Nematic Shells Research Articles

Topological Stabilization and Dynamics of Self-Propelling Nematic Shells.

Liquid shells (e.g., double emulsions, vesicles, etc.) are susceptible to interfacial instability and rupturing when driven out of mechanical equilibrium. This poses a significant challenge for the design of liquid-shell-based micromachines, where the goal is to maintain stability and dynamical control in combination with motility. Here, we present our solution to this problem with controllable self-propelling liquid shells, which we have stabilized using the soft topological constraints imposed by a nematogen oil. We demonstrate, through experiments and simulations, that anisotropic elasticity can counterbalance the destabilizing effect of viscous drag induced by shell motility and inhibit rupturing. We analyze their propulsion dynamics and identify a peculiar meandering behavior driven by a combination of topological and chemical spontaneously broken symmetries. Based on our understanding of these symmetry breaking mechanisms, we provide routes to control shell motion via topology, chemical signaling, and hydrodynamic interactions.

Stabilising the tetrahedral defect configuration in nematic shells

ABSTRACTMonte Carlo simulations are used to investigate the director configurations and the arrangements of the disclination lines that form in nematic shells surrounding interior spherical and polyhedral particles. For nematic shells formed around an interior spherical particle, a tetrahedral arrangement of the disclination lines is observed but this arrangement is lost if the interior sphere is shifted from the centre. The simulations indicate that a narrower angular distribution is observed for interior tetrahedral particles and, for non-centrosymmetric shells, the distribution is significantly less broadened compared to similar spherical inclusions. Other shape polyhedral particles are shown to give disclination geometries of different symmetries.

Lifting ordered surfaces: Ellipsoidal nematic shells.

When a material surface is functionalized so as to acquire some type of order, functionalization of which soft condensed matter systems have recently provided many interesting examples, the modeler faces an alternative. Either the order is described on the curved, physical surface where it belongs, or it is described on a flat surface that is unrolled as preimage of the physical surface under a suitable height function. This paper applies a general method that pursues the latter avenue by lifting whatever order tensor is deemed appropriate from a flat to a curved surface. We specialize this method to nematic shells, for which it also provides a simple but perhaps convincing interpretation of the outcomes of some molecular dynamics experiments on ellipsoidal shells.

Sub-second dynamic phototuning of alignment in azodendrimer-doped nematic liquid crystal shells

The alignment of nematic liquid crystal 5CB in micron-thick shells, suspended in and containing aqueous liquid phases, can be rapidly switched between radial (homeotropic) and tangential (planar) director field by doping them with a photoresponsive dendrimer with multiple azobenzene moieties in the branches. The dendrimer spontaneously segregates to the inner as well as outer shell interfaces, folding into an amphiphilic conformation irrespective of the sign of interface curvature. The branches are directed into the liquid crystal, inducing a homeotropic ground state. Upon UV irradiation, the trans-cis isomerization of azobenzene triggers immediate switching to planar alignment. The very fast realignment and the simultaneous response throughout the shell leads to an initially random planar director field, with many topological defects of both positive and negative signs becoming visible within a second of irradiation. All but two +1 defects quickly annihilate, and the remaining defect pair moves up towards the thinnest part of the shell to form the planar steady state. By illuminating with visible light the homeotropic alignment is quickly recovered. By exchanging the solvent used for assisting the dendrimer dissolution, also dynamic phase separation phenomena can be studied in the shells, revealing that the dendrimer solubility in 5CB is greater in the UV-induced cis state than in the trans ground state.

Defects in Nematic Shells: A Γ-Convergence Discrete-to-Continuum Approach

In this paper we rigorously investigate the emergence of defects on Nematic Shells with a genus different from one. This phenomenon is related to a non-trivial interplay between the topology of the shell and the alignment of the director field. To this end, we consider a discrete XY system on the shell M, described by a tangent vector field with unit norm sitting at the vertices of a triangulation of the shell. Defects emerge when we let the mesh size of the triangulation go to zero, namely in the discrete-to-continuum limit. In this paper we investigate the discrete-to-continuum limit in terms of Γ-convergence in two different asymptotic regimes. The first scaling promotes the appearance of a finite number of defects whose charges are in accordance with the topology of shell M, via the Poincare–Hopf Theorem. The second scaling produces the so called Renormalized Energy that governs the equilibrium of the configurations with defects.

Cylindrical nematic liquid crystal shell: effect of saddle-splay elasticity.

This study introduces cylindrical nematic liquid crystal (LC) shells. Shells as confinement can provide soft matter with intriguing topology and geometry. Indeed, in spherical shells of LCs, rich defect structures have been reported. Avoiding the inherent Plateau-Rayleigh instability of cylindrical liquid-liquid interfaces, we realize the cylindrical nematic LC shell by two different methods: the phase separation in the nematic-isotropic coexistence phase and a cylindrical cavity with a glass rod suspended in the middle. Specifically, the director configurations of lyotropic chromonic LCs (LCLCs) in the cylindrical shell and their energetics are investigated theoretically and experimentally. Unusual elastic properties of LCLCs, i.e., a large saddle-splay modulus, and a shell geometry with both concave and convex curvatures, result in a double-twist director configuration.

Vortex formation and dynamics of defects in active nematic shells

We present a hydrodynamic model for a thin spherical shell of active nematic liquid crystal with an arbitrary configuration of defects. The active flows generated by defects in the director lead to the formation of stable vortices, analogous to those seen in confined systems in flat geometries, which generate effective dynamics for four +1/2 defects that reproduces the tetrahedral to planar oscillations observed in experiments. As the activity is increased and two counterrotating vortices dominate the flow, the defects are drawn more tightly into pairs, rotating about antipodal points. We extend this situation to also describe the dynamics of other configurations of defects. For example, two +1 defects are found to attract or repel according to the local geometric character of the director field around them and the extensile or contractile nature of the material, while additional pairs of opposite charge defects can give rise to flow states containing more than two vortices. Finally, we describe the generic relationship between defects in the orientation and singular points of the flow, and suggest implications for the three-dimensional nature of the flow and deformation in the shape of the shell.

Bistable curvature potential at hyperbolic points of nematic shells

Nematic shells are colloidal particles coated with nematic liquid crystal molecules which may freely glide and rotate on the colloid's surface while keeping their long axis on the local tangent plane. We describe the nematic order on a shell by a unit director field on an orientable surface. Equilibrium fields can then be found by minimising the elastic energy, which in general is a function of the surface gradient of the director field. We learn how to extract systematically out of this energy a fossil component, related only to the surface and its curvatures, which expresses a curvature potential for the molecular torque. At hyperbolic points on the colloid's surface, and only there, the alignment preferred by the curvature potential may fail to be a direction of principal curvature. There the fossil energy becomes bistable.

Frame, metric and geodesic evolution in shape-changing nematic shells.

Non-uniform director fields in flat, responsive, glassy nematic sheets lead to the induction of shells with non-trivial topography on the application of light or heat. Contraction along the director causes metric change, with, in general, the induction of Gaussian curvature, that drives the topography change. We describe the metric change, the evolution of the director field, and the transformation of reference state material curves, e.g. spirals into radii, as curvature develops. The non-isometric deformations associated with heat or light change the geodesics of the surface, intriguingly even in regions where no Gaussian curvature results.

Impact of curvature on topological defects

We analyze the impact of extrinsic and intrinsic curvature on positions of topological defects (TDs) in two-dimensional (2D) nematic films. We demonstrate that both these curvature contributions are commonly present and are expected to be weighted by comparable elastic constants. A simple Landau-de Gennes approach in terms of tensor nematic order parameter is used to numerically demonstrate impact of the curvatures on position of TDs on 2D ellipsoidal nematic shells. In particular, in oblate ellipsoids the extrinsic and intrinsic elastic terms enforce conflicting tendencies to positions of TDs.

Spherical nematic shells with a prolate ellipsoidal core.

Liquid crystal shells have attracted considerable attention in recent years. In such systems, a combination of confinement and curvature generates topological defect structures that do not exist in the bulk. Past studies, however, have largely focused on perfectly spherical shells, and little attention has been devoted to the impact of core geometry on the configuration and arrangement of topological defects. In this work, a microfluidic glass capillary device is used to encapsulate spherical and prolate ellipsoidal particles in nematic liquid crystal (LC) droplets dispersed in aqueous media. Our experimental studies show that, when trapped inside a radial LC droplet, spherical particles with both homeotropic and planar anchoring are highly localized at the droplet's center. While the radial configuration of the LC droplets is not altered by a homeotropic particle, polystyrene particles with strong planar anchoring disturb the radial ordering, leading to a twisted structure. Experiments indicate that off-center particle positions can also arise, in which defects are displaced towards the vicinity of the droplet's surface. In contrast, when prolate ellipsoidal particles are encapsulated in a thick radial LC shell, the minimum free energy corresponds to configurations where the particle is positioned at the droplet center. In this case, defects arise at the two ends of the prolate ellipsoid, where the curvature of the particle is maximal, leading to the formation of peculiar hybrid and twisted structures.

Elastic interactions between topological defects in chiral nematic shells.

We present a self-consistent and robust theoretical model to investigate elastic interactions between topological defects in liquid crystal shells. Accounting for the nonconcentric nature of the shell in a simple manner, we are able to successfully and accurately explain and predict the positions of the defects, most relevant in the context of colloidal self-assembly. We calibrate and test our model on existing experimental data and extend it to all observed defects configurations in chiral nematic shells. We perform experiments to check further and confirm the validity of the present model. Moreover, we are able to obtain quantitative estimates of the energies of +1 or +3/2 disclination lines in cholesterics, whose intricate nature was only reported recently [A. Darmon, et al. Proc. Natl. Acad. Sci. USA 113, 9469 (2016)10.1073/pnas.1525059113].

Dynamic structure of active nematic shells.

When a thin film of active, nematic microtubules and kinesin motor clusters is confined on the surface of a vesicle, four +1/2 topological defects oscillate in a periodic manner between tetrahedral and planar arrangements. Here a theoretical description of nematics, coupled to the relevant hydrodynamic equations, is presented here to explain the dynamics of active nematic shells. In extensile microtubule systems, the defects repel each other due to elasticity, and their collective motion leads to closed trajectories along the edges of a cube. That motion is accompanied by oscillations of their velocities, and the emergence and annihilation of vortices. When the activity increases, the system enters a chaotic regime. In contrast, for contractile systems, which are representative of some bacterial suspensions, a hitherto unknown static structure is predicted, where pairs of defects attract each other and flows arise spontaneously.

Taming Liquid Crystal Self-Assembly: The Multifaceted Response of Nematic and Smectic Shells to Polymerization.

By photopolymerizing liquid crystal shells, their rich variety of self-assembled structures can be rendered permanent and the lifetime extended from days to months, without removing the characteristic responsiveness. If polymerization is carried out close to either boundary of the nematic phase, the process triggers the transition into the adjacent phase, to higher or to lower degree of order.

Analysis of a variational model for nematic shells

We analyze an elastic surface energy which was recently introduced by G. Napoli and L. Vergori to model thin films of nematic liquid crystals. We show how a novel approach in modeling the surface’s extrinsic geometry leads to considerable differences with respect to the classical intrinsic energy. Our results concern three connected aspects: (i) using methods of the calculus of variations, we establish a relation between the existence of minimizers and the topology of the surface; (ii) we prove, by a Ginzburg–Landau approximation, the well-posedness of the gradient flow of the energy; (iii) in the case of a parametrized torus we obtain a stronger characterization of global and local minimizers, which we supplement with numerical experiments.

Hydrodynamic theory for nematic shells: The interplay among curvature, flow, and alignment.

We derive the hydrodynamic equations for nematic liquid crystals lying on curved substrates. We invoke the Lagrange-Rayleigh variational principle to adapt the Ericksen-Leslie theory to two-dimensional nematics in which a degenerate anchoring of the molecules on the substrate is enforced. The only constitutive assumptions in this scheme concern the free-energy density, given by the two-dimensional Frank potential, and the density of dissipation which is required to satisfy appropriate invariance requirements. The resulting equations of motion couple the velocity field, the director alignment, and the curvature of the shell. To illustrate our findings, we consider the effect of a simple shear flow on the alignment of a nematic lying on a cylindrical shell.

Spherical nematic shells with a threefold valence.

We present a theoretical study of the energetics of thin nematic shells with two charge-one-half defects and one charge-one defect. We determine the optimal arrangement: the defects are located on a great circle at the vertices of an isosceles triangle with angles of 66^{∘} at the charge-one-half defects and a distinct angle of 48^{∘}, consistent with experimental findings. We also analyze thermal fluctuations around this ground state and estimate the energy as a function of thickness. We find that the energy of the three-defect shell is close to the energy of other known configurations having two charge-one and four charge-one-half defects. This finding, together with the large energy barriers separating one configuration from the others, explains their observation in experiments as well as their long-time stability.

Close packing of rods on spherical surfaces.

We study the optimal packing of short, hard spherocylinders confined to lie tangential to a spherical surface, using simulated annealing and molecular dynamics simulations. For clusters of up to twelve particles, we map out the changes in the geometry of the closest-packed configuration as a function of the aspect ratio L/D, where L is the cylinder length and D the diameter of the rods. We find a rich variety of cluster structures. For larger clusters, we find that the best-packed configurations up to around 100 particles are highly dependent on the exact number of particles and aspect ratio. For even larger clusters, we find largely disordered clusters for very short rods (L/D = 0.25), while slightly longer rods (L/D = 0.5 or 1) prefer a global baseball-like geometry of smectic-like domains, similar to the behavior of large-scale nematic shells. Intriguingly, we observe that when compared to their optimal flat-plane packing, short rods adapt to the spherical geometry more efficiently than both spheres and longer rods. Our results provide predictions for experimentally realizable systems of colloidal rods trapped at the interface of emulsion droplets.

Tunable topological valence in nematic shells on spherocylindrical colloidal particles.

We perform molecular dynamics simulations of the orientational ordering on nematic shells delimited by spherocylindrical nanoscopic colloidal particles. We show that under conditions of degenerate planar anchoring, the equilibrium director field structure in these shells exhibits pairs of +1/2 topological defects at the poles of spherical cups in the absence of an external electric field. In addition, a certain number of pairs of ±1/2 defects occurs on the spherical cups far from the poles, thus resulting in a total of eight valence spots. A strong field applied along the main spherocylindrical axis removes the ±1/2 defect pairs while it coalesces the polar ones into a single +1 topological defect. A strong transverse field destroys all defects on the spherical cups but generates four +1/2 defects in the cylindrical part. Therefore, an external field can be used to control the number of valence centers in spherocylindrical nematic shells, thus unveiling their capability of acting as multivalent building blocks for nanophotonic devices.

Topological defects in cholesteric liquid crystal shells.

We investigate experimentally and numerically the defect configurations emerging when a cholesteric liquid crystal is confined to a spherical shell. We uncover a rich scenario of defect configurations, some of them non-existent in nematic shells, where new types of defects are stabilized by the helical ordering of the liquid crystal. In contrast to nematic shells, here defects are not simple singular points or lines, but have a large structured core. Specifically, we observe five different types of cholesteric shells. We study the statistical distribution of the different types of shells as a function of the two relevant geometrical dimensionless parameters of the system. By playing with these parameters, we are able to induce transitions between different types of shells. These transitions involve interesting topological transformations in which the defects recombine to form new structures. Surprisingly, the defects do not approach each other by taking the shorter distance route (geodesic), but by following intricate paths.