Abstract
Self-oscillators that sustain periodic dynamics under constant input are ubiquitous in natural and engineered systems, where their interactions enable spatiotemporal coordination among many individual units. New forms of organization can emerge when these self-oscillating units are free to move and rotate, linking their spatial arrangement and orientation with their oscillation frequencies and phases. Here, we report experiments and simulations on populations of Quincke colloids that behave as self-oscillating units characterized by position, orientation, frequency, and phase. Hydrodynamic interactions among these colloids drive temporal synchronization and spatial alignment of their phases and orientations, giving rise to a new form of collective order that we term synchronematic. Within finite-size crystalline clusters, these non-reciprocal interactions promote global synchronization and circular alignment, with a collective frequency that increases with cluster size. Using the theory of weakly coupled oscillators, we derive a reduced-order model that captures the coupled evolution of phase and orientation and explains how synchronematic order depends sensitively on the particle configuration. Our results establish Quincke colloids as a model system for active oscillatory matter and reveal fundamental principles by which synchronization, alignment, and structure co-emerge—offering a framework for designing adaptive, frequency-tunable materials.
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Data availability
The microscopy videos generated and analyzed in this study are available on figshare at https://doi.org/10.6084/m9.figshare.30946217. Source data are provided with this paper.
Code availability
Swarmalator simulation code is available on GitHub at https://github.com/slevinskygra/ReducedSynchronematic.
References
Jenkins, A. Self-oscillation. Phys. Rep. 525, 167–222 (2013).
Strogatz, S. H. From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Phys. D 143, 1–20 (2000).
Samatas, S. & Lintuvuori, J. Hydrodynamic synchronization of chiral microswimmers. Phys. Rev. Lett. 130, 024001 (2023).
Dou, Y., Pandey, S., Cartier, C. A., Miller, O. & Bishop, K. J. M. Emergence of traveling waves in linear arrays of electromechanical oscillators. Commun. Phys. 1, 85 (2018).
Hickey, D. J., Golestanian, R. & Vilfan, A. Nonreciprocal interactions give rise to fast cilium synchronization in finite systems. Proc. Natl. Acad. Sci. USA 120, e2307279120 (2023).
Kuramoto, Y. Chemical turbulence. In Chemical Oscillations, Waves, and Turbulence (Springer, 1984) https://doi.org/10.1007/978-3-642-69689-3.
Uchida, N. & Golestanian, R. Synchronization and collective dynamics in a carpet of microfluidic rotors. Phys. Rev. Lett. 104, 178103 (2010).
Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The Kuramoto model: a simple paradigm for synchronization phenomena. Rev. Mod. Phys. 77, 137–185 (2005).
Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge University Press, 1995).
Rouzaire, Y. & Levis, D. Defect superdiffusion and unbinding in a 2D XY model of self-driven rotors. Phys. Rev. Lett. 127, 088004 (2021).
Pargellis, A. N., Green, S. & Yurke, B. Planar xy-model dynamics in a nematic liquid crystal system. Phys. Rev. E 49, 4250 (1994).
Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143 (2013).
Howse, J. R. et al. Self-motile colloidal particles: from directed propulsion to random walk. Phys. Rev. Lett. 99, 048102 (2007).
Caprini, L., Marini Bettolo Marconi, U. & Puglisi, A. Spontaneous velocity alignment in motility-induced phase separation. Phys. Rev. Lett. 124, 078001 (2020).
Soto, R. & Golestanian, R. Self-assembly of catalytically active colloidal molecules: tailoring activity through surface chemistry. Phys. Rev. Lett. 112, 068301 (2014).
Sakaguchi, H., Shinomoto, S. & Kuramoto, Y. Mutual entrainment in oscillator lattices with nonvariational type interaction. Prog. Theor. Exp. Phys. 79, 1069–1079 (1988).
O’Keeffe, K. P., Hong, H. & Strogatz, S. H. Oscillators that sync and swarm. Nat. Commun. 8, 1504 (2017).
Ceron, S., O’Keeffe, K. & Petersen, K. Diverse behaviors in non-uniform chiral and non-chiral swarmalators. Nat. Commun. 14, 940 (2023).
Riedel, I. H., Kruse, K. & Howard, J. A self-organized vortex array of hydrodynamically entrained sperm cells. Science 309, 300–303 (2005).
Yan, J., Bloom, M., Bae, S. C., Luijten, E. & Granick, S. Linking synchronization to self-assembly using magnetic janus colloids. Nature 491, 578–581 (2012).
Bechinger, C. et al. Active particles in complex and crowded environments. Rev. Mod. Phys. 88, 045006 (2016).
Bishop, K. J. M., Biswal, S. L. & Bharti, B. Active colloids as models, materials, and machines. Annu. Rev. Chem. Biomol. Eng. 14, 1–30 (2023).
Bricard, A., Caussin, J.-B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).
Bricard, A. et al. Emergent vortices in populations of colloidal rollers. Nat. Commun. 6, 7470 (2015).
Karani, H., Pradillo, G. E. & Vlahovska, P. M. Tuning the random walk of active colloids: from individual run-and-tumble to dynamic clustering. Phys. Rev. Lett. 123, 208002 (2019).
Zhang, B., Yuan, H., Sokolov, A., de la Cruz, M. O. & Snezhko, A. Polar state reversal in active fluids. Nat. Phys. 18, 154–159 (2022).
Jones, T. B. Quincke rotation of spheres. IEEE Trans. Ind. Appl. IA-20, 845–849 (1984).
Das, D. & Saintillan, D. Electrohydrodynamic interaction of spherical particles under Quincke rotation. Phys. Rev. E 87, 043014 (2013).
Peters, F., Lobry, L. & Lemaire, E. Experimental observation of Lorenz chaos in the quincke rotor dynamics. Chaos 15, 13102 (2005).
Zhang, B., Sokolov, A. & Snezhko, A. Reconfigurable emergent patterns in active chiral fluids. Nat. Commun. 11, 4401 (2020).
Zhang, Z., Yuan, H., Dou, Y., Olvera de la Cruz, M. & Bishop, K. J. M. Quincke oscillations of colloids at planar electrodes. Phys. Rev. Lett. 126, 258001 (2021).
Kotar, J., Leoni, M., Bassetti, B., Lagomarsino, M. C. & Cicuta, P. Hydrodynamic synchronization of colloidal oscillators. Proc. Natl. Acad. Sci. USA 107, 7669–7673 (2010).
Kakoty, H., Huang, Y., Banerjee, R., Dasgupta, C. & Ghosh, A. Colloidal crystallites under external oscillation. Soft Matter 16, 5770–5776 (2020).
Kato, A. N., Takeuchi, K. A. & Sano, M. Active colloid with externally induced periodic bipolar motility and its cooperative motion. Soft Matter 18, 5435–5445 (2022).
Brady, J. F. & Bossis, G. Stokesian Dynamics. Annu. Rev. Fluid Mech. 20, 111–157 (1988).
Swan, J. W. & Brady, J. F. Simulation of hydrodynamically interacting particles near a no-slip boundary. Phys. Fluid 19, 113306 (2007).
Sprinkle, B., van der Wee, E. B., Luo, Y., Driscoll, M. M. & Donev, A. Driven dynamics in dense suspensions of microrollers. Soft Matter 16, 7982–8001 (2020).
Dabelow, L., Bo, S. & Eichhorn, R. Irreversibility in active matter systems: fluctuation theorem and mutual information. Phys. Rev. X 9, 021009 (2019).
Zhang, Z. & Bishop, K. J. M. Synchronization and alignment of model oscillators based on quincke rotation. Phys. Rev. E 107, 054603 (2023).
Trau, M., Saville, D. A. & Aksay, I. A. Field-induced layering of colloidal crystals. Science 272, 706–709 (1996).
Solomentsev, Y., Böhmer, M. & Anderson, J. L. Particle clustering and pattern formation during electrophoretic deposition: a hydrodynamic model. Langmuir 13, 6058–6068 (1997).
Sapozhnikov, M. V., Tolmachev, Y. V., Aranson, I. S. & Kwok, W.-K. Dynamic self-assembly and patterns in electrostatically driven granular media. Phys. Rev. Lett. 90, 114301 (2003).
Ristenpart, W. D., Aksay, I. A. & Saville, D. A. Electrically driven flow near a colloidal particle close to an electrode with a faradaic current. Langmuir 23, 4071–4080 (2007).
Zhang, B., Glatz, A., Aranson, I. S. & Snezhko, A. Spontaneous shock waves in pulse-stimulated flocks of quincke rollers. Nat. Commun. 14, 7050 (2023).
Blake, J. R. & Chwang, A. T. Fundamental singularities of viscous flow: Part I: the image systems in the vicinity of a stationary no-slip boundary. J. Eng. Math. 8, 23–29 (1974).
Kim, S. & Karrila, S. J. Microhydrodynamics: Principles and Selected Applications (Dover Publications, 2005).
Schwemmer, M. A. & Lewis, T. J. The theory of weakly coupled oscillators. In Phase Response Curves in Neuroscience 3–31 (eds Nathan W. Schultheiss, Astrid A. Prinz & Robert J. Butera) (Springer, 2012) https://doi.org/10.1007/978-1-4614-0739-3.
Hunter, I. et al. Pattern formation in a four-ring reaction-diffusion network with heterogeneity. Phys. Rev. E 105, 024310 (2022).
Shankar, S., Souslov, A., Bowick, M. J., Marchetti, M. C. & Vitelli, V. Topological active matter. Nat. Rev. Phys. 4, 380–398 (2022).
Niedermayer, T., Eckhardt, B. & Lenz, P. Synchronization, phase locking, and metachronal wave formation in ciliary chains. Chaos 18, 037128 (2008).
Xu, H., Huang, Y., Zhang, R. & Wu, Y. Autonomous waves and global motion modes in living active solids. Nat. Phys. 19, 46–51 (2023).
Baconnier, P. et al. Selective and collective actuation in active solids. Nat. Phys. 18, 1234–1239 (2022).
Caprini, L., Liebchen, B. & Löwen, H. Self-reverting vortices in chiral active matter. Commun. Phys. 7, 153 (2024).
Igoshin, O. A., Mogilner, A., Welch, R. D., Kaiser, D. & Oster, G. Pattern formation and traveling waves in myxobacteria: theory and modeling. Proc. Nat. Acad. Sci. USA 98, 14913–14918 (2001).
Bishop, C. M. Pattern Recognition and Machine Learning (Springer, 2006).
Kalman, R. E. A new approach to linear filtering and prediction problems. J. Basic Eng. 382, 35–45 (1960).
Rauch, H. E., Tung, F. & Striebel, C. T. Maximum likelihood estimates of linear dynamic systems. AIAA J. 3, 1445–1450 (1965).
Saville, D. A. Electrohydrodynamics: the Taylor-Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64 (1997).
Acknowledgements
This work was supported as part of the Center for Bio-Inspired Energy Science, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0000989. S.G.L. and M.O.d.l.C. were supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Contract DE-FG02-08ER46539.
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K.J.M.B. conceived the project. Z.Z. performed the experiments and analyzed the experimental data. S.G.L. developed and performed the Stokesian Dynamics simulations. K.J.M.B. and S.G.L. developed the reduced-order model and performed the swarmalator simulations. All authors contributed to data interpretation. S.G.L. drafted the manuscript with contributions from K.J.M.B. and M.O.d.l.C. K.J.M.B. and M.O.d.l.C. supervised the research and secured funding. All authors reviewed and approved the final manuscript.
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Leyva, S.G., Zhang, Z., Olvera de la Cruz, M. et al. Self-oscillating synchronematic colloids. Nat Commun (2026). https://doi.org/10.1038/s41467-026-68552-8
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DOI: https://doi.org/10.1038/s41467-026-68552-8
