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Spinodal - Wikipedia

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class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/File:LCST-UCST_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/LCST-UCST_plot.svg/350px-LCST-UCST_plot.svg.png" decoding="async" width="350" height="350" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/LCST-UCST_plot.svg/525px-LCST-UCST_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/LCST-UCST_plot.svg/700px-LCST-UCST_plot.svg.png 2x" data-file-width="430" data-file-height="430" /></a><figcaption>A phase diagram displaying spinodal curves, within the <a href="/wiki/Binodal" title="Binodal">binodal</a> coexistence curves and two <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical points</a>: an <a href="/wiki/Upper_critical_solution_temperature" title="Upper critical solution temperature">upper</a> and <a href="/wiki/Lower_critical_solution_temperature" title="Lower critical solution temperature">lower critical solution temperature</a>.</figcaption></figure> <p>In <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, the limit of local stability against <a href="/wiki/Phase_separation" title="Phase separation">phase separation</a> with respect to small fluctuations is clearly defined by the condition that the <a href="/wiki/Second_derivative" title="Second derivative">second derivative</a> of <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> is zero. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\operatorname {d} ^{2}\!G \over \operatorname {d} \!x^{2}}{=}0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">d</mi> <mspace width="negativethinmathspace" /> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\operatorname {d} ^{2}\!G \over \operatorname {d} \!x^{2}}{=}0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ab3a053942e08c3bd6a59004d448d96cc75ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.98ex; height:6.009ex;" alt="{\displaystyle {\operatorname {d} ^{2}\!G \over \operatorname {d} \!x^{2}}{=}0}"></span></dd></dl> <p>The <a href="/wiki/Locus_(mathematics)" title="Locus (mathematics)">locus</a> of these points (the inflection point within a G-x or G-c curve, <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> as a function of <a href="/wiki/Chemical_composition" title="Chemical composition">composition</a>) is known as the <b>spinodal</b> curve.<sup id="cite_ref-astarita1990_1-0" class="reference"><a href="#cite_note-astarita1990-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-Sandler571_2-0" class="reference"><a href="#cite_note-Sandler571-2"><span class="cite-bracket">&#91;</span>2<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-PPD12_3-0" class="reference"><a href="#cite_note-PPD12-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup> For compositions within this curve, infinitesimally small fluctuations in composition and density will lead to phase separation via <a href="/wiki/Spinodal_decomposition" title="Spinodal decomposition">spinodal decomposition</a>. Outside of the curve, the solution will be at least metastable with respect to fluctuations.<sup id="cite_ref-PPD12_3-1" class="reference"><a href="#cite_note-PPD12-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup> In other words, outside the spinodal curve some careful process may obtain a single phase system.<sup id="cite_ref-PPD12_3-2" class="reference"><a href="#cite_note-PPD12-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup> Inside it, only processes far from <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>, such as <a href="/wiki/Physical_vapor_deposition" title="Physical vapor deposition">physical vapor deposition</a>, will enable one to prepare single phase compositions.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">&#91;</span>4<span class="cite-bracket">&#93;</span></a></sup> The local points of coexisting compositions, defined by the common tangent construction, are known as a <a href="/wiki/Binodal" title="Binodal">binodal coexistence curve</a>, which denotes the minimum-energy equilibrium state of the system. Increasing temperature results in a decreasing difference between mixing entropy and mixing enthalpy, and thus the coexisting compositions come closer. The binodal curve forms the basis for the <a href="/wiki/Miscibility_gap" title="Miscibility gap">miscibility gap</a> in a phase diagram. The free energy of mixing changes with temperature and concentration, and the binodal and spinodal meet at the <a href="/wiki/Upper_critical_solution_temperature" title="Upper critical solution temperature">critical or consolute temperature</a> and composition.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">&#91;</span>5<span class="cite-bracket">&#93;</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Criterion">Criterion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spinodal&amp;action=edit&amp;section=1" title="Edit section: Criterion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For binary solutions, the thermodynamic criterion which defines the spinodal curve is that the second derivative of free energy with respect to density or some composition variable is zero.<sup id="cite_ref-PPD12_3-3" class="reference"><a href="#cite_note-PPD12-3"><span class="cite-bracket">&#91;</span>3<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-Sandler557_6-0" class="reference"><a href="#cite_note-Sandler557-6"><span class="cite-bracket">&#91;</span>6<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-PPD46_7-0" class="reference"><a href="#cite_note-PPD46-7"><span class="cite-bracket">&#91;</span>7<span class="cite-bracket">&#93;</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Critical_point">Critical point</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spinodal&amp;action=edit&amp;section=2" title="Edit section: Critical point"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Extrema of the spinodal in a temperature vs composition plot coincide with those of the binodal curve, and are known as <a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">critical points</a>.<sup id="cite_ref-PPD46_7-1" class="reference"><a href="#cite_note-PPD46-7"><span class="cite-bracket">&#91;</span>7<span class="cite-bracket">&#93;</span></a></sup> The spinodal itself can be thought of as a line of pseudocritical points, with the <a href="/wiki/Correlation_function_(statistical_mechanics)" title="Correlation function (statistical mechanics)">correlation function</a> taking a scaling form with non-classical <a href="/wiki/Critical_exponent" title="Critical exponent">critical exponents</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">&#91;</span>8<span class="cite-bracket">&#93;</span></a></sup> Strictly speaking, a spinodal is defined as a <a href="/wiki/Mean_field_theory" class="mw-redirect" title="Mean field theory">mean field theoretic object</a>. As such, the spinodal does not exist in real systems,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">&#91;</span>9<span class="cite-bracket">&#93;</span></a></sup> but one can extrapolate to infer the existence of a pseudospinodal that exhibits critical-like behavior such as <a href="/wiki/Critical_transition#Critical_slow_down" title="Critical transition">critical slowing down</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">&#91;</span>10<span class="cite-bracket">&#93;</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Isothermal_liquid-liquid_equilibria">Isothermal liquid-liquid equilibria</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spinodal&amp;action=edit&amp;section=3" title="Edit section: Isothermal liquid-liquid equilibria"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the case of ternary isothermal liquid-liquid equilibria, the spinodal curve (obtained from the Hessian matrix) and the corresponding critical point can be used to help the experimental data correlation process.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">&#91;</span>11<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">&#91;</span>12<span class="cite-bracket">&#93;</span></a></sup><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">&#91;</span>13<span class="cite-bracket">&#93;</span></a></sup> </p> <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=Spinodal&amp;action=edit&amp;section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-astarita1990-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-astarita1990_1-0">^</a></b></span> <span class="reference-text">G. 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title="Liquid">Liquid</a></li> <li><a href="/wiki/Gas" title="Gas">Gas</a> / <a href="/wiki/Vapor" title="Vapor">Vapor</a></li> <li><a href="/wiki/Supercritical_fluid" title="Supercritical fluid">Supercritical fluid</a></li> <li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Phase_change_-_en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/150px-Phase_change_-_en.svg.png" decoding="async" width="150" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/225px-Phase_change_-_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Phase_change_-_en.svg/300px-Phase_change_-_en.svg.png 2x" data-file-width="493" data-file-height="524" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Low energy</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li> <li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li> <li><a href="/wiki/Degenerate_matter" title="Degenerate matter">Degenerate matter</a></li> <li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall</a></li> <li><a href="/wiki/Rydberg_matter" title="Rydberg matter">Rydberg matter</a></li> <li><a href="/wiki/Strange_matter" title="Strange matter">Strange matter</a></li> <li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluid</a></li> <li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li> <li><a href="/wiki/Photonic_molecule" title="Photonic molecule">Photonic molecule</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">High energy</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/QCD_matter" title="QCD matter">QCD matter</a></li> <li><a href="/wiki/Quark%E2%80%93gluon_plasma" title="Quark–gluon plasma">Quark–gluon plasma</a></li> <li><a href="/wiki/Color-glass_condensate" title="Color-glass condensate">Color-glass condensate</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other states</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li> <li><a href="/wiki/Crystal" title="Crystal">Crystal</a></li> <li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li> <li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li> <li><a href="/wiki/Quantum_spin_liquid" title="Quantum spin liquid">Quantum spin liquid</a></li> <li><a href="/wiki/Exotic_matter" title="Exotic matter">Exotic matter</a></li> <li><a href="/wiki/Programmable_matter" title="Programmable matter">Programmable matter</a></li> <li><a href="/wiki/Dark_matter" title="Dark matter">Dark matter</a></li> <li><a href="/wiki/Antimatter" title="Antimatter">Antimatter</a></li> <li>Magnetically ordered <ul><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a></li> <li><a href="/wiki/Ferrimagnetism" title="Ferrimagnetism">Ferrimagnet</a></li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li></ul></li> <li><a href="/wiki/String-net_liquid" title="String-net liquid">String-net liquid</a></li> <li><a href="/wiki/Superglass" title="Superglass">Superglass</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Phase_transitions" class="mw-redirect" title="Phase transitions">Phase transitions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boiling" title="Boiling">Boiling</a></li> <li><a href="/wiki/Boiling_point" title="Boiling point">Boiling point</a></li> <li><a href="/wiki/Condensation" title="Condensation">Condensation</a></li> <li><a href="/wiki/Critical_line_(thermodynamics)" title="Critical line (thermodynamics)">Critical line</a></li> <li><a href="/wiki/Critical_point_(thermodynamics)" title="Critical point (thermodynamics)">Critical point</a></li> <li><a href="/wiki/Crystallization" title="Crystallization">Crystallization</a></li> <li><a href="/wiki/Deposition_(phase_transition)" title="Deposition (phase transition)">Deposition</a></li> <li><a href="/wiki/Evaporation" title="Evaporation">Evaporation</a></li> <li><a href="/wiki/Flash_evaporation" title="Flash evaporation">Flash evaporation</a></li> <li><a href="/wiki/Freezing" title="Freezing">Freezing</a></li> <li><a href="/wiki/Chemical_ionization" title="Chemical ionization">Chemical ionization</a></li> <li><a href="/wiki/Ionization" title="Ionization">Ionization</a></li> <li><a href="/wiki/Lambda_point" title="Lambda point">Lambda point</a></li> <li><a href="/wiki/Melting" title="Melting">Melting</a></li> <li><a href="/wiki/Melting_point" title="Melting point">Melting point</a></li> <li><a href="/wiki/Plasma_recombination" title="Plasma recombination">Recombination</a></li> <li><a href="/wiki/Regelation" title="Regelation">Regelation</a></li> <li><a href="/wiki/Vapor%E2%80%93liquid_equilibrium" title="Vapor–liquid equilibrium">Saturated fluid</a></li> <li><a href="/wiki/Sublimation_(phase_transition)" title="Sublimation (phase transition)">Sublimation</a></li> <li><a href="/wiki/Supercooling" title="Supercooling">Supercooling</a></li> <li><a href="/wiki/Triple_point" title="Triple point">Triple point</a></li> <li><a href="/wiki/Vaporization" title="Vaporization">Vaporization</a></li> <li><a href="/wiki/Vitrification" title="Vitrification">Vitrification</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Quantities</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Enthalpy_of_fusion" title="Enthalpy of fusion">Enthalpy of fusion</a></li> <li><a href="/wiki/Enthalpy_of_sublimation" title="Enthalpy of sublimation">Enthalpy of sublimation</a></li> <li><a href="/wiki/Enthalpy_of_vaporization" title="Enthalpy of vaporization">Enthalpy of vaporization</a></li> <li><a href="/wiki/Latent_heat" title="Latent heat">Latent heat</a></li> <li><a href="/wiki/Latent_internal_energy" title="Latent internal energy">Latent internal energy</a></li> <li><a href="/wiki/Trouton%27s_rule" title="Trouton&#39;s rule">Trouton's rule</a></li> <li><a href="/wiki/Volatility_(chemistry)" title="Volatility (chemistry)">Volatility</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baryonic_matter" class="mw-redirect" title="Baryonic matter">Baryonic matter</a></li> <li><a href="/wiki/Binodal" title="Binodal">Binodal</a></li> <li><a href="/wiki/Compressed_fluid" title="Compressed fluid">Compressed fluid</a></li> <li><a href="/wiki/Cooling_curve" title="Cooling curve">Cooling curve</a></li> <li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Leidenfrost_effect" title="Leidenfrost effect">Leidenfrost effect</a></li> <li><a href="/wiki/Macroscopic_quantum_phenomena" title="Macroscopic quantum phenomena">Macroscopic quantum phenomena</a></li> <li><a href="/wiki/Mpemba_effect" title="Mpemba effect">Mpemba effect</a></li> <li><a href="/wiki/Order_and_disorder_(physics)" class="mw-redirect" title="Order and disorder (physics)">Order and disorder (physics)</a></li> <li><a class="mw-selflink selflink">Spinodal</a></li> <li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductivity</a></li> <li><a href="/wiki/Superheated_water" title="Superheated water">Superheated vapor</a></li> <li><a href="/wiki/Superheating" title="Superheating">Superheating</a></li> <li><a href="/wiki/Thermo-dielectric_effect" title="Thermo-dielectric effect">Thermo-dielectric effect</a></li></ul> </div></td></tr></tbody></table></div> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐cdd4f87bc‐8hscg Cached time: 20241120213522 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.267 seconds Real time usage: 0.346 seconds Preprocessor visited node count: 649/1000000 Post‐expand include size: 27891/2097152 bytes Template argument size: 130/2097152 bytes Highest 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