Colloidal liquid crystals in rectangular confinement: theory and experiment

Prismatic Confinement Induces Tunable Orientation in Plasmonic Supercrystals

Throughout history scientists have looked to Nature for inspiration and attempted to replicate intricate complex structures formed by self-assembly. In the context of synthetic supercrystals, achieving such complexity remains a challenge due to the highly symmetric nature of most nanoparticles (NPs). Previous works have shown intricate coupling between the self-assembly of NPs and confinement in templates, such as emulsion droplets (spherical confinement) or tubes (cylindrical confinement). This study focuses on the interplay between anisotropic NP shape and tunable "prismatic confinement" leading to the self-assembly of supercrystals in cavities featuring polygonal cross sections. A multiscale characterization strategy is employed to investigate the orientation and structure of the supercrystals locally and at the ensemble level. Our findings highlight the role of the mold interface in guiding the growth of distinct crystal domains: each side of the mold directs the formation of a monodomain that extends until it encounters another, leading to the creation of grain boundaries. Computer simulations in smaller prismatic cavities were conducted to predict the effect of an increased confinement. Comparison between prismatic and cylindrical confinements shows that flat interfaces are key to orienting the growth of supercrystals. This work shows a method of inducing orientation in plasmonic supercrystals and controlling their textural defects, thus offering insight into the design of functional metasurfaces and hierarchically structured devices.

Self-assembly of cellulose nanocrystals confined to square capillaries

Biological systems exploit restricted degrees of freedom to drive self-assembly of nano- and microarchitectures. Simplified systems, such as colloidal nanoparticles that behave as lyotropic liquid crystalline mesophases in confined geometric spaces, may be used to mimic biological structures. Cellulose nanocrystals (CNCs) are colloidally stable nanoparticles that self-assemble into chiral nematic (ChN) liquid crystalline mesophases. To date, the self-assembly of ChN mesophases of CNCs has been studied under confinement conditions within curved surfaces or under drying conditions that impose curvatures that can be exploited to control ChN ordering; however, their self-assembly has not been investigated in geometries with square cross-sections under static conditions. Here, we show that because of surface anchoring on perpendicular surfaces, the ChN CNC phase is unable to bend with the 90° angle of the square capillary under increasing confinement. Instead, the ChN phase forms radial layers in the shape of concentric squircle shells. With increasing layer distance from the capillary wall, the squircles transition into concentric cylinder shells. In larger capillaries, the radial shell layers appear as a continuous spiral pattern that engulfs fragmented ChN pseudolayers, a defect to accommodate the cylindrical confinement of the mesophase. These results are useful for understanding the fundamentals of self-assembling systems and development of new technologies.

Weak-anchoring effects in a thin pinned ridge of nematic liquid crystal

A theoretical investigation of weak-anchoring effects in a thin two-dimensional pinned static ridge of nematic liquid crystal resting on a flat solid substrate in an atmosphere of passive gas is performed. Specifically, we solve a reduced version of the general system of governing equations recently derived by Cousins et al. [Proc. R. Soc. A 478, 20210849 (2022)10.1098/rspa.2021.0849] valid for a symmetric thin ridge under the one-constant approximation of the Frank-Oseen bulk elastic energy with pinned contact lines to determine the shape of the ridge and the behavior of the director within it. Numerical investigations covering a wide range of parameter values indicate that the energetically preferred solutions can be classified in terms of the Jenkins-Barratt-Barbero-Barberi critical thickness into five qualitatively different types of solution. In particular, the theoretical results suggest that anchoring breaking occurs close to the contact lines. The theoretical predictions are supported by the results of physical experiments for a ridge of the nematic 4^{'}-pentyl-4-biphenylcarbonitrile (5CB). In particular, these experiments show that the homeotropic anchoring at the gas-nematic interface is broken close to the contact lines by the stronger rubbed planar anchoring at the nematic-substrate interface. A comparison between the experimental values of and the theoretical predictions for the effective refractive index of the ridge gives a first estimate of the anchoring strength of an interface between air and 5CB to be (9.80±1.12)×10^{-6}Nm^{-1} at a temperature of (22±1.5)^{∘}C.

Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals

Observing and characterizing the complex ordering phenomena of liquid crystals subjected to external constraints constitutes an ongoing challenge for chemists and physicists alike. To elucidate the delicate balance appearing when the intrinsic positional order of smectic liquid crystals comes into play, we perform Monte-Carlo simulations of rod-like particles in a range of cavities with a cylindrical symmetry. Based on recent insights into the topology of smectic orientational grain boundaries in two dimensions, we analyze the emerging three-dimensional defect structures from the perspective of tetratic symmetry. Using an appropriate three-dimensional tetratic order parameter constructed from the Steinhardt order parameters, we show that those grain boundaries can be interpreted as a pair of tetratic disclination lines that are located on the edges of the nematic domain boundary. Thereby, we shed light on the fine structure of grain boundaries in three-dimensional confined smectics.

Defect patterns of two-dimensional nematic liquid crystals in confinement

A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.e., following the homogeneous boundary condition). A general theory, based on geometric argument, is presented for the relationship between the sum of all winding numbers in the system (the total winding number) and the type of confinement angles and curved segments. The conclusion is validated by comparing the theoretical defect rule with existing nematic textures observed experimentally and theoretically in recent years.

Topology of Orientational Defects in Confined Smectic Liquid Crystals

We propose a general formalism to characterize orientational frustration of smectic liquid crystals in confinement by interpreting the emerging networks of grain boundaries as objects with a topological charge. In a formal idealization, this charge is distributed in pointlike units of quarter-integer magnitude, which we identify with tetratic disclinations located at the end points and nodes. This coexisting nematic and tetratic order is analyzed with the help of extensive Monte Carlo simulations for a broad range of two-dimensional confining geometries as well as colloidal experiments, showing how the observed defect networks can be universally reconstructed from simple building blocks. We further find that the curvature of the confining wall determines the anchoring behavior of grain boundaries, such that the number of nodes in the emerging networks and the location of their end points can be tuned by changing the number and smoothness of corners, respectively.

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution C ± with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

Particle-resolved topological defects of smectic colloidal liquid crystals in extreme confinement

Confined samples of liquid crystals are characterized by a variety of topological defects and can be exposed to external constraints such as extreme confinements with nontrivial topology. Here we explore the intrinsic structure of smectic colloidal layers dictated by the interplay between entropy and an imposed external topology. Considering an annular confinement as a basic example, a plethora of competing states is found with nontrivial defect structures ranging from laminar states to multiple smectic domains and arrays of edge dislocations, which we refer to as Shubnikov states in formal analogy to the characteristic of type-II superconductors. Our particle-resolved results, gained by a combination of real-space microscopy of thermal colloidal rods and fundamental-measure-based density functional theory of hard anisotropic bodies, agree on a quantitative level.

Rodlike molecules in extreme confinement

A unique feature of colloid particles and biopolymers is the molecule's intrinsic rigidity characterized by a molecular-level length scale. Under extreme confinement conditions at cellular scales or in nanodevices, these molecules can display orientational ordering accompanied by severe density depletion. Conventional liquid-crystal theories, such as the Oseen-Frank or Landau-de Gennes theories, cannot capture the essential molecular-level properties: the boundary effects, which extend to a distance of the rigidity length scale, and the drastic variations of the inhomogeneous molecular density. Here we show, based on a simple interpretation of the Onsager model, that rodlike molecules in extreme annular confinement produce unusual liquid-crystal defect structures that are independent phases from the patterns usually seen in a weaker confinement environment.

Nematic ordering of hard rods under strong confinement in a dense array of nanoposts

The effect of confinement on the behavior of liquid crystals is interesting from a fundamental and practical standpoint. In this work, we report Monte Carlo simulations of hard rods in an array of hard nanoposts, where the surface-to-surface separations between nanoposts are comparable to or less than the length of hard rods. This particular system shows promise as a means of generating large-scale organization of the nematic liquid by introducing an entropic external field set by the alignment of nanoposts. The simulations show that nematic ordering of hard rods is enhanced in the nanopost arrays compared with that in bulk, in the sense that the nematic order is significant even at low concentrations at which hard rods remain isotropic in bulk, and the enhancement becomes more significant as the passage width between two nearest nanoposts decreases. An analysis of local distribution of hard-rod orientations at low concentrations with weak nematic ordering reveals that hard rods are preferentially aligned along nanoposts in the narrowing regions between two curved surfaces of nearest nanoposts; hard rods are less ordered in the passages and in the centers of interpost spaces. It is concluded that at low concentrations the confinement in a dense array of nanoposts induces the localized nematic order first in the narrowing regions and, as the concentration further increases, the nematic order spreads over the whole region. The formation of a well-ordered phase at low concentrations of hard rods in a dense array of nanoposts can provide a new route to the low-concentration preparation of nematic liquid crystals that can be used as anisotropic dispersion media.

Tailored morphologies in two-dimensional ferronematic wells

We focus on a dilute uniform suspension of magnetic nanoparticles in a nematic-filled micron-sized shallow well with tangent boundary conditions as a paradigm system with two coupled order parameters. This system exhibits spontaneous magnetization without magnetic fields. We numerically obtain the stable nematic and associated magnetization morphologies, induced purely by the geometry, the boundary conditions, and the coupling between the magnetic nanoparticles and the host nematic medium. Our most striking observations pertain to domain walls in the magnetization profile, whose location can be manipulated by the coupling and material properties, and stable interior and boundary nematic defects, whose location and multiplicity can be tailored by the coupling too. These tailored morphologies are not accessible in uncoupled systems and can be used for multistable systems with singularities and stable interfaces.

Construction of a Pathway Map on a Complicated Energy Landscape

How do we search for the entire family tree of possible intermediate states, without unwanted random guesses, starting from a stationary state on the energy landscape all the way down to energy minima? Here we introduce a general numerical method that constructs the pathway map, which guides our understanding of how a physical system moves on the energy landscape. The method identifies the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau-de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in a square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

Highly confined mixtures of parallel hard squares: A density-functional-theory study

Using the fundamental-measure density-functional theory, we study theoretically the phase behavior of extremely confined mixtures of parallel hard squares in slit geometry. The pore width is chosen such that configurations consisting of two consecutive big squares, or three small squares, in the transverse direction, perpendicular to the walls, are forbidden. We analyze two different mixtures with edge lengths of species selected so as to allow or forbid one big plus one small square to fit into the channel. For the first mixture we obtain first-order transitions between symmetric and asymmetric packings of particles: Small and big squares are preferentially adsorbed at different walls. Asymmetric configurations are shown to lead to more efficient packing at finite pressures. We argue that the stability region of the asymmetric phase in the pressure-composition plane is bounded so that the symmetric phase is stable at low and very high pressure. For the second mixture, we observe strong demixing between phases which are rich in different species. Demixing occurs in the lateral direction, i.e., the dividing interface is perpendicular to the walls, and phases exhibit symmetric density profiles. The possible experimental realization of this behavior (which in practical terms is precluded by jamming) in strictly two-dimensional systems is discussed. Finally, the phase behavior of a mixture with periodic boundary conditions is analyzed and the differences and similarities between the latter and the confined system are discussed. We claim that, although exact calculations exclude the existence of true phase transitions in (1+ε)-dimensional systems, density-functional theory is still successful in describing packing properties of large clusters of particles.

Colloidal Liquid Crystals Confined to Synthetic Tactoids

When a liquid crystal forming particles are confined to a spatial volume with dimensions comparable to that of their own size, they face a complex trade-off between their global tendency to align and the local constraints imposed by the boundary conditions. This interplay may lead to a non-trivial orientational patterns that strongly depend on the geometry of the confining volume. This novel regime of liquid crystalline behavior can be probed with colloidal particles that are macro-aggregates of biomolecules. Here we study director fields of filamentous fd-viruses in quasi-2D lens-shaped chambers that mimic the shape of tactoids, the nematic droplets that form during isotropic-nematic phase separation. By varying the size and aspect ratio of the chambers we force these particles into confinements that vary from circular to extremely spindle-like shapes and observe the director field using fluorescence microscopy. In the resulting phase diagram, next to configurations predicted earlier for 3D tactoids, we find a number of novel configurations. Using Monte Carlo Simulations, we show that these novel states are metastable, yet long-lived. Their multiplicity can be explained by the co-existence of multiple dynamic relaxation pathways leading to the final stable states.

Machine learning topological defects of confined liquid crystals in two dimensions

Supervised machine learning can be used to classify images with spatially correlated physical features. We demonstrate the concept by using the coordinate files generated from an off-lattice computer simulation of rodlike molecules confined in a square box as an example. Because of the geometric frustrations at high number density, the nematic director field develops an inhomogeneous pattern containing various topological defects as the main physical feature. We describe two machine-learning procedures that can be used to effectively capture the correlation between the defect positions and the nematic directors around them and hence classify the topological defects. First is a feedforward neural network, which requires the aid of presorting the off-lattice simulation data in a coarse-grained fashion. Second is a recurrent neural network, which needs no such sorting and can be directly used for finding spatial correlations. The issues of when to presort a simulation data file and how the network structures affect such a decision are addressed.

Synthesis of Colloidal SU-8 Polymer Rods Using Sonication

The bulk synthesis of fluorescent colloidal SU-8 polymer rods with tunable dimensions is described. The colloidal SU-8 rods are prepared by shearing an emulsion of SU-8 polymer droplets and then exposing the resulting non-Brownian rods to ultrasonic waves, which breaks them into colloidal rods with typical lengths of 3.5-10 µm and diameters of 0.4-1 µm. The rods are stable in both aqueous and apolar solvents, and by varying the composition of apolar solvent mixtures both the difference in refractive index and mass density between particles and solvent can be independently controlled. Consequently, these colloidal SU-8 rods can be used in both 3D confocal microscopy and optical trapping experiments while carefully tuning the effect of gravity. This is demonstrated by using confocal microscopy to image the liquid crystalline phases and the isotropic-nematic interface formed by the colloidal SU-8 rods and by optically trapping single rods in water. Finally, the simultaneous confocal imaging and optical manipulation of multiple SU-8 rods in the isotropic phase is shown.

Topological defects in two-dimensional liquid crystals confined by a box

When a spatially uniform system that displays a liquid-crystal ordering on a two-dimensional surface is confined inside a rectangular box, the liquid crystal direction field develops inhomogeneous textures accompanied by topological defects because of the geometric frustrations. We show that the rich variety of nematic textures and defect patterns found in recent experimental and theoretical studies can be classified by the solutions of the rather fundamental, extended Onsager model. This is critically examined based on the determined free energies of different defect states, as functions of a few relevant, dimensionless geometric parameters.

Uniform phases in fluids of hard isosceles triangles: One-component fluid and binary mixtures

We formulate the scaled particle theory for a general mixture of hard isosceles triangles and calculate different phase diagrams for the one-component fluid and for certain binary mixtures. The fluid of hard triangles exhibits a complex phase behavior: (i) the presence of a triatic phase with sixfold symmetry, (ii) the isotropic-uniaxial nematic transition is of first order for certain ranges of aspect ratios, and (iii) the one-component system exhibits nematic-nematic transitions ending in critical points. We found the triatic phase to be stable not only for equilateral triangles but also for triangles of similar aspect ratios. We focus the study of binary mixtures on the case of symmetric mixtures: equal particle areas with aspect ratios (κ_{i}) symmetric with respect to the equilateral one, κ_{1}κ_{2}=3. For these mixtures we found, aside from first-order isotropic-nematic and nematic-nematic transitions (the latter ending in a critical point): (i) a region of triatic phase stability even for mixtures made of particles that do not form this phase at the one-component limit, and (ii) the presence of a Landau point at which two triatic-nematic first-order transitions and a nematic-nematic demixing transition coalesce. This phase behavior is analogous to that of a symmetric three-dimensional mixture of rods and plates.

Ordering transitions of weakly anisotropic hard rods in narrow slitlike pores

The effect of strong confinement on the positional and orientational ordering is examined in a system of hard rectangular rods with length L and diameter D (L>D) using the Parsons-Lee modification of the second virial density-functional theory. The rods are nonmesogenic (L/D<3) and confined between two parallel hard walls, where the width of the pore (H) is chosen in such a way that both planar (particle's long axis parallel to the walls) and homeotropic (particle's long axis perpendicular to the walls) orderings are possible and a maximum of two layers is allowed to form in the pore. In the extreme confinement limit of H≤2D, where only one-layer structures appear, we observe a structural transition from a planar to a homeotropic fluid layer with increasing density, which becomes sharper as L→H. In wider pores (2D<H<3D) planar order with two layers, homeotropic order, and even combined bilayer structures (one layer is homeotropic, while the other is planar) can be stabilized at high densities. Moreover, first-order phase transitions can be seen between different structures. One of them emerges between a monolayer and a bilayer with planar orders at relatively low packing fractions.

Nematic liquid crystal director structures in rectangular regions

We consider a shallow rectangular well of nematic liquid crystal subject to weak anchoring on the sides of the well. By considering weak anchoring instead of infinitely strong anchoring, we are able to analyze nematic equilibria in the well without the need to exclude point defects at the corners, as done in previous work in the area. For relatively weak anchoring, we are able to derive analytic expressions for the director alignment angle in terms of an infinite series of modes, involving roots of a transcendental equation. The analytic forms of the director configuration are then used to calculate critical anchoring strengths at which uniform and distorted director structures exchange stability. We also consider the asymptotic behavior of the director structure and energy for very strong anchoring. We show that in both cases-for the transitions from uniform to distorted states and the limit of infinitely strong anchoring-the approximate analytic expansions agree very well with corresponding numerical calculations of the full model.

The Landau-de Gennes approach revisited: A minimal self-consistent microscopic theory for spatially inhomogeneous nematic liquid crystals

We design a novel microscopic mean-field theory of inhomogeneous nematic liquid crystals formulated entirely in terms of the tensor order parameter field. It combines the virtues of the Landau-de Gennes approach in allowing both the direction and magnitude of the local order to vary, with a self-consistent treatment of the local free-energy valid beyond the small order parameter limit. As a proof of principle, we apply this theory to the well-studied problem of a colloid dispersed in a nematic liquid crystal by including a tunable wall coupling term. For the two-dimensional case, we investigate the organization of the liquid crystal and the position of the point defects as a function of the strength of the coupling constant.

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.

Colloidal liquid crystals in square confinement: isotropic, nematic and smectic phases

We report on the confinement of colloidal liquid crystals in three dimensional chambers with a square footprint. To this end we use colloidal silica rods and exploit their relatively large density difference with respect to the dispersing solvent to study isotropic, nematic and smectic phases confined into a single chamber. Combining laser scanning confocal microscopy and soft-lithography techniques enables us to characterize the configurations down to the single particle level. We will focus on the smectic phase and compare to recent theories and simulations.

Finite particle size drives defect-mediated domain structures in strongly confined colloidal liquid crystals

When liquid crystals are confined to finite volumes, the competition between the surface anchoring imposed by the boundaries and the intrinsic orientational symmetry-breaking of these materials gives rise to a host of intriguing phenomena involving topological defect structures. For synthetic molecular mesogens, like the ones used in liquid-crystal displays, these defect structures are independent of the size of the molecules and well described by continuum theories. In contrast, colloidal systems such as carbon nanotubes and biopolymers have micron-sized lengths, so continuum descriptions are expected to break down under strong confinement conditions. Here, we show, by a combination of computer simulations and experiments with virus particles in tailor-made disk- and annulus-shaped microchambers, that strong confinement of colloidal liquid crystals leads to novel defect-stabilized symmetrical domain structures. These finite-size effects point to a potential for designing optically active microstructures, exploiting the as yet unexplored regime of highly confined liquid crystals.

Liquid crystals confined to micrometre-sized geometries can be well described by a continuum theory, where the size effect of constituent mesogens is negligible. Here, the authors show how the continuum theory breaks down in colloidal liquid crystal, leading to the formation of defect-mediated domains.

Analytical description of 2D magnetic Freedericksz transition in a rectangular cell of a nematic liquid crystal

We study the Freedericksz transition induced by a magnetic field in a rectangular cell filled with a nematic liquid crystal. In the initial state the director of the nematic liquid crystal is uniformly aligned in the cross section plane of the cell with rigid anchoring of the director at cell walls: planar on the top and bottom walls, and homeotropic on the left and right ones. The magnetic field is directed perpendicular to the cell cross section plane. We consider two-dimensional (2D) orientational deformations of the nematic liquid crystal in the rectangular cell and determine the critical value of the Freedericksz transition field above which these orientational deformations occur. The 2D expression for the director alignment profile above the threshold of Freedericksz transition is analytically found and the profile shapes as functions of cell sizes, values of the Frank elastic constants of the nematic liquid crystal and the magnetic field are studied.

Stability of a double twisted structure in spherical cholesteric droplets

We found for the first time the stabilization of a double twisted structure in cholesteric liquid crystals confined to small spherical droplets under weak anchoring conditions. The direct observation of the droplets using a polarized microscope revealed the physical properties of the structure. The experimental results showed that the stability of the double twisted structure is determined by the relationship between the helical pitch length and the droplet size. We theoretically analyzed the structural stability by the calculation of the Frank elastic free energy including the surface elastic term, and succeeded in explaining the experimental results. In this paper, we concluded that the stability of the double twisted structure is determined by the competition between the surface and the bulk elasticity.

Hydrodynamic confinement and capillary alignment of gold nanorods

Controlling the alignment and orientation of nanorods on various surfaces poses major challenges. In this work, we investigate hydrodynamic confinement and capillary alignment of gold nanorod assembly on chemically stripe-patterned substrates. The surface patterns consist of alternating hydrophilic and hydrophobic micrometer wide stripes; a macroscopic wettability gradient enables controlling the dynamics of deposited suspension droplets. We show that drying of residual liquid on the hydrophilic stripes gives rise to spatially localized deposition and alignment of the nanorods. Moreover, a universal relation between the extent of order within the single layers of nanoparticles and the lateral dimension of the deposits is presented and discussed.

Viral nematics in confined geometries

Motivated by recent experiments on the rod-like virus bacteriophage fd, confined to circular and annular domains, we present a theoretical study of structural transitions in these geometries. Using the continuum theory of nematic liquid crystals, we examine the competition between bulk elasticity and surface anchoring, mediated by the formation of topological defects. We show analytically that bulk defects are unstable with respect to defects sitting at the boundary. In the case of an annulus, whose topology does not require the presence of topological defects, we find that nematic textures with boundary defects are stable compared to defect-free configurations when the anchoring is weak. Our simple approach, with no fitting parameters, suggests a possible symmetry breaking mechanism responsible for the formation of one-, two- and three-fold textures under annular confinement.

Surveying the free energy landscapes of continuum models: Application to soft matter systems

A variety of methods are developed for characterising the free energy landscapes of continuum, Landau-type free energy models. Using morphologies of lipid vesicles and a multistable liquid crystal device as examples, I show that the methods allow systematic study of not only the most relevant minimum energy configurations, but also the transition pathways between any two minima, as well as their corresponding energy barriers and transition state configurations. A global view of the free energy landscapes can then be visualized using either a disconnectivity graph or a network representation. Different forms of free energy functionals and boundary conditions can be readily implemented, thus allowing these tools to be utilised for a broad range of problems.

Free energy pathways of a multistable liquid crystal device

The planar bistable device [Tsakonas et al., Appl. Phys. Lett., 2007, 90, 111913] is known to have two distinct classes of stable equilibria: the diagonal and rotated solutions. We model this device within the two-dimensional Landau-de Gennes theory, with a surface potential and without any external fields. We systematically compute a special class of transition pathways, referred to as minimum energy pathways, between the stable equilibria that provide new information about how the equilibria are connected in the Landau-de Gennes free energy landscape. These transition pathways exhibit an intermediate transition state, which is a saddle point of the Landau-de Gennes free energy. We numerically compute the structural details of the transition states, the optimal transition pathways and the free energy barriers between the equilibria, as a function of the surface anchoring strength. For strong anchoring, the transition pathways are mediated by defects whereas we get defect-free transition pathways for moderate and weak anchoring. In the weak anchoring limit, we recover a cusp catastrophe situation for which the rotated state acts as a transition state connecting two different diagonal states.

Defect structures mediate the isotropic-nematic transition in strongly confined liquid crystals

Using Monte Carlo simulations, we study rod-like lyotropic liquid crystals confined to a square slab-like geometry with lateral dimensions comparable to the length of the particles. We observe that this system develops linear defect structures upon entering the planar nematic phase. These defect structures flank a lens-shaped nematic region oriented along a diagonal of the square box. We interpret these structures as a compromise between the 2-fold order of the bulk nematic phase and the 4-fold order imposed by the lateral boundaries. A simple Onsager-type theory that effectively implements these competing tendencies is used to model the phase behavior in the center of the box and shows that the free-energy cost of forming the defect structures strongly offsets the transition-inducing effects of both the transverse and lateral confinement.