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Sedimentation profiles and phase stacking diagrams in polydisperse hard rounded rectangle fluids

Tobias Eckert

Daniel de las Heras*

Enrique Velasco

Yuri Martínez-Ratón

  • Remote Sensing Technology Institute, Earth Observation Center, German Aerospace Center (DLR), Oberpfaffenhofen, D-82234 Weßling, Germany and Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

  • Departamento de Física Teórica de la Materia Condensada, Instituto de Física de la Materia Condensada (IFIMAC) and Instituto de Ciencia de Materiales Nicolás Cabrera, Universidad Autónoma de Madrid, E-28049 Madrid, Spain

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Phys. Rev. E 112, 065410 – Published 2 December, 2025

DOI: https://doi.org/10.1103/kd2c-5hkf

Abstract

We analyze the sedimentation behavior of a polydisperse two-dimensional liquid-crystal fluid using a local density-functional theory based on scaled particle theory. Polydispersity is incorporated through variations in the roundness of hard rectangular particles interacting solely via excluded area effects. Despite its simplicity, the model displays a rich phenomenology. In bulk, the fluid exhibits isotropic, nematic, and tetratic phases. In sedimentation, we obtain complex phase stacking diagrams featuring multiphasic stacking sequences with up to four stacks of different bulk phases, inverted stacking sequences such as top isotropic and bottom nematic together with top nematic and bottom isotopic, as well as stacking sequences with reentrant stacks such as tetratic and nematic stacks floating between two isotropic stacks. This phenomenology arises as a result of an intricate coupling between particle polydispersity and the effect of gravity. Our approach can be easily adapted to investigate the sedimentation behavior of other polydisperse colloidal systems.

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References (67)

  1. G. Lagaly, Handbook of Clay Science (Elsevier, New York, 2006), p. 141.
  2. R. P. Murphy, K. Hong, and N. J. Wagner, Synthetic control of the size, shape, and polydispersity of anisotropic silica colloids, J. Colloid Interface Sci. 501, 45 (2017).
  3. J.-H. Kim, H. J. Hwang, J. S. Oh, S. Sacanna, and G.-R. Yi, Monodisperse magnetic silica hexapods, J. Am. Chem. Soc. 140, 9230 (2018).
  4. J. Roller, J. D. Geiger, M. Voggenreiter, J.-M. Meijer, and A. Zumbusch, Formation of nematic order in 3D systems of hard colloidal ellipsoids, Soft Matter 16, 1021 (2020).
  5. M. Voggenreiter, J. Roller, J. Geiger, L. Ebner, A. Zumbusch, and J.-M. Meijer, Preparation and tracking of oblate core-shell polymethyl-methacrylate ellipsoids, Langmuir 36, 13087 (2020).
  6. J. C. de la Vega, P. Elischer, T. Schneider, and U. O. Häfeli, Uniform polymer microspheres: Monodispersity criteria, methods of formation and applications, Nanomedicine 8, 265 (2013).
  7. B. N. Khlebtsov and A. M. Burov, Synthesis of monodisperse silica particles by controlled regrowth, Colloid J. 85, 456 (2023).
  8. H. N. W. Lekkerkerker and G. J. Vroege, Liquid crystal phase transitions in suspensions of mineral colloids: New life from old roots, Phil. Trans. R. Soc. A. 371, 20120263 (2013).
  9. H. Almohammadi, S. A. Khadem, P. Azzari, Y. Yuan, A. Guerra, A. D. Rey, and R. Mezzenga, Liquid-liquid crystalline phase separation of filamentous colloids and semiflexible polymers: Experiments, theory and simulations, Rep. Prog. Phys. 88, 036601 (2025).
  10. P. Sollich, Predicting phase equilibria in polydisperse systems, J. Phys.: Condens. Matter 14, R79 (2002).
  11. Y. Martínez-Ratón and J. A. Cuesta, Enhancement by polydispersity of the biaxial nematic phase in a mixture of hard rods and plates, Phys. Rev. Lett. 89, 185701 (2002).
  12. C. A. De Filippo, S. Del Galdo, P. Corsi, C. De Michele, and B. Capone, On the role of polydispersity on the phase diagram of colloidal rods, Soft Matter 19, 1732 (2023).
  13. Y. Martínez-Ratón and E. Velasco, Effect of polydispersity, bimodality, and aspect ratio on the phase behavior of colloidal platelet suspensions, J. Chem. Phys. 137, 134906 (2012).
  14. E. Velasco and Y. Martínez-Ratón, Interplay between columnar and smectic stability in suspensions of polydisperse colloidal platelets, Phys. Chem. Chem. Phys. 16, 765 (2014).
  15. A. Díaz-De Armas and Y. Martínez-Ratón, Role of length polydispersity in the phase behavior of freely rotating hard-rectangle fluids, Phys. Rev. E 95, 052702 (2017).
  16. D. V. Byelov, M. C. D. Mourad, I. Snigireva, A. Snigirev, A. V. Petukhov, and H. N. W. Lekkerkerker, Experimental observation of fractionated crystallization in polydisperse platelike colloids, Langmuir 26, 6898 (2010).
  17. P. Pusey, The effect of polydispersity on the crystallization of hard spherical colloids, J. Phys. France 48, 709 (1987).
  18. D. A. Kofke and P. G. Bolhuis, Freezing of polydisperse hard spheres, Phys. Rev. E 59, 618 (1999).
  19. S. Auer and D. Frenkel, Suppression of crystal nucleation in polydisperse colloids due to increase of the surface free energy, Nature (London) 413, 711 (2001).
  20. M. A. Bates and D. Frenkel, Influence of polydispersity on the phase behavior of colloidal liquid crystals: A Monte Carlo simulation study, J. Chem. Phys. 109, 6193 (1998).
  21. A. Kuijk, D. V. Byelov, A. V. Petukhov, A. van Blaaderen, and A. Imhof, Phase behavior of colloidal silica rods, Faraday Discuss. 159, 181 (2012).
  22. A. V. Petukhov, D. van der Beek, R. P. A. Dullens, I. P. Dolbnya, G. J. Vroege, and H. N. W. Lekkerkerker, Observation of a hexatic columnar liquid crystal of polydisperse colloidal disks, Phys. Rev. Lett. 95, 077801 (2005).
  23. D. Sun, H.-J. Sue, Z. Cheng, Y. Martínez-Ratón, and E. Velasco, Stable smectic phase in suspensions of polydisperse colloidal platelets with identical thickness, Phys. Rev. E 80, 041704 (2009).
  24. R. Kotni, A. Grau-Carbonell, M. Chiappini, M. Dijkstra, and A. van Blaaderen, Splay-bend nematic phases of bent colloidal silica rods induced by polydispersity, Nat. Commun. 13, 7264 (2022).
  25. T. Biben, J.-P. Hansen, and J.-L. Barrat, Density profiles of concentrated colloidal suspensions in sedimentation equilibrium, J. Chem. Phys. 98, 7330 (1993).
  26. R. Piazza, T. Bellini, and V. Degiorgio, Equilibrium sedimentation profiles of screened charged colloids: A test of the hard-sphere equation of state, Phys. Rev. Lett. 71, 4267 (1993).
  27. D. van der Beek, T. Schilling, and H. N. W. Lekkerkerker, Gravity-induced liquid crystal phase transitions of colloidal platelets, J. Chem. Phys. 121, 5423 (2004).
  28. S. V. Savenko and M. Dijkstra, Sedimentation and multiphase equilibria in suspensions of colloidal hard rods, Phys. Rev. E 70, 051401 (2004).
  29. T. Eckert, M. Schmidt, and D. de las Heras, Effect of sample height and particle elongation in the sedimentation of colloidal rods, Soft Matter 19, 2214 (2023).
  30. F. M. van der Kooij and H. N. W. Lekkerkerker, Liquid-crystalline phase behavior of a colloidal rod-plate mixture, Phys. Rev. Lett. 84, 781 (2000).
  31. F. M. van der Kooij and H. N. W. Lekkerkerker, Liquid-crystal phases formed in mixed suspensions of rod- and platelike colloids, Langmuir 16, 10144 (2000).
  32. L. Luan, W. Li, S. Liu, and D. Sun, Phase behavior of mixtures of positively charged colloidal platelets and nonadsorbing polymer, Langmuir 25, 6349 (2009).
  33. T. Nakato, Y. Yamashita, E. Mouri, and K. Kuroda, Multiphase coexistence and destabilization of liquid crystalline binary nanosheet colloids of titanate and clay, Soft Matter 10, 3161 (2014).
  34. M. Chen, H. Li, Y. Chen, A. F. Mejia, X. Wang, and Z. Cheng, Observation of isotropic-isotropic demixing in colloidal platelet-sphere mixtures, Soft Matter 11, 5775 (2015).
  35. T. Eckert, M. Schmidt, and D. de las Heras, Gravity-induced phase phenomena in plate-rod colloidal mixtures, Commun. Phys. 4, 202 (2021).
  36. D. de las Heras, N. Doshi, T. Cosgrove, J. Phipps, D. I. Gittins, J. S. V. Duijneveldt, and M. Schmidt, Floating nematic phase in colloidal platelet-sphere mixtures, Sci. Rep. 2, 789 (2012).
  37. G. Vroege, D. Thies-Weesie, A. Petukhov, B. Lemaire, and P. Davidson, Smectic liquid-crystalline order in suspensions of highly polydisperse goethite nanorods, Adv. Mater. 18, 2565 (2006).
  38. E. van den Pol, D. M. E. Thies-Weesie, A. V. Petukhov, G. J. Vroege, and K. Kvashnina, Influence of polydispersity on the phase behavior of colloidal goethite, J. Chem. Phys. 129, 164715 (2008).
  39. Z. X. Zhang and J. S. van Duijneveldt, Isotropic-nematic phase transition of nonaqueous suspensions of natural clay rods, J. Chem. Phys. 124, 154910 (2006).
  40. F. M. van der Kooij, D. van der Beek, and H. N. W. Lekkerkerker, Isotropic-nematic phase separation in suspensions of polydisperse colloidal platelets, J. Phys. Chem. B 105, 1696 (2001).
  41. D. de las Heras and M. Schmidt, The phase stacking diagram of colloidal mixtures under gravity, Soft Matter 9, 8636 (2013).
  42. T. Geigenfeind and D. de las Heras, The role of sample height in the stacking diagram of colloidal mixtures under gravity, J. Phys.: Condens. Matter 29, 064006 (2017).
  43. T. Drwenski, P. Hooijer, and R. van Roij, Sedimentation stacking diagrams of binary mixtures of thick and thin hard rods, Soft Matter 12, 5684 (2016).
  44. D. de las Heras, L. L. Treffenstädt, and M. Schmidt, Reentrant network formation in patchy colloidal mixtures under gravity, Phys. Rev. E 93, 030601(R) (2016).
  45. G. Avvisati, T. Dasgupta, and M. Dijkstra, Fabrication of colloidal laves phases via hard tetramers and hard spheres: Bulk phase diagram and sedimentation behavior, ACS Nano 11, 7702 (2017).
  46. T. Dasgupta and M. Dijkstra, Towards the colloidal laves phase from binary hard-sphere mixtures via sedimentation, Soft Matter 14, 2465 (2018).
  47. R. Braz Teixeira, D. de las Heras, J. M. Tavares, and M. M. Telo da Gama, Phase behavior of a binary mixture of patchy colloids: Effect of particle size and gravity, J. Chem. Phys. 155, 044903 (2021).
  48. T. Eckert, M. Schmidt, and D. de las Heras, Sedimentation of colloidal plate-sphere mixtures and inference of particle characteristics from stacking sequences, Phys. Rev. Res. 4, 013189 (2022).
  49. D. de las Heras and M. Schmidt, Sedimentation stacking diagram of binary colloidal mixtures and bulk phases in the plane of chemical potentials, J. Phys.: Condens. Matter 27, 194115 (2015).
  50. T. Eckert, M. Schmidt, and D. de las Heras, Sedimentation path theory for mass-polydisperse colloidal systems, J. Chem. Phys 157, 234901 (2022).
  51. L. Bellier-Castella and H. Xu, Sedimentation profiles of polydisperse fluids, J. Phys.: Condens. Matter 15, 5417 (2003).
  52. Y. Martínez-Ratón and E. Velasco, Effect of combined roundness and polydispersity on the phase behavior of hard-rectangle fluids, Phys. Rev. E 106, 034602 (2022).
  53. E. Helfand, H. Reiss, H. L. Frisch, and J. L. Lebowitz, Scaled particle theory of fluids, J. Chem. Phys. 33, 1379 (1960).
  54. S. Pollock, L. G. Rebholz, and M. Xiao, Anderson-accelerated convergence of Picard iterations for incompressible Navier–Stokes equations, SIAM J. Numer. Anal. 57, 615 (2019).
  55. A. Speranza and P. Sollich, Isotropic-nematic phase equilibria of polydisperse hard rods: The effect of fat tails in the length distribution, J. Chem. Phys. 118, 5213 (2003).
  56. A. L. Thorneywork, J. L. Abbott, D. G. A. L. Aarts, and R. P. A. Dullens, Two-dimensional melting of colloidal hard spheres, Phys. Rev. Lett. 118, 158001 (2017).
  57. R. Roth, Fundamental measure theory for hard-sphere mixtures: A review, J. Phys.: Condens. Matter 22, 063102 (2010).
  58. Y. Martínez-Ratón, S. Varga, and E. Velasco, Biaxial nematic phases in fluids of hard board-like particles, Phys. Chem. Chem. Phys. 13, 13247 (2011).
  59. R. Wittmann, M. Marechal, and K. Mecke, Fundamental measure theory for non-spherical hard particles: Predicting liquid crystal properties from the particle shape, J. Phys.: Condens. Matter 28, 244003 (2016).
  60. R. Wittmann, C. E. Sitta, F. Smallenburg, and H. Löwen, Phase diagram of two-dimensional hard rods from fundamental mixed measure density functional theory, J. Chem. Phys. 147, 134908 (2017).
  61. A. El Moumane, M. te Vrugt, H. Löwen, and R. Wittmann, Biaxial nematic order in fundamental measure theory, J. Chem. Phys. 160, 094903 (2024).
  62. R. Zwanzig, First-order phase transition in a gas of long thin rods, J. Chem. Phys. 39, 1714 (1963).
  63. M. Buzzacchi, I. Pagonabarraga, and N. B. Wilding, Polydisperse hard spheres at a hard wall, J. Chem. Phys. 121, 11362 (2004).
  64. Y.-X. Yu, J. Wu, Y.-X. Xin, and G.-H. Gao, Structures and correlation functions of multicomponent and polydisperse hard-sphere mixtures from a density functional theory, J. Chem. Phys. 121, 1535 (2004).
  65. F. Sammüller, S. Hermann, D. de las Heras, and M. Schmidt, Neural functional theory for inhomogeneous fluids: Fundamentals and applications, Proc. Natl. Acad. Sci. USA 120, e2312484120 (2023).
  66. T. Zimmermann, F. Sammüller, S. Hermann, M. Schmidt, and D. de las Heras, Neural force functional for non-equilibrium many-body colloidal systems, Mach. Learn.: Sci. Technol. 5, 035062 (2024).
  67. L. Ding and C. Do, Deciphering the small-angle scattering of polydisperse hard spheres using deep learning, APL Mach. Learn. 3, 036112 (2025).

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Phys. Rev. E 112, 065410– Published 2 December, 2025
  • Received 1 September 2025
  • Accepted 6 November 2025
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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.

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