Growth of tactoidal droplets during the first-order isotropic to nematic phase transition of F-actin

Single-Step Control of Liquid-Liquid Crystalline Phase Separation by Depletion Gradients

Fine-tuning nucleation and growth of colloidal liquid crystalline (LC) droplets, also known as tactoids, is highly desirable in both fundamental science and technological applications. However, the tactoid structure results from the trade-off between thermodynamics and nonequilibrium kinetics effects, and controlling liquid-liquid crystalline phase separation (LLCPS) in these systems is still a work in progress. Here, a single-step strategy is introduced to obtain a rich palette of morphologies for tactoids formed via nucleation and growth within an initially isotropic phase exposed to a gradient of depletants. The simultaneous appearance is shown of rich LC structures along the depleting potential gradient, where the position of each LC structure is correlated with the magnitude of the depleting potential. Changing the size (nanoparticles) or the nature (polymers) of the depleting agent provides additional, precise control over the resulting LC structures through a size-selective mechanism, where the depletant may be found both within and outside the LC droplets. The use of depletion gradients from depletants of varying sizes and nature offers a powerful toolbox for manipulation, templating, imaging, and understanding heterogeneous colloidal LC structures.

Curvature and confinement effects on chiral liquid crystal morphologies

Chiral liquid crystals (ChLCs) exhibit an inherent twist that originates at the molecular scale and can extend over multiple length scales when unconstrained. Under confinement, the twist is thwarted, leading to formation of defects in the molecular order that offer distinct optical responses and opportunities for colloidal driven assembly. Past studies have explored spheroidal confinement down to the nanoscopic regime, where curved boundaries produce surface defects to accommodate topological constraints and restrict the propagation of cuboidal defect networks. Similarly, strict confinement in channels and shells has been shown to give rise to escaped configurations and skyrmions. However, little is known about the role of extrinsic curvature in the development of cholesteric textures and Blue Phases (BP). In this paper, we examine the palette of morphologies that arises when ChLCs are confined in toroidal and cylindrical cavities. The equilibrium morphologies are obtained following an annealing strategy of a Landau-de Gennes free energy functional. Three dimensionless groups are identified to build phase diagrams: the natural twist, the ratio of elastic energies, and the circumscription of a BP cell. Curvature is shown to introduce helical features that are first observed as a Double Twist, and progress to Chiral Ribbons and, ultimately, Helical BP and BP. Chiral ribbons are examined as useful candidates for driven assembly given their tunability and robustness.

Evaporation-Driven Liquid-Liquid Crystalline Phase Separation in Droplets of Anisotropic Colloids

Drying a colloidal droplet involves complex physics that is often accompanied by evaporation-induced concentration gradients inside of the droplet, offering a platform for fundamental and technological opportunities, including self-assembly, thin film deposition, microfabrication, and DNA stretching. Here, we investigate the drying, liquid crystalline structures, and deposit patterns of colloidal liquid crystalline droplets undergoing liquid-liquid crystalline phase separation (LLCPS) during evaporation. We show that evaporation-induced progressive up-concentration inside the drying droplets makes it possible to cross, at different speeds, various thermodynamic stability states in solutions of amyloid fibril rigid filamentous colloids, thus allowing access to both metastable states, where phase separation occurs via nucleation and growth, as well as to unstable states, where phase separation occurs via the more elusive spinodal decomposition, leading to the formation of liquid crystalline microdroplets (or tactoids) of different shapes. We present the tactoids "phase diagram" as a function of the position within the droplet and elucidate their hydrodynamics. Furthermore, we demonstrate that the presence of the amyloid fibrils not only does not enhance the pinning behavior during droplet evaporation but also slightly suppresses it, thus minimizing the coffee-ring effect. We observed that microsize domains with cholesteric structure emerge in the drying droplet close to the droplet's initial edge, yet such domains are not connected to form a uniform cholesteric dried film. Finally, we demonstrate that a fully cholesteric dried layer can be generated from the drying droplets by regulating the kinetics of the evaporation process.

Disentangling kinetics from thermodynamics in heterogeneous colloidal systems

In Nucleation and Growth, the process by which most heterogeneous systems form, thermodynamics sets the asymptotic boundaries toward which the system must evolve, while kinetics tries to cope with it by imposing the transport rates. In all heterogeneous colloidal systems observed in nature, composition, shape, structure and physical properties result from the trade-off between thermodynamics and kinetics. Here we show, by carefully selecting colloidal systems and controlling phase separation in microfluidic devices, that it becomes possible to disentangle kinetics effects from thermodynamics. Using amyloids and nanocellulose filamentous colloids, we demonstrate that decoupling kinetics from thermodynamics in the phase separation process unveils new physical phenomena, such as orders of magnitude shorter timescales, a wider phase diagram, and structures that are not observable via conventional liquid-liquid phase separation. Our approach enables on-demand fabrication of multicomponent heterogeneous liquid crystals, enhancing their potential, and introducing original fundamental and technological directions in multicomponent structured fluids.

Cellulose Nanocrystal Aqueous Colloidal Suspensions: Evidence of Density Inversion at the Isotropic-Liquid Crystal Phase Transition

The colloidal suspensions of aqueous cellulose nanocrystals (CNCs) are known to form liquid crystalline (LC) systems above certain critical concentrations. From an isotropic phase, tactoid formation, growth, and sedimentation have been determined as the genesis of a high-density cholesteric phase, which, after drying, originates solid iridescent films. Herein, the coexistence of a liquid crystal upper phase and an isotropic bottom phase in CNC aqueous suspensions at the isotropic-nematic phase separation is reported. Furthermore, isotropic spindle-like domains are observed in the low-density LC phase and high-density LC phases are also prepared. The CNCs isolated from the low- and high-density LC phases are found to have similar average lengths, diameters, and surface charges. The existence of an LC low-density phase is explained by the presence of air dissolved in the water present within the CNCs. The air dissolves out when the water solidifies into ice and remains within the CNCs. The self-adjustment of the cellulose chain conformation enables the entrapment of air within the CNCs and CNC buoyancy in aqueous suspensions.

Shape and structural relaxation of colloidal tactoids

Facile geometric-structural response of liquid crystalline colloids to external fields enables many technological advances. However, the relaxation mechanisms for liquid crystalline colloids under mobile boundaries remain still unexplored. Here, by combining experiments, numerical simulations and theory, we describe the shape and structural relaxation of colloidal liquid crystalline micro-droplets, called tactoids, where amyloid fibrils and cellulose nanocrystals are used as model systems. We show that tactoids shape relaxation bears a universal single exponential decay signature and derive an analytic expression to predict this out of equilibrium process, which is governed by liquid crystalline anisotropic and isotropic contributions. The tactoids structural relaxation shows fundamentally different paths, with first- and second-order exponential decays, depending on the existence of splay/bend/twist orientation structures in the ground state. Our findings offer a comprehensive understanding on dynamic confinement effects in liquid crystalline colloidal systems and may set unexplored directions in the development of novel responsive materials.

Tactoids, consisting of micro-confined liquid crystalline colloids with self-selected shape, bear both fundamental and technological significance. The authors show that the shape relaxation of tactoids follows an exponential decay and develop a model to predict this out-of-the-equilibrium process.

Structure of nematic tactoids of hard rods

We study by means of Monte Carlo simulations the internal structure of nematic droplets or tactoids formed by hard, rod-like particles in a gas of spherical ghost particles that act as depletion agents for the rods. We find that the shape and internal structure of tactoids are strongly affected by the size of the droplets. The monotonically increasing degree of nematic order with increasing particle density that characterizes the bulk nematic phase is locally violated and more so the smaller the tactoid. We also investigate the impact of an external quadrupolar alignment field on tactoids and find that this tends to make the director field more uniform, but not to very significantly increase the tactoid’s aspect ratio. This agrees with recent theoretical predictions yet is at variance with experimental observations and dynamical simulations. We explain this discrepancy in terms of competing relaxation times.

Shape bistability in 2D chromonic droplets

An extensive experimental study of the shapes of two-dimensional bipolar droplets of the chromonic nematic phase of disodium cromoglycate (DSCG) sandwiched between glass plates, by Kimet alwas published in (2013J. Phys.: Condens. Matter25404202). The paper includes a mathematical model of this system. We have extended this study by further theoretical modelling. Our results are in good, quantitative agreement with the experimental data. The model has produced what promises to be a more accurate estimate for the isotropic surface tension at the nematic/isotropic solution interface-and predicts a regime of shape bistability (which has not yet been observed) for larger droplets, where tactoids (pointed, zeppelin-shaped droplets) and smooth-edged discoids can coexist in equilibrium. The general method presented in this paper is also applied to the tactoids formed by F-actin filaments in solution, for which an estimate is given for the value of the isotropic surface tension at the nematic/isotropic interface.

Effect of electric fields on the director field and shape of nematic tactoids

Tactoids are spindle-shaped droplets of a uniaxial nematic phase suspended in the coexisting isotropic phase. They are found in dispersions of a wide variety of elongated colloidal particles, including actin, fd virus, carbon nanotubes, vanadium peroxide, and chitin nanocrystals. Recent experiments on tactoids of chitin nanocrystals in water show that electric fields can very strongly elongate tactoids even though the dielectric properties of the coexisting isotropic and nematic phases differ only subtly. We develop a model for partially bipolar tactoids, where the degree of bipolarness of the director field is free to adjust to optimize the sum of the elastic, surface, and Coulomb energies of the system. By means of a combination of a scaling analysis and a numerical study, we investigate the elongation and director field's behavior of the tactoids as a function of their size, the strength of the electric field, the surface tension, anchoring strength, the elastic constants, and the electric susceptibility anisotropy. We find that tactoids cannot elongate significantly due to an external electric field, unless the director field is bipolar or quasibipolar and somehow frozen in the field-free configuration. Presuming that this is the case, we find reasonable agreement with experimental data.

Liquid-liquid crystalline phase separation in biological filamentous colloids: nucleation, growth and order-order transitions of cholesteric tactoids

The process of liquid-liquid crystalline phase separation (LLCPS) in filamentous colloids is at the very core of multiple biological, physical and technological processes of broad significance. However, the complete theoretical understanding of the process is still missing. LLCPS involves the nucleation, growth and up-concentration of anisotropic droplets from a continuous isotropic phase, until a state of equilibrium is reached. Herein, by combining the thermodynamic extremum principle with the Onsager theory, we describe the nucleation and growth of liquid crystalline droplets, and the evolution of their size and concentration during phase separation, eventually leading to a multitude of order-order phase transitions. Furthermore, a decreasing pitch behaviour can be predicted using this combined theory during tactoid growth, already observed experimentally but not yet explained by present theories. The results of this study are compared with the experimental data of cholesteric pitch, observed in three different systems of biological chiral liquid crystals. These findings give an important framework for predicting the formation, growth and phase behaviour of biological filamentous colloids undergoing LLCPS, advancing our understanding of liquid-liquid phase separation and self-assembly mechanisms in biological systems, and provide a valuable rationale for developing nanomaterials and applications in nanotechnology.

Crowder and surface effects on self-organization of microtubules

Microtubules are an essential physical building block of cellular systems. They are organized using specific crosslinkers, motors, and influencers of nucleation and growth. With the addition of antiparallel crosslinkers, microtubule self-organization patterns go through a transition from fanlike structures to homogeneous tactoid condensates in vitro. Tactoids are reminiscent of biological mitotic spindles, the cell division machinery. To create these organizations, we previously used polymer crowding agents. Here we study how altering the properties of the crowders, such as type, size, and molecular weight, affects microtubule organization. Comparing simulations with experiments, we observe a scaling law associated with the fanlike patterns in the absence of crosslinkers. Tactoids formed in the presence of crosslinkers show variable length, depending on the crowders. We correlate the subtle differences to filament contour length changes, affected by nucleation and growth rate changes induced by the polymers in solution. Using quantitative image analysis, we deduce that the tactoids differ from traditional liquid crystal organization, as they are limited in width irrespective of crowders and surfaces, and behave as solidlike condensates.

Nematic tactoid population

Tactoids are pointed, spindlelike droplets of nematic liquid crystal in an isotropic fluid. They have long been observed in inorganic and organic nematics, in thermotropic phases as well as lyotropic colloidal aggregates. The variational problem of determining the optimal shape of a nematic droplet is formidable and has only been attacked in selected classes of shapes and director fields. Here, by considering a special class of admissible solutions for a bipolar droplet, we study the prevalence in the population of all equilibrium shapes of each of the three that may be optimal (tactoids primarily among them). We show how the prevalence of a shape is affected by a dimensionless measure α of the drop's volume and the ratios k_{24} and k_{3} of the saddle-splay constant K_{24} and the bending constant K_{33} of the material to the splay constant K_{11}. Tactoids, in particular, prevail for α⪅16.2+0.3k_{3}-(14.9-0.1k_{3})k_{24}. Our class of shapes (and director fields) is sufficiently different from those employed so far to unveil a rather different role of K_{24}.

Elastic constants of biological filamentous colloids: estimation and implications on nematic and cholesteric tactoid morphologies

Biological liquid crystals, originating from the self-assembly of biological filamentous colloids, such as cellulose and amyloid fibrils, show a complex lyotropic behaviour that is extremely difficult to predict and characterize. Here we analyse the liquid crystalline phases of amyloid fibrils, and sulfated and carboxylated cellulose nanocrystals and measure their Frank-Oseen elastic constants K1, K2 and K3 by four different approaches. The first two approaches are based on the benchmark of the predictions of: (i) a scaling form and (ii) a variational form of the Frank-Oseen energy functional with the experimental critical volumes at order-order liquid crystalline transitions of the tactoids. The third and the fourth methods imply: (iii) the direct scaling equations of elastic constants and (iv) a molecular theory predicting the elastic constants from the experimentally accessible contour length distributions of the filamentous colloids. These three biological systems exhibit diverse liquid crystalline behaviour, governed by the distinct elastic constants characterizing each colloid. Differences and similarities among the three systems are highlighted and interpreted based on the present analysis, providing a general framework to study dispersed liquid crystalline systems.

Flow-induced order-order transitions in amyloid fibril liquid crystalline tactoids

Liquid crystalline droplets, also known as tactoids, forming by nucleation and growth within the phase diagram region where isotropic and nematic phases coexist, challenge our understanding of liquid crystals under confinement due to anisotropic surface boundaries at vanishingly small interfacial tension, resulting in complex, non-spherical shapes. Little is known about their dynamical properties, since they are mostly studied under quiescent, quasi-equilibrium conditions. Here we show that different classes of amyloid based nematic and cholesteric tactoids undergo order–order transitions by flow-induced deformations of their shape. Tactoids align under extensional flow, undergoing extreme deformation into highly elongated prolate shapes, with the cholesteric pitch decreasing as an inverse power-law of the tactoids aspect ratio. Free energy functional theory and experimental measurements are combined to rationalize the critical elongation above which the director-field configuration of tactoids transforms from bipolar and uniaxial cholesteric to homogenous and to debate on the thermodynamic nature of these transitions.

Tactoids are liquid crystal droplets with nearly vanishing interfacial tension. Almohammadi et al. show using a microfluidic focusing device how to manipulate them gently enough to facilitate the study of amyloid liquid crystal phase transitions subject to non-equilibirum forcing and shape changes.

Active forces shape the metaphase spindle through a mechanical instability

The metaphase spindle is a dynamic structure orchestrating chromosome segregation during cell division. Recently, soft matter approaches have shown that the spindle behaves as an active liquid crystal. Still, it remains unclear how active force generation contributes to its characteristic spindle-like shape. Here we combine theory and experiments to show that molecular motor-driven forces shape the structure through a barreling-type instability. We test our physical model by titrating dynein activity in Xenopus egg extract spindles and quantifying the shape and microtubule orientation. We conclude that spindles are shaped by the interplay between surface tension, nematic elasticity, and motor-driven active forces. Our study reveals how motor proteins can mold liquid crystalline droplets and has implications for the design of active soft materials.

Tuning shape and internal structure of protein droplets via biopolymer filaments

Macromolecules can phase separate to form liquid condensates, which are emerging as critical compartments in fields as diverse as intracellular organization and soft materials design. A myriad of macromolecules, including the protein FUS, form condensates which behave as isotropic liquids. Here, we investigate the influence of filament dopants on the material properties of protein liquids. We find that the short, biopolymer filaments of actin spontaneously partition into FUS droplets to form composite liquid droplets. As the concentration of the filament dopants increases, the coalescence time decreases, indicating that the dopants control viscosity relative to surface tension. The droplet shape is tunable and ranges from spherical to tactoid as the filament length or concentration is increased. We find that the tactoids are well described by a model of a quasi bipolar liquid crystal droplet, where nematic order from the anisotropic actin filaments competes with isotropic interfacial energy from the FUS, controlling droplet shape in a size-dependent manner. Our results demonstrate a versatile approach to construct tunable, anisotropic macromolecular liquids.

The Actin Cytoskeleton as an Active Adaptive Material

Actin is the main protein used by biological cells to adapt their structure and mechanics to their needs. Cellular adaptation is made possible by molecular processes that strongly depend on mechanics. The actin cytoskeleton is also an active material that continuously consumes energy. This allows for dynamical processes that are possible only out of equilibrium and opens up the possibility for multiple layers of control that have evolved around this single protein.Here we discuss the actin cytoskeleton from the viewpoint of physics as an active adaptive material that can build structures superior to man-made soft matter systems. Not only can actin be used to build different network architectures on demand and in an adaptive manner, but it also exhibits the dynamical properties of feedback systems, like excitability, bistability, or oscillations. Therefore, it is a prime example of how biology couples physical structure and information flow and a role model for biology-inspired metamaterials.

Nucleation and shape dynamics of model nematic tactoids around adhesive colloids

Recent experiments have shown how nematically ordered tactoid shaped actin droplets can be reorganized and divided by the action of myosin molecular motors. In this paper, we consider how similar morphological changes can potentially be achieved under equilibrium conditions. Using simulations, both atomistic and continuum, and a simple macroscopic model, we explore how the nucleation dynamics, shape changes, and the final steady state of a nematic tactoid droplet can be modified by interactions with model adhesive colloids that mimic a myosin motor cluster. We show how tactoid reorganization may occur in an equilibrium colloidal-nematic setting. We then suggest based on the simple macroscopic model how the simulation models may be extended to potentially stabilize divided tactoids.

Anisotropic viscoelastic phase separation in polydisperse hard rods leads to nonsticky gelation

Spinodal demixing into two phases having very different viscosities leads to viscoelastic networks-i.e., gels-usually as a result of attractive particle interactions. Here, however, we demonstrate demixing in a colloidal system of polydisperse, rod-like clay particles that is driven by particle repulsions instead. One of the phases is a nematic liquid crystal with a highly anisotropic viscosity, allowing flow along the director, but suppressing it in other directions. This phase coexists with a dilute isotropic phase. Real-space analysis and molecular-dynamics simulations both reveal a long-lived network structure that is locally anisotropic, yet macroscopically isotropic. We show that our system exhibits the characteristics of colloidal gelation, leading to nonsticky gels.

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.

Organization of associating or crosslinked actin filaments in confinement

A key factor of actin cytoskeleton organization in cells is the interplay between the dynamical properties of actin filaments and cell geometry, which restricts, confines and directs their orientation. Crosslinking interactions among actin filaments, together with geometrical cues and regulatory proteins can give rise to contractile rings in dividing cells and actin rings in neurons. Motivated by recent in vitro experiments, in this work we performed computer simulations to study basic aspects of the interplay between confinement and attractive interactions between actin filaments. We used a spring-bead model and Brownian dynamics to simulate semiflexible actin filaments that polymerize in a confining sphere with a rate proportional to the monomer concentration. We model crosslinking, or attraction through the depletion interaction, implicitly as an attractive short-range potential between filament beads. In confining geometries smaller than the persistence length of actin filaments, we show rings can form by curving of filaments of length comparable to, or longer than the confinement diameter. Rings form for optimal ranges of attractive interactions that exist in between open bundles, irregular loops, aggregated, and unbundled morphologies. The probability of ring formation is promoted by attraction to the confining sphere boundary and decreases for large radii and initial monomer concentrations, in agreement with prior experimental data. The model reproduces ring formation along the flat plane of oblate ellipsoids.

Liquid crystalline tactoids: ordered structure, defective coalescence and evolution in confined geometries

Tactoids are liquid crystalline microdroplets that spontaneously nucleate from isotropic dispersions, and transform into macroscopic anisotropic phases. These intermediate structures have been found in a range of molecular, polymeric and colloidal liquid crystals. Typically only studied by polarized optical microscopy, these ordered but easily deformable microdroplets are now emerging as interesting components for structural investigations and developing new materials. In this review, we highlight the structure, property and transformation of tactoids in different compositions, but especially cellulose nanocrystals. We have selected references that illustrate the diversity and most exciting developments in tactoid research, while capturing the historical development of this field.This article is part of a discussion meeting issue 'New horizons for cellulose nanotechnology'.

Purification and Dissolution of Carbon Nanotube Fibers Spun from the Floating Catalyst Method

In this study, we apply a simple but effective oxidative purification method to purify carbon nanotube (CNT) fibers synthesized via a floating catalyst technique. After the purification treatment, the resulting CNT fibers exhibited significant improvements in mechanical and electrical properties with an increase in strength, Young's modulus, and electrical conductivity by approximately 81, 230, and 100%, respectively. With the successful dissolution of the CNT fibers in superacid, an extensional viscosity method could be applied to measure the aspect ratio of the CNTs constituting the fibers, whereas high-purity CNT thin films could be produced with a low resistance of 720 Ω/sq at a transmittance of 85%. This work suggests that the oxidative purification approach and dissolution process are promising methods to improve the purity and performance of CNT macroscopic structures.

Line Tension of Twist-Free Carbon Nanotube Lyotropic Liquid Crystal Microdroplets on Solid Surfaces

Line tension, i.e., the force on a three-phase contact line, has been a subject of extensive research due to its impact on technological applications including nanolithography and nanofluidics. However, there is no consensus on the sign and magnitude of the line tension, mainly because it only affects the shape of small droplets, below the length scale dictated by the ratio of line tension to surface tension σ/τ. This ratio is related to the size of constitutive molecules in the system, which translates to a nanometer for conventional fluids. Here, we show that this ratio is orders of magnitude larger in lyotropic liquid crystal systems comprising micrometer-long colloidal particles. Such systems are known to form spindle-shaped elongated liquid crystal droplets in coexistence with the isotropic phase, with the droplets flattening when in contact with flat solid surfaces. We propose a method to characterize the line tension by fitting measured droplet shape to a macroscopic theoretical model that incorporates interfacial forces and elastic deformation of the nematic phase. By applying this method to hundreds of droplets of carbon nanotubes dissolved in chlorosulfonic acid, we find that σ/τ ∼ -0.84 ± 0.06 μm. This ratio is 2 orders of magnitude larger than what has been reported for conventional fluids, in agreement with theoretical scaling arguments.

Electric-field-induced shape transition of nematic tactoids

The occurrence of new textures of liquid crystals is an important factor in tuning their optical and photonics properties. Here, we show, both experimentally and by numerical computation, that under an electric field chitin tactoids (i.e., nematic droplets) can stretch to aspect ratios of more than 15, leading to a transition from a spindlelike to a cigarlike shape. We argue that the large extensions occur because the elastic contribution to the free energy is dominated by the anchoring. We demonstrate that the elongation involves hydrodynamic flow and is reversible: the tactoids return to their original shapes upon removing the field.

Using chiral tactoids as optical probes to study the aggregation behavior of chromonics

Tactoids are nuclei of an orientationally ordered nematic phase that emerge upon cooling the isotropic phase. In addition to providing a natural setting for exploring chromonics under confinement, we show that tactoids can also serve as optical probes to delineate the role of temperature and concentration in the aggregation behavior of chromonics. For high concentrations, we observe the commonly reported elongated bipolar tactoids. As the concentration is lowered, breaking of achiral symmetry in the director configuration is observed with a predominance of twisted bipolar tactoids. On further reduction of concentration, a remarkable transformation of the director configuration occurs, wherein it conforms to a unique splay-minimizing configuration. Based on a simple model, we arrive at an interesting result that lower concentrations have longer aggregates at the same reduced temperature. Hence, the splay deformation that scales linearly with the aggregate length becomes prohibitive for lower concentrations and is relieved via twist and bend deformations in this unique configuration. Raman scattering measurements of the order parameters independently verify the trend in aggregate lengths and provide a physical picture of the nematic-biphasic transition.

Liquid behavior of cross-linked actin bundles

The actin cytoskeleton is a critical regulator of cytoplasmic architecture and mechanics, essential in a myriad of physiological processes. Here we demonstrate a liquid phase of actin filaments in the presence of the physiological cross-linker, filamin. Filamin condenses short actin filaments into spindle-shaped droplets, or tactoids, with shape dynamics consistent with a continuum model of anisotropic liquids. We find that cross-linker density controls the droplet shape and deformation timescales, consistent with a variable interfacial tension and viscosity. Near the liquid-solid transition, cross-linked actin bundles show behaviors reminiscent of fluid threads, including capillary instabilities and contraction. These data reveal a liquid droplet phase of actin, demixed from the surrounding solution and dominated by interfacial tension. These results suggest a mechanism to control organization, morphology, and dynamics of the actin cytoskeleton.

Interfacial and morphological features of a twist-bend nematic drop

In this experimental and theoretical study, we examine the equilibrium shapes of quasi-two-dimensional twist-bend nematic (Ntb) drops formed within a planarly aligned nematic layer of the liquid crystal CB7CB. Initially, at the setting point of the Ntb phase, the drops assume a nonequilibrium cusped elliptical geometry with the major axis orthogonal to the director of the surrounding nematic fluid; this growth is governed principally by anisotropic heat diffusion. The drops attain equilibrium through thermally driven dynamical evolution close to their melting temperature. They are associated with a characteristic twin-striped morphology that transforms into the familiar focal conic texture as the temperature is lowered. At equilibrium, large millimetric drops are tactoidlike, elongated along the director of the surrounding nematic fluid. This geometry is explained using a mathematical model that features two dimensionless parameters, of which one is the structural cone angle of the Ntb phase and the other is the relative strength of mismatch elastic energy at the drop's interface. Both parameters are extracted from the observations by measuring the aspect ratio of the equilibrium shapes and the inner corner angle of the cusps.

Condensation and dissolution of nematic droplets in dispersions of colloidal rods with thermo-sensitive depletants

Nematic droplets are droplets composed of elongated molecules that tend to point in the same direction but do not have any positional order. Such droplets are well known to adopt a spindle shape called tactoid. How such droplets condensate or melt and how the orientational symmetry is broken remains however unclear. Here we use a colloidal system composed of filamentous viruses as model rod-like colloids and pnipam microgel particles to induce thermo-sensitive depletion attraction between the rods. Microscopy experiments coupled to particle tracking reveal that the condensation of a nematic droplet is preceded by the formation of a new phase, an isotropic droplet. As the viruses constitute an excellent experimental realization of hard rods, it follows that the phenomenology we describe should be relevant to diverse micro- and nano-sized rods that interact through excluded volume interactions. This transition between isotropic and nematic droplets provides a new and reversible pathway to break the symmetry and order colloidal rods within a droplet with an external stimulus, and could constitute a benchmark experiment for a variety of technologies relying on reconfigurable control of rods.

Experimental realization of crossover in shape and director field of nematic tactoids

Spindle-shaped nematic droplets (tactoids) form in solutions of rod-like molecules at the onset of the liquid crystalline phase. Their unique shape and internal structure result from the interplay of the elastic deformation of the nematic and anisotropic surface forces. The balance of these forces dictates that tactoids must display a continuous variation in aspect ratio and director-field configuration. Yet, such continuous transition has eluded observation for decades: tactoids have displayed either a bipolar configuration with particles aligned parallel to the droplet interface or a homogeneous configuration with particles aligned parallel to the long axis of the tactoid. Here, we report the first observation of the continuous transition in shape and director-field configuration of tactoids in true solutions of carbon nanotubes in chlorosulfonic acid. This observation is possible because the exceptional length of carbon nanotubes shifts the transition to a size range that can be visualized by optical microscopy. Polarization micrographs yield the interfacial and elastic properties of the system. Absorbance anisotropy measurements provide the highest nematic order parameter (S=0.79) measured to date for a nematic phase of carbon nanotubes at coexistence with its isotropic phase.

Morphogenesis of defects and tactoids during isotropic-nematic phase transition in self-assembled lyotropic chromonic liquid crystals

We explore the structure of nuclei and topological defects in the first-order phase transition between the nematic (N) and isotropic (I) phases in lyotropic chromonic liquid crystals (LCLCs). The LCLCs are formed by self-assembled molecular aggregates of various lengths and show a broad biphasic region. The defects emerge as a result of two mechanisms: (1) surface-anisotropy that endows each N nucleus ('tactoid') with topological defects thanks to preferential (tangential) orientation of the director at the closed I-N interface, and (2) Kibble mechanism with defects forming when differently oriented N tactoids merge with each other. Different scenarios of phase transition involve positive (N-in-I) and negative (I-in-N) tactoids with nontrivial topology of the director field and also multiply connected tactoid-in-tactoid configurations. The closed I-N interface limiting a tactoid shows a certain number of cusps; the lips of the interface on the opposite sides of the cusp make an angle different from π. The N side of each cusp contains a point defect-boojum. The number of cusps shows how many times the director becomes perpendicular to the I-N interface when one circumnavigates the closed boundary of the tactoid. We derive conservation laws that connect the number of cusps c to the topological strength m of defects in the N part of the simply connected and multiply connected tactoids. We demonstrate how the elastic anisotropy of the N phase results in non-circular shape of the disclination cores. A generalized Wulff construction is used to derive the shape of I and N tactoids as a function of I-N interfacial tension anisotropy in the approximation of frozen director field of various topological charges m. The complex shapes and structures of tactoids and topological defects demonstrate an important role of surface anisotropy in morphogenesis of phase transitions in liquid crystals.

Texture and shape of two-dimensional domains of nematic liquid crystals

We present a generalized approach to compute the shape and internal structure of two-dimensional nematic domains. By using conformal mappings, we are able to compute the director field for a given domain shape that we choose from a rich class, which includes drops with large and small aspect ratios and sharp domain tips as well as smooth ones. Results are assembled in a phase diagram that for given domain size, surface tension, anchoring strength, and elastic constant shows the transitions from a homogeneous to a bipolar director field, from circular to elongated droplets, and from sharp to smooth domain tips. We find a previously unaccounted for regime, where the drop is nearly circular, the director field bipolar, and the tip rounded. We also find that bicircular director fields, with foci that lie outside the domain, provide a remarkably accurate description of the optimal director field for a large range of values of the various shape parameters.

Self-assembly of 2D membranes from mixtures of hard rods and depleting polymers()

We combine simulations and experiments to elucidate the molecular forces leading to the assembly of two dimensional membrane-like structures composed of a one rod-length thick monolayer of aligned rods from an immiscible suspension of hard rods and depleting polymers. Computer simulations predict that monolayer membranes are thermodynamically stable above a critical rod aspect ratio and below a critical depletion interaction length scale. Outside of these conditions alternative structures such as stacked smectic columns or nematic droplets are thermodynamically stable. These predictions are confirmed by subsequent experiments using a model system of virus rod-like molecules and non-adsorbing polymer. Our work demonstrates that collective molecular protrusion fluctuations alone are sufficient to stabilize membranes composed of homogenous rods with simple excluded volume interactions.

Theoretical calculation of the phase behavior of colloidal membranes

We formulate a density functional theory that describes the phase behavior of hard rods and depleting polymers, as realized in recent experiments on suspensions of fd virus and nonadsorbing polymer. The theory predicts the relative stability of nematic droplets, stacked smectic columns, and a recently discovered phase of isolated monolayers of rods, or colloidal membranes. We find that a minimum rod aspect ratio is required for stability of colloidal membranes and that collective protrusion undulations are the dominant effect that stabilizes this phase. The theoretical predictions are shown to be qualitatively consistent with experimental and computational results.

Chiral symmetry breaking by spatial confinement in tactoidal droplets of lyotropic chromonic liquid crystals

In many colloidal systems, an orientationally ordered nematic (N) phase emerges from the isotropic (I) melt in the form of spindle-like birefringent tactoids. In cases studied so far, the tactoids always reveal a mirror-symmetric nonchiral structure, sometimes even when the building units are chiral. We report on chiral symmetry breaking in the nematic tactoids formed in molecularly nonchiral polymer-crowded aqueous solutions of low-molecular weight disodium cromoglycate. The parity is broken by twisted packing of self-assembled molecular aggregates within the tactoids as manifested by the observed optical activity. Fluorescent confocal microscopy reveals that the chiral N tactoids are located at the boundaries of cells. We explain the chirality induction as a replacement of energetically costly splay packing of the aggregates within the curved bipolar tactoidal shape with twisted packing. The effect represents a simple pathway of macroscopic chirality induction in an organic system with no molecular chirality, as the only requirements are orientational order and curved shape of confinement.

Tactoids of plate-like particles: size, shape, and director field

We studied, by means of polarized light microscopy, the shape and director field of nematic tactoids as a function of their size in dispersions of colloidal gibbsite platelets in polar and apolar solvents. Because of the homeotropic anchoring of the platelets to the interface, we found large tactoids to be spherical with a radial director field, whereas small tactoids turn out to have an oblate shape and a homogeneous director field, in accordance with theoretical predictions. The transition from a radial to a homogeneous director field seems to proceed via two different routes depending in our case on the solvent. In one route, the what presumably is a hedgehog point defect in the center of the tactoid transforms into a ring defect with a radius that presumably goes to infinity with decreasing drop size. In the other route, the hedgehog defect is displaced from the center to the edge of the tactoid, where it becomes virtual again going to infinity with decreasing drop size. Furthermore, quantitative analysis of the tactoid properties provides us with useful information on the ratio of the splay elastic constant and the anchoring strength and the ratio of the anchoring strength and the surface tension.

Nematic droplets in aqueous dispersions of carbon nanotubes

Aqueous dispersions of exfoliated, bile-salt stabilized single-wall carbon nanotubes exhibit a first order transition to a nematic liquid-crystalline phase. The nematic phase presents itself in the form of micron-sized nematic droplets also known as tactoids, freely floating in the isotropic host dispersion. The nematic droplets are spindle shaped and have an aspect ratio of about four, irrespective of their size. We attribute this to a director field that is uniform rather than bipolar, which is confirmed by polarization microscopy. It follows that the ratio of the anchoring strength and the surface tension must be about four, which is quite larger than predicted theoretically but in line with earlier observations of bipolar tactoids. From the scatter in the data we deduce that the surface tension of the coexisting isotropic and nematic phases must be extremely low, that is, of the order of nN/m.

Liquid crystalline phase in ethanol suspension of anisometric 4-cyano-4-n-hexylbiphenyl microcrystals: Witnessed by dielectric spectroscopy

The suspensions of 4-cyano-4-n-hexylbiphenyl (6CB) anisometric microcrystal were obtained by quenching homogeneous 6CB/ethanol mixtures with different 6CB concentrations. Such suspensions were strongly suggested from the differential scanning calorimetry examinations and the image observations of the samples during the quench and heating processes. The crystallization process for the mixtures with higher 6CB concentration looked like the isotropic-nematic phase transition of bulk liquid crystal. Dielectric measurement was carried out on the mixtures during the heating process after quench. Distinct dielectric relaxation was observed in the frequency range between kHz and 100 kHz, which exhibited obvious dependence on temperature and 6CB concentration. Curve fitting on the complex conductivity spectra indicates that three Debye type relaxations exist in this narrow frequency range. Based on the dependences of relaxation parameters on temperature and 6CB concentration, the possible relaxation mechanisms and the phase conformation of the mixtures were discussed. It was concluded that the relaxations, from high to low relaxation frequency, originate from the Maxwell-Wagner polarization, the rotation of 6CB microcrystal around its long axis, and the reorientation of 6CB microcrystal around its short axis, respectively. It was also confirmed that the quenched 6CB/ethanol mixtures show isotropic-nematic phase transition with the increase of 6CB concentration.

Supersaturated dispersions of rodlike viruses with added attraction

The kinetics of isotropic-nematic (I-N) and nematic-isotropic (N-I) phase transitions in dispersions of rodlike fd viruses are studied. Concentration quenches were applied using pressure jumps in combination with polarization microscopy, birefringence, and turbidity measurements. The full biphasic region could be accessed, resulting in the construction of an experimental analog of the bifurcation diagram. The N-I spinodal points for dispersions of rods with varying concentrations of depletion agent (dextran) were obtained from orientation quenches using cessation of shear flow in combination with small-angle light scattering. We found that the location of the N-I spinodal point is independent of the attraction, which was confirmed by theory. Surprisingly, the experiments showed that also the absolute induction time, the critical nucleus, and the growth rate are insensitive of the attraction if the concentration is scaled to the distance to the phase boundaries.

Buckling-induced zebra stripe patterns in nematic F-actin

Rather than forming a simple and uniform nematic liquid crystal, concentrated solutions of semiflexible polymers, such as F-actin, have been observed to display a spatially periodic switching of the nematic director. When observed with polarization microscopy, these patterns appear as alternating light and dark bands, often referred to as zebra stripe patterns. Zebra stripe patterns, although not fully characterized, are due to periodic orientation distortions in the nematic order. We characterize such patterns by using a combination of two techniques. Using polarization microscopy, we quantify the periodic orientation distortions and show that the magnitude of the order parameter also varies periodically in the striped domains. When using fluorescently labeled filaments as markers, filaments spanning the striped domains are seen to undergo large angle bends. With fluorescence, clear density differences between adjacent stripes are also observed with domains of lesser density corresponding to strongly bent filaments. By directly comparing patterned areas with both polarization and fluorescence techniques, we show that periodic variation in the orientation, order parameter, filament bending, and density are correlated. We propose that these effects originate from the coupling of orientation and density that occurs for highly concentrated solutions of long semiflexible polymers subject to shear flows, as previously proposed [P. de Gennes, Mol. Cryst. Liq. Cryst. (Phila. Pa.) 34, 177 (1977)]. After cessation of shearing, strong interfilament interactions and high compressibility can lead to periodic buckling from the relaxation of filaments stretched during flows. The characterization of zebra stripe patterns presented here provides evidence that buckling in confined F-actin nematics produces strong periodic bending that is responsible for the observed features.