The prospect of mimicking molecular chemistry with colloidal rather than molecular building blocks could enable unprecedented control over the properties of microstructured materials 1 . The usual absence of directionality to the interaction between colloids has limited the complexity of the structures they can spontaneously form. One way to address this is to coat spherical colloid particles with a thin layer of nematic liquid crystal 2 and functionalize 3 the unavoidable defects or bold spots that arise when nematic order is established on the surface of a sphere 4,5 . The number and arrangement of these defects can vary 2,6-16 , providing flexibility for tuning directional interactions that are more difficult to achieve by other methods [17][18][19][20][21][22][23][24][25][26] . Yet, many theoretically predicted structures have not been observed and control over defect location remains elusive. In this work, we show that varying the thickness of a nematic liquid crystal shell enables us to systematically control the number and orientation of defects formed. For thin shells, these defects can be engineered to emulate the linear, trigonal and tetrahedral geometries of sp, sp 2 and sp 3 carbon bonds, respectively. Such control opens up the possibility to engineer particles with tunable-valence and directional-binding capabilities.To fabricate spherical nematic shells, we generate double emulsions with a microcapillary device 27 ; these consist of a nematic drop that contains a smaller aqueous drop, all inside an aqueous continuous phase. Both the inner and outer water phases contain 1 wt% polyvinyl alcohol, which stabilizes the emulsion against coalescence and enforces tangential anchoring of the rod-like molecules of the nematic liquid crystal, pentylcyanobiphenyl. The resulting double-emulsion drops are characterized by an outer radius, R, of around 50 µm and an inner radius, a, that are varied to produce shells of different average thicknesses,h = R − a, as schematically shown in Fig. 1a. With this microfluidic method the thinnest shells that we can generate haveh ≈ 1 µm. However, it is possible to significantly reduce this value by increasing the volume of the inner drop once the double emulsion is formed. We achieve this by inducing a difference in osmotic pressure between the inner and outer water phases through the addition of a salt, CaCl 2 . As pentylcyanobiphenyl has a finite permeability to water, an incoming flow of water from the outer phase can be established if the inner drop contains a higher salt concentration than the outer phase. By controlling this difference, we can control the kinetics of the process and ultimately the thickness of the shells.The thinnest shells have four defects, each with a topological charge s = 1/2, reflecting the π rotation experienced by the local nematic direction along a path encircling each defect. As a result, the total topological charge on the sphere is equal to 4 × 1/2 = 2; this is consistent with a mathematical theorem due to Poincaré and Hopf, which establish...

We stabilize nematic droplets with handles against surface tension-driven instabilities, using a yield-stress material as outer fluid, and study the complex nematic textures and defect structures that result from the competition between topological constraints and the elasticity of the nematic liquid crystal. We uncover a surprisingly persistent twisted configuration of the nematic director inside the droplets when tangential anchoring is established at their boundaries, which we explain after considering the influence of saddle splay on the elastic free energy. For toroidal droplets, we find that the saddle-splay energy screens the twisting energy, resulting in a spontaneous breaking of mirror symmetry; the chiral twisted state persists for aspect ratios as large as ∼20. For droplets with additional handles, we observe in experiments and computer simulations that there are two additional −1 surface defects per handle; these are located in regions with local saddle geometry to minimize the nematic distortions and hence the corresponding elastic free energy.geometric frustration | topology | torus | double twist | boojum T he liquid crystal in a common display is twisted due to the orientation of the molecules at the confining glass plates. By manipulating this twist using electric fields, an image can be generated. More exotic structures can emerge when the liquid crystal is confined by curved rather than flat surfaces. The topology and geometry of the bounding surface can drive the system into structures that would not be achieved without the presence of external fields. In this sense, the shape of the surface plays a role akin to that of an external field. Thus, under confinement by curved surfaces, the molecules can self-assemble into complex hierarchical structures with emergent macroscopic properties not observed for flat liquid crystal cells. However, the design principles and properties of structures generated by this geometric route are still largely unknown.The lowest energy state of an ordered material, such as a liquid crystal or a simple crystal, is typically defect-free because any disruption of the order will raise the elastic energy. However, the situation can be very different if the material is encapsulated within a confining volume and there is strong alignment of the molecules at the bounding surfaces. In this case, the preferred local order cannot be maintained throughout space. Such a material will be geometrically frustrated and its ground state could contain topological defects, which are spatial regions where the characteristic order of the material is lost. For nematic liquid crystals, the molecules tend to align along a common director, n. The presence of defects at the boundaries, which we characterize with their topological charge, s, giving the amount of n-rotation at the boundary as we encircle the defect, raises the energy of the system. Thus, the formation of defects is normally disfavored due to this increase in energy. However, when an orientationally ordered material is confine...

We present a theoretical study of director fields in toroidal geometries with degenerate planar boundary conditions. We find spontaneous chirality: despite the achiral nature of nematics the director configuration show a handedness if the toroid is thick enough. In the chiral state the director field displays a double twist, whereas in the achiral state there is only bend deformation. The critical thickness increases as the difference between the twist and saddle-splay moduli grows. A positive saddle-splay modulus prefers alignment along the short circle of the bounding torus, and hence stimulates promotes a chiral configuration. The chiral-achiral transition mimics the order-disorder transition of the mean-field Ising model. The role of the magnetisation in the Ising model is played by the degree of twist. The role of the temperature is played by the aspect ratio of the torus. Remarkably, an external field does not break the chiral symmetry explicitly, but shifts the transition. In the case of toroidal cholesterics, we do find a preference for one chirality over the other -the molecular chirality acts as a field in the Ising analogy.

We present a theoretical study of the director fields and energetics of nematic liquid crystal shells with two pairs of surface defects. The pairs of defects can undergo abrupt transitions between a configuration of maximum separation to at state in which the defects are confined to the thinnest hemisphere. We construct a phase diagram that maps out the stability and coexistence of these two configurations as a function of shell thickness and thickness inhomogeneity. Our results compare favorably with the experimentally observed transitions in nematic double emulsion droplets and explain their hysteretic character.

Conforming materials to rigid substrates with Gaussian curvature-positive for spheres and negative for saddles-has proven a versatile tool to guide the self-assembly of defects such as scars, pleats, folds, blisters, and liquid crystal ripples. Here, we show how curvature can likewise be used to control material failure and guide the paths of cracks. In our experiments, and unlike in previous studies on cracked plates and shells, we constrained flat elastic sheets to adopt fixed curvature profiles. This constraint provides a geometric tool for controlling fracture behaviour: curvature can stimulate or suppress the growth of cracks and steer or arrest their propagation. A simple analytical model captures crack behaviour at the onset of propagation, while a two-dimensional phase-field model with an added curvature term successfully captures the crack's path. Because the curvature-induced stresses are independent of material parameters for isotropic, brittle media, our results apply across scales.