Tunable three-dimensional architecture of nematic disclination lines.

Disclination lines play a key role in many physical processes, from the fracture of materials to the formation of the early universe. Achieving versatile control over disclinations is key to developing novel electro-optical devices, programmable origami, directed colloidal assembly, and controlling active matter. Here, we introduce a theoretical framework to tailor three-dimensional disclination architecture in nematic liquid crystals experimentally. We produce quantitative predictions for the connectivity and shape of disclination lines found in nematics confined between two thinly spaced glass substrates with strong patterned planar anchoring. By drawing an analogy between nematic liquid crystals and magnetostatics, we find that i) disclination lines connect defects with the same topological charge on opposite surfaces and ii) disclination lines are attracted to regions of the highest twist. Using polarized light to pattern the in-plane alignment of liquid crystal molecules, we test these predictions experimentally and identify critical parameters that tune the disclination lines' curvature. We verify our predictions with computer simulations and find nondimensional parameters enabling us to match experiments and simulations at different length scales. Our work provides a powerful method to understand and practically control defect lines in nematic liquid crystals.

Shape Morphing of Planar Liquid Crystal Elastomers.

We consider planar liquid crystal elastomers: two-dimensional objects made of anisotropic responsive materials that remain flat when stimulated, however change their planar shape. We derive a closed form, analytical solution based on the implicit linearity featured by this subclass of deformations. Our solution provides the nematic director field on an arbitrary domain starting with two initial director curves. We discuss the different gauge choices for this problem and the inclusion of disclinations in the nematic order. Finally, we propose several applications and useful design principles based on this theoretical framework.

Energetics of topographically designed Smectic-A oily streaks.

A thin Smectic-A liquid crystal (LC) film is deposited on a polymer vinyl alcohol-coated substrate that had been scribed with a uniform easy axis pattern over a square of side length L ≤ 85 µm. The small size of the patterned region facilitates material distribution to form either a hill (for a thin film) or divot (for a thick film) above the scribed square and having an oily streak (OS) texture. Optical profilometry measurements vs. film thickness suggest that the OS structure aims to adopt a preferred thickness z0 that depends on the nature of the molecule, the temperature, and the surface tension at the air interface. We present a phenomenological model that estimates the energy cost of the OS layer as its thickness deviates from z0.

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.

Curved Geometries from Planar Director Fields: Solving the Two-Dimensional Inverse Problem.

Thin nematic elastomers, composite hydrogels, and plant tissues are among many systems that display uniform anisotropic deformation upon external actuation. In these materials, the spatial orientation variation of a local director field induces intricate global shape changes. Despite extensive recent efforts, to date there is no general solution to the inverse design problem: How to design a director field that deforms exactly into a desired surface geometry upon actuation, or whether such a field exists. In this work, we phrase this inverse problem as a hyperbolic system of differential equations. We prove that the inverse problem is locally integrable, provide an algorithm for its integration, and derive bounds on global solutions. We classify the set of director fields that deform into a given surface, thus paving the way to finding optimized fields.

Twist renormalization in molecular crystals driven by geometric frustration.

Symmetry considerations preclude the possibility of twist or continuous helical symmetry in bulk crystalline structures. However, as has been shown nearly a century ago, twisted molecular crystals are ubiquitous and can be formed by about 1/4 of organic substances. Despite its ubiquity, this phenomenon has so far not been satisfactorily explained. In this work we study twisted molecular crystals as geometrically frustrated assemblies. We model the molecular constituents as uniaxially twisted cubes and examine their crystalline assembly. We exploit a renormalization group (RG) approach to follow the growth of the rod-like twisted crystals these constituents produce, inquiring in every step into the evolution of their morphology, response functions and residual energy. The gradual untwisting of the rod-like frustrated crystals predicted by the RG approach is verified experimentally using silicone rubber models of similar geometry. Our theory provides a mechanism for the conveyance of twist across length-scales observed experimentally and reconciles the apparent paradox of a twisted single crystal as a finite size effect.

Geometry and mechanics of two-dimensional defects in amorphous materials.

We study the geometry of defects in amorphous materials and their elastic interactions. Defects are defined and characterized by deviations of the material's intrinsic metric from a Euclidian metric. This characterization makes possible the identification of localized defects in amorphous materials, the formulation of a corresponding elastic problem, and its solution in various cases of physical interest. We present a multipole expansion that covers a large family of localized 2D defects. The dipole term, which represents a dislocation, is studied analytically and experimentally. Quadrupoles and higher multipoles correspond to fundamental strain-carrying entities. The interactions between those entities, as well as their interaction with external stress fields, are fundamental to the inelastic behavior of solids. We develop analytical tools to study those interactions. The model, methods, and results presented in this work are all relevant to the study of systems that involve a distribution of localized sources of strain. Examples are plasticity in amorphous materials and mechanical interactions between cells on a flexible substrate.

Composite Dislocations in Smectic Liquid Crystals.

Smectic liquid crystals are characterized by layers that have a preferred uniform spacing and vanishing curvature in their ground state. Dislocations in smectics play an important role in phase nucleation, layer reorientation, and dynamics. Typically modeled as possessing one line singularity, the layer structure of a dislocation leads to a diverging compression strain as one approaches the defect center, suggesting a large, elastically determined melted core. However, it has been observed that for large charge dislocations, the defect breaks up into two disclinations [C. E. Williams, Philos. Mag. 32, 313 (1975)PHMAA40031-808610.1080/14786437508219956]. Here we investigate the topology of the composite core. Because the smectic cannot twist, transformations between different disclination geometries are highly constrained. We demonstrate the geometric route between them and show that despite enjoying precisely the topological rules of the three-dimensional nematic, the additional structure of line disclinations in three-dimensional smectics localizes transitions to higher-order point singularities.

Internal Stresses Lead to Net Forces and Torques on Extended Elastic Bodies.

A geometrically frustrated elastic body will develop residual stresses arising from the mismatch between the intrinsic geometry of the body and the geometry of the ambient space. We analyze these stresses for an ambient space with gradients in its intrinsic curvature, and show that residual stresses generate effective forces and torques on the center of mass of the body. We analytically calculate these forces in two dimensions, and experimentally demonstrate their action by the migration of a non-Euclidean gel disc in a curved Hele-Shaw cell. An extension of our analysis to higher dimensions shows that these forces are also generated in three dimensions, but are negligible compared to gravity.

Geometry of thin nematic elastomer sheets.

A thin sheet of nematic elastomer attains 3D configurations depending on the nematic director field upon heating. In this Letter, we describe the intrinsic geometry of such a sheet and derive an expression for the metric induced by general nematic director fields. Furthermore, we investigate the reverse problem of constructing a director field that induces a specified 2D geometry. We provide an explicit recipe for how to construct any surface of revolution using this method. Finally, we show that by inscribing a director field gradient across the sheet's thickness, one can obtain a nontrivial hyperbolic reference curvature tensor, which together with the prescription of a reference metric allows dictation of actual configurations for a thin sheet of nematic elastomer.

Shape selection in chiral ribbons: from seed pods to supramolecular assemblies.

We provide a geometric-mechanical model for calculating equilibrium configurations of chemical systems that self-assemble into chiral ribbon structures. The model is based on incompatible elasticity and uses dimensionless parameters to determine the equilibrium configurations. As such, it provides universal curves for the shape and energy of self-assembled ribbons. We provide quantitative predictions for the twisted-to-helical transition, which was observed experimentally in many systems, and demonstrate it with synthetic ribbons made of responsive gels. In addition, we predict the bi-stability of wide ribbons and also show how geometrical frustration can cause arrest of ribbon widening. Finally, we show that the model's predictions provide explanations for experimental observations in different chemical systems.

Emergence of spontaneous twist and curvature in non-euclidean rods: application to Erodium plant cells.

We present a limiting model for thin non-euclidean elastic rods. Originating from the three-dimensional (3D) reference metric of the rod, which is determined by its internal material structure, we derive a 1D reduced rod theory. Specifically, we show how the spontaneous twist and curvature of a rod emerge from the reference metric derivatives. Thus, the model allows calculating the unconstrained equilibrium configuration of a thin rod directly from its internal structure. The model is applied to the study of cells from members of the Geraniaceae plant family and their configurational response to dehydration. We show how the geometrical arrangement of cellulose fibrils on the cell walls determines the helical shapes of isolated cells.

Direct observation of the temporal and spatial dynamics during crumpling.

Crumpling occurs when a thin deformable sheet is crushed under an external load or grows within a confining geometry. Crumpled sheets have large resistance to compression and their elastic energy is focused into a complex network of localized structures. Different aspects of crumpling have been studied theoretically, experimentally and numerically. However, very little is known about the dynamic evolution of three-dimensional spatial configurations of crumpling sheets. Here we present direct measurements of the configurations of a fully elastic sheet evolving during the dynamic process of crumpling under isotropic confinement. We observe the formation of a network of ridges and vertices into which the energy is localized. The network is dynamic. Its evolution involves movements of ridges and vertices. Although the characteristics of ridges agree with theoretical predictions, the measured accumulation of elastic energy within the entire sheet is considerably slower than predicted. This could be a result of the observed network rearrangement during crumpling.

The smectic order of wrinkles.

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. These features are responsible for the liquid crystalline smectic-like behaviour of such systems at intermediate length scales. This insight allows better understanding of the wrinkling patterns seen in such systems, with which we explain pattern breaking into domains, the properties of domain walls and wrinkle undulation. We compare our predictions with numerical simulations and with experimental observations.