Phase transitions and topological defects in discotic liquid crystal droplets with planar anchoring: a Monte Carlo simulation study.

In this work we present the results of Monte Carlo (MC) simulations at the isothermal-isobaric ensemble for a discotic liquid crystal (DLC) droplet whose surface promotes edge-on (planar) anchoring. For a given pressure, we simulate an annealing process that enables observation of phase transitions within the spherical droplet. In particular, we report a first order isotropic-nematic transition as well as a nematic-columnar transition at the center of the droplet. We found the appearance of topological defects consisting of two disclination lines with ends at the surface of the sphere. We also observed that both transitions, isotropic-nematic and nematic-columnar, occur at lower temperatures as compared to the unconfined system.

A General Path-Integral Monte Carlo Method for Exact Simulations of Chemical Reaction Networks.

Chemical Reaction Networks (CRNs) are stochastic many-body systems used to model real-world chemical systems through a differential Master Equation (ME); analytical solutions to these equations are only known for the simplest systems. In this paper, we construct a path-integral inspirited framework for studying CRNs. Under this scheme, the time-evolution of a reaction network can be encoded in a Hamiltonian-like operator. This operator yields a probability distribution which can be sampled, using Monte Carlo Methods, to generate exact numerical simulations of a reaction network. We recover the grand probability function used in the Gillespie Algorithm as an approximation to our probability distribution, which motivates the addition of a leapfrog correction step. To assess the utility of our method in forecasting real-world phenomena, and to contrast it with the Gillespie Algorithm, we simulated a COVID-19 epidemiological model using parameters from the United States for the Original Strain and the Alpha, Delta and Omicron Variants. By comparing the results of these simulations with official data, we found that our model closely agrees with the measured population dynamics, and given the generality of this framework it can also be applied to study the spread dynamics of other contagious diseases.

Collective dynamics and self-motions in the van der Waals liquid tetrahydrofuran from meso- to inter-molecular scales disentangled by neutron spectroscopy with polarization analysis.

By using time-of-flight neutron spectroscopy with polarization analysis, we have separated coherent and incoherent contributions to the scattering of deuterated tetrahydrofuran in a wide scattering vector (Q)-range from meso- to inter-molecular length scales. The results are compared with those recently reported for water to address the influence of the nature of inter-molecular interactions (van der Waals vs hydrogen bond) on the dynamics. The phenomenology found is qualitatively similar in both systems. Both collective and self-scattering functions are satisfactorily described in terms of a convolution model that considers vibrations, diffusion, and a Q-independent mode. We observe a crossover in the structural relaxation from being dominated by the Q-independent mode at the mesoscale to being dominated by diffusion at inter-molecular length scales. The characteristic time of the Q-independent mode is the same for collective and self-motions and, contrary to water, faster and with a lower activation energy (≈1.4 Kcal/mol) than the structural relaxation time at inter-molecular length scales. This follows the macroscopic viscosity behavior. The collective diffusive time is well described by the de Gennes narrowing relation proposed for simple monoatomic liquids in a wide Q-range entering the intermediate length scales, in contraposition to the case of water.

Director Distortion and Phase Modulation in Deformable Nematic and Smectic Liquid Crystal Spheroids.

The growing interest in integrating liquid crystals (LCs) into flexible and miniaturized technologies brings about the need to understand the interplay between spatially curved geometry, surface anchoring, and the order associated with these materials. Here, we integrate experimental methods and computational simulations to explore the competition between surface-induced orientation and the effects of deformable curved boundaries in uniaxially and biaxially stretched nematic and smectic microdroplets. We find that the director field of the nematic LCs upon uniaxial strain reorients and forms a larger twisted defect ring to adjust to the new deformed geometry of the stretched droplet. Upon biaxial extension, the director field initially twists in the now oblate geometry and subsequently transitions into a uniform vertical orientation at high strains. In smectic microdroplets, on the other hand, LC alignment transforms from a radial smectic layering to a quasi-flat layering in a compromise between interfacial and dilatation forces. Upon removing the mechanical strain, the smectic LC recovers its initial radial configuration; however, the oblate geometry traps the nematic LC in the metastable vertical state. These findings offer a basis for the rational design of LC-based flexible devices, including wearable sensors, flexible displays, and smart windows.

Mesoscale martensitic transformation in single crystals of topological defects.

Liquid-crystal blue phases (BPs) are highly ordered at two levels. Molecules exhibit orientational order at nanometer length scales, while chirality leads to ordered arrays of double-twisted cylinders over micrometer scales. Past studies of polycrystalline BPs were challenged by the existence of grain boundaries between randomly oriented crystalline nanodomains. Here, the nucleation of BPs is controlled with precision by relying on chemically nanopatterned surfaces, leading to macroscopic single-crystal BP specimens where the dynamics of mesocrystal formation can be directly observed. Theory and experiments show that transitions between two BPs having a different network structure proceed through local reorganization of the crystalline array, without diffusion of the double-twisted cylinders. In solid crystals, martensitic transformations between crystal structures involve the concerted motion of a few atoms, without diffusion. The transformation between BPs, where crystal features arise in the submicron regime, is found to be martensitic in nature when one considers the collective behavior of the double-twist cylinders. Single-crystal BPs are shown to offer fertile grounds for the study of directed crystal nucleation and the controlled growth of soft matter.

Blue-phase liquid crystal droplets.

Blue phases of liquid crystals represent unique ordered states of matter in which arrays of defects are organized into striking patterns. Most studies of blue phases to date have focused on bulk properties. In this work, we present a systematic study of blue phases confined into spherical droplets. It is found that, in addition to the so-called blue phases I and II, several new morphologies arise under confinement, with a complexity that increases with the chirality of the medium and with a nature that can be altered by surface anchoring. Through a combination of simulations and experiments, it is also found that one can control the wavelength at which blue-phase droplets absorb light by manipulating either their size or the strength of the anchoring, thereby providing a liquid-state analog of nanoparticles, where dimensions are used to control absorbance or emission. The results presented in this work also suggest that there are conditions where confinement increases the range of stability of blue phases, thereby providing intriguing prospects for applications.

Sharp Morphological Transitions from Nanoscale Mixed-Anchoring Patterns in Confined Nematic Liquid Crystals.

Liquid crystals are known to be particularly sensitive to orientational cues provided at surfaces or interfaces. In this work, we explore theoretically, computationally, and experimentally the behavior of liquid crystals on isolated nanoscale patterns with controlled anchoring characteristics at small length scales. The orientation of the liquid crystal is controlled through the use of chemically patterned polymer brushes that are tethered to a surface. This system can be engineered with remarkable precision, and the central question addressed here is whether a characteristic length scale exists at which information encoded on a surface is no longer registered by a liquid crystal. To do so, we adopt a tensorial description of the free energy of the hybrid liquid-crystal-surface system, and we investigate its morphology in a systematic manner. For long and narrow surface stripes, it is found that the liquid crystal follows the instructions provided by the pattern down to 100 nm widths. This is accomplished through the creation of line defects that travel along the sides of the stripes. We show that a "sharp" morphological transition occurs from a uniform undistorted alignment to a dual uniform/splay-bend morphology. The theoretical and numerical predictions advanced here are confirmed by experimental observations. Our combined analysis suggests that nanoscale patterns can be used to manipulate the orientation of liquid crystals at a fraction of the energetic cost that is involved in traditional liquid crystal-based devices. The insights presented in this work have the potential to provide a new fabrication platform to assemble low power bistable devices, which could be reconfigured upon application of small external fields.

Strain-induced alignment and phase behavior of blue phase liquid crystals confined to thin films.

We report on the influence of surface confinement on the phase behavior and strain-induced alignment of thin films of blue phase liquid crystals (BPs). Confining surfaces comprised of bare glass, dimethyloctadecyl [3-(trimethoxysilyl)propyl] ammonium chloride (DMOAP)-functionalized glass, or polyvinyl alcohol (PVA)-coated glass were used with or without mechanically rubbing to influence the azimuthal anchoring of the BPs. These experiments reveal that confinement can change the phase behavior of the BP films. For example, in experiments performed with rubbed-PVA surfaces, we measured the elastic strain of the BPs to change the isotropic-BPII phase boundary, suppressing formation of BPII for film thicknesses incommensurate with the BPII lattice. In addition, we observed strain-induced alignment of the BPs to exhibit a complex dependence on both the surface chemistry and azimuthal alignment of the BPs. For example, when using bare glass surfaces causing azimuthally degenerate and planar anchoring, BPI oriented with (110) planes of the unit cell parallel to the contacting surfaces for thicknesses below 3 µm but transitioned to an orientation with (200) planes aligned parallel to the contacting surfaces for thicknesses above 4 µm. In contrast, BPI aligned with (110) planes parallel to confining surfaces for all other thicknesses and surface treatments, including bare glass with uniform azimuthal alignment. Complementary simulations based on minimization of the total free energy (Landau-de Gennes formalism) confirmed a thickness-dependent reorientation due to strain of BPI unit cells within a window of surface anchoring energies and in the absence of uniform azimuthal alignment. In contrast to BPI, BPII did not exhibit thickness-dependent orientations but did exhibit orientations that were dependent on the surface chemistry, a result that was also captured in simulations by varying the anchoring energies. Overall, the results in this paper reveal that the orientations assumed by BPs in thin films reflect a complex interplay of surface interactions and elastic energies associated with strain of the BP lattice. The results also provide new principles and methods to control the structure and properties of BP thin films, which may find use in BP-templated material synthesis, and BP-based optical and electronic devices.

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.

Patterned surface anchoring of nematic droplets at miscible liquid-liquid interfaces.

We report on the internal configurations of droplets of nematic liquid crystals (LCs; 10-50 µm-in-diameter; comprised of 4-cyano-4'-pentylbiphenyl and 4-(3-acryloyloxypropyloxy)benzoic acid 2-methyl-1,4-phenylene ester) sedimented from aqueous solutions of sodium dodecyl sulfate (SDS) onto interfaces formed with pure glycerol. We observed a family of internal LC droplet configurations and topological defects consistent with a remarkably abrupt transition from homeotropic (perpendicular) to tangential anchoring on the surface of the LC droplets in the interfacial environment. Calculations of the interdiffusion of water and glycerol at the aqueous-glycerol interface revealed the thickness of the diffuse interfacial region of the two miscible liquids to be small (0.2-0.5 µm) compared to the diameters of the LC droplets on the experimental time-scale (15-120 minutes), leading us to hypothesize that the patterned surface anchoring was induced by gradients in concentration of SDS and glycerol across the diameter of the LC droplets in the interfacial region. This hypothesis received additional support from experiments in which the time of sedimentation of the LC droplets onto the interface was systematically increased and the droplets were photo-polymerized to preserve their configurations: the configurations of the LC droplets were consistent with a time-dependent decrease in the fraction of the surface area of each droplet exhibiting homeotropic anchoring. Specifically, LC droplets with <10% surface area with tangential anchoring exhibited a bulk point defect within the LC droplet, whereas droplets with >10% surface area with tangential anchoring exhibited a boojum defect within the tangential region and a disclination loop separated the regions with tangential and homeotropic anchoring. The topological charge of these LC droplet configurations was found to be consistent with the geometrical theorems of Poincaré and Gauss and also well-described by computer simulations performed by minimization of a Landau-de Gennes free energy. Additional experimental observations (e.g., formation of "Janus-like" particles with one hemisphere exhibiting tangential anchoring and the other perpendicular anchoring) and simulations (e.g., a size-dependent set of LC droplet configurations with <10% surface area exhibiting tangential anchoring) support our general conclusion that placement of LC droplets into miscible liquid-liquid interfacial environments with compositional gradients can lead to a rich set of LC droplet configurations with symmetries and optical characteristics that are not encountered in LC droplet systems in homogeneous, bulk environments. Our results also reveal that translocation of LC droplets across liquid-liquid interfaces can define new transition pathways that connect distinct configurations of LC droplets.

Mesoscale structure of chiral nematic shells.

There is considerable interest in understanding and controlling topological defects in nematic liquid crystals (LCs). Confinement, in the form of droplets, has been particularly effective in that regard. Here, we employ a Landau-de Gennes formalism to explore the geometrical frustration of nematic order in shell geometries, and focus on chiral materials. By varying the chirality and thickness in uniform shells, we construct a phase diagram that includes tetravalent structures, bipolar structures (BS), bent structures and radial spherical structures (RSS). It is found that, in uniform shells, the BS-to-RSS structural transition, in response to both chirality and shell geometry, is accompanied by an abrupt change of defect positions, implying a potential use for chiral nematic shells as sensors. Moreover, we investigate thickness heterogeneity in shells and demonstrate that non-chiral and chiral nematic shells exhibit distinct equilibrium positions of their inner core that are governed by shell chirality c.

Positioning colloids at the surfaces of cholesteric liquid crystal droplets.

We report on the internal configurations of aqueous dispersions of droplets of cholesteric liquid crystals (LCs; 5-50 µm-in-diameter; comprised of 4-cyano-4'-pentylbiphenyl and 4-(1-methylheptyloxycarbonyl)phenyl-4-hexyloxybenzoate) and their influence on the positioning of surface-adsorbed colloids (0.2 or 1 µm-in-diameter polystyrene (PS)). When N = 2D/P was less than 4, where D is the droplet diameter and P is the cholesteric pitch, the droplets adopted a twisted bipolar structure (TBS) and colloids were observed to assume positions at either the poles or equator of the droplets. A statistical analysis of the distribution of locations of the colloids revealed a potential well of depth 2.7 kBT near the equator, a conclusion that was supported by computer simulations performed via the minimization of the Landau-de Gennes free energy (well depth of 7 kBT from simulation). In contrast, for N > 4, a majority of the droplets exhibited a radial spherical structure (RSS) characterized by a pair of closely spaced surface defects (angle of separation with respect to the center of the droplet θ < 5°) connected by a disclination winding to/from the droplet center, which led to the positioning of pairs of colloids with well-defined spacing at these surface defects. The separation of the pairs of surface-adsorbed colloids was colloid size-dependent, ranging from 1.11 ± 0.04 µm for 1 µm-in-diameter colloids to 1.7 ± 0.2 µm for 200 nm-in-diameter colloids. We also observed long-lived metastable configurations in which the two surface point defects were separated by much larger distances (corresponding to populations with angles of θ = 20 ± 10° and 85 ± 10° with respect to the center), and observed these pairs of defects to also position pairs of colloids. A third configuration, the diametrical spherical structure (DSS) was also observed. Consistent with the predictions of computer simulations, we found experimentally that the DSS is indeed composed of disconnected defect rings positioned along the diameter of the droplet. Overall, these results reveal that the rich palette of defects exhibited by confined cholesteric LC systems (equilibrium and metastable) provide the basis of a versatile class of templates that enable the surface positioning of colloids in ways that are not possible with achiral LC droplets.

Directed self-assembly of nematic liquid crystals on chemically patterned surfaces: morphological states and transitions.

The morphology and through-film optical properties of nematic liquid crystals (LCs) confined between two surfaces may be engineered to create switches that respond to external electric fields, thereby enabling applications in optoelectronics that require fast responses and low power. Interfacial properties between the confining surfaces and the LC play a central role in device design and performance. Here we investigate the morphology of LCs confined in hybrid cells with a top surface that exhibits uniform homeotropic anchoring and a bottom surface that is chemically patterned with sub-micron and micron- wide planar anchoring stripes in a background of homeotropic anchoring. In a departure from past work, we first investigate isolated stripes, as opposed to dense periodic arrays of stripes, thereby allowing for an in-depth interpretation of the effects of patterning on LC morphology. We observe three LC morphologies and sharp transitions between them as a function of stripe width in the submicron and micron regimes. Numerical simulations and theory help explain the roles of anchoring energy, elastic deformation, entropy, pattern geometry, and coherence length of the LC in the experimentally observed behavior. The knowledge and models developed from an analysis of results generated on isolated features are then used to design dense patterned substrates for high-contrast and efficient orientational switching of LCs in response to applied fields.

Functional soft materials from blue phase liquid crystals.

Blue phase (BP) liquid crystals are chiral fluids wherein millions of molecules self-assemble into cubic lattices that are on the order of hundred nanometers. As the unit cell sizes of BPs are comparable to the wavelength of light, they exhibit selective Bragg reflections in the visible. The exploitation of the photonic properties of BPs for technological applications is made possible through photopolymerization, a process that renders mechanical robustness and thermal stability. We review here the preparation and characterization of stimuli-responsive, polymeric photonic crystals based on BPs. We highlight recent studies that demonstrate the promise that polymerized BP photonic crystals hold for colorimetric sensing and dynamic light control. We review using Landau-de Gennes simulations for predicting the self-assembly of BPs and the potential for using theory to guide experimental design. Finally, opportunities for using BPs to synthesize new soft materials, such as highly structured polymer meshes, are discussed.

Photonic features of blue phase liquid crystals under curved confinement.

Blue phase (BP) liquid crystals represent a fascinating state of soft matter that showcases unique optical and electro-optical properties. Existing between chiral nematic and isotropic phases, BPs are characterized by a three-dimensional cubic lattice structure resulting in selective Bragg reflections of light and consequent vivid structural colors. However, the practical realization of these material systems is hampered by their narrow thermal stability and multi-domain crystalline nature. This feature article provides an overview of the efforts devoted to stabilizing these phases and creating monodomain structures. In particular, it delves into the complex relationship between geometrical confinement, induced curvature, and the structural stability and photonic features of BPs. Understanding the interaction of curved confinement and structural stability of BPs proves crucially important for the integration of these materials into flexible and miniaturized devices. By shedding light on these critical aspects, this feature review aims to highlight the significance of understanding the coupling effects of physical and mechanical forces on the structural stability of these systems, which can pave the way for the development of efficient and practical devices based on BP liquid crystals.

Chiral Liquid Crystal Microdroplets for Sensing Phospholipid Amphiphiles.

Designing simple, sensitive, fast, and inexpensive readout devices to detect biological molecules and biomarkers is crucial for early diagnosis and treatments. Here, we have studied the interaction of the chiral liquid crystal (CLC) and biomolecules at the liquid crystal (LC)-droplet interface. CLC droplets with high and low chirality were prepared using a microfluidic device. We explored the reconfiguration of the CLC molecules confined in droplets in the presence of 1,2-diauroyl-sn-glycero3-phosphatidylcholine (DLPC) phospholipid. Cross-polarized optical microscopy and spectrometry techniques were employed to monitor the effect of droplet size and DLPC concentration on the structural reorganization of the CLC molecules. Our results showed that in the presence of DLPC, the chiral LC droplets transition from planar to homeotropic ordering through a multistage molecular reorientation. However, this reconfiguration process in the low-chirality droplets happened three times faster than in high-chirality ones. Applying spectrometry and image analysis, we found that the change in the chiral droplets' Bragg reflection can be correlated with the CLC-DLPC interactions.

Structural properties and ring defect formation in discotic liquid crystal nanodroplets.

In this work, we performedNpTMonte Carlo simulations of a Gay-Berne discotic liquid crystal confined in a spherical droplet under face-on anchoring and fixed pressure. We find that, in contrast to the unbounded system, a plot of the order parameter as function of temperature does not show a clear evidence of a first-order isotropic-nematic transition. We also find that the impossibility of simultaneously satisfy the uniform director field requirement of a nematic phase with the radial boundary conditions, results in the appearance of a ring disclination line as a stress release mechanism in the interior of the droplet. Under further cooling, a columnar phase appears at the center of the droplet.

Elastic Instability of Cubic Blue Phase Nano Crystals in Curved Shells.

Many crystallization processes, including biomineralization and ice-freezing, occur in small and curved volumes, where surface curvature can strain the crystal, leading to unusual configurations and defect formation. The role of curvature on crystallization, however, remains poorly understood. Here, we study the crystallization of blue phase (BP) liquid crystals under curved confinement, which provides insights into the mechanism by which BPs reconfigure their three-dimensional lattice structure to adapt to curvature. BPs are a three-dimensional assembly of high-chirality liquid crystal molecules arranged into body-centered (BPI) or simple cubic (BPII) symmetries. BPs with submicrometer cubic-crystalline lattices exhibit tunable Bragg reflection and submillisecond response time to external stimuli such as an electric field, making them attractive for advanced photonic materials. In this work, we have systematically studied BPs confined in spherical shells with well-defined curvature and boundary conditions. The optical behavior of shells has also been examined at room temperature, where the cholesteric structure forms. In the cholesteric phase, perpendicular anchoring generates focal conic domains on the shell's surface, which transition into stripe patterns as the degree of curvature increases. Our results demonstrate that both higher degrees of curvature and strong spatial confinement destabilize BPI and reconfigure that phase to adopt the structure and optical features of BPII. We also show that the coupling of curvature and confinement nucleates skyrmions at greater thicknesses than those observed for a flat geometry. These findings are particularly important for integrating BPs into miniaturized and curved/flexible devices, including flexible displays, wearable sensors, and smart fabrics.

Control of Monodomain Polymer-Stabilized Cuboidal Nanocrystals of Chiral Nematics by Confinement.

Liquid crystals are important components of optical technologies. Cuboidal crystals consisting of chiral liquid crystals-the so-called blue phases (BPs), are of particular interest due to their crystalline structures and fast response times, but it is critical that control be gained over their phase behavior as well as the underlying dislocations and grain boundaries that arise in such systems. Blue phases exhibit cubic crystalline symmetries with lattice parameters in the 100 nm range and a network of disclination lines that can be polymerized to widen the range of temperatures over which they occur. Here, we introduce the concept of strain-controlled polymerization of BPs under confinement, which enables formation of strain-correlated stabilized morphologies that, under some circumstances, can adopt perfect single-crystal monodomain structures and undergo reversible crystal-to-crystal transformations, even if their disclination lines are polymerized. We have used super-resolution laser confocal microscopy to reveal the periodic structure and the lattice planes of the strain and polymerization stabilized BPs in 3D real space. Our experimental observations are supported and interpreted by relying on theory and computational simulations in terms of a free energy functional for a tensorial order parameter. Simulations are used to determine the orientation of the lattice planes unambiguously. The findings presented here offer opportunities for engineering optical devices based on single-crystal, polymer-stabilized BPs whose inherent liquid nature, fast dynamics, and long-range crystalline order can be fully exploited.

Soft crystal martensites: An in situ resonant soft x-ray scattering study of a liquid crystal martensitic transformation.

Liquid crystal blue phases (BPs) are three-dimensional soft crystals with unit cell sizes orders of magnitude larger than those of classic, atomic crystals. The directed self-assembly of BPs on chemically patterned surfaces uniquely enables detailed in situ resonant soft x-ray scattering measurements of martensitic phase transformations in these systems. The formation of twin lamellae is explicitly identified during the BPII-to-BPI transformation, further corroborating the martensitic nature of this transformation and broadening the analogy between soft and atomic crystal diffusionless phase transformations to include their strain-release mechanisms.