Nanocrystal Assemblies: Current Advances and Open Problems.

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.

Cumulative geometric frustration and superextensive energy scaling in a nonlinear classical XY-spin model.

Geometric frustration results from a discrepancy between the locally favored arrangement of the constituents of a system and the geometry of the embedding space. Geometric frustration can be either noncumulative, which implies an extensive energy growth, or cumulative, which implies superextensive energy scaling and highly cooperative ground-state configurations which may depend on the dimensions of the system. Cumulative geometric frustration was identified in a variety of continuous systems including liquid crystals, filament bundles, and molecular crystals. However, a spin-lattice model which clearly demonstrates cumulative geometric frustration was lacking. In this paper we describe a nonlinear variation of the XY-spin model on a triangular lattice that displays cumulative geometric frustration. The model is studied numerically and analyzed in three distinct parameter regimes, which are associated with different energy minimizing configurations. We show that, despite the difference in the ground-state structure in the different regimes, in all cases the superextensive power-law growth of the frustration energy for small domains grows with the same universal exponent that is predicted from the structure of the underlying compatibility condition.

Cumulative geometric frustration in physical assemblies.

Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that cannot be realized globally. Recently, such frustrated assemblies were shown to exhibit filamentation, size limitation, large morphological variations and other exotic response properties. While these unique characteristics can be shown to be a direct outcome of the geometric frustration, some geometrically frustrated systems do not exhibit any of the above phenomena. In this work we exploit the intrinsic approach to provide a framework for directly addressing the frustration in physical assemblies. The framework highlights the role of the compatibility conditions associated with the intrinsic fields describing the physical assembly. We show that the structure of the compatibility conditions determines the behavior of small assemblies and in particular predicts their superextensive energy growth exponent. We illustrate the use of this framework to several well-known frustrated assemblies.

Construction of exact minimal parking garages: nonlinear helical motifs in optimally packed lamellar structures.

Minimal surfaces arise as energy minimizers for fluid membranes and are thus found in a variety of biological systems. The tight lamellar structures of the endoplasmic reticulum and plant thylakoids are comprised of such minimal surfaces in which right- and left-handed helical motifs are embedded in stoichiometry suggesting global pitch balance. So far, the analytical treatment of helical motifs in minimal surfaces was limited to the small-slope approximation where motifs are represented by the graph of harmonic functions. However, in most biologically and physically relevant regimes the inter-motif separation is comparable with its pitch, and thus this approximation fails. Here, we present a recipe for constructing exact minimal surfaces with an arbitrary distribution of helical motifs, showing that any harmonic graph can be deformed into a minimal surface by exploiting lateral displacements only. We analyse in detail pairs of motifs of the similar and of opposite handedness and also an infinite chain of identical motifs with similar or alternating handedness. Last, we study the second variation of the area functional for collections of helical motifs with asymptotic helicoidal structure and show that in this subclass of minimal surfaces stability requires that the collection of motifs is pitch balanced.

Predicting delayed instabilities in viscoelastic solids.

Determining the stability of a viscoelastic structure is a difficult task. Seemingly stable conformations of viscoelastic structures may gradually creep until their stability is lost, while a discernible creeping in viscoelastic solids does not necessarily lead to instability. In lieu of theoretical predictive tools for viscoelastic instabilities, we are presently limited to numerical simulation to predict future stability. In this work, we describe viscoelastic solids through a temporally evolving instantaneous reference metric with respect to which elastic strains are measured. We show that for incompressible viscoelastic solids, this transparent and intuitive description allows to reduce the question of future stability to static calculations. We demonstrate the predictive power of the approach by elucidating the subtle mechanism of delayed instability in thin elastomeric shells, showing quantitative agreement with experiments.

Crystals of Benzamide, the First Polymorphous Molecular Compound, Are Helicoidal.

The growth of spontaneously twisted crystals is a common but poorly understood phenomenon. An analysis of the formation of twisted crystals of a metastable benzamide polymorph (form II) crystallizing from highly supersaturated aqueous and ethanol solutions is given here. Benzamide, the first polymorphic molecular crystal reported (1832), would have been the first helicoidal crystal observed had the original authors undertaken an analysis by light microscopy. Polymorphism and twisting frequently concur as they are both associated with high thermodynamic driving forces for crystallization. Optical and electron microscopies as well as electron and powder X-ray diffraction reveal a complex lamellar structure of benzamide form II needle-like crystals. The internal stress produced by the overgrowth of lamellae is shown to be able to create a twist moment that is responsible for the observed non-classical morphologies.

Regular regimes of the harmonic three-mass system.

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 Lévy 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.

Fundamental helical geometry consolidates the plant photosynthetic membrane.

Plant photosynthetic (thylakoid) membranes are organized into complex networks that are differentiated into 2 distinct morphological and functional domains called grana and stroma lamellae. How the 2 domains join to form a continuous lamellar system has been the subject of numerous studies since the mid-1950s. Using different electron tomography techniques, we found that the grana and stroma lamellae are connected by an array of pitch-balanced right- and left-handed helical membrane surfaces of different radii and pitch. Consistent with theoretical predictions, this arrangement is shown to minimize the surface and bending energies of the membranes. Related configurations were proposed to be present in the rough endoplasmic reticulum and in dense nuclear matter phases theorized to exist in neutron star crusts, where the right- and left-handed helical elements differ only in their handedness. Pitch-balanced helical elements of alternating handedness may thus constitute a fundamental geometry for the efficient packing of connected layers or sheets.

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.

Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System.

In flat space, changing a system's velocity requires the presence of an external force. However, an isolated nonrigid system can freely change its orientation due to the nonholonomic nature of the angular momentum conservation law. Such nonrigid isolated systems may thus manifest their internal dynamics as rotations. In this work, we show that for such systems chaotic internal dynamics may lead to macroscopic rotational random walk resembling thermally induced motion. We do so by studying the classical harmonic three-mass system in the strongly nonlinear regime, the simplest physical model capable of zero angular momentum rotation as well as chaotic dynamics. At low energies, the dynamics are regular and the system rotates at a constant rate with zero angular momentum. For sufficiently high energies a rotational random walk is observed. For intermediate energies the system performs ballistic bouts of constant rotation rates interrupted by unpredictable orientation reversal events, and the system constitutes a simple physical model for Lévy walks. The orientation reversal statistics in this regime lead to a fractional rotational diffusion that interpolates smoothly between the ballistic and regular diffusive regimes.

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.

Correction: Geometric frustration and compatibility conditions for two-dimensional director fields.

Correction for 'Geometric frustration and compatibility conditions for two-dimensional director fields' by Idan Niv et al., Soft Matter, 2018, DOI: .

Geometric frustration and compatibility conditions for two-dimensional director fields.

Bent core (or banana shaped) liquid-crystal-forming-molecules locally favor an ordered state of zero splay and constant bend. Such a state, however, cannot be realized in the plane and the resulting liquid-crystalline phase is frustrated and must exhibit some compromise of these two mutually contradicting local intrinsic tendencies. This constitutes one of the most well-studied examples in which the intrinsic geometry of the constituents of a material gives rise to a geometrically frustrated assembly. Such geometric frustration is not only natural and ubiquitous but also leads to a striking variety of morphologies of ground states and exotic response properties. In this work we establish the necessary and sufficient conditions for two scalar functions, s and b to describe the splay and bend of a director field in the plane. We generalize these compatibility conditions for geometries with non-vanishing constant Gaussian curvature, and provide a reconstruction formula for the director field depending only on the splay and bend fields and their derivatives. Finally, we discuss optimal compromises for simple incompatible cases where the locally preferred values of the splay and bend cannot be simultaneously achieved.

Strong Resistance to Bending Observed for Nanoparticle Membranes.

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.

Furrows in the wake of propagating d-cones.

A crumpled sheet of paper displays an intricate pattern of creases and point-like singular structures, termed d-cones. It is typically assumed that elongated creases form when ridges connecting two d-cones fold beyond the material yielding threshold, and scarring is thus a by-product of the folding dynamics that seek to minimize elastic energy. Here we show that rather than merely being the consequence of folding, plasticity can act as its instigator. We introduce and characterize a different type of crease that is inherently plastic and is formed by the propagation of a single point defect. When a pre-existing d-cone is strained beyond a certain threshold, the singular structure at its apex sharpens abruptly. The resulting focusing of strains yields the material just ahead of the singularity, allowing it to propagate, leaving a furrow-like scar in its wake. We suggest an intuitive fracture analogue to explain the creation of furrows.

Confined disclinations: exterior versus material constraints in developable thin elastic sheets.

We examine the shape change of a thin disk with an inserted wedge of material when it is pushed against a plane, using analytical, numerical, and experimental methods. Such sheets occur in packaging, surgery, and nanotechnology. We approximate the sheet as having vanishing strain, so that it takes a conical form in which straight generators converge to a disclination singularity. Then, its shape is that which minimizes elastic bending energy alone. Real sheets are expected to approach this limiting shape as their thickness approaches zero. The planar constraint forces a sector of the sheet to buckle into the third dimension. We find that the unbuckled sector is precisely semicircular, independent of the angle δ of the inserted wedge. We generalize the analysis to include conical as well as planar constraints and thereby establish a law of corresponding states for shallow cones of slope ε and thin wedges. In this regime, the single parameter δ/ε^{2} determines the shape. We discuss the singular limit in which the cone becomes a plane, and the unexpected slow convergence to the semicircular buckling observed in real sheets.

Why do sleeping nematodes adopt a hockey-stick-like posture?

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.

Three-dimensional geometry of the Heineke-Mikulicz strictureplasty.

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.

Geometry and mechanics in the opening of chiral seed pods.

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.