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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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.
The authors declare no competing financial interests.
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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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