- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

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

Ordering and association of patchy particles in quasi-one-dimensional channels

Péter Gurin* and Szabolcs Varga†

  • Physics Department, Centre for Natural Sciences, University of Pannonia, PO Box 158, Veszprém H-8201, Hungary

  • *Contact author: peter.gurin@mk.uni-pannon.hu
  • †Contact author: szabolcs.varga@mk.uni-pannon.hu
PDF

Phys. Rev. Research 7, 033094 – Published 28 July, 2025

DOI: https://doi.org/10.1103/1xp1-mnqx

Abstract

We show that the formalism of Wertheim's first-order thermodynamic perturbation theory can be generalized for the fluid of rotating sticky particles with anisotropic hard core confined to a quasi-one-dimensional channel. Using the transfer-matrix method, we prove that the theory is exact if the hard-body interaction is additive, only the first neighbors interact, and the particles can stick together only along the channel. We show that the most convenient treatment of association in narrow channels is to work in the NPT ensemble, where all structural and thermodynamic quantities can be expressed as a function of pressure and fraction of sites unbonded.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (75)

  1. S. C. Glotzer and M. J. Solomon, Anisotropy of building blocks and their assembly into complex structures, Nat. Mater. 6, 557 (2007).
  2. S.-Y. Zhang, M. D. Regulacio, and M.-Y. Han, Self-assembly of colloidal one-dimensional nanocrystals, Chem. Soc. Rev. 43, 2301 (2014).
  3. I. Coluzza, P. D. J. van Oostrum, B. Capone, E. Reimhult, and C. Dellago, Design and folding of colloidal patchy polymers, Soft Matter 9, 938 (2013).
  4. A. B. G. M. Leferink op Reinink, E. van den Pol, A. V. Petukhov, G. J. Vroege, and H. N. W. Lekkerkerker, Phase behaviour of lyotropic liquid crystals in external fields and confinement, Eur. Phys. J. Spec. Top. 222, 3053 (2013).
  5. C. P. Royall, P. Charbonneau, M. Dijkstra, J. Russo, F. Smallenburg, T. Speck, and C. Valeriani, Colloidal hard spheres: Triumphs, challenges, and mysteries, Rev. Mod. Phys. 96, 045003 (2024).
  6. J. T. Jack and P. C. Millett, Numerical study of the phase behavior of rod-like colloidal particles with attractive tips, AIP Adv. 11, 025030 (2021).
  7. Y. V. Kalyuzhnyi, T. Patsahan, M. Holovko, and P. T. Cummings, Phase behavior of patchy colloids confined in patchy porous media, Nanoscale 16, 4668 (2024).
  8. T. Vo and S. C. Glotzer, A theory of entropic bonding, Proc. Natl. Acad. Sci. USA 119, e2116414119 (2022).
  9. K. F. Herzfeld and M. Goeppert-Mayer, On the states of aggregation, J. Chem. Phys. 2, 38 (1934).
  10. J. Brader and R. Evans, An exactly solvable model for a colloid–polymer mixture in one-dimension, Physica A 306, 287 (2002).
  11. R. Fantoni and A. Santos, One-dimensional fluids with second nearest–neighbor interactions, J. Stat. Phys. 169, 1171 (2017).
  12. A. M. Montero and A. Santos, Triangle-well and ramp interactions in one-dimensional fluids: A fully analytic exact solution, J. Stat. Phys. 175, 269 (2019).
  13. S. Meddour, L. Bouzar, and R. Messina, Equation of state and universal solid phase of one-dimensional dipolar fluids, J. Phys.: Condens. Matter 37, 075101 (2025).
  14. M. Kac, On the partition function of a one-dimensional gas, Phys. Fluids 2, 8 (1959).
  15. C. Carraro, Continuous freezing in an infinite-range one-dimensional model, Phys. Rev. E 67, 051502 (2003).
  16. R. Fantoni, M. A. G. Maestre, and A. Santos, Finite-size effects and thermodynamic limit in one-dimensional Janus fluids, J. Stat. Mech. (2021) 103210.
  17. J. L. Lebowitz, J. K. Percus, and J. Talbot, On the orientational properties of some one-dimensional model systems, J. Stat. Phys. 49, 1221 (1987).
  18. Y. Kantor and M. Kardar, Universality in the jamming limit for elongated hard particles in one dimension, Europhys. Lett. 87, 60002 (2009).
  19. M. N. Joswiak, M. F. Doherty, and B. Peters, Critical length of a one-dimensional nucleus, J. Chem. Phys. 145, 211916 (2016).
  20. R. K. Bowles, A thermodynamic description of the glass transition: An exact one-dimensional example, Physica A 275, 217 (2000).
  21. M. J. Godfrey and M. A. Moore, Understanding the ideal glass transition: Lessons from an equilibrium study of hard disks in a channel, Phys. Rev. E 91, 022120 (2015).
  22. S. S. Ashwin and R. K. Bowles, Complete jamming landscape of confined hard discs, Phys. Rev. Lett. 102, 235701 (2009).
  23. M. Zarif, R. J. Spiteri, and R. K. Bowles, Inherent structure landscape of hard spheres confined to narrow cylindrical channels, Phys. Rev. E 104, 064602 (2021).
  24. Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood, Molecular distribution functions in a one-dimensional fluid, J. Chem. Phys. 21, 1098 (1953).
  25. L. M. Casey and L. K. Runnels, Model for correlated molecular rotation, J. Chem. Phys. 51, 5070 (1969).
  26. D. A. Kofke and A. J. Post, Hard particles in narrow pores. Transfer-matrix solution and the periodic narrow box, J. Chem. Phys. 98, 4853 (1993).
  27. A. Santos, A Concise Course on the Theory of Classical Liquids. Basics and Selected Topics, Lecture Notes in Physics Vol. 923 (Springer, New York, 2016).
  28. J. K. Percus, Density functional theory of single-file classical fluids, Mol. Phys. 100, 2417 (2002).
  29. Theory and Simulation of Hard-Sphere Fluids and Related Systems, edited by Á. Mulero, Lecture Notes in Physics Vol. 753 (Springer, New York, 2008).
  30. L. Mederos, E. Velasco, and Y. Martínez-Ratón, Hard-body models of bulk liquid crystals, J. Phys.: Condens. Matter 26, 463101 (2014).
  31. J. K. Percus, Equilibrium state of a classical fluid of hard rods in an external field, J. Stat. Phys. 15, 505 (1976).
  32. J. K. Percus, One-dimensional classical fluid with nearest-neighbor interaction in arbitrary external field, J. Stat. Phys. 28, 67 (1982).
  33. E. Kierlik and M. L. Rosinberg, The classical fluid of associating hard rods in an external field, J. Stat. Phys. 68, 1037 (1992).
  34. T. K. Vanderlick, H. T. Davis, and J. K. Percus, The statistical mechanics of inhomogeneous hard rod mixtures, J. Chem. Phys. 91, 7136 (1989).
  35. G. R. Brannock and J. K. Percus, Wertheim cluster development of free energy functionals for general nearest-neighbor interactions in d=1, J. Chem. Phys. 105, 614 (1996).
  36. J. K. Percus, Nonuniform classical fluid mixture in one-dimensional space with next neighbor interactions, J. Stat. Phys. 89, 249 (1997).
  37. C. Tutschka and J. A. Cuesta, Overcomplete free energy functional for D=1 particle systems with next neighbor interactions, J. Stat. Phys. 111, 1125 (2003).
  38. A. Y. Sahnoun, M. Djebbar, T. Benmessabih, and B. Bakhti, One dimensional lattice fluid mixture with nearest neighbour interactions, J. Phys. A: Math. Theor. 57, 325007 (2024).
  39. Y. Rosenfeld, Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing, Phys. Rev. Lett. 63, 980 (1989).
  40. S.-C. Lin and M. Oettel, A classical density functional from machine learning and a convolutional neural network, SciPost Phys. 6, 025 (2019).
  41. J. Wu and Z. Li, Density-functional theory for complex fluids, Annu. Rev. Phys. Chem. 58, 85 (2007).
  42. M. Schmidt, Fundamental measure density functional theory for nonadditive hard-core mixtures: The one-dimensional case, Phys. Rev. E 76, 031202 (2007).
  43. A. El Moumane, M. te Vrugt, H. Löwen, and R. Wittmann, Biaxial nematic order in fundamental measure theory, J. Chem. Phys. 160, 094903 (2024).
  44. B. D. Marshall, Thermodynamic perturbation theory for associating fluids confined in a one-dimensional pore, J. Chem. Phys. 142, 234906 (2015).
  45. B. D. Marshall, Equilibrium adsorption and self-assembly of patchy colloids in microchannels, Phys. Rev. E 94, 012615 (2016).
  46. J. Percus and M. Zhang, The quasi-one dimensional hard square gas, Mol. Phys. 69, 347 (1990).
  47. P. Gurin and S. Varga, Beyond the single-file fluid limit using transfer matrix method: Exact results for confined parallel hard squares, J. Chem. Phys. 142, 224503 (2015).
  48. B. Lin, D. Valley, M. Meron, B. Cui, H. M. Ho, and S. A. Rice, The quasi-one-dimensional colloid fluid revisited, J. Phys. Chem. B 113, 13742 (2009).
  49. S. Gorantla, F. Börrnert, A. Bachmatiuk, M. Dimitrakopoulou, R. Schönfelder, F. Schäffel, J. Thomas, T. Gemming, E. Borowiak-Palen, J. H. Warner, B. I. Yakobson, J. Eckert, B. Büchner, and M. H. Rümmeli, In situ observations of fullerene fusion and ejection in carbon nanotubes, Nanoscale 2, 2077 (2010).
  50. S. S. Ashwin, M. Z. Yamchi, and R. K. Bowles, Inherent structure landscape connection between liquids, granular materials, and the jamming phase diagram, Phys. Rev. Lett. 110, 145701 (2013).
  51. J. F. Robinson, M. J. Godfrey, and M. A. Moore, Glasslike behavior of a hard-disk fluid confined to a narrow channel, Phys. Rev. E 93, 032101 (2016).
  52. A. M. Montero, A. Santos, P. Gurin, and S. Varga, Ordering properties of anisotropic hard bodies in one-dimensional channels, J. Chem. Phys. 159, 154507 (2023).
  53. P. Gurin, S. Mizani, and S. Varga, Orientational ordering and correlation in quasi-one-dimensional hard-body fluids due to close-packing degeneracy, Phys. Rev. E 110, 014702 (2024).
  54. S. Mizani, M. Oettel, P. Gurin, and S. Varga, Universality of the close packing properties and markers of isotropic-to-tetratic crossover in quasi-one-dimensional superdisk fluid, SciPost Phys. Core 8, 008 (2025).
  55. S. Mizani, P. Gurin, S. Varga, and M. Oettel, Competition between shape anisotropy and deformation in the ordering and close packing properties of quasi-one-dimensional hard superellipse fluids, Phys. Rev. E 111, 064121 (2025).
  56. J. L. Lebowitz and J. K. Percus, One-dimensional models of anisotropic fluids, Ann. N.Y. Acad. Sci. 410, 351 (1983).
  57. P. Gurin and S. Varga, Anomalous phase behavior of quasi-one-dimensional attractive hard rods, Phys. Rev. E 106, 044606 (2022).
  58. R. J. Baxter, Percus–Yevick equation for hard spheres with surface adhesion, J. Chem. Phys. 49, 2770 (1968).
  59. M. S. Wertheim, Fluids with highly directional attractive forces. I. Statistical thermodynamics, J. Stat. Phys. 35, 19 (1984).
  60. M. S. Wertheim, Fluids with highly directional attractive forces. II. Thermodynamic perturbation theory and integral equations, J. Stat. Phys. 35, 35 (1984).
  61. M. S. Wertheim, Fluids with highly directional attractive forces. III. Multiple attraction sites, J. Stat. Phys. 42, 459 (1986).
  62. M. S. Wertheim, Fluids with highly directional attractive forces. IV. Equilibrium polymerization, J. Stat. Phys. 42, 477 (1986).
  63. M. S. Wertheim, Thermodynamic perturbation theory of polymerization, J. Chem. Phys. 87, 7323 (1987).
  64. J. Cuesta and A. Sanchez, General non-existence theorem for phase transitions in one-dimensional systems with short range interactions, and physical examples of such transitions, J. Stat. Phys. 115, 869 (2004).
  65. P. Gurin and S. Varga, Pair correlation functions of two- and three-dimensional hard-core fluids confined into narrow pores: Exact results from transfer-matrix method, J. Chem. Phys. 139, 244708 (2013).
  66. G. Jackson, W. G. Chapman, and K. E. Gubbins, Phase equilibria of associating fluids: Spherical molecules with multiple bonding sites, Mol. Phys. 65, 1 (1988).
  67. P. Paricaud, A. Galindo, and G. Jackson, Recent advances in the use of the SAFT approach in describing electrolytes, interfaces, liquid crystals and polymers, Fluid Phase Equilib. 194-197, 87 (2002).
  68. H. J. Jonas, P. Schall, and P. G. Bolhuis, Extended Wertheim theory predicts the anomalous chain length distributions of divalent patchy particles under extreme confinement, J. Chem. Phys. 157, 094903 (2022).
  69. P. Teixeira and J. Tavares, Phase behaviour of pure and mixed patchy colloids–theory and simulation, Curr. Opin. Colloid Interface Sci. 30, 16 (2017).
  70. A. Barthes, T. Bernet, D. Grégoire, and C. Miqueu, A molecular density functional theory for associating fluids in 3D geometries, J. Chem. Phys. 160, 054704 (2024).
  71. B. D. Marshall and W. G. Chapman, A density functional theory for patchy colloids based on Wertheim's association theory: Beyond the single bonding condition, J. Chem. Phys. 138, 044901 (2013).
  72. J. Algaba, B. Mendiboure, P. Gómez-Álvarez, and F. J. Blas, Density functional theory for the prediction of interfacial properties of molecular fluids within the SAFT-γ coarse-grained approach, RSC Adv. 12, 18821 (2022).
  73. S. C. McGrother, R. P. Sear, and G. Jackson, The liquid crystalline phase behavior of dimerizing hard spherocylinders, J. Chem. Phys. 106, 7315 (1997).
  74. J. L. Lebowitz and D. Zomick, Mixtures of hard spheres with nonadditive diameters: Some exact results and solution of PY equation, J. Chem. Phys. 54, 3335 (1971).
  75. A. Gil-Villegas, A. Galindo, P. J. Whitehead, S. J. Mills, G. Jackson, and A. N. Burgess, Statistical associating fluid theory for chain molecules with attractive potentials of variable range, J. Chem. Phys. 106, 4168 (1997).

Outline

Information

Phys. Rev. Research 7, 033094– Published 28 July, 2025
  • Received 22 April 2025
  • Accepted 26 June 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