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Spinodal
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<h2 id="criterion">Criterion</h2> <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.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="critical-point">Critical point</h2> <p>Extrema of the spinodal in a temperature vs composition plot coincide with those of the binodal curve, and are known as <a href="/facts/Critical_point_(thermodynamics)/pkM7RLJK">critical points</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> The spinodal itself can be thought of as a line of pseudocritical points, with the <a href="/facts/Correlation_function_(statistical_mechanics)/eWP84mzu">correlation function</a> taking a scaling form with non-classical <a href="/facts/Critical_exponent/RHnV8E3P">critical exponents</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Strictly speaking, a spinodal is defined as a <a href="/facts/Mean_field_theory/tJrwSURE">mean field theoretic object</a>. As such, the spinodal does not exist in real systems,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> but one can extrapolate to infer the existence of a pseudospinodal that exhibits critical-like behavior such as <a href="/facts/Critical_transition/Mv10GCHt">critical slowing down</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h3>Isothermal liquid-liquid equilibria</h3> <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.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a><a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>G. Astarita: Thermodynamics: An Advanced Textbook for Chemical Engineers (Springer 1990), chaps 4, 8, 9, 12. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Sandler S. I., Chemical and Engineering Thermodynamics. 1999 John Wiley & Sons, Inc., p 571. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Koningsveld K., Stockmayer W. H., Nies, E., Polymer Phase Diagrams: A Textbook. 2001 Oxford, p 12. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Koningsveld K., Stockmayer W. H., Nies, E., Polymer Phase Diagrams: A Textbook. 2001 Oxford, p 12. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Koningsveld K., Stockmayer W. H., Nies, E., Polymer Phase Diagrams: A Textbook. 2001 Oxford, p 12. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>P.H. Mayrhofer et al. Progress in Materials Science 51 (2006) 1032-1114 doi:10.1016/j.pmatsci.2006.02.002 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Cahn RW, Haasen P. Physical metallurgy. 4th ed. Cambridge: Univ Press; 1996 <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Koningsveld K., Stockmayer W. H., Nies, E., Polymer Phase Diagrams: A Textbook. 2001 Oxford, p 12. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Sandler S. I., Chemical and Engineering Thermodynamics. 1999 John Wiley & Sons, Inc., p 557. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Koningsveld K., Stockmayer W. H., Nies, E., Polymer Phase Diagrams: A Textbook. 2001 Oxford, pp 46-47. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Koningsveld K., Stockmayer W. H., Nies, E., Polymer Phase Diagrams: A Textbook. 2001 Oxford, pp 46-47. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Saito, Y. (1978-02-01). "Pseudocritical Phenomena near the Spinodal Point". Progress of Theoretical Physics. 59 (2): 375–385. Bibcode:1978PThPh..59..375S. doi:10.1143/PTP.59.375. ISSN 0033-068X. <a href="https://academic.oup.com/ptp/article-lookup/doi/10.1143/PTP.59.375" target="_blank">https://academic.oup.com/ptp/article-lookup/doi/10.1143/PTP.59.375</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>MONETTE, L. (1994-05-30). "Spinodal Nucleation". International Journal of Modern Physics B. 08 (11n12): 1417–1527. Bibcode:1994IJMPB...8.1417M. doi:10.1142/s0217979294000646. ISSN 0217-9792. <a href="https://dx.doi.org/10.1142/s0217979294000646" target="_blank">https://dx.doi.org/10.1142/s0217979294000646</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Unger, Chris; Klein, W. (1984-03-01). "Nucleation theory near the classical spinodal". Physical Review B. 29 (5): 2698–2708. Bibcode:1984PhRvB..29.2698U. doi:10.1103/physrevb.29.2698. ISSN 0163-1829. <a href="https://dx.doi.org/10.1103/physrevb.29.2698" target="_blank">https://dx.doi.org/10.1103/physrevb.29.2698</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Marcilla, A.; Serrano, M.D.; Reyes-Labarta, J.A.; Olaya, M.M. (2012). "Checking Liquid-Liquid Critical Point Conditions and their Application in Ternary Systems". Industrial & Engineering Chemistry Research. 51 (13): 5098–5102. doi:10.1021/ie202793r. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Marcilla, A.; Reyes-Labarta, J.A.; Serrano, M.D.; Olaya, M.M. (2011). "GE Models and Algorithms for Condensed Phase Equilibrium Data Regression in Ternary Systems: Limitations and Proposals". The Open Thermodynamics Journal. 5: 48–62. doi:10.2174/1874396X01105010048. hdl:10045/19865. <a href="https://doi.org/10.2174%2F1874396X01105010048" target="_blank">https://doi.org/10.2174%2F1874396X01105010048</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Labarta, Juan A.; Olaya, Maria del Mar; Marcilla, Antonio (2015-11-27). "GMcal_TieLinesLL: Graphical User Interface (GUI) for the Topological Analysis of Calculated GM Surfaces and Curves, including Tie-Lines, Hessian Matrix, Spinodal Curve, Plait Point Location, etc. for Binary and Ternary Liquid -Liquid Equilibrium (LLE) Data". University of Alicante. hdl:10045/51725. <a href="https://hdl.handle.net/10045/51725" target="_blank">https://hdl.handle.net/10045/51725</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> </ol>