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  • Open Access

Density-selected topological pathways in the melting of single-particle-thick stripes

José Rafael Bordin*

  • *Contact author: jrbordin@ufpel.edu.br
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Phys. Rev. E 114, 045412 – Published 27 October, 2026

DOI: https://doi.org/10.1103/w3lf-h7nt

Abstract

The melting of stripe-forming systems involves changes in connectivity that are not fully captured by conventional structural and orientational descriptors. We investigate a two-dimensional model with competing interactions whose low-temperature phase consists of one-particle-thick stripes. Molecular dynamics simulations along seven heating isochores are analyzed using thermodynamic, orientational, dynamical, and graph-based observables. Heating produces a multistage reconstruction in which the loss of stripe alignment and the reorganization of filament connectivity occur over distinct temperature ranges. Density controls whether the disordered filaments fragment into finite polymerlike clusters or remain joined in a dynamically fluid, system-spanning network. The topological observables distinguish these outcomes, which are not resolved by the thermodynamic and orientational responses alone. Thus, the same ordered stripe microphase can melt into topologically distinct fluids selected by density.

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References (70)

  1. M. Seul and D. Andelman, Domain shapes and patterns: The phenomenology of modulated phases, Science 267, 476 (1995).
  2. A. Ciach, J. Pekalski, and W. T. Góźdź, Origin of similarity of phase diagrams in amphiphilic and colloidal systems with competing interactions, Soft Matter 9, 6301 (2013).
  3. Y. Zhuang, K. Zhang, and P. Charbonneau, Equilibrium phase behavior of a continuous-space microphase former, Phys. Rev. Lett. 116, 098301 (2016).
  4. C. P. Royall, Hunting mermaids in real space: Known knowns, known unknowns and unknown unknowns, Soft Matter 14, 4020 (2018).
  5. Y. Liu and Y. Xi, Colloidal systems with a short-range attraction and long-range repulsion: Phase diagrams, structures, and dynamics, Curr. Opin. Colloid Interface Sci. 39, 123 (2019).
  6. E. A. Jagla, Phase behavior of a system of particles with core collapse, Phys. Rev. E 58, 1478 (1998).
  7. E. A. Jagla, Minimum energy configurations of repelling particles in two dimensions, J. Chem. Phys. 110, 451 (1999).
  8. G. Malescio and G. Pellicane, Stripe phases from isotropic repulsive interactions, Nat. Mater. 2, 97 (2003).
  9. M. A. Glaser, G. M. Grason, R. D. Kamien, A. Košmrlj, C. D. Santangelo, and P. Ziherl, Soft spheres make more mesophases, Europhys. Lett. 78, 46004 (2007).
  10. H. Pattabhiraman and M. Dijkstra, On the formation of stripe, sigma, and honeycomb phases in a core–corona system, Soft Matter 13, 4418 (2017).
  11. W. R. C. Somerville, A. D. Law, M. Rey, N. Vogel, A. J. Archer, and D. M. A. Buzza, Pattern formation in two-dimensional hard-core/soft-shell systems with variable soft shell profiles, Soft Matter 16, 3564 (2020).
  12. A. Ciach, Universal sequence of ordered structures obtained from mesoscopic description of self-assembly, Phys. Rev. E 78, 061505 (2008).
  13. A. Imperio and L. Reatto, Microphase separation in two-dimensional systems with competing interactions, J. Chem. Phys. 124, 164712 (2006).
  14. A. J. Archer and N. B. Wilding, Phase behavior of a fluid with competing attractive and repulsive interactions, Phys. Rev. E 76, 031501 (2007).
  15. M. Edelmann and R. Roth, Gyroid phase of fluids with spherically symmetric competing interactions, Phys. Rev. E 93, 062146 (2016).
  16. J. R. Bordin, A DPD model of soft spheres with waterlike anomalies and poly(a)morphism, Soft Matter 19, 7613 (2023).
  17. D. McDermott, C. J. Olson Reichhardt, and C. Reichhardt, Stripe systems with competing interactions on quasi-one dimensional periodic substrates, Soft Matter 10, 6332 (2014).
  18. J. Pekalski and A. Ciach, Orientational ordering of lamellar structures on closed surfaces, J. Chem. Phys. 148, 174902 (2018).
  19. A. De Virgiliis, A. G. Meyra, and A. Ciach, Lattice model results for pattern formation in a mixture with competing interactions, Molecules 29, 1512 (2024).
  20. T. P. O. Nogueira and J. R. Bordin, Patterns in 2D core-softened systems: From sphere to dumbbell colloids, Physica A 605, 128048 (2022).
  21. C. Reichhardt and C. J. O. Reichhardt, Peak effect and dynamics of stripe- and pattern-forming systems on a periodic one-dimensional substrate, Phys. Rev. E 109, 054606 (2024).
  22. A. Stradner, H. Sedgwick, F. Cardinaux, W. C. K. Poon, S. U. Egelhaaf, and P. Schurtenberger, Equilibrium cluster formation in concentrated protein solutions and colloids, Nature (London) 432, 492 (2004).
  23. A. I. Campbell, V. J. Anderson, J. S. van Duijneveldt, and P. Bartlett, Dynamical arrest in attractive colloids: The effect of long-range repulsion, Phys. Rev. Lett. 94, 208301 (2005).
  24. A. Al Harraq, A. A. Hymel, E. Lin, T. M. Truskett, and B. Bharti, Dual nature of magnetic nanoparticle dispersions enables control over short-range attraction and long-range repulsion interactions, Commun. Chem. 5, 72 (2022).
  25. H. M. Gauri, Z. M. Sherman, A. Al Harraq, T. M. Truskett, and B. Bharti, Magnetic field enabled in situ control over the structure and dynamics of colloids interacting via SALR potentials, Soft Matter 19, 4439 (2023).
  26. M. Rey, T. Yu, K. Bley, K. Landfester, D. M. A. Buzza, and N. Vogel, Amphiphile-induced anisotropic colloidal self-assembly, Langmuir 34, 9990 (2018).
  27. S. Ciarella, M. Rey, J. Harrer, N. Holstein, M. Ickler, H. Löwen, N. Vogel, and L. M. C. Janssen, Soft particles at liquid interfaces: From molecularparticle architecture to collective phase behavior, Langmuir 37, 5364 (2021).
  28. S. Haddadi, M. Skepö, P. Jannasch, S. Manner, and J. Forsman, Building polymer-like clusters from colloidal particles with isotropic interactions, in aqueous solution, J. Colloid Interface Sci. 581, 669 (2021).
  29. S. Haddadi, H. Lu, M. Bäcklund, C. E. Woodward, and J. Forsman, Polymer-like self-assembled structures from particles with isotropic interactions: Dependence upon the range of the attraction, Langmuir 37, 6052 (2021).
  30. J. Huang, A. R. Tao, S. Connor, R. He, and P. Yang, A general method for assembling single colloidal particle lines, Nano Lett. 6, 524 (2006).
  31. F. Sciortino, P. Tartaglia, and E. Zaccarelli, One-dimensional cluster growth and branching gels in colloidal systems with short-range depletion attraction and screened electrostatic repulsion, J. Phys. Chem. B 109, 21942 (2005).
  32. M. D. Haw, Growth kinetics of colloidal chains and labyrinths, Phys. Rev. E 81, 031402 (2010).
  33. D. Banerjee, B. A. Lindquist, R. B. Jadrich, and T. M. Truskett, Assembly of particle strings via isotropic potentials, J. Chem. Phys. 150, 124903 (2019).
  34. F. Sciortino, S. Mossa, E. Zaccarelli, and P. Tartaglia, Equilibrium cluster phases and low-density arrested disordered states: The role of short-range attraction and long-range repulsion, Phys. Rev. Lett. 93, 055701 (2004).
  35. E. Mani, W. Lechner, W. K. Kegel, and P. G. Bolhuis, Equilibrium and non-equilibrium cluster phases in colloids with competing interactions, Soft Matter 10, 4479 (2014).
  36. P. Douglas Godfrin, N. E. Valadez-Pérez, R. Castañeda-Priego, N. J. Wagner, and Y. Liu, Generalized phase behavior of cluster formation in colloidal dispersions with competing interactions, Soft Matter 10, 5061 (2014).
  37. J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973).
  38. B. I. Halperin and D. R. Nelson, Theory of two-dimensional melting, Phys. Rev. Lett. 41, 121 (1978).
  39. D. R. Nelson and B. I. Halperin, Dislocation-mediated melting in two dimensions, Phys. Rev. B 19, 2457 (1979).
  40. A. P. Young, Melting and the vector Coulomb gas in two dimensions, Phys. Rev. B 19, 1855 (1979).
  41. K. Zahn, R. Lenke, and G. Maret, Two-stage melting of paramagnetic colloidal crystals in two dimensions, Phys. Rev. Lett. 82, 2721 (1999).
  42. H. H. von Grünberg, P. Keim, K. Zahn, and G. Maret, Elastic behavior of a two-dimensional crystal near melting, Phys. Rev. Lett. 93, 255703 (2004).
  43. U. Gasser, C. Eisenmann, G. Maret, and P. Keim, Melting of crystals in two dimensions, ChemPhysChem 11, 963 (2010).
  44. A. D. Stoycheva and S. J. Singer, Stripe melting in a two-dimensional system with competing interactions, Phys. Rev. Lett. 84, 4657 (2000).
  45. C. J. Olson Reichhardt, C. Reichhardt, and A. R. Bishop, Structural transitions, melting, and intermediate phases for stripe- and clump-forming systems, Phys. Rev. E 82, 041502 (2010).
  46. E. A. Jagla, Numerical simulations of two-dimensional magnetic domain patterns, Phys. Rev. E 70, 046204 (2004).
  47. T. Puccinelli, A. V. Ilha, and J. R. Bordin, Competing length scales and symmetry frustration govern nonuniversal melting in two-dimensional core-softened colloidal crystals, Phys. Rev. E 112, 055415 (2025).
  48. I. Donato, M. Gori, M. Pettini, G. Petri, S. De Nigris, R. Franzosi, and F. Vaccarino, Persistent homology analysis of phase transitions, Phys. Rev. E 93, 052138 (2016).
  49. A. Cole, G. J. Loges, and G. Shiu, Quantitative and interpretable order parameters for phase transitions from persistent homology, Phys. Rev. B 104, 104426 (2021).
  50. I. Membrillo Solis, T. Orlova, K. Bednarska, P. Lesiak, T. R. Woliński, G. D'Alessandro, J. Brodzki, and M. Kaczmarek, Tracking the time evolution of soft matter systems via topological structural heterogeneity, Commun. Mater. 3, 1 (2022).
  51. J. Colombo and E. Del Gado, Self-assembly and cooperative dynamics of a model colloidal gel network, Soft Matter 10, 4003 (2014).
  52. A. Barros de Oliveira, P. A. Netz, T. Colla, and M. C. Barbosa, Thermodynamic and dynamic anomalies for a three-dimensional isotropic core-softened potential, J. Chem. Phys. 124, 084505 (2006).
  53. J. R. Bordin and L. B. Krott, Solid-amorphous transition is related to the waterlike anomalies in a fluid without liquid–liquid phase transition, J. Chem. Phys. 158, 134501 (2023).
  54. D. S. Cardoso, V. F. Hernandes, T. Nogueira, and J. R. Bordin, Structural behavior of a two length scale core-softened fluid in two dimensions, Physica A 566, 125628 (2021).
  55. A. Barros de Oliveira, E. B. Neves, C. Gavazzoni, J. Z. Paukowski, P. A. Netz, and M. C. Barbosa, Liquid crystal phase and waterlike anomalies in a core-softened shoulder-dumbbells system, J. Chem. Phys. 132, 164505 (2010).
  56. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, 2nd ed. (Oxford University Press, Oxford, 2017).
  57. F. Weik, R. Weeber, K. Szuttor, K. Breitsprecher, J. de Graaf, M. Kuron, J. Landsgesell, H. Menke, D. Sean, and C. Holm, ESPResSo 4.0 – an extensible software package for simulating soft matter systems, Eur. Phys. J.: Spec. Top. 227, 1789 (2019).
  58. See Supplemental Material at http://link.aps.org/supplemental/10.1103/w3lf-h7nt for complete definitions of the observables, the robustness against the graph connectivity cutoff, the thermodynamic response along the heating isochores, and the system-size analysis, which includes Refs. [29, 51, 52, 54, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67].
  59. P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Bond-orientational order in liquids and glasses, Phys. Rev. B 28, 784 (1983).
  60. Y. Hiraoka, T. Nakamura, A. Hirata, E. G. Escolar, K. Matsue, and Y. Nishiura, Hierarchical structures of amorphous solids characterized by persistent homology, Proc. Natl. Acad. Sci. USA 113, 7035 (2016).
  61. A. Wang and G. C. Sosso, Graph-based descriptors for condensed matter, Phys. Rev. E 111, 064302 (2025).
  62. S. Mertens and R. M. Ziff, Percolation in finite matching lattices, Phys. Rev. E 94, 062152 (2016).
  63. S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, k-core organization of complex networks, Phys. Rev. Lett. 96, 040601 (2006).
  64. K. Paton, An algorithm for finding a fundamental set of cycles of a graph, Commun. ACM 12, 514 (1969).
  65. B. Hamprecht and H. Kleinert, End-to-end distribution function of stiff polymers for all persistence lengths, Phys. Rev. E 71, 031803 (2005).
  66. N. E. Valadez-Pérez, Y. Liu, and R. Castañeda-Priego, Cluster morphology of colloidal systems with competing interactions, Front. Phys. 9, 637138 (2021).
  67. J. R. Bordin and M. C. Barbosa, Waterlike anomalies in a two-dimensional core-softened potential, Phys. Rev. E 97, 022604 (2018).
  68. A. D. Vyas, P. W. A. Schönhöfer, T. M. Hopkins, A. D. Hollingsworth, S. Sacanna, S. C. Glotzer, and P. Chaikin, Two-dimensional non-equilibrium melting of charged colloids, Nat. Phys. 22, 287 (2026).
  69. J. Opdam, M. Tateno, and H. Tanaka, Unraveling the impact of competing interactions on nonequilibrium colloidal gelation, ACS Nano 19, 21515 (2025).
  70. J. R. Bordin, Bordin-Lab/linear_stripes_espresso: Version 1.0.0 – Code accompanying the manuscript, Zenodo, 2026, https://doi.org/10.5281/zenodo.21512438.

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Phys. Rev. E 114, 045412– Published 27 October, 2026
  • Received 12 August 2026
  • Accepted 14 October 2026
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