Topological colloids

Abstract

Smoke, fog, jelly, paints, milk and shaving cream are common everyday examples of colloids 1 , 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 glasses 2 to fabrication of third-generation quantum-dot solar cells 3 . Because naturally occurring colloids have a shape that is typically determined by minimization of interfacial tension (for example, during phase separation) or faceted crystal growth 1 , 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 exist 4-7 and soft matter defects can be shaped as rings 8 and knots 9 , 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 threedimensional 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 theorems [11][12][13] . This allows us to establish and experimentally test the procedure for assignment and summation of topological charges in threedimensional director fields. Our findings lay the groundwork for new applications of colloids and liquid crystals that range from topological memory devices 14 , through new types of self-assembly [15][16][17][18][19][20][21][22][23] , to the experimental study of low-dimensional topology 6,7,[11][12][13] .Although a coffee mug and a doughnut look different to most of us, they are topologically equivalent solid tori or handlebodies of genus g = 1, both being different from, say, balls and solid cylinders of genus g = 0, to which they cannot be smoothly morphed without cutting 11,12 . In a similar way, molecules can form topologically distinct structures including rings, knots and other molecular configurations satisfying the constraints imposed by chemical bonds 24 . Although the topology of shapes, fields and defects is important in many phenomena and in theories ranging from the nature of elementary particles to early-Universe cosmology 25,26 , topological aspects of colloidal systems (composed of particles larger than molecules and atoms but much smaller than the objects that we encounter in our everyday life) are rarely explored. Typically dealing with...

Transport of particles in liquid crystals

“…The ring structure can be stabilized in a thin cell of thickness close to 2 R 34 , for relatively small R or for relatively weak anchoring strength 35 , under a high electric or magnetic field 36,37 . Besides the configurations shown in Fig.1, more complex structures, including twisted ones, around particles of various shapes have been considered, see, for example, [38][39][40][41][42][43] and references therein. Figure 2 shows the optical microscopy textures of a sphere with normal anchoring and accompanying hyperbolic hedgehog and its modification by a strong alternating current (AC) electric field.…”

“…If the sphere spins at a distance h   from the wall, the velocity gradient between the wall and the sphere is much steeper (and thus the viscous stress is larger) than in the rest of the space, Fig.18, so there is a force pushing the sphere along the wall, perpendicular to the axis of spinning. By balancing the torques and forces acting on the Quincke rotator near the wall, one finds the velocity of translation in the direction perpendicular to both the applied electric field and the spinning axis 60 , , the velocities might be very high, (40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50) V/μm . In smectic A samples with air bubbles, the particles are strongly trapped in the meniscus region, at the grain boundary separating the regions of differently tilted smectic layers.…”

Self-assembly of colloidal particles in deformation landscapes of electrically driven layer undulations in cholesteric liquid crystals

“…Compared to isotropic fluid hosts, LCs bring many new possibilities in designing guided colloidal self-assembly arising from elasticity-mediated anisotropic interactions [3], nanoparticle localization by topological defects [11,19,26], topography [27][28][29] and patterning [30,31] of confinement, and structural switching enabled by the LC's facile response to external fields [10,32]. However, the focus of these studies of LC colloidal dispersions is primarily on using ground-state phases [11], predesigned molecular fields modulations [30,31], or topological defect arrays to mediate self-assembly.…”

Knots and nonorientable surfaces in chiral nematics

“…For orientable surfaces with normal anchoring, the director can be given the orientation of the Gauss map, so that G describes precisely the molecular orientation at the surface. The degree of this map--the number of times every point on S 2 is visited, counted with sign--is a homotopy invariant (43) characterizing the type of defect that the surface generates (38,44). Although experimentally the same surface can produce seemingly different defects, they are always characterized by this same element of π 2 ðRP 2 Þ.…”

“…Closed, orientable surfaces are known to induce defects corresponding to the element 1 − g = χ=2 of π 2 ðRP 2 Þ, where g is the genus of the surface and χ is the Euler characteristic (38,43,44).…”