Topological colloids

Abstract

Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids1, a type of soft matter consisting of tiny particles dispersed in chemically distinct host media. Being abundant in nature, colloids also find increasingly important applications in science and technology, ranging from direct probing of kinetics in crystals and glasses2 to fabrication of third-generation quantum-dot solar cells3. Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth1, their surfaces tend to have minimum-area spherical or topologically equivalent shapes such as prisms and irregular grains (all continuously deformable—homeomorphic—to spheres). Although toroidal DNA condensates and vesicles with different numbers of handles can exist4,5,6,7 and soft matter defects can be shaped as rings8 and knots9, the role of particle topology in colloidal systems remains unexplored. Here we fabricate and study colloidal particles with different numbers of handles and genus g ranging from 1 to 5. When introduced into a nematic liquid crystal—a fluid made of rod-like molecules that spontaneously align along the so-called ‘director’10—these particles induce three-dimensional director fields and topological defects dictated by colloidal topology. Whereas electric fields, photothermal melting and laser tweezing cause transformations between configurations of particle-induced structures, three-dimensional nonlinear optical imaging reveals that topological charge is conserved and that the total charge of particle-induced defects always obeys predictions of the Gauss–Bonnet and Poincaré–Hopf index theorems11,12,13. This allows us to establish and experimentally test the procedure for assignment and summation of topological charges in three-dimensional director fields. Our findings lay the groundwork for new applications of colloids and liquid crystals that range from topological memory devices14, through new types of self-assembly15,16,17,18,19,20,21,22,23, to the experimental study of low-dimensional topology6,7,11,12,13.

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Figure 1: Colloidal handlebodies aligned orthogonal to the far-field director.
Figure 2: Colloidal g handlebodies aligned parallel to the far-field director.
Figure 3: Director field and point defects around colloidal handlebodies.
Figure 4: Diffusion and electric control of colloidal handlebodies in a nematic liquid crystal.

References

  1. 1

    Hunter, R. J. Foundations of Colloid Science (Oxford Univ. Press, 2001)

    Google Scholar 

  2. 2

    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)

    ADS  CAS  Article  Google Scholar 

  3. 3

    Semonin, O. E. et al. Peak external photocurrent quantum efficiency exceeding 100% via MEG in a quantum dot solar cell. Science 334, 1530–1533 (2011)

    ADS  CAS  Article  Google Scholar 

  4. 4

    Hud, N. V., Downing, K. H. & Balhord, R. A constant radius of curvature model for the organization of DNA in toroidal condensates. Proc. Natl Acad. Sci. USA 92, 3581–3585 (1995)

    ADS  CAS  Article  Google Scholar 

  5. 5

    Michalet, X. & Bensimon, D. Observation of stable shapes and conformal diffusion in genus 2 vesicles. Science 269, 666–668 (1995)

    ADS  CAS  Article  Google Scholar 

  6. 6

    Hsu, L., Kusner, R. & Sullivan, J. Minimizing the squared mean curvature integral for surfaces in space forms. Exp. Math. 1, 191–207 (1992)

    MathSciNet  Article  Google Scholar 

  7. 7

    Kamien, R. D. The geometry of soft materials: a primer. Rev. Mod. Phys. 74, 953–971 (2002)

    ADS  Article  Google Scholar 

  8. 8

    Lavrentovich, O. D. & Terentjev, E. M. Phase transition altering the symmetry of topological point defects (hedgehogs) in a nematic liquid crystal. Sov. Phys. JETP 64, 1237–1244 (1986)

    Google Scholar 

  9. 9

    Tkaleč, U., Ravnik, M., Čopar, S., Žumer, S. & Muševič, I. Reconfigurable knots and links in chiral nematic colloids. Science 33, 62–65 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10

    Kleman, M. & Lavrentovich, O. D. Topological point defects in nematic liquid crystals. Phil. Mag. 86, 4117–4137 (2006)

    ADS  Article  Google Scholar 

  11. 11

    Montiel, S. & Ros, A. Curves and Surfaces 2nd edn (American Mathematical Society, 2009)

    Google Scholar 

  12. 12

    Milnor, J. W. Topology from the Differentiable Viewpoint (Univ. Press of Virginia, 1965)

    Google Scholar 

  13. 13

    Hopf, H. Vektorfelder in n-dimensionalen Mannigfaltigkeiten. Math. Ann. 96, 225–249 (1927)

    MathSciNet  Article  Google Scholar 

  14. 14

    Araki, T., Buscaglia, M., Bellini, T. & Tanaka, H. Memory and topological frustration in nematic liquid crystals confined in porous materials. Nature Mater. 10, 303–309 (2011)

    ADS  CAS  Article  Google Scholar 

  15. 15

    Poulin, P., Holger, S., Lubensky, T. C. & Weitz, D. A. Novel colloidal interactions in anisotropic fluids. Science 275, 1770–1773 (1997)

    CAS  Article  Google Scholar 

  16. 16

    Irvine, W. T. M., Vitelli, V. & Chaikin, P. M. Pleats in crystal on curved surfaces. Nature 468, 947–951 (2010)

    ADS  CAS  Article  Google Scholar 

  17. 17

    Jones, M. R. et al. DNA-nanoparticle superlattices formed from anisotropic building blocks. Nature Mater. 9, 913–917 (2010)

    ADS  CAS  Article  Google Scholar 

  18. 18

    Lapointe, C. P., Mason, T. G. & Smalyukh, I. I. Shape-controlled colloidal interactions in nematic liquid crystals. Science 326, 1083–1086 (2009)

    ADS  CAS  Article  Google Scholar 

  19. 19

    Sacanna, S., Irvine, W. T. M., Chaikin, P. M. & Pine, D. J. Lock and key colloids. Nature 464, 575–578 (2010)

    ADS  CAS  Article  Google Scholar 

  20. 20

    Wood, T. A., Lintuvuori, J. S., Schofield, A. B., Marenduzzo, D. & Poon, W. C. K. A self-quenched defect glass in a colloid–nematic liquid crystal composite. Science 334, 79–83 (2011)

    ADS  CAS  Article  Google Scholar 

  21. 21

    Poulin, P., Cabuil, V. & Weitz, D. A. Direct measurement of colloidal forces in an anisotropic solvent. Phys. Rev. Lett. 79, 4862–4865 (1997)

    ADS  CAS  Article  Google Scholar 

  22. 22

    Tkalec, U., Ravnik, M., Žumer, S. & Muševič, I. Vortexlike topological defects in nematic colloids: chiral colloidal dimers and 2D crystals. Phys. Rev. Lett. 103, 127801 (2009)

    ADS  CAS  Article  Google Scholar 

  23. 23

    Trivedi, R. P., Klevets, I. I., Senyuk, B., Lee, T. & Smalyukh, I. I. Multi-scale interactions and three-dimensional patterning of colloidal particles and defects in lamellar soft media. Proc. Natl Acad. Sci. USA 109, 4744–4749 (2012)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Herges, R. Topology in chemistry: designing Möbius molecules. Chem. Rev. 106, 4820–4842 (2006)

    CAS  Article  Google Scholar 

  25. 25

    Mosseri, R. Geometrical frustration and defects in condensed matter systems. C. R. Chim. 11, 192–197 (2008)

    CAS  Article  Google Scholar 

  26. 26

    Bowick, M. J., Chandar, L., Schiff, E. A. & Srivastava, A. M. The cosmological Kibble mechanism in the laboratory: string formation in liquid crystals. Science 263, 943–945 (1994)

    ADS  CAS  Article  Google Scholar 

  27. 27

    Han, Y. et al. Brownian motion of an ellipsoid. Science 314, 626–630 (2006)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  28. 28

    Solomon, M. J. Directions for targeted self-assembly of anisotropic colloids from statistical thermodynamics. Curr. Opin. Colloid Interface Sci. 16, 158–167 (2011)

    CAS  Article  Google Scholar 

  29. 29

    Merkel, T. J. et al. Scalable, shape-specific, top-down fabrication methods for the synthesis of engineered colloidal particles. Langmuir 26, 13086–13096 (2010)

    CAS  Article  Google Scholar 

  30. 30

    Lee, T., Trivedi, R. P. & Smalyukh, I. I. Multimodal nonlinear optical polarizing microscopy of long-range molecular order in liquid crystals. Opt. Lett. 35, 3447–3449 (2010)

    ADS  CAS  Article  Google Scholar 

  31. 31

    Pettey, D. & Lubensky, T. C. Stability of texture and shape of circular domains of Langmuir monolayers. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 59, 1834–1845 (1999)

    CAS  Google Scholar 

  32. 32

    Alexander, G. P., Chen, B. G., Matsumoto, E. A. & Kamien, R. D. Colloquium: Disclination loops, point defects, and all that in nematic liquid crystals. Rev. Mod. Phys. 84, 497–514 (2012)

    ADS  CAS  Article  Google Scholar 

  33. 33

    Chaikin, P. M. & Lubensky, T. C. Principles of Condensed Matter Physics (Cambridge Univ. Press, 2000)

    Google Scholar 

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Acknowledgements

We thank P. Chen, N. Clark, J.-i. Fukuda and S. Žumer for discussions. This work was supported by the International Institute for Complex Adaptive Matter and the National Science Foundation grants DMR-0844115 (Q.L., S.H. and I.I.S.), DMR-0820579 (B.S. and I.I.S.), DMR-0847782 (B.S., Q.L. and I.I.S.), PHY11-25915 (R.D.K., R.B.K., T.C.L. and I.I.S.) and DMR-1120901 (R.D.K. and T.C.L.). R.B.K., R.D.K., T.C.L. and I.I.S. thank the Kavli Institute for Theoretical Physics for their hospitality while this work was being discussed and prepared for publication.

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B.S., Q.L. and I.I.S. performed experimental work. Q.L., S.H. and I.I.S. designed and fabricated particles. B.S. and I.I.S. reconstructed director fields induced by colloids. T.C.L. and I.I.S. characterized topological charges of defects in particle-induced director fields. R.B.K., R.D.K., T.C.L. and I.I.S. proposed models of field transformations satisfying topological constraints and explained the relations between genus of colloids and the net topological charge of liquid crystal defects. I.I.S. conceived the project, designed experiments, provided funding and wrote the manuscript. All authors edited and commented on the manuscript.

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Correspondence to Ivan I. Smalyukh.

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The authors declare no competing financial interests.

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This file contains Supplementary Text and Data 1-4, Supplementary References and Supplementary Figures 1-12. (PDF 5542 kb)

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Senyuk, B., Liu, Q., He, S. et al. Topological colloids. Nature 493, 200–205 (2013). https://doi.org/10.1038/nature11710

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