Publications

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

The resolution of geometric frustration in systems with continuous degrees of freedom often involves a cooperative inhomogeneous response and superextensive energy scaling. In contrast, the frustration in frustrated Ising-like spin systems is resolved uniformly. In this work we bridge between these two extremes by studying a frustrated model composed of N-state spins, and varying N. The expected cooperative response, observed for large N, is strongly attenuated as N is reduced, in a nontrivial way. Moderate N values show unique topological-like phases not observed before in frustrated models.

The geometry and topology of the region in which a director field is embedded impose limitations on the kind of supported orientational order. These limitations manifest as compatibility conditions that relate the quantities describing the director field to the geometry of the embedding space. For example, in two dimensions the splay and bend fields suffice to determine a director uniquely (up to rigid motions) and must comply with one relation linear in the Gaussian curvature of the embedding manifold. In 3D there are additional local fields describing the director, i.e. fields available to a local observer residing within the material, and a number of distinct ways to yield geometric frustration. So far it was unknown how many such local fields are required to uniquely describe a 3D director field, nor what are the compatibility relations they must satisfy. In this work, we address these questions directly. We employ the method of moving frames to show that a director field is fully determined by five local fields. These fields are shown to be related to each other and to the curvature of the embedding space through six differential relations. As an application of our method, we characterize all uniform distortion director fields, i.e., directors for which all the local characterizing fields are constant in space, in manifolds of constant curvature. The classification of such phases has been recently provided for directors in Euclidean space, where the textures correspond to foliations of space by parallel congruent helices. For non-vanishing curvature, we show that the pure twist phase is the only solution in positively curved space, while in the hyperbolic space uniform distortion fields correspond to foliations of space by (non-necessarily parallel) congruent helices. Further analysis of the obtained compatibility fields is expected to allow to also construct new non-uniform director fields.

Unlike Lego bricks that perfectly assemble next to one another, solid assemblies of organic compounds often include some inevitable misfit between constituents, giving rise to geometric frustration. In order to fit into the assembly the molecular building blocks must distort, at some finite energetic cost. In cases where this distortion at the ground state is uniform across all the units in the assembly, the associated geometric frustration is said to be locally resolved. Such locally resolved frustration carries little implications on the morphology and response properties of the assembled structure. However, in many cases, for small enough assemblies there are non-local compromises that are more energetically favorable. These conformations are associated with non-uniform distortions and highly cooperative response between the molecular constituents. The cooperative nature of frustrated assemblies may result in growth arrest, tendency to form filaments, exotic response properties and large morphological variations during the growth of the assembly. Almost a century ago German mineralogist Ferdinand Bernauer discovered that a large fraction of small organic compounds could form twisted molecular crystals. These are straight and narrow needle-like structures with mesoscopic pitch, a crystalographically impossible structure. Recent revived interest in twisted molecular crystals discovered even more compounds that form these exotic assemblies and led to their study by modern means. Electron microscopy revealed straight faceted structures with sharp diffraction peaks in selected area electron diffraction, much like regular crystals. Moreover, the pitch of the molecular crystals varied with size, with thicker crystals exhibiting less twist. In this work we review twisted molecular crystals as frustrated assemblies. In this approach twist emerges from the preferred morphology at the constituent scale, and gets attenuated with size by the incompatibility of twist and large-scale crystalline order. We discuss two distinct mechanisms that produce twisted molecular crystals, and provide a prediction for the twist decay as a function of the crystals spatial dimensions.

More than one-quarter of molecular crystals that are able to be melted can be made to grow in the form of twisted lamellae or fibers. The mechanisms leading to such unusual crystal morphologies lacking long-range translational symmetry on the mesoscale are poorly understood. Benzil (C6H5C(O)-C(O)C6H5) is one such crystal. Here, we calculate the morphology of rod-shaped benzil nanocrystals and other related structures. The ground states of these ensembles were twisted by 0.05-0.75°/Å for rods with cross sections of 50-10 nm2, respectively; the degree of twisting decreased inversely proportional to the crystal cross-sectional area. In the aggregate, our computational studies, combined with earlier observations by light microscopy, suggest that in some cases very small crystals acquire 3D translational periodicity only after reaching a certain size. Twisting is accompanied by conformational changes of molecules on the {101¯ 0} surfaces of the six-sided rods, although it is not easily answered from our data whether such changes are causes of the twisting, consequences of surface stress where symmetry is broken, or consequences of intrinsic dissymmetry when two or more geometric tendencies are in conflict. Nevertheless, it has become clear that, in some cases, the development of a crystal with a lattice having long-range translational symmetry is not foretold in the thermodynamics of aggregates of molecules. Rather, a lattice is sometimes a device for allowing a growing crystal to take advantage of the thermodynamic driving force of growth, the best compromise for a large number of molecules, which on a smaller scale would be dissymmetric (have a point symmetry only). The relationship between these calculations and the ubiquity of crystal twisting on the mesoscale are discussed.

The symmetric harmonic three-mass system with finite rest lengths, despite its apparent simplicity, displays a wide array of interesting dynamics for different energy values. At low energy the system shows regular behavior that produces a deformation-induced rotation with a constant averaged angular velocity. As the energy is increased this behavior makes way to a chaotic regime with rotational behavior statistically resembling Levy walks and random walks. At high enough energies, where the rest lengths become negligible, the chaotic signature vanishes and the system returns to regularity, with a single dominant frequency. The transition to and from chaos, as well as the anomalous power-law statistics measured for the angular displacement of the harmonic three-mass system are largely governed by the structure of regular solutions of this mixed Hamiltonian system. Thus, a deeper understating of the system's irregular behavior requires mapping out its regular solutions. In this work we provide a comprehensive analysis of the system's regular regimes of motion, using perturbative methods to derive analytical expressions of the system as almost-integrable in its low- and high-energy extremes. The compatibility of this description with the full system is shown numerically. In the low-energy regime, the Birkhoff normal form method is utilized to circumvent the low-order 1:1 resonance of the system, and the conditions for Kolmogorov-Arnold-Moser theory are shown to hold. The integrable approximations provide the back-bone structure around which the behavior of the full nonlinear system is organized and provide a pathway to understanding the origin of the power-law statistics measured in the system.

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.

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.

The authors regret the following errors in the original paper. In Section 2 of the original paper, an expression contained errors. The correct expression should read as follows: (Formula Presented). The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.

We demonstrate how gold nanoparticle monolayers can be curled up into hollow scrolls that make it possible to extract both bending and stretching moduli from indentation by atomic force microscopy. We find a bending modulus that is 2 orders of magnitude larger than predicted by standard continuum elasticity, an enhancement we associate with nonlocal microstructural constraints. This finding opens up new opportunities for independent control of resistance to bending and stretching at the nanoscale.

This paper presents a generalization of the Sadowsky functional, which measures the bending energy of narrow ribbons, for the case of incompatible ribbons (having no stress-free configuration). Specific solutions to special cases where the reference normal curvatures vanish, and for a naturally curved developable ribbon are presented and the resulting twist-stretch relations are discussed.The classical theory of ribbons as developed by Sadowsky and Wunderlich has received much attention in recent years. It concerns the equilibrium conformations of thin and narrow ribbons whose intrinsic structure favors a rectangular and flat state. However, the intrinsic structure of naturally formed ribbons will often be more complicated; Spatial variations in the in-plane distance metric can give rise to both geodesic curvature and Gaussian curvature, curving the ribbon in and out of its plane. Moreover, metric variation across the thickness of the ribbon may result in nontrivial reference normal curvatures. The resulting geometric structure is likely to have no zero-energy (stress-free) realizations in Euclidean space.

A characteristic posture is considered one of the behavioral hallmarks of sleep, and typically includes functional features such as support for the limbs and shielding of sensory organs. The nematode C. elegans exhibits a sleep-like state during a stage termed lethargus, which precedes ecdysis at the transition between larval stages. A hockey-stick-like posture is commonly observed during lethargus. What might its function be? It was previously noted that during lethargus, C. elegans nematodes abruptly rotate about their longitudinal axis. Plausibly, these "flips" facilitate ecdysis by assisting the disassociation of the old cuticle from the new one. We found that body-posture during lethargus was established using a stereotypical motor program and that body bends during lethargus quiescence were actively maintained. Moreover, flips occurred almost exclusively when the animals exhibited a single body bend, preferentially in the anterior or mid section of the body. We describe a simple biomechanical model that imposes the observed lengths of the longitudinally directed body-wall muscles on an otherwise passive elastic rod. We show that this minimal model is sufficient for generating a rotation about the anterior-posterior body axis. Our analysis suggests that posture during lethargus quiescence may serve a developmental role in facilitating flips and that the control of body wall muscles in anterior and posterior body regions are distinct.

Chirality occupies a central role in fields ranging from biological self-assembly to the design of optical metamaterials. The definition of chirality, as given by Lord Kelvin, associates chirality with the lack of mirror symmetry: the inability to superpose an object on its mirror image. While this definition has guided the classification of chiral objects for over a century, the quantification of handed phenomena based on this definition has proven elusive, if not impossible, as manifest in the paradox of chiral connectedness. In this work, we put forward a quantification scheme in which the handedness of an object depends on the direction in which it is viewed. While consistent with familiar chiral notions, such as the right-hand rule, this framework allows objects to be simultaneously right and left handed. We demonstrate this orientation dependence in three different systems-a biomimetic elastic bilayer, a chiral propeller, and optical metamaterial-and find quantitative agreement with chirality pseudotensors whose form we explicitly compute. The use of this approach resolves the existing paradoxes and naturally enables the design of handed metamaterials from symmetry principles.

Background: The objective of this study was to assess the regional geometry of the Heineke-Mikulicz (HM) strictureplasty. The HM intestinal strictureplasty is commonly performed for the treatment of stricturing Crohn's disease of the small intestine. This procedure shifts relatively normal proximal and distal tissue to the point of narrowing and thus increases the luminal diameter. The overall effect on the regional geometry of the HM strictureplasty, however, has not been previously described in detail. Methods: HM strictureplasties were created in latex tubing and cast with an epoxy resin. The resultant casts of the lumens were then imaged using computed tomography. Using 3-dimensional vascular reconstruction software, the cross-sectional areas were determined and the surface geometry was examined. Results: The HM strictureplasty, while increasing the lumen at the point of the stricture, also results in a counterproductive luminal narrowing proximal and distal to the strictureplasty. Within the model used, cross-sectional area was diminished 25% to 50% below baseline. This effect is enhanced when 2 strictureplasties are placed in close proximity to each other. Conclusions: The HM strictureplasty results in alterations in the regional geometry that may result in a compromise of the lumen proximal and distal to the location of the strictureplasty. When 2 HM strictureplasties are created in close proximity to each other, care should be undertaken to assure that the lumen of the intervening segment is adequate.

We studied the mechanical process of seed pods opening in Bauhinia variegate and found a chirality-creating mechanism, which turns an initially flat pod valve into a helix. We studied configurations of strips cut from pod valve tissue and from composite elastic materials that mimic its structure. The experiments reveal various helical configurations with sharp morphological transitions between them. Using the mathematical framework of "incompatible elasticity," we modeled the pod as a thin strip with a flat intrinsic metric and a saddle-like intrinsic curvature. Our theoretical analysis quantitatively predicts all observed configurations, thus linking the pod's microscopic structure and macroscopic conformation. We suggest that this type of incompatible strip is likely to play a role in the self-assembly of chiral macromolecules and could be used for the engineering of synthetic self-shaping devices.

Non-Euclidean plates are plates ("stacks" of identical surfaces) whose two-dimensional intrinsic geometry is not Euclidean, i.e. cannot be realized in a flat configuration. They can be generated via different mechanisms, such as plastic deformation, natural growth or differential swelling. In recent years there has been a concurrent theoretical and experimental progress in describing and fabricating non-Euclidean plates (NEP). In particular, an effective plate theory was derived and experimental methods for a controlled fabrication of responsive NEP were developed. In this paper we review theoretical and experimental works that focus on shape selection in NEP and provide an overview of this new field. We made an effort to focus on the governing principles, rather than on details and to relate the main observations to known mechanical behavior of ordinary plates. We also point out to open questions in the field and to its applicative potential.

We present an experimental study of the three-dimensional (3D) configurations that result from non-uniform lateral growth/shrinking of thin elastic sheets. We build gel sheets that undergo inducible differential shrinking. The non-uniform shrinking prescribes a non-Euclidean metric on a disc, and thus a non-zero Gaussian curvature. To minimize their elastic energy the free sheets form three-dimensional structures that approximate the imposed metric. We show how both large scale buckling and wrinkling-type structures can be generated, depending on the nature of possible embeddings of the imposed metric in Euclidean space.