Elastic Instability of a Crystal Growing on a Curved Surface

Meng, Guangnan; Paulose, Jayson; Nelson, David R.; Manoharan, Vinothan N.

doi:10.1126/science.1244827

Cited by 231 publications

(329 citation statements)

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“…In this work we provide an explicit formula for the Gaussian curvature associated with given splay and bend fields, thus mapping the geometrically frustrated problem of bent-core liquid crystals (or any other local splay/bend tendency) in two dimensions to the much studied realm of optimal embedding of manifolds of mismatched Gaussian curvatures. A problem encountered in elasticity [13][14][15], crystal growth of curved surfaces [11,12] and the bundling of twisted filaments [16,17].…”

Section: Fig 1 (A)mentioning

confidence: 99%

“…The case of bent core liquid crystals constitutes a particular example of what has been recently termed frustrated assemblies [3] and has seen numerous experimental realizations using colloidal particles [10,11]. While one could pursue the individual equilibria shapes through direct molecular simulation, unravelling the general principles governing the assembly of such frustrated structures requires a coarse grained continuous geometric description.…”

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confidence: 99%

“…In general, in such a coarse grained description the geometric frustration is associated with a mismatched geometric charge [11,12], and one of the central challenges in formulating such a description is to properly identify the geometric charge associated with the mutually incompatible local preferences of the building blocks of the assembly.…”

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confidence: 99%

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Geometric frustration and compatibility conditions for two-dimensional director fields

Niv

Efrati

2018

Soft Matter

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The uniform director field obtained for the nematic ground state of the hard-rod model of liquid crystals in two dimensions reflects the high symmetry of the constituents of the liquid [1]; It is a manifestation of the constituents' local tendency to avoid splaying and bending with respect to one another. In contrast, bent-core (or banana shaped) liquid-crystal-forming-molecules locally favor a state of zero splay and constant bend. However, such a structure cannot be realized in the plane [2] and the resulting liquid-crystalline phase is frustrated and must exhibit some compromise of these two mutually contradicting local intrinsic tendencies. The generation of geometric frustration from the intrinsic geometry of the constituents of a material is not only natural and ubiquitous but also leads to a striking variety of morphologies of ground states [3,4] and exotic response properties [5].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. Last, we discuss optimal compromises for simple incompatible cases where the locally preferred values of the splay and bend cannot be globally achieved.

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confidence: 99%

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Geometric frustration and compatibility conditions for two-dimensional director fields

Niv

Efrati

2018

Soft Matter

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“…1 B and C, shows colloids ordered in a polycrystalline, close-packed lattice, previously assigned as the (111) lattice plane of the 3D crystal formed in the spherical confinement (23,24). However, the curvature in the confinement impedes the formation of a perfectly ordered crystal, resulting in characteristic defects and assembly structures (20,21). A focused ion beam (FIB) was used to cross-section photonic balls and uncover their internal crystal structure, which we show in Fig.…”

Section: Resultsmentioning

confidence: 92%

“…Curvature imposed by the spherical confinement is a unique structural element of photonic balls. It is known that curvature induces different types of defect structures in 2D (20,21) and 3D assemblies (22), which can affect the resulting optical properties. Photonic balls have recently gained attention in various optical applications, including pigments (23)(24)(25)(26), sensors (27), magnetically switchable colorants (25,28), and color-coded substrates for biomaterials evaluation (29,30).…”

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Color from hierarchy: Diverse optical properties of micron-sized spherical colloidal assemblies

Vogel

Utech

England

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

270

351

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Materials in nature are characterized by structural order over multiple length scales have evolved for maximum performance and multifunctionality, and are often produced by self-assembly processes. A striking example of this design principle is structural coloration, where interference, diffraction, and absorption effects result in vivid colors. Mimicking this emergence of complex effects from simple building blocks is a key challenge for manmade materials. Here, we show that a simple confined selfassembly process leads to a complex hierarchical geometry that displays a variety of optical effects. Colloidal crystallization in an emulsion droplet creates micron-sized superstructures, termed photonic balls. The curvature imposed by the emulsion droplet leads to frustrated crystallization. We observe spherical colloidal crystals with ordered, crystalline layers and a disordered core. This geometry produces multiple optical effects. The ordered layers give rise to structural color from Bragg diffraction with limited angular dependence and unusual transmission due to the curved nature of the individual crystals. The disordered core contributes nonresonant scattering that induces a macroscopically whitish appearance, which we mitigate by incorporating absorbing gold nanoparticles that suppress scattering and macroscopically purify the color. With increasing size of the constituent colloidal particles, grating diffraction effects dominate, which result from order along the crystal's curved surface and induce a vivid polychromatic appearance. The control of multiple optical effects induced by the hierarchical morphology in photonic balls paves the way to use them as building blocks for complex optical assemblies-potentially as more efficient mimics of structural color as it occurs in nature.self-assembly | colloids | photonic crystal | structural color | hierarchy H ierarchical design principles, i.e., the structuration of material over multiple length scales, are ubiquitously used in nature to maximize functionality from a limited choice of available components. Hierarchically structured materials often provide better performance than their unstructured counterparts and novel properties can arise solely from the multiscale structural arrangement. Examples can be found in the extreme water repellency of the lotus leaf (1); the outstanding mechanical stability and toughness of sea creatures such as sea sponges (2) and abalone shells (3); and the bright coloration found in beetles, birds, and butterflies (4, 5).To achieve the strongest visual effects, many organisms combine optical effects arising from light interacting with structured matter at different length scales (6). Structural periodicity on the scale of visible light wavelengths can result in regular optical density variations that give rise to bright, iridescent colors due to pronounced interference effects (4). At the micron scale, regular structural features act as diffraction gratings that produce vivid, rainbow coloration (7) and are used to control scatteri...

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Melting of Colloidal Crystals

Wang

Zhou

Han

2016

Adv Funct Materials

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8903wileyonlinelibrary.com ClassificationCrystal melting can be categorized into two types: homogeneous melting in which the melting is equally likely to occur anywhere in the crystal, and heterogeneous melting in which the melting is more likely to occur at the surfaces and defects than the defectfree regions. Heterogeneous melting can be divided into surface melting from a free surface (i.e., the solid-vapor interface), interfacial melting from the interface of two materials (e.g., a crystal and its substrate), grain-boundary melting and melting at weaker defects such as dislocations and vacancies. Homogeneous melting exists in defect-free crystals whose surface melting is suppressed. This type of melting is of theoretical importance but is difficult to achieve experimentally because most crystals melt from surfaces and are not defect-free, which leads to heterogeneous melting. However, homogeneous melting can be conveniently studied by simulation using defect-free crystals under periodic boundary conditions.In heterogeneous melting, particles at the surfaces or defects usually have a higher free energy than those in a perfect crystal. If the free energy is high enough, melting can take place even before the melting point is reached, which is known as premelting. If none of the surfaces and defects has a high enough free energy, then the crystal can be superheated to form a metastable crystal above the bulk melting point. The bulk melting point is the point at which the solid and liquid phases share the same chemical potential, and surfaces are negligible at the thermodynamic limit, i.e., the crystals are infinitely large. For 3D crystals, the likelihood of inducing melting in decreasing order is roughly as follows: 0D corners and 1D edges (intersection of three or two free surfaces) > 1D surface grain boundaries (intersection of a free surface and a bulk grain boundary) > free surfaces > bulk triple junctions > high-angle grain boundaries > the interface between two solids (e.g., on a flat substrate) > lowangle grain boundaries > dislocations > partial dislocations > vacancies > interstitials. [4] Melting starts from the strongest defects and spreads rapidly into the crystal, and thus usually preempts the melting from other weaker defects. Note that this order may not always hold. For example, the energy of free surfaces and grain boundaries varies with their orientations and the interfacial energy at the substrate depends on the affinity between the two solids. Consequently, most free surfaces, some grain boundaries and some solid interfaces will premelt below the bulk melting point T m , while other solid interfaces and grain boundaries and a few free surfaces will only induce melting above T m , i.e., they can be superheated.Crystal melting is affected by many factors such as defects, surfaces, dimensionality, lattice structure, and particle interaction, and thus exhibits rich phenomenology. It is usually a first-order phase transition which currently lacks a fundamental theory. Despite numerous studies ove...

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Elastic Instability of a Crystal Growing on a Curved Surface

Cited by 231 publications

References 51 publications

Geometric frustration and compatibility conditions for two-dimensional director fields

Geometric frustration and compatibility conditions for two-dimensional director fields

Color from hierarchy: Diverse optical properties of micron-sized spherical colloidal assemblies

Melting of Colloidal Crystals

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