_condensation_nuclei" title="Cloud condensation nuclei">Cloud condensation nuclei</a></li> <li><a href="/wiki/Crystal_structure" title="Crystal structure">Crystal structure</a></li> <li><a href="/wiki/Czochralski_process" class="mw-redirect" title="Czochralski process">Czochralski process</a></li> <li><a href="/wiki/Dendrite_(metal)" title="Dendrite (metal)">Dendrite (metal)</a></li> <li><a href="/wiki/Diana%27s_Tree" title="Diana's Tree">Diana's Tree</a></li> <li><a href="/wiki/Fractional_crystallization_(chemistry)" title="Fractional crystallization (chemistry)">Fractional crystallization</a></li> <li><a href="/wiki/Ice_nucleus" title="Ice nucleus">Ice nucleus</a></li> <li><a href="/wiki/Laser-heated_pedestal_growth" title="Laser-heated pedestal growth">Laser-heated pedestal growth</a></li> <li><a href="/wiki/Manganese_nodule" title="Manganese nodule">Manganese nodule</a></li> <li><a href="/wiki/Micro-pulling-down" title="Micro-pulling-down">Micro-pulling-down</a></li> <li><a href="/wiki/Monocrystalline_whisker" title="Monocrystalline whisker">Monocrystalline whisker</a></li> <li><a href="/wiki/Protocrystalline" title="Protocrystalline">Protocrystalline</a></li> <li><a href="/wiki/Recrystallization_(chemistry)" title="Recrystallization (chemistry)">Recrystallization (chemistry)</a></li> <li><a href="/wiki/Seed_crystal" title="Seed crystal">Seed crystal</a></li> <li><a href="/wiki/Single_crystal" title="Single crystal">Single crystal</a></li> <li><a href="/wiki/Whisker_(metallurgy)" title="Whisker (metallurgy)">Whisker (metallurgy)</a></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Simulation">Simulation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Crystal_growth&action=edit&section=10" title="Edit section: Simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Kinetic_Monte_Carlo_surface_growth_method" class="mw-redirect" title="Kinetic Monte Carlo surface growth method">Kinetic Monte Carlo surface growth method</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=Crystal_growth&action=edit&section=11" 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 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.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="CITEREFMarkov2016" class="citation book cs1">Markov, Ivan (2016). <i>Crystal Growth For Beginners: Fundamentals Of Nucleation, Crystal Growth And Epitaxy</i> (Third ed.). Singapore: World Scientific. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F10127">10.1142/10127</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-3143-85-2" title="Special:BookSources/978-981-3143-85-2"><bdi>978-981-3143-85-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Crystal+Growth+For+Beginners%3A+Fundamentals+Of+Nucleation%2C+Crystal+Growth+And+Epitaxy&rft.place=Singapore&rft.edition=Third&rft.pub=World+Scientific&rft.date=2016&rft_id=info%3Adoi%2F10.1142%2F10127&rft.isbn=978-981-3143-85-2&rft.aulast=Markov&rft.aufirst=Ivan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystal+growth" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPimpinelliVillain2010" class="citation book cs1">Pimpinelli, Alberto; Villain, Jacques (2010). <i>Physics of Crystal Growth</i>. 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C. (1949). "The influence of dislocations on crystal growth". <i>Discussions of the Faraday Society</i>. <b>5</b>: 48. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FDF9490500048">10.1039/DF9490500048</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discussions+of+the+Faraday+Society&rft.atitle=The+influence+of+dislocations+on+crystal+growth&rft.volume=5&rft.pages=48&rft.date=1949&rft_id=info%3Adoi%2F10.1039%2FDF9490500048&rft.aulast=Frank&rft.aufirst=F.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystal+growth" class="Z3988"></span></span> </li> <li id="cite_note-nguyen17-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-nguyen17_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNguyenKhanBruceForbes2017" class="citation journal cs1">Nguyen, Thai; Khan, Azeem; Bruce, Layla; Forbes, Clarissa; o'Leary, Richard; Price, Chris (2017). <a rel="nofollow" class="external text" href="https://doi.org/10.3390%2Fcryst7100294">"The effect of ultrasound on the crystallisation of paracetamol in the presence of structurally similar impurities"</a>. <i>Crystals</i>. <b>7</b> (10): 294. <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/10.3390%2Fcryst7100294">10.3390/cryst7100294</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Crystals&rft.atitle=The+effect+of+ultrasound+on+the+crystallisation+of+paracetamol+in+the+presence+of+structurally+similar+impurities&rft.volume=7&rft.issue=10&rft.pages=294&rft.date=2017&rft_id=info%3Adoi%2F10.3390%2Fcryst7100294&rft.aulast=Nguyen&rft.aufirst=Thai&rft.au=Khan%2C+Azeem&rft.au=Bruce%2C+Layla&rft.au=Forbes%2C+Clarissa&rft.au=o%27Leary%2C+Richard&rft.au=Price%2C+Chris&rft_id=https%3A%2F%2Fdoi.org%2F10.3390%252Fcryst7100294&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystal+growth" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Volmer, M., "Kinetic der Phasenbildung", T. Steinkopf, Dresden (1939)</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurtonCabrera1949" class="citation journal cs1">Burton, W. K.; Cabrera, N. (1949). "Crystal growth and surface structure. 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K.; Cabrera, N. (1949). "Crystal growth and surface structure. Part II". <i>Discuss. Faraday Soc</i>. <b>5</b>: 40–48. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1039%2FDF9490500040">10.1039/DF9490500040</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discuss.+Faraday+Soc.&rft.atitle=Crystal+growth+and+surface+structure.+Part+II&rft.volume=5&rft.pages=40-48&rft.date=1949&rft_id=info%3Adoi%2F10.1039%2FDF9490500040&rft.aulast=Burton&rft.aufirst=W.+K.&rft.au=Cabrera%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystal+growth" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">E.M. Aryslanova, A.V.Alfimov, S.A. Chivilikhin, <a rel="nofollow" class="external text" href="http://nanojournal.ifmo.ru/en/articles-2/volume4/4-5/physics/paper01/">"Model of porous aluminum oxide growth in the initial stage of anodization"</a>, Nanosystems: physics, chemistry, mathematics, October 2013, Volume 4, Issue 5, pp 585</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurtonCabreraFrank1951" class="citation journal cs1">Burton, W. K.; Cabrera, N.; Frank, F. C. (1951). "The Growth of Crystals and the Equilibrium Structure of their Surfaces". <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society_A" title="Philosophical Transactions of the Royal Society A">Philosophical Transactions of the Royal Society A</a></i>. <b>243</b> (866): 299. <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/1951RSPTA.243..299B">1951RSPTA.243..299B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1951.0006">10.1098/rsta.1951.0006</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:119643095">119643095</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+A&rft.atitle=The+Growth+of+Crystals+and+the+Equilibrium+Structure+of+their+Surfaces&rft.volume=243&rft.issue=866&rft.pages=299&rft.date=1951&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119643095%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frsta.1951.0006&rft_id=info%3Abibcode%2F1951RSPTA.243..299B&rft.aulast=Burton&rft.aufirst=W.+K.&rft.au=Cabrera%2C+N.&rft.au=Frank%2C+F.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACrystal+growth" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Jackson, K.A. (1958) in <i>Growth and Perfection of Crystals</i>, Doremus, R.H., Roberts, B.W. and Turnbull, D. (eds.). Wiley, New York.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCabrera1959" class="citation journal cs1">Cabrera, N. (1959). 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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/PMC6290125">6290125</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/30582044">30582044</a>.</cite><span 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