Escape from the second dimension: A topological distinction between edge and screw dislocations.

Volterra's definition of dislocations in crystals geometrically distinguishes edge and screw defects according to whether the Burgers vector is perpendicular or parallel to the defect. A homotopy-theoretic analysis of dislocations as topological defects fails to differentiate edge and screw. Here we bridge the gap between the geometric and topological descriptions by demonstrating that there is a topological difference between screw and edge defects. Our construction distinguishes edge and screw based on the disclination-line pairs at the core of smectic dislocations. By exploiting the connection between topology and geometry in the form of Gaussian curvature, this analysis results in an invariant for dislocations in the saddle-splay vector. This construction can be generalized to crystals with triply periodic order.

Geometry of focal conics in sessile cholesteric droplets.

Focal conic domains are defects characteristic of layered liquid crystal phases. Their association can build flowers where petals are the ellipses of the Dupin cyclides involved in these defects. We report here the observation of focal conic flowers in cholesteric droplets sessile on a glass surface and surrounded by glycerol. The observation of the droplets in different directions helps to solve the three dimensional architecture of the flower. The effects of the droplet size and of the pitch value are also reported.

What promotes smectic order: Applying mean-field theory to the ends.

Not every particle that forms a nematic liquid crystal makes a smectic. The particle tip is critical for this behavior. Ellipsoids do not make a smectic, but spherocylinders do. Similarly, only those N-CB alkylcyanobiphenyls with sufficiently long (N≥8 carbons) alkane tails form smectics. We understand the role of the particle tip in the smectic transition by means of a simple two-dimensional model. We model spherocylinders by "boubas" with rounded tips, and ellipsoids by "kikis" with pointed tips. The N-CB molecules are modeled by a small body with a polymer tail. We find that rounded tips and longer polymer tails lead to a smectic at lower densities by making the space between layers less accessible, destabilizing the nematic.

Expression of Concern: Colossal Density-Driven Resistance Response in the Negative Charge Transfer Insulator MnS_{2} [Phys. Rev. Lett. 127, 016401 (2021)].

This corrects the article DOI: 10.1103/PhysRevLett.127.016401.

Natural Shaping of Acellular Dermal Matrices for Implant-Based Breast Reconstruction via Expansile Kirigami.

To complete a successful and aesthetic breast reconstruction for breast cancer survivors, tissue reinforcing acellular dermal matrices (ADMs) are widely utilized to create slings/pockets to keep breast implants or autologous tissue transfer secured against the chest wall in the desired location. However, ADM sheets are 2D and cannot completely cover the entire implant without wrinkles. Here, guided by finite element modeling, a kirigami strategy is presented to cut the ADM sheets with locally and precisely controlled stretchability, curvature, and elasticity. Upon expansion, a single kirigami ADM sheet can conformably wrap the implant regardless of the shape and size, forming a natural teardrop shape; contour cuts prescribe the topographical height and fractal cuts in the center ensures horizontal expandability and thus conformability. This kirigami ADM can provide support to the reconstructed breast in the desired regions, potentially offering optimal outcomes and patient-specific reconstruction, while minimizing operative time and cost.

Ratchetlike motion of helical bilayers induced by boundary constraints.

We show that application of boundary constraints generates unusual folding behaviors in responsive (swellable) helical bilayer strips. Unlike the smooth folding trajectories typical of free helical bilayers, the boundary-constrained bilayers exhibit intermittent folding behaviors characterized by rapid, steplike movements. We experimentally study bilayer strips as they swell and fold, and we propose a simple model to explain the emergence of ratchetlike behavior. Experiments and model predictions are then compared to simulations, which enable calculation of elastic energy during swelling. We investigate the dependence of this steplike behavior as a function of elastic boundary condition strength, strip length, and strip shape; interestingly, "V-shape" strips with the same boundary conditions fold smoothly.

Twisted loxodromes in spindle-shaped polymer nematics.

We develop an energetic model that captures the twisting behavior of spindle-shaped polymer microparticles with nematic ordering, which display remarkably different twisting behavior to ordinary nematics confined to spindles. We have previously developed a geometric model of the twisting, based on experimental observations, in which we showed that the twist pattern follows loxodrome spirals [Ansell, et al., Phys. Rev. Lett., 2019, 123, 157801]. In this study, we first consider a spindle-shaped surface and show that the loxodrome twisting behavior of our system can be captured by the Frank free energy of the nematic with an additional term constraining the length of the integral curves of the system. We then extend the ideas of this model to the bulk and explore the parameter space for which the twisted loxodrome solution is energetically favorable.

Geodesic fibrations for packing diabolic domains.

We describe a theory of packing hyperboloid "diabolic" domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts," yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

Gnomonious projections for bend-free textures: thoughts on the splay-twist phase.

The Hopf fibration has inspired any number of geometric structures in physical systems, in particular, in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, S 3 , some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic-preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new liquid crystalline texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.

Threading the Spindle: A Geometric Study of Chiral Liquid Crystal Polymer Microparticles.

Polymeric particles are strong candidates for designing artificial materials capable of emulating the complex twisting-based functionality observed in biological systems. In this Letter, we provide the first detailed investigation of the swelling behavior of bipolar polymer liquid crystalline microparticles. Deswelling from the spherical bipolar configuration causes the microparticles to contract anisotropically and twist in the process, resulting in a twisted spindle-shaped structure. We propose a model to describe the observed spiral patterns and twisting behavior.

Mechanisms to splay-bend nematic phases.

While twist-bend nematic phases have been extensively studied, the experimental observation of two dimensional, oscillating splay-bend phases is recent. We consider two theoretical models that have been used to explain the formation of twist-bend phases-flexoelectricity and bond orientational order-as mechanisms to induce splay-bend phases. Flexoelectricity is a viable mechanism, and splay and bend flexoelectric couplings can lead to splay-bend phases with different modulations. We show that while bond orientational order circumvents the need for higher order terms in the free energy, the important role of nematic symmetry and phase chirality rules it out as a basic mechanism.

Shaping nanoparticle fingerprints at the interface of cholesteric droplets.

The ordering of nanoparticles into predetermined configurations is of importance to the design of advanced technologies. Here, we balance the interfacial energy of nanoparticles against the elastic energy of cholesteric liquid crystals to dynamically shape nanoparticle assemblies at a fluid interface. By adjusting the concentration of surfactant that plays the dual role of tuning the degree of nanoparticle hydrophobicity and altering the molecular anchoring of liquid crystals, we pattern nanoparticles at the interface of cholesteric liquid crystal emulsions. In this system, interfacial assembly is tempered by elastic patterns that arise from the geometric frustration of confined cholesterics. Patterns are tunable by varying both surfactant and chiral dopant concentrations. Adjusting the particle hydrophobicity more finely by regulating the surfactant concentration and solution pH further modifies the rigidity of assemblies, giving rise to surprising assembly dynamics dictated by the underlying elasticity of the cholesteric. Because particle assembly occurs at the interface with the desired structures exposed to the surrounding water solution, we demonstrate that particles can be readily cross-linked and manipulated, forming structures that retain their shape under external perturbations. This study serves as a foundation for better understanding inter-nanoparticle interactions at interfaces by tempering their assembly with elasticity and for creating materials with chemical heterogeneity and linear, periodic structures, essential for optical and energy applications.

Edges impose planar alignment in nematic monolayers by directing cell elongation and enhancing migration.

Boundaries play an important role in the emergence of nematic order in classical liquid crystal systems; we explore their importance in adhesive cells that form active nematics. In particular, we study how cells are affected by an edge, which in our experiments is a boundary between adhesive and non-adhesive domains on a planar surface. We find that such edges induce elongation and direct the migration of isolated fibroblasts. In confluent monolayers, these elongated cells co-align and migrate to form an active, two-dimensional nematic structure in which edges enforce planar alignment and provide local slip to streams of cells that move along them. On an adhesive square island of dimensions 1 mm × 1 mm, cells near the edges in confluent nematic monolayers have enhanced alignment and velocity. The corners of the adhesive island seed defects with signs that depend on the direction of the motion of the streams of cells that meet there. Distortions emerge with rotations of -π/2 to form a -1/4 defect for streams that move clockwise or counterclockwise, and +π/2 to form a +1/4 defect for converging streams. We explore how cells transmit alignment information to each other in the absence of an edge by studying cell pairs and find that while such pairs do co-align, this alignment is only transient and short lived. These results shed light on the importance of edges in imposing nematic order in confluent monolayers and how edges can be used as tools to pattern the long-range organization of cells for tissue engineering applications.

Universal inverse design of surfaces with thin nematic elastomer sheets.

Programmable shape-shifting materials can take different physical forms to achieve multifunctionality in a dynamic and controllable manner. Although morphing a shape from 2D to 3D via programmed inhomogeneous local deformations has been demonstrated in various ways, the inverse problem-finding how to program a sheet in order for it to take an arbitrary desired 3D shape-is much harder yet critical to realize specific functions. Here, we address this inverse problem in thin liquid crystal elastomer (LCE) sheets, where the shape is preprogrammed by precise and local control of the molecular orientation of the liquid crystal monomers. We show how blueprints for arbitrary surface geometries can be generated using approximate numerical methods and how local extrinsic curvatures can be generated to assist in properly converting these geometries into shapes. Backed by faithfully alignable and rapidly lockable LCE chemistry, we precisely embed our designs in LCE sheets using advanced top-down microfabrication techniques. We thus successfully produce flat sheets that, upon thermal activation, take an arbitrary desired shape, such as a face. The general design principles presented here for creating an arbitrary 3D shape will allow for exploration of unmet needs in flexible electronics, metamaterials, aerospace and medical devices, and more.

Aspects of nucleation on curved and flat surfaces.

We investigate the energetics of droplets sourced by the thermal fluctuations in a system undergoing a first-order transition. In particular, we confine our studies to two dimensions with explicit calculations in the plane and on the sphere. Using an isoperimetric inequality from the differential geometry literature and a theorem on the inequality's saturation, we show how geometry informs the critical droplet size and shape. This inequality establishes a "mean field" result for nucleated droplets. We then study the effects of fluctuations on the interfaces of droplets in two dimensions, treating the droplet interface as a fluctuating line. We emphasize that care is needed in deriving the line curvature energy from the Landau-Ginzburg energy functional and in interpreting the scalings of the nucleation rate with the size of the droplet. We end with a comparison of nucleation in the plane and on a sphere.

Gaussian Curvature Directs Stress Fiber Orientation and Cell Migration.

We show that substrates with nonzero Gaussian curvature influence the organization of stress fibers and direct the migration of cells. To study the role of Gaussian curvature, we developed a sphere-with-skirt surface in which a positive Gaussian curvature spherical cap is seamlessly surrounded by a negative Gaussian curvature draping skirt, both with principal radii similar to cell-length scales. We find significant reconfiguration of two subpopulations of stress fibers when fibroblasts are exposed to these curvatures. Apical stress fibers in cells on skirts align in the radial direction and avoid bending by forming chords across the concave gap, whereas basal stress fibers bend along the convex direction. Cell migration is also strongly influenced by the Gaussian curvature. Real-time imaging shows that cells migrating on skirts repolarize to establish a leading edge in the azimuthal direction. Thereafter, they migrate in that direction. This behavior is notably different from migration on planar surfaces, in which cells typically migrate in the same direction as the apical stress fiber orientation. Thus, this platform reveals that nonzero Gaussian curvature not only affects the positioning of cells and alignment of stress fiber subpopulations but also directs migration in a manner fundamentally distinct from that of migration on planar surfaces.

Elastocapillary Driven Assembly of Particles at Free-Standing Smectic-A Films.

Colloidal particles at complex fluid interfaces and within films assemble to form ordered structures with high degrees of symmetry via interactions that include capillarity, elasticity, and other fields like electrostatic charge. Here we study microparticle interactions within free-standing smectic-A films, in which the elasticity arising from the director field distortion and capillary interactions arising from interface deformation compete to direct the assembly of motile particles. New colloidal assemblies and patterns, ranging from 1D chains to 2D aggregates, sensitive to the initial wetting conditions of particles at the smectic film, are reported. This work paves the way to exploiting LC interfaces as a means to direct spontaneously formed, reconfigurable, and optically active materials.