Key-lock mechanism in nematic colloidal dispersions

Lock-key microfluidics: simulating nematic colloid advection along wavy-walled channels

Liquid crystalline media mediate interactions between suspended particles and confining geometries, which not only has potential to guide patterning and bottom-up colloidal assembly, but can also control colloidal migration in microfluidic devices. However, simulating such dynamics is challenging because nemato-elasticity, diffusivity and hydrodynamic interactions must all be accounted for within complex boundaries. We model the advection of colloids dispersed in flowing and fluctuating nematic fluids confined within 2D wavy channels. A lock-key mechanism between homeotropic colloids and troughs is found to be stronger for planar anchoring on the wavy walls compared to homeotropic anchoring on the wavy walls due to the relative location of the colloid-associated defects. Sufficiently large amplitudes result in stick-slip trajectories and even permanent locking of colloids in place. These results demonstrate that wavy walls not only have potential to direct colloids to specific docking sites but also to control site-specific resting duration and intermittent elution.

Wetting of Nematic Liquid Crystals on Crenellated Substrates: A Frank–Oseen Approach

We revisit the wetting of nematic liquid crystals in contact with crenellated substrates, studied previously using the Landau–de Gennes formalism. However, due to computational limitations, the characteristic length scales of the substrate relief considered in that study limited to less than 100 nematic correlation lengths. The current work uses an extended Frank–Oseen formalism, which includes not only the free-energy contribution due to the elastic deformations but also the surface tension contributions and, if disclinations or other orientational field singularities are present, their core contributions. Within this framework, which was successfully applied to the anchoring transitions of a nematic liquid crystal in contact with structured substrates, we extended the study to much larger length scales including the macroscopic scale. In particular, we analyzed the interfacial states and the transitions between them at the nematic–isotropic coexistence.

Deck the Walls with Anisotropic Colloids in Nematic Liquid Crystals

Nematic liquid crystals (NLCs) offer remarkable opportunities to direct colloids to form complex structures. The elastic energy field that dictates colloid interactions is determined by the NLC director field, which is sensitive to and can be controlled by boundaries including vessel walls and colloid surfaces. By molding the director field via liquid-crystal alignment on these surfaces, elastic energy landscapes can be defined to drive structure formation. We focus on colloids in otherwise defect-free director fields formed near undulating walls. Colloids can be driven along prescribed paths and directed to well-defined docking sites on such wavy boundaries. Colloids that impose strong alignment generate topologically required companion defects. Configurations for homeotropic colloids include a dipolar structure formed by the colloid and its companion hedgehog defect or a quadrupolar structure formed by the colloid and its companion Saturn ring. Adjacent to wavy walls with wavelengths larger than the colloid diameter, spherical particles are attracted to locations along the wall with distortions in the nematic director field that complement those from the colloid. This is the basis of lock-and-key interactions. Here, we study ellipsoidal colloids with homeotropic anchoring near complex undulating walls. The walls impose distortions that decay with distance from the wall to a uniform director in the far field. Ellipsoids form dipolar defect configurations with the colloid's major axis aligned with the far field director. Two distinct quadrupolar defect structures also form, stabilized by confinement; these include the Saturn I configuration with the ellipsoid's major axis aligned with the far field director and the Saturn II configuration with the major axis perpendicular to the far field director. The ellipsoid orientation varies only weakly in bulk and near undulating walls. All configurations are attracted to walls with long, shallow waves. However, for walls with wavelengths that are small compared to the colloid length, Saturn II is repelled, allowing selective docking of aligned objects. Deep, narrow wells prompt the insertion of a vertical ellipsoid. By introducing an opening at the bottom of such a deep well, we study colloids within pores that connect two domains. Ellipsoids with different aspect ratios find different equilibrium positions. An ellipsoid of the right dimension and aspect ratio can plug the pore, creating a class of 2D selective membranes.

Colloids in confined liquid crystals: a plot twist in the lock-and-key mechanism

By confining soft materials within tailored boundaries it is possible to design energy landscapes to address and control colloidal dynamics. This provides unique opportunities to create reconfigurable, hierarchically organized structures, a leading challenge in materials science. Example soft matter systems include liquid crystals. For instance, when nematic liquid crystals (NLCs) are confined in a vessel with undulated boundaries, bend and splay distortions can be used to position particles. Here we confine this system in a twist cell. We also study cholesteric liquid crystals, which have an "intrinsic" twist distortion which adds to the ones imposed by the solid boundaries. The cholesteric pitch competes with the other length scales in the system (colloid radius, vessel thickness, wavelength of boundary undulations), enriching the possible configurations. Depending on the pitch-to-radius and pitch-to-thickness ratios the interaction can be attractive or repulsive. By tuning the pitch (i.e. changing the concentration of the chiral dopant), it is possible to selectively promote or inhibit particle trapping at the docking sites.

Tunable colloid trajectories in nematic liquid crystals near wavy walls

The ability to dictate the motion of microscopic objects is an important challenge in fields ranging from materials science to biology. Field-directed assembly drives microparticles along paths defined by energy gradients. Nematic liquid crystals, consisting of rod-like molecules, provide new opportunities in this domain. Deviations of nematic liquid crystal molecules from uniform orientation cost elastic energy, and such deviations can be molded by bounding vessel shape. Here, by placing a wavy wall in a nematic liquid crystal, we impose alternating splay and bend distortions, and define a smoothly varying elastic energy field. A microparticle in this field displays a rich set of behaviors, as this system has multiple stable states, repulsive and attractive loci, and interaction strengths that can be tuned to allow reconfigurable states. Microparticles can transition between defect configurations, move along distinct paths, and select sites for preferred docking. Such tailored landscapes have promise in reconfigurable systems and in microrobotics applications.

Nematic liquid crystals have a rich energy landscape which can define elastic fields to guide colloidal assembly. Here the authors show controllable trapping of colloidal particles by placing them in a system with wavy walls which are exploited to obtain stable, metastable and unstable equilibria.

Colloidal transport within nematic liquid crystals with arrays of obstacles

We have investigated the gravity-driven transport of spherical colloids suspended in the nematic liquid crystal 4-cyano-4'-pentylbiphenyl (5CB) within microfluidic arrays of cylindrical obstacles arranged in a square lattice. Homeotropic anchoring at the surfaces of the obstacles created periodic director-field patterns that strongly influenced the motion of the colloids, whose surfaces had planar anchoring. When the gravitational force was oriented parallel to a principal axis of the lattice, the particles moved along channels between columns of obstacles and displayed pronounced modulations in their velocity. Quantitative analysis indicates that this modulation resulted from a combination of a spatially varying effective drag viscosity and elastic interactions engendered by the periodic director field. The interactions differed qualitatively from a sum of pair-wise interactions between the colloids and isolated obstacles, reflecting the distinct nematic environment created by confinement within the array. As the angle α between the gravitational force and principal axis of the lattice was varied, the velocity did not follow the force but instead locked into a discrete set of directions commensurate with the lattice. The transitions between these directions occurred at values of α that were different from those observed when the spheres were in an isotropic liquid, indicating the ability of the liquid crystal forces to tune the lateral displacement behavior in such devices.

Pattern-induced anchoring transitions in nematic liquid crystals

In this paper we revisit the problem of a nematic liquid crystal in contact with patterned substrates. The substrate is modelled as a periodic array of parallel infinite grooves of well-defined cross-section sculpted on a chemically homogeneous substrate which favours local homeotropic anchoring of the nematic. We consider three cases: a sawtooth, a crenellated and a sinusoidal substrate. We analyse this problem within the modified Frank-Oseen formalism. We argue that, for substrate periodicities much larger than the extrapolation length, the existence of different nematic textures with distinct far-field orientations, as well as the anchoring transitions between them, are associated with the presence of topological defects either on or close to the substrate. For the sawtooth and sinusoidal cases, we observe a homeotropic to planar anchoring transition as the substrate roughness increases. On the other hand, a homeotropic to oblique anchoring transition is observed for crenellated substrates. In this case, the anchoring phase diagram shows a complex dependence on the substrate roughness and substrate anchoring strength.

Key-lock colloids in a nematic liquid crystal

The Landau-de Gennes free energy is used to study theoretically the effective interaction of spherical "key" and anisotropic "lock" colloidal particles. We assume identical anchoring properties of the surfaces of the key and of the lock particles, and we consider planar degenerate and perpendicular anchoring conditions separately. The lock particle is modeled as a spherical particle with a spherical dimple. When such a particle is introduced into a nematic liquid crystal, it orients its dimple at an oblique angle θ_{eq} with respect to the far field director n_{∞}. This angle depends on the depth of the dimple. Minimization results show that the free energy of a pair of key and lock particles exhibits a global minimum for the configuration when the key particle is facing the dimple of the lock colloidal particle. The preferred orientation ϕ_{eq} of the key-lock composite doublet relative to n_{∞} is robust against thermal fluctuations. The preferred orientation θ_{eq}^{(2)} of the dimple particle in the doublet is different from the isolated situation. This is related to the "direct" interaction of defects accompanying the key particle with the edge of the dimple. We propose that this nematic-amplified key-lock interaction can play an important role in self-organization and clustering of mixtures of colloidal particles with dimple colloids present.

Experimental realization of the "lock-and-key" mechanism in liquid crystals

The ability to control the movement and assembly of particles in liquid crystals is not only an important route to design functional materials, but also sheds light on the mechanisms of colloidal interactions. In this study we place micron-sized particles with "Saturn ring" defects near a wall with hills and dales that impose perpendicular (homeotropic) molecular anchoring. The strong splay distortion at the wall interacts with the distortion around the particles in the near field and favors their migration towards the dales via the so-called "lock-and-key" mechanism. We demonstrate experimentally that the lock-and-key mechanism can robustly localize a particle at specific topographical features. We observe the complex trajectories traced by the particles as they dock on the dales, estimate the binding energy, and explore a range of parameters that favor or disfavor the docking event, thus exploiting the capabilities of our experimental system. We extend the study to colloids with homeotropic anchoring but with an associated point defect instead of a Saturn ring and show that they find a different preferred location, i.e. we can place otherwise identical particles at well defined sites according to their topological defect structure. Finally, for deep enough wells, confinement drives topological transitions of Saturn rings to dipoles. This ability to tailor wall geometry to guide colloids to well defined sites within nematic liquid crystals represents an important new tool in directed assembly.

Around the corner: Colloidal assembly and wiring in groovy nematic cells

We study colloids suspended in nematic liquid crystal in grooves with homeotropic anchoring. We observe "eyelashes", topological dipole chains that follow the local, curved director field. These beget wires that connect the groove corners to topographical features on the cell lid to yield oriented, curvilinear colloidal wires spanning the cell, formed in a nonsingular director field. As the groove aspect ratio changes, we find different ground states and corroborate our observation with numerics. Our results rely upon on the scale of topographical features, the sharpness of edges, and the colloid-sourced distortions; all these elements can be exploited to guide the formation of reconfigurable structures in nematics.

Particle selection through topographic templates in nematic colloids

Liquid crystal colloids have been proposed as suitable candidates for responsive photonic crystals. Large scale growth of such colloidal systems is, however, a challenge and recently template-assisted assembly has been proposed to guide the growth of colloidal crystals, with controlled symmetries, in nematic liquid crystals. Known for their long-range anisotropic interactions, these colloidal systems are stabilized typically at the center of the cells due to strong particle-wall repulsion from the confining substrates. This behaviour is dramatically changed in the presence of topographic patterning. Here we propose the use of topographic modulation of surfaces to select and localize particles in nematic colloids. By considering convex and concave deformations of one of the confining surfaces we show that the colloid-flat surface repulsion may be enhanced or switched into an attraction. In particular, we find that when the colloidal particles have the same anchoring conditions as the patterned surfaces, they are strongly attracted to concave dimples, while if they exhibit different anchoring conditions they are pinned at the top of convex protrusions. Although dominated by elastic interactions the first mechanism is reminiscent of the depletion induced attraction or of the key-lock mechanism, while the second is specific to liquid crystal colloids. These long-ranged, highly tunable, surface-colloid interactions contribute to the development of template-assisted assembly of large colloidal crystals, with well defined symmetries, as required for applications.

Towards template-assisted assembly of nematic colloids

Colloidal crystals belong to a new class of materials with unusual properties in which the big challenge is to grow large-scale structures of a given symmetry in a well-controlled and inexpensive way. Recently, template-assisted crystallization was successfully exploited experimentally in the case of colloidal particles dispersed in isotropic fluids. In liquid crystal (LC) colloids, particles are subjected to long-range anisotropic elastic forces originating from the anisotropic deformation of the underlying order parameter. These effective interactions are easily tunable by external electric or magnetic fields, light, temperature, or confinement and, thus, provide additional handles for better control of colloidal assembly. Here we use the coupling between microsculptured bounding surfaces and LC elasticity in order to guide the self-assembly of large-scale colloidal structures. We present explicit numerical calculations of the free energy landscape of colloidal particles in the presence of convex protrusions modeled as squared pyramids comparable to the size of the particles. We show the existence of strong trapping potentials that are able to efficiently localize the colloidal particles and withstand thermal fluctuations. Three-dimensional optical imaging experiments support the theoretical predictions.

Generalized Berreman's model of the elastic surface free energy of a nematic liquid crystal on a sawtoothed substrate

In this paper we present a generalization of Berreman's model for the elastic contribution to the surface free-energy density of a nematic liquid crystal in presence of a sawtooth substrate which favors homeotropic anchoring as a function of the wave number of the surface structure q, the tilt angle α, and the surface anchoring strength w. In addition to the previously reported nonanalytic contribution proportional to -q ln q, due to the nucleation of disclination lines at the wedge bottoms and apexes of the substrate, the next-to-leading contribution is proportional to q for a given substrate roughness, in agreement with Berreman's predictions. We characterize this term, finding that it has two contributions: the deviations of the nematic director field with respect to a reference field corresponding to the isolated disclination lines and their associated core free energies. Comparison with the results obtained from the Landau-de Gennes model shows that our model is quite accurate in the limit wL>1, when strong anchoring conditions are effectively achieved.

Sterically stabilized lock and key colloids: a self-consistent field theory study

A self-consistent field theory study of lock and key type interactions between sterically stabilized colloids in polymer solution is performed. Both the key particle and the lock cavity are assumed to have cylindrical shape and their surfaces are uniformly grafted with polymer chains. The lock-key potential of mean force is computed for various model parameters, such as length of free and grafted chains, lock and key size matching, free chain volume fraction, grafting density, and various enthalpic interactions present in the system. The lock-key interaction is found to be highly tunable, which is important in the rapidly developing field of particle self-assembly.

Manipulation and organization of ferromagnetic nanowires by patterned nematic liquid crystals

We introduce a method to manipulate and organize ferromagnetic nanowires using the elastic forces imposed on nanowires suspended in nematic liquid crystals via patterned variations in the nematic director. As a test case for the technique, we investigate nematic environments consisting of stripes of alternating director orientations formed by lithographically patterned substrates. Nanowires oriented by small external magnetic fields are driven by the liquid crystal to specific locations of the pattern. The observed forces on the nanowires agree with calculations based on nematic elasticity.

Liquid-crystal-mediated force between a cylindrical nanoparticle and substrate

Using classical density functional theory, the structure of a molecular fluid around a cylindrical nanoparticle near a solid substrate is studied. The solvent-mediated force between the nanoparticle and the substrate is calculated in both the nematic and isotropic phases of the solvent. In the nematic phase, the force is short ranged and arises due to interaction between high-density regions near the substrate and nanoparticle. In the isotropic phase, the formation of a nematic bridge between the substrate and nanoparticle gives rise to an attractive force between them. The potential between the nanoparticle and substrate as a function of separation calculated numerically is compared to that calculated from the Derjaguin approximation. In the isotropic phase these are found to be in reasonable agreement at low separations, while the agreement is poorer in the nematic phase.

Nanoparticles in nematic liquid crystals: Interactions with nanochannels

A mesoscale theory for the tensor order parameter Q is used to investigate the structures that arise when spherical nanoparticles are suspended in confined nematic liquid crystals (NLCs). The NLC is "sandwiched" between a wall and a small channel. The potential of mean force is determined between particles and the bottom of the channels or between several particles. Our results suggest that strong NLC-mediated interactions between the particles and the sidewalls of the channels, on the order of hundreds of k(B)T, arise when the colloids are inside the channels. The magnitude of the channel-particle interactions is dictated by a combination of two factors, namely, the type of defect structures that develop when a nanoparticle is inside a channel, and the degree of ordering of the nematic in the region between the colloid and the nanochannel. The channel-particle interactions become stronger as the nanoparticle diameter becomes commensurate with the nanochannel width. Nanochannel geometry also affects the channel-particle interactions. Among the different geometries considered, a cylindrical channel seems to provide the strongest interactions. Our calculations suggest that small variations in geometry, such as removing the sharp edges of the channels, can lead to important reductions in channel-particle interactions. Our calculations for systems of several nanoparticles indicate that linear arrays of colloids with Saturn ring defects, which for some physical conditions are not stable in a bulk system, can be stabilized inside the nanochannels. These results suggest that nanochannels and NLCs could be used to direct the assembly of nanoparticles into ordered arrays with unusual morphologies.