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Motile dislocations knead odd crystals into whorls


The competition between thermal fluctuations and potential forces governs the stability of matter in equilibrium, in particular the proliferation and annihilation of topological defects. However, driving matter out of equilibrium allows for a new class of forces that are neither attractive nor repulsive, but rather transverse. The possibility of activating transverse forces raises the question of how they affect basic principles of material self-organization and control. Here we show that transverse forces organize colloidal spinners into odd elastic crystals crisscrossed by motile dislocations. These motile topological defects organize into a polycrystal made of grains with tunable length scale and rotation rate. The self-kneading dynamics drive super-diffusive mass transport, which can be controlled over orders of magnitude by varying the spinning rate. Simulations of both a minimal model and fully resolved hydrodynamics establish the generic nature of this crystal whorl state. Using a continuum theory, we show that both odd and Hall stresses can destabilize odd elastic crystals, giving rise to a generic state of crystalline active matter. Adding rotations to a material’s constituents has far-reaching consequences for continuous control of structures and transport at all scales.

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Fig. 1: A crystal whorl state.
Fig. 2: Motile dislocations.
Fig. 3: Transport in the crystalline whorl state.
Fig. 4: Odd response in the steady state.
Fig. 5: Measuring an elasto-hydrodynamic instability.

Data availability

The data contained in the plots within this paper and other findings of this study are available from the corresponding author on reasonable request.

Code availability

The minimal model simulations were performed using freely available HOOMD-Blue codes38. Hydrodynamic simulations were carried out using codes based on the publicly available code at Input files are available on request to the authors.


  1. Jiang, W. et al. Direct observation of the skyrmion Hall effect. Nat. Phys. 13, 162–169 (2017).

    Article  Google Scholar 

  2. Jülicher, F., Ajdari, A. & Prost, J. Modeling molecular motors. Rev. Mod. Phys. 69, 1269 (1997).

    Article  ADS  Google Scholar 

  3. Chaikin, P. M., Lubensky, T. C. & Witten, T. A. Principles of Condensed Matter Physics Vol. 10 (Cambridge Univ. Press, 1995).

  4. Strandburg, K. J. Two-dimensional melting. Rev. Mod. Phys. 60, 161–207 (1988).

    Article  ADS  Google Scholar 

  5. Bishop, D. J. & Reppy, J. D. Study of the superfluid transition in two-dimensional 4He films. Phys. Rev. Lett. 40, 1727–1730 (1978).

    Article  ADS  Google Scholar 

  6. Zahn, K., Lenke, R. & Maret, G. Two-stage melting of paramagnetic colloidal crystals in two dimensions. Phys. Rev. Lett. 82, 2721–2724 (1999).

    Article  ADS  Google Scholar 

  7. Alsayed, A. M., Islam, M. F., Zhang, J., Collings, P. J. & Yodh, A. G. Premelting at defects within bulk colloidal crystals. Science 309, 1207–1210 (2005).

    Article  ADS  Google Scholar 

  8. Meng, G., Paulose, J., Nelson, D. R. & Manoharan, V. N. Elastic instability of a crystal growing on a curved surface. Science 343, 634–637 (2014).

    Article  ADS  Google Scholar 

  9. Thorneywork, A. L., Abbott, J. L., Aarts, D. G. A. L. & Dullens, R. P. A. Two-dimensional melting of colloidal hard spheres. Phys. Rev. Lett. 118, 158001 (2017).

    Article  ADS  Google Scholar 

  10. Cafiero, R., Luding, S. & Herrmann, H. J. Rotationally driven gas of inelastic rough spheres. Europhys. Lett. 60, 854 (2002).

    Article  ADS  Google Scholar 

  11. Grzybowski, B. A., Stone, H. A. & Whitesides, G. M. Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid–air interface. Nature 405, 1033–1036 (2000).

    Article  ADS  Google Scholar 

  12. Yan, J., Bae, S. C. & Granick, S. Rotating crystals of magnetic Janus colloids. Soft Matter 11, 147–153 (2014).

    Article  ADS  Google Scholar 

  13. Nguyen, N. H., Klotsa, D., Engel, M. & Glotzer, S. C. Emergent collective phenomena in a mixture of hard shapes through active rotation. Phys. Rev. Lett. 112, 075701 (2014).

    Article  ADS  Google Scholar 

  14. Yeo, K., Lushi, E. & Vlahovska, P. M. Collective dynamics in a binary mixture of hydrodynamically coupled microrotors. Phys. Rev. Lett. 114, 188301 (2015).

    Article  ADS  Google Scholar 

  15. Kokot, G., Piet, D., Whitesides, G. M., Aranson, I. S. & Snezhko, A. Emergence of reconfigurable wires and spinners via dynamic self-assembly. Sci. Rep. 5, 9528 (2015).

    Article  ADS  Google Scholar 

  16. Soni, V. et al. The odd free surface flows of a colloidal chiral fluid. Nat. Phys. 15, 1188–1194 (2019).

    Article  Google Scholar 

  17. Shen, Z. & Lintuvuori, J. S. Two-phase crystallization in a carpet of inertial spinners. Phys. Rev. Lett. 125, 228002 (2020).

    Article  ADS  Google Scholar 

  18. Liu, P. et al. Oscillating collective motion of active rotors in confinement. Proc. Natl Acad. Sci. USA 117, 11901–11907 (2020).

    Article  MathSciNet  Google Scholar 

  19. Armitage, P. J. Turbulence and angular momentum transport in a global accretion disk simulation. Astrophys. J. Lett. 501, L189 (1998).

    Article  ADS  Google Scholar 

  20. Petroff, A. P., Wu, X.-L. & Libchaber, A. Fast-moving bacteria self-organize into active two-dimensional crystals of rotating cells. Phys. Rev. Lett. 114, 158102 (2015).

    Article  ADS  Google Scholar 

  21. Oppenheimer, N., Stein, D. B. & Shelley, M. J. Rotating membrane inclusions crystallize through hydrodynamic and steric interactions. Phys. Rev. Lett. 123, 148101 (2019).

    Article  ADS  Google Scholar 

  22. van Zuiden, B. C., Paulose, J., Irvine, W. T. M., Bartolo, D. & Vitelli, V. Spatiotemporal order and emergent edge currents in active spinner materials. Proc. Natl Acad. Sci. USA 113, 12919–12924 (2016).

    Article  ADS  Google Scholar 

  23. Aubret, A., Youssef, M., Sacanna, S. & Palacci, J. Targeted assembly and synchronization of self-spinning microgears. Nat. Phys. 14, 1114–1118 (2018).

    Article  Google Scholar 

  24. Snezhko, A. Complex collective dynamics of active torque-driven colloids at interfaces. Curr. Opin. Colloid Interface Sci. 21, 65–75 (2016).

    Article  Google Scholar 

  25. Scholz, C., Engel, M. & Pöschel, T. Rotating robots move collectively and self-organize. Nat. Commun. 9, 931 (2018).

    Article  ADS  Google Scholar 

  26. Kokot, G. et al. Active turbulence in a gas of self-assembled spinners. Proc. Natl Acad. Sci. USA 114, 12870–12875 (2017).

    Article  ADS  Google Scholar 

  27. Lim, M. X., Souslov, A., Vitelli, V. & Jaeger, H. M. Cluster formation by acoustic forces and active fluctuations in levitated granular matter. Nat. Phys. 15, 460–464 (2019).

    Article  Google Scholar 

  28. Bouchet, F. & Venaille, A. Statistical mechanics of two-dimensional and geophysical flows. Phys. Rep. 515, 227–295 (2012).

    Article  ADS  MathSciNet  Google Scholar 

  29. Baroud, C. N., Plapp, B. B., She, Z.-S. & Swinney, H. L. Anomalous self-similarity in a turbulent rapidly rotating fluid. Phys. Rev. Lett. 88, 114501 (2002).

    Article  ADS  Google Scholar 

  30. Korving, J., Hulsman, H., Knaap, H. & Beenakker, J. Transverse momentum transport in viscous flow of diatomic gases in a magnetic field. Phys. Lett. 21, 5–7 (1966).

    Article  ADS  Google Scholar 

  31. Hoyos, C., Moroz, S. & Son, D. T. Effective theory of chiral two-dimensional superfluids. Phys. Rev. B 89, 174507 (2014).

    Article  ADS  Google Scholar 

  32. Wiegmann, P. & Abanov, A. G. Anomalous hydrodynamics of two-dimensional vortex fluids. Phys. Rev. Lett. 113, 034501 (2014).

    Article  ADS  Google Scholar 

  33. Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. Topological waves in fluids with odd viscosity. Phys. Rev. Lett. 122, 128001 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  34. Banerjee, D., Souslov, A., Abanov, A. G. & Vitelli, V. Odd viscosity in chiral active fluids. Nat. Commun. 8, 1573 (2017).

    Article  ADS  Google Scholar 

  35. Han, M. et al. Fluctuating hydrodynamics of chiral active fluids. Nat. Phys. 17, 1260–1269 (2021).

    Article  Google Scholar 

  36. Eyink, G. L. & Sreenivasan, K. R. Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. Lavergne, F. A., Aarts, D. G. & Dullens, R. P. Anomalous grain growth in a polycrystalline monolayer of colloidal hard spheres. Phys. Rev. X 7, 041064 (2017).

    Google Scholar 

  38. Anderson, J. A., Glaser, J. & Glotzer, S. C. HOOMD-blue: a Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations. Comput. Mater. Sci. 173, 109363 (2020).

    Article  Google Scholar 

  39. Peierls, R. Reports on progress of physics. Phys. Soc. VI, 78 (1939).

    Google Scholar 

  40. Nabarro, F. Dislocations in a simple cubic lattice. Proc. Phys. Soc. 59, 256 (1947).

    Article  ADS  Google Scholar 

  41. Schall, P., Cohen, I., Weitz, D. A. & Spaepen, F. Visualization of dislocation dynamics in colloidal crystals. Science 305, 1944–1948 (2004).

    Article  ADS  Google Scholar 

  42. Weinberger, C. R. & Cai, W. Surface-controlled dislocation multiplication in metal micropillars. Proc. Natl Acad. Sci. USA 105, 14304–14307 (2008).

    Article  ADS  Google Scholar 

  43. Irvine, W. T. M., Hollingsworth, A. D., Grier, D. G. & Chaikin, P. M. Dislocation reactions, grain boundaries and irreversibility in two-dimensional lattices using topological tweezers. Proc. Natl Acad. Sci. USA 110, 15544–15548 (2013).

    Article  ADS  Google Scholar 

  44. Amir, A. & Nelson, D. R. Dislocation-mediated growth of bacterial cell walls. Proc. Natl Acad. Sci. USA 109, 9833–9838 (2012).

    Article  ADS  Google Scholar 

  45. Deutschländer, S., Dillmann, P., Maret, G. & Keim, P. Kibble-Zurek mechanism in colloidal monolayers. Proc. Natl Acad. Sci. USA 112, 6925–6930 (2015).

    Article  ADS  Google Scholar 

  46. Braverman, L., Scheibner, C. & Vitelli, V. Topological defects in non-reciprocal active solids with odd elasticity. Preprint at (2020).

  47. Sanchez, T., Chen, D. T., DeCamp, S. J., Heymann, M. & Dogic, Z. Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–434 (2012).

    Article  ADS  Google Scholar 

  48. Dahler, J. S. & Scriven, L. E. Theory of structured continua. I. General consideration of angular momentum and polarization. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 275, 504–527 (1963).

    ADS  Google Scholar 

  49. Tsai, J.-C., Ye, F., Rodriguez, J., Gollub, J. P. & Lubensky, T. C. A chiral granular gas. Phys. Rev. Lett. 94, 214301 (2005).

    Article  ADS  Google Scholar 

  50. Scheibner, C. et al. Odd elasticity. Nat. Phys. 16, 475–480 (2020).

    Article  Google Scholar 

  51. Epstein, J. M. & Mandadapu, K. K. Time-reversal symmetry breaking in two-dimensional nonequilibrium viscous fluids. Phys. Rev. E 101, 052614 (2020).

    Article  ADS  MathSciNet  Google Scholar 

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We acknowledge discussions with P. Wiegmann, A. Abanov, D. Nelson, C. Scheibner, M. Han, M. Fruchart, S. Gokhale, N. Fakhri and J. Dunkel. We thank V. Vitelli for an insightful discussion on the importance of odd stress on defect motility. We thank W. Yan for useful conversations. This work was primarily supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by the National Science Foundation (NSF) under award no. DMR-2011854. Additional support was provided by NSF DMR-1905974, NSF EFRI NewLAW 1741685 and the Packard Foundation. M.J.S. acknowledges support from NSF grants DMR-1420073 (NYU-MRSEC) and DMR-2004469. D.B. acknowledges support from ARN grant WTF and IdexLyon Tore. E.S.B. was supported by the National Science Foundation Graduate Research Fellowship under grant no. 1746045. D.B. and W.T.M.I. gratefully acknowledge support from the Chicago-France FACCTS programme. F.B.U. acknowledges support from ‘la Caixa’ Foundation (ID 100010434), fellowship LCF/BQ/PI20/11760014 and from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement no. 847648. The University of Chicago’s Research Computing Center and the University of Chicago’s GPU-based high-performance computing system (NSF DMR-1828629) are acknowledged for access to computational resources and the Chicago MRSEC (US NSF grant no. DMR-2011854) for access to its shared experimental facilities.

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Authors and Affiliations



E.S.B. designed and performed experiments and analysed data. Y.A.G. designed and performed minimal model simulations and elastic theory. F.B.U. designed and performed fully hydrodynamic simulations. V.S. and S.M. contributed to experiments and analytical tools. A.P., D.B., Y.A.G., E.S.B., W.T.M.I. and M.J.S. performed continuum modelling. W.T.M.I., D.B. and M.J.S. designed and supervised research. All authors discussed the results and analysis.

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Correspondence to Michael J. Shelley, Denis Bartolo or William T. M. Irvine.

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Peer review information Nature Physics thanks Juho Lintuvuori and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Bililign, E.S., Balboa Usabiaga, F., Ganan, Y.A. et al. Motile dislocations knead odd crystals into whorls. Nat. Phys. 18, 212–218 (2022).

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