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  • Open Access

Formation of binary cluster crystals by ultrasoft particles

Vanessa Schweidler*, Felix Tscharnutter*, and Gerhard Kahl†

  • Institut für Theoretische Physik, TU Wien, Wiedner Hauptstraße 8–10, A-1040 Vienna, Austria

  • *These authors contributed equally to this work.
  • †Contact author: gerhard.kahl@tuwien.ac.at
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Phys. Rev. Research 8, 033259 – Published 1 September, 2026

DOI: https://doi.org/10.1103/w54t-nnlf

Abstract

We provide, to our knowledge, first and unambiguous evidence of the existence of cluster crystals formed by a binary mixture of ultrasoft, cluster-forming particles. In an effort to reduce the high dimensionality of the parameter space of binary mixtures, we have restricted ourselves to a symmetric, equimolar system, introducing as the pivotal parameter the scaling factor ζ that characterizes the cross interaction. Our investigations are based on a commonly used model for ultrasoft particles and on classical density functional theory, which, within a mean-field framework, allows for mixed clusters composed of particles of both species. Minimizing the free energy functional predicts three competing cluster crystal structures, namely, a bcc and a tetragonal lattice that reduces under certain conditions to an fcc lattice. The emerging phase diagram is rather intricate, but is well understood in terms of an energetic competition between the two sublattices. Complementary molecular dynamics simulations confirm the stability of the predicted structures and allow to trace the particles in their migration processes through the crystal.

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References (38)

  1. B. M. Mladek, D. Gottwald, G. Kahl, M. Neumann, and C. N. Likos, Formation of polymorphic cluster phases for a class of models of purely repulsive soft spheres, Phys. Rev. Lett. 96, 045701 (2006).
  2. B. M. Mladek, P. Charbonneau, and D. Frenkel, Phase coexistence of cluster crystals: Beyond the Gibbs phase rule, Phys. Rev. Lett. 99, 235702 (2007).
  3. K. Zhang, P. Charbonneau, and B. M. Mladek, Reentrant and isostructural transitions in a cluster-crystal former, Phys. Rev. Lett. 105, 245701 (2010).
  4. A. Nikoubashman, G. Kahl, and C. N. Likos, Cluster crystals under shear, Phys. Rev. Lett. 107, 068302 (2011).
  5. A. Scacchi, M. Sammalkorpi, and T. Ala-Nissila, Self-assembly of binary solutions to complex structures, J. Chem. Phys. 155, 014904 (2021).
  6. M. Maharana, D. Frenkel, and D. J, Monte-Carlo study of compositional heterogeneity in multicomponent cluster crystals, arXiv:2606.03235.
  7. A. R. Denton, Effective interactions in soft materials, in Nanostructured Soft Matter: Experiment, Theory, Simulation and Perspectives, edited by A. V. Zvelindovsky (Springer Netherlands, Dordrecht, 2007), pp. 395–433
  8. B. M. Mladek, D. Gottwald, G. Kahl, M. Neumann, and C. N. Likos, Clustering in the absence of attractions: Density functional theory and computer simulations, J. Phys. Chem. B 111, 12799 (2007).
  9. C. N. Likos, B. M. Mladek, D. Gottwald, and G. Kahl, Why do ultrasoft repulsive particles cluster and crystallize? Analytical results from density-functional theory, J. Chem. Phys. 126, 224502 (2007).
  10. C. N. Likos, A. Lang, M. Watzlawek, and H. Löwen, Criterion for determining clustering versus reentrant melting behavior for bounded interaction potentials, Phys. Rev. E 63, 031206 (2001).
  11. F. H. Stillinger, Phase transitions in the Gaussian core system, J. Chem. Phys. 65, 3968 (1976).
  12. F. H. Stillinger and D. K. Stillinger, Negative thermal expansion in the Gaussian core model, Physica A 244, 358 (1997).
  13. R. Evans, The nature of the liquid-vapour interface and other topics in the statistical mechanics of non-uniform, classical fluids, Adv. Phys. 28, 143 (1979).
  14. R. Evans, Density functionals in the theory of nonuniform fluids, in Fundamentals of Inhomogeneous Fluids, edited by D. Henderson (Taylor & Francis, Boca Raton, FL, 1992), Chap. 3, pp. 85–175.
  15. R. Evans, M. Oettel, R. Roth, and G. Kahl, New developments in classical density functional theory, J. Phys.: Condens. Matter 28, 240401 (2016).
  16. A. J. Archer and R. Evans, Binary Gaussian core model: Fluid-fluid phase separation and interfacial properties, Phys. Rev. E 64, 041501 (2001).
  17. A. J. Archer, M. C. Walters, U. Thiele, and E. Knobloch, Solidification in soft-core fluids: Disordered solids from fast solidification fronts, Phys. Rev. E 90, 042404 (2014).
  18. D. Pini, A. Parola, and L. Reatto, An unconstrained DFT approach to microphase formation and application to binary Gaussian mixtures, J. Chem. Phys. 143, 034902 (2015).
  19. A. J. Archer, C. N. Likos, and R. Evans, Soft-core binary fluid exhibiting a λ-line and freezing to a highly delocalized crystal, J. Phys.: Condens. Matter 16, L297 (2004).
  20. D. Gottwald, Genetic algorithms in condensed matter theory, Ph.D. thesis, TU Wien, 2005.
  21. D. Gottwald, G. Kahl, and C. N. Likos, Predicting equilibrium structures in freezing processes, J. Chem. Phys. 122, 204503 (2005).
  22. F. Tscharnutter, Cluster crystal formation in binary solutions with bounded interaction potentials, Master’s thesis, TU Wien, 2024.
  23. F. Tscharnutter, V. Schweidler, and G. Kahl, Binary cluster crystals formed by ultrasoft particles: Classical density functional theory, J. Chem. Phys. 164, 064911 (2026).
  24. V. Schweidler, F. Tscharnutter, and G. Kahl, Binary cluster crystals formed by ultrasoft particles: Molecular dynamics simulations (unpublished).
  25. Note that this parametrization captures the simulation-based observation that we find for all clusters (irrespective of the sublattice they pertain to) follows the same probability distribution.
  26. J. H. Holland, Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence (MIT Press, Reading, MA, 1992).
  27. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, 1st ed. (Addison-Wesley Longman Publishing Co., Inc., USA, 1989).
  28. A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in ‘t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS—A flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022).
  29. T. Soddemann, B. Dünweg, and K. Kremer, Dissipative particle dynamics: A useful thermostat for equilibrium and nonequilibrium molecular dynamics simulations, Phys. Rev. E 68, 046702 (2003).
  30. T. Biswas, G. Kahl, and G. P. Shrivastav, Static and dynamic properties of a binary, symmetric mixture of ultrasoft particles in the vicinity of criticality, J. Chem. Phys. 163, 224904 (2025).
  31. T. Biswas, G. Kahl, and G. P. Shrivastav, Phase separation dynamics in a symmetric binary mixture of ultrasoft particles, J. Chem. Phys. 160, 214901 (2024).
  32. B. M. Mladek, Exotic phenomena in the phase behaviour of soft matter systems, Ph.D. thesis, TU Wien, 2007.
  33. A. Scacchi, S. J. Nikkhah, M. Sammalkorpi, and T. Ala-Nissila, Self-assembly in soft matter with multiple length scales, Phys. Rev. Res. 3, L022008 (2021).
  34. D. Pini, M. Weißenhofer, and G. Kahl, A finite-temperature study of the degeneracy of the crystal phases in systems of soft aspherical particles, J. Chem. Phys. 161, 134903 (2024).
  35. M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, in Proceedings of the 2nd International Conference on Knowledge Discovery and Data Mining (KDD-96) (AAAI Press, Portland, OR, 1996), pp. 226–231.
  36. E. Schubert, J. Sander, M. Ester, H.-P. Kriegel, and X. Xu, DBSCAN revisited, revisited: Why and how you should (still) use DBSCAN, ACM Trans. Database Syst. 42, 1 (2017).
  37. scikit-learn developers, sklearn.cluster.dbscan—scikit-learn 1.3.1 documentation, https://scikit-learn.org/dev/index.html, accessed: [18.09.2025].
  38. J.-P. Hansen and I. R. McDonald, Theory of Simple Liquids: With Applications to Soft Matter, 4th ed. (Elsevier-Academic Press, 2013).

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Phys. Rev. Research 8, 033259– Published 1 September, 2026
  • Received 28 September 2025
  • Accepted 9 June 2026
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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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