Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams
Phase diagrams are powerful tools to understand the multi-scale behaviour of complex systems. Yet, their determination requires in practice both experiments and computations, which quickly becomes a daunting task. Here, we propose a geometrical approach to simplify the numerical computation of liqui...
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| Format: | Article |
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MDPI AG
2023-09-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/9/1329 |
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| author | Nataliya Shcherbakova Vincent Gerbaud Kevin Roger |
| author_facet | Nataliya Shcherbakova Vincent Gerbaud Kevin Roger |
| author_sort | Nataliya Shcherbakova |
| collection | DOAJ |
| description | Phase diagrams are powerful tools to understand the multi-scale behaviour of complex systems. Yet, their determination requires in practice both experiments and computations, which quickly becomes a daunting task. Here, we propose a geometrical approach to simplify the numerical computation of liquid–liquid ternary phase diagrams. We show that using the intrinsic geometry of the binodal curve, it is possible to formulate the problem as a simple set of ordinary differential equations in an extended 4D space. Consequently, if the thermodynamic potential, such as Gibbs free energy, is known from an experimental data set, the whole phase diagram, including the spinodal curve, can be easily computed. We showcase this approach on four ternary liquid–liquid diagrams, with different topological properties, using a modified Flory–Huggins model. We demonstrate that our method leads to similar or better results comparing those obtained with other methods, but with a much simpler procedure. Acknowledging and using the intrinsic geometry of phase diagrams thus appears as a promising way to further develop the computation of multiphase diagrams. |
| first_indexed | 2024-03-10T22:47:18Z |
| format | Article |
| id | doaj.art-6e497fe13fb0408991b7f7c3b8b2073c |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T22:47:18Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-6e497fe13fb0408991b7f7c3b8b2073c2023-11-19T10:36:05ZengMDPI AGEntropy1099-43002023-09-01259132910.3390/e25091329Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase DiagramsNataliya Shcherbakova0Vincent Gerbaud1Kevin Roger2Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INP, UPS, 31432 Toulouse, FranceLaboratoire de Génie Chimique, Université de Toulouse, CNRS, INP, UPS, 31432 Toulouse, FranceLaboratoire de Génie Chimique, Université de Toulouse, CNRS, INP, UPS, 31432 Toulouse, FrancePhase diagrams are powerful tools to understand the multi-scale behaviour of complex systems. Yet, their determination requires in practice both experiments and computations, which quickly becomes a daunting task. Here, we propose a geometrical approach to simplify the numerical computation of liquid–liquid ternary phase diagrams. We show that using the intrinsic geometry of the binodal curve, it is possible to formulate the problem as a simple set of ordinary differential equations in an extended 4D space. Consequently, if the thermodynamic potential, such as Gibbs free energy, is known from an experimental data set, the whole phase diagram, including the spinodal curve, can be easily computed. We showcase this approach on four ternary liquid–liquid diagrams, with different topological properties, using a modified Flory–Huggins model. We demonstrate that our method leads to similar or better results comparing those obtained with other methods, but with a much simpler procedure. Acknowledging and using the intrinsic geometry of phase diagrams thus appears as a promising way to further develop the computation of multiphase diagrams.https://www.mdpi.com/1099-4300/25/9/1329LLE ternary diagramsbinodal curvesspinodal curvesdifferential path-following methodordinary differential equations |
| spellingShingle | Nataliya Shcherbakova Vincent Gerbaud Kevin Roger Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams Entropy LLE ternary diagrams binodal curves spinodal curves differential path-following method ordinary differential equations |
| title | Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams |
| title_full | Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams |
| title_fullStr | Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams |
| title_full_unstemmed | Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams |
| title_short | Using the Intrinsic Geometry of Binodal Curves to Simplify the Computation of Ternary Liquid–Liquid Phase Diagrams |
| title_sort | using the intrinsic geometry of binodal curves to simplify the computation of ternary liquid liquid phase diagrams |
| topic | LLE ternary diagrams binodal curves spinodal curves differential path-following method ordinary differential equations |
| url | https://www.mdpi.com/1099-4300/25/9/1329 |
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