- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Ensemble inequivalence in the design of mixtures with super-Gibbs phase coexistence

Filipe C. Thewes1,2,* and Peter Sollich1,3

  • *Contact author: filipe.thewes@ds.mpg.de
PDF

Phys. Rev. E 112, 024131 – Published 25 August, 2025

DOI: https://doi.org/10.1103/z6v8-pmm9

Abstract

Designing the phase behavior of multicomponent mixtures is a rich area with many potential applications. One key question is how more than M+1 phases, as would normally be allowed by Gibbs' phase rule at generic temperature in a mixture of M molecular species, can be made to coexist in equilibrium. While such “super-Gibbs” phase coexistence is possible in the grand-canonical ensemble by tuning interactions among the M species, there is no straightforward equivalence in the canonical ensemble: Only a subset of the grand-canonical phases will generically be realized. Here, we show that, upon further design of interface tensions, it is possible to stabilize a super-Gibbs number of phases also in the experimentally relevant canonical ensemble, thus effectively restoring equivalence to the grand-canonical one. Using a graph-theoretical approach, we determine a sufficient set of inequalities for the interfacial tensions for which all grand-canonical phases are realized. We illustrate the design method for a two-component mixture with four coexisting phases and point out the route for generalizing this to a higher number of components.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (36)

  1. J. Meunier and A. Morel, Condensation and metastability in the 2D Potts model, Eur. Phys. J. B 13, 341 (2000).
  2. D. Zwicker and L. Laan, Evolved interactions stabilize many coexisting phases in multicomponent liquids, Proc. Natl. Acad. Sci. USA 119, e2201250119 (2022).
  3. S. Mao, D. Kuldinow, M. P. Haataja, and A. Košmrlj, Phase behavior and morphology of multicomponent liquid mixtures, Soft. Matter. 15, 1297 (2019).
  4. P. Sollich, Predicting phase equilibria in polydisperse systems, J. Phys.: Condens. Matter 14, R79 (2002).
  5. A. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43, 357 (1994).
  6. S. Mao, M. S. Chakraverti-Wuerthwein, H. Gaudio, and A. Košmrlj, Designing the morphology of separated phases in multicomponent liquid mixtures, Phys. Rev. Lett. 125, 218003 (2020).
  7. W. M. Jacobs, Self-assembly of biomolecular condensates with shared components, Phys. Rev. Lett. 126, 258101 (2021).
  8. F. Chen and W. M. Jacobs, Programmable phase behavior in fluids with designable interactions, J. Chem. Phys. 158, (2023).
  9. V. F. D. Peters, A. González García, H. H. Wensink, M. Vis, and R. Tuinier, Multiphase coexistences in rod–polymer mixtures, Langmuir 37, 11582 (2021).
  10. M. F. Haase and J. Brujic, Tailoring of high-order multiple emulsions by the liquid-liquid phase separation of ternary mixtures, Angew. Chem. Int. Ed. 53, 11793 (2014).
  11. M. Feric, N. Vaidya, T. S. Harmon, D. M. Mitrea, L. Zhu, T. M. Richardson, R. W. Kriwacki, R. V. Pappu, and C. P. Brangwynne, Coexisting liquid phases underlie nucleolar subcompartments, Cell 165, 1686 (2016).
  12. J. R. Simon, N. J. Carroll, M. Rubinstein, A. Chilkoti, and G. P. López, Programming molecular self-assembly of intrinsically disordered proteins containing sequences of low complexity, Nat. Chem. 9, 509 (2017).
  13. T. Lu and E. Spruijt, Multiphase complex coacervate droplets, J. Am. Chem. Soc. 142, 2905 (2020).
  14. T. Kaur, M. Raju, I. Alshareedah, R. B. Davis, D. A. Potoyan, and P. R. Banerjee, Sequence-encoded and composition-dependent protein-RNA interactions control multiphasic condensate morphologies, Nat. Commun. 12, 872 (2021).
  15. A. J. Mathijssen, M. Lisicki, V. N. Prakash, and E. J. Mossige, Culinary fluid mechanics and other currents in food science, Rev. Mod. Phys. 95, 025004 (2023).
  16. C. Vega and P. A. Monson, Plastic crystal phases of hard dumbbells and hard spherocylinders, J. Chem. Phys. 107, 2696 (1997).
  17. K. Akahane, J. Russo, and H. Tanaka, A possible four-phase coexistence in a single-component system, Nat. Commun. 7, 12599 (2016).
  18. V. F. D. Peters, M. Vis, A. G. García, H. H. Wensink, and R. Tuinier, Defying the Gibbs phase rule: Evidence for an entropy-driven quintuple point in colloid-polymer mixtures, Phys. Rev. Lett. 125, 127803 (2020).
  19. H.-O. Georgii, The equivalence of ensembles for classical systems of particles, J. Stat. Phys. 80, 1341 (1995).
  20. P. Sollich and M. E. Cates, Projected free energies for polydisperse phase equilibria, Phys. Rev. Lett. 80, 1365 (1998).
  21. D. S. Dean and A. Gopinathan, The non-equilibrium behavior of pseudo-Casimir forces, J. Stat. Mech. (2009) L08001.
  22. M. Krüger, A. Solon, V. Démery, C. M. Rohwer, and D. S. Dean, Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory, J. Chem. Phys. 148, 084503 (2018).
  23. F. C. Thewes, M. Krüger, and P. Sollich, Mobility-induced kinetic effects in multicomponent mixtures, Europhys. Lett. 147, 27001 (2024).
  24. This is strictly true only for an infinite system. In the Hamiltonian setup in 1d one can only use the conserved value of the Hamiltonian to change the trajectories. This allows one to change the total time, i.e., the system size L, but not the fractional volumes of the two phases. For the latter one needs to add a small tilt to −Veff by changing the chemical potential by ∼1/L. So in the generic case the maxima are not perfectly at the same height; nonetheless, their heights either decrease or increase monotonically with density so that periodic trajectories remain constrained to a single valley between adjacent maxima.
  25. K. Kawasaki and T. Nagai, Statistical dynamics of interacting kinks I, Physica A 121, 175 (1983).
  26. K. Kawasaki and T. Nagai, Quasi-one-dimensional kinetics of ordering processes in layered Ising magnets, J. Phys C: Solid State Phys. 19, L551 (1986).
  27. K. Kawasaki, A. Ogawa, and T. Nagai, Dynamical behavior of domain walls in quenched quasi-one-dimensional systems, Physica B+C 149, 97 (1988).
  28. C. Varea, Spinodal decomposition, power laws, and wetting at a triple point, Phys. Rev. E 67, 011508 (2003).
  29. The direction of the walk is unimportant here because time can be reversed in Hamiltonian dynamics or, physically, the density profiles can be mirrored, x→−x.
  30. D. B. West, Introduction to Graph Theory, 2nd ed. (Pearson, Upper Saddle River, NJ, 2000).
  31. D. S. Dean, Langevin equation for the density of a system of interacting Langevin processes, J. Phys. A: Math. Gen. 29, L613 (1996).
  32. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids, 3rd ed. (Academic Press, San Diego, CA, 2006).
  33. M. Vis, K. J. H. Brouwer, A. González García, A. V. Petukhov, O. Konovalov, and R. Tuinier, Quantification of the structure of colloidal gas–liquid interfaces, J. Phys. Chem. Lett. 11, 8372 (2020).
  34. V. Mandalaparthy and W. G. Noid, A simple theory for interfacial properties of dilute solutions, J. Chem. Phys. 157, 034703 (2022).
  35. C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (MIT Press, Cambridge, MA, 2005).
  36. P. Bogacki and L. Shampine, A 3(2) pair of Runge-Kutta formulas, Appl. Math. Lett. 2, 321 (1989).

Outline

Information

Phys. Rev. E 112, 024131– Published 25 August, 2025
  • Received 15 January 2025
  • Accepted 30 July 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation