Phyllotactic description of hard sphere packing in cylindrical channels

Flexural rigidity of pressurized model notochords in regular packing patterns

The biomechanics of embryonic notochords are studied using an elastic membrane model. An initial study varying internal pressure and stiffness ratio determines tension and geometric ratios as a function of internal pressure, membrane stiffness ratio, and cell packing pattern. A subsequent three-point bending study determines flexural rigidity as a function of internal pressure, configuration, and orientation. Flexural rigidity is found to be independent of membrane stiffness ratio. Controlling for number and volume of cells and their internal pressure, the eccentric staircase pattern of cell packing has more than double the flexural rigidity of the radially symmetric bamboo pattern. Moreover, the eccentric staircase pattern is found to be more than twice as stiff in lateral bending than in dorsoventral bending. This suggests a mechanical advantage to the eccentric WT staircase pattern of the embryonic notochord, over patterns with round cross-section.

Chiral photonic crystals from sphere packing

Inspired by recent developments in self-assembled chiral nanostructures, we have explored the possibility of using spherical particles packed in cylinders as building blocks for chiral photonic crystals. In particular, we focused on an array of parallel cylinders arranged in a perfect triangular lattice, each containing an identical densest sphere packing structure. Despite the non-chirality of both the spheres and cylinders, the self-assembled system can exhibit chirality due to spontaneous symmetry breaking during the assembly process. We have investigated the circular dichroism effects of the system and have found that, for both perfect electric conductor and dielectric spheres, the system can display dual-polarization photonic band gaps for circularly polarized light at normal incidence along the axis of the helix. We have also examined how the polarization band gap size depends on the dielectric constant of the spheres and the packing fraction of the cylinders. Furthermore, we have explored the effects of non-ideality and found that the polarization gap persists even in the presence of imperfections and heterogeneity. Our study suggests that a cluster formed by spheres self-assembling inside parallel cylinders with appropriate material parameters can be a promising approach to creating chiral photonic crystals.

Densest packings from size segregation of particles in geometric confinement

A correlation between density maximization and size segregation for packings of polydisperse particles in geometric confinement has been discovered, through the derivation of a general solution for the densest-packed zigzag arrangements of polydisperse particles. This solution is a size-graded structure in which the larger a particle the closer it is located to either end of the system, such that the smaller particles in the interior are encapsulated by the larger ones away from it. Any pair of different-sized adjacent particles is a grain-boundary-like configuration that reduces the overall packing efficiency of the system, and this solution corresponds to a minimization of excess-volume contributions from grain-boundary-like configurations of different-sized particles as a result of the clustering of equal- or like-sized particles. Our findings provide new insights into how structural order of polydisperse particles emerges in confined settings.

Shape multistability in flexible tubular crystals through interactions of mobile dislocations

Crystalline sheets rolled up into cylinders occur in diverse biological and synthetic systems, including carbon nanotubes, biofilaments of the cellular cytoskeleton, and packings of colloidal particles. In this work, we show, computationally, that such tubular crystals can be programmed with reconfigurable shapes, due to motions of defects that interrupt the periodicity of the crystalline lattice. By identifying and exploiting stable patterns of these defects, we cause tubular crystals to relax into desired target geometries, a design principle that could guide the creation of versatile colloidal analogues to nanotubes. Our results suggest routes to tunable and switchable material properties in ordered, soft materials on deformable surfaces.

We study avenues to shape multistability and shape morphing in flexible crystalline membranes of cylindrical topology, enabled by glide mobility of dislocations. Using computational modeling, we obtain states of mechanical equilibrium presenting a wide variety of tubular crystal deformation geometries, due to an interplay of effective defect interactions with out-of-tangent-plane deformations that reorient the tube axis. Importantly, this interplay often stabilizes defect configurations quite distinct from those predicted for a two-dimensional crystal confined to the surface of a rigid cylinder. We find that relative and absolute stability of competing states depend strongly on control parameters such as bending rigidity, applied stress, and spontaneous curvature. Using stable dislocation pair arrangements as building blocks, we demonstrate that targeted macroscopic three-dimensional conformations of thin crystalline tubes can be programmed by imposing certain sparse patterns of defects. Our findings reveal a broad design space for controllable and reconfigurable colloidal tube geometries, with potential relevance also to architected carbon nanotubes and microtubules.

Inherent structure landscape of hard spheres confined to narrow cylindrical channels

The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbors, is studied using an analytical model that extends previous results [Phys. Rev. Lett. 115, 025702 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.025702] to provide a comprehensive picture of jammed packings over a range of packing densities. In the model, a packing is described as an arrangement of k helical sections, separated by defects, that have alternating helical twist directions and where all spheres satisfy local jamming constraints. The structure of each helical section is determined by a single helical twist angle, and a jammed packing is obtained by minimizing the length of the channel per particle with respect to the k helical section angles. An analysis of a small system of N=20 spheres shows that the basins on the inherent structure landscape associated with these helical arrangements split into a number of distinct jammed states separated by low barriers giving rise to a degree of hierarchical organization. The model accurately predicts the geometric properties of packings generated using the Lubachevsky and Stillinger compression scheme (N=10^{4}) and provides insight into the nature of the probability distribution of helical section lengths.

Flow-induced surface crystallization of granular particles in cylindrical confinement

An interesting phenomenon that a layer of crystallized shell formed at the container wall during an orifice flow in a cylinder is observed experimentally and is investigated in DEM simulation. Different from shear or vibration driven granular crystallization, our simulation shows during the flow the shell layer is formed spontaneously from stagnant zone at the base and grows at a constant rate to the top with no external drive. Roughness of the shell surface is defined as a standard deviation of the surface height and its development is found to disobey existed growth models. The growth rate of the shell is found linearly proportional to the flow rate. This shell is static and served as a rough wall in an orifice flow with frictionless sidewall, which changes the flow profiles and its stress properties, and in turn guarantees a constant flow rate.

Hierarchical Self-Assembly of Nanowires on the Surface by Metallo-Supramolecular Truncated Cuboctahedra

Parastichy, the spiral arrangement of plant organs, is an example of the long-range apparent order seen in biological systems. These ordered arrangements provide scientists with both an aesthetic challenge and a mathematical inspiration. Synthetic efforts to replicate the regularity of parastichy may allow for molecular-scale control over particle arrangement processes. Here we report the packing of a supramolecular truncated cuboctahedron (TCO) into double-helical (DH) nanowires on a graphite surface with a non-natural parastichy pattern ascribed to the symmetry of the TCOs and interactions between TCOs. Such a study is expected to advance our understanding of the design inputs needed to create complex, but precisely controlled, hierarchical materials. It is also one of the few reported helical packing structures based on Platonic or Archimedean solids since the discovery of the Boerdijk-Coxeter helix. As such, it may provide experimental support for studies of packing theory at the molecular level.

Depletion-Induced Chiral Chain Formation of Magnetic Spheres

Experimental evidence is presented for the spontaneous formation of chiral configurations in bulk dispersions of magnetized colloids that interact by a combination of anisotropic dipolar interactions and isotropic depletion attractions. The colloids are superparamagnetic silica spheres, magnetized and aligned by a carefully tuned uniform external magnetic field; isotropic attractions are induced by using poly(ethylene oxide) polymers as depleting agents. At specific polymer concentrations, sphere chains wind around each other to form helical structures–of the type that previously have only been observed in simulations on small sets of unconfined dipolar spheres with additional isotropic interactions.

Shape-Anisotropy-Induced Ordered Packings in Cylindrical Confinement

Densest possible packings of identical spheroids in cylindrical confinement have been obtained through Monte Carlo simulations. By varying the shape anisotropy of spheroids and also the cylinder-to-spheroid size ratio, a variety of densest possible crystalline structures have been discovered, including achiral structures with specific orientations of particles and chiral helical structures with rotating orientations of particles. Our findings reveal a transition between confinement-induced chiral ordering and shape-anisotropy-induced orientational ordering and would serve as a guide for the fabrication of crystalline wires using anisotropic particles.

Boosting Specific Energy and Power of Carbon-Ionic Liquid Supercapacitors by Engineering Carbon Pore Structures

Carbon-ionic liquid (C-IL) supercapacitors (SCs) promise to provide high capacitance and high operating voltage, and thus high specific energy. It is still highly demanding to enhance the capacitance in order to achieve high power and energy density. We synthesized a high-pore-volume and specific-surface-area activated carbon material with a slit mesoporous structure by two-step processes of carbonization and the activation from polypyrrole. The novel slit-pore-structured carbon materials provide a specific capacity of 310 F g⁻¹ at 0.5 A g⁻¹ for C-IL SCs, which is among one of the highest recorded specific capacitances. The slit mesoporous activated carbons have a maximum ion volume utilization of 74%, which effectively enhances ion storage, and a better interaction with ions in ionic liquid electrolyte, thus providing superior capacitance. We believe that this work provides a new strategy of engineering pore structure to enhance specific capacitance and rate performance of C-IL SCs.

Polymer-guided assembly of inorganic nanoparticles

The self-assembly of inorganic nanoparticles is of great importance in realizing their enormous potentials for broad applications due to the advanced collective properties of nanoparticle ensembles. Various molecular ligands (e.g., small molecules, DNAs, proteins, and polymers) have been used to assist the organization of inorganic nanoparticles into functional structures at different hierarchical levels. Among others, polymers are particularly attractive for use in nanoparticle assembly, because of the complex architectures and rich functionalities of assembled structures enabled by polymers. Polymer-guided assembly of nanoparticles has emerged as a powerful route to fabricate functional materials with desired mechanical, optical, electronic or magnetic properties for a broad range of applications such as sensing, nanomedicine, catalysis, energy storage/conversion, data storage, electronics and photonics. In this review article, we summarize recent advances in the polymer-guided self-assembly of inorganic nanoparticles in both bulk thin films and solution, with an emphasis on the role of polymers in the assembly process and functions of resulting nanostructures. Precise control over the location/arrangement, interparticle interaction, and packing of inorganic nanoparticles at various scales are highlighted.

Densest versus jammed packings of bent-core trimers

We identify putatively maximally dense packings of tangent-sphere trimers with fixed bond angles (θ=θ_{0}), and contrast them to the disordered jammed states they form under quasistatic and dynamic athermal compression. Incommensurability of θ_{0} with three-dimensional (3D) close packing does not by itself inhibit formation of dense 3D crystals; all θ_{0} allow formation of crystals with ϕ_{max}(θ_{0})>0.97ϕ_{cp}. Trimers are always able to arrange into periodic structures composed of close-packed bilayers or trilayers of triangular-lattice planes, separated by "gap layers" that accommodate the incommensurability. All systems have ϕ_{J} significantly below the monomeric value, indicating that trimers' quenched bond-length and bond-angle constraints always act to promote jamming. ϕ_{J} varies strongly with θ_{0}; straight (θ_{0}=0) trimers minimize ϕ_{J} while closed (θ_{0}=120^{∘}) trimers maximize it. Marginally jammed states of trimers with lower ϕ_{J}(θ_{0}) exhibit quantifiably greater disorder, and the lower ϕ_{J} for small θ_{0} is apparently caused by trimers' decreasing effective configurational freedom as they approach linearity.

Theory of rotational columnar structures of soft spheres

There is a growing interest in cylindrical structures of hard and soft particles. A promising new method to assemble such structures has recently been introduced by Lee et al. [Lee, Gizynski, and Grzybowski, Adv. Mater. 29, 1704274 (2017)ADVMEW0935-964810.1002/adma.201704274]. They used rapid rotations around a central axis to drive spheres of lower density than the surrounding fluid towards the axis. This resulted in different structures as the number of spheres is varied. Here, we present comprehensive analytic energy calculations for such self-assembled structures, based on a generic soft sphere model, from which we obtain a phase diagram. It displays interesting features, including peritectoid points. These analytic calculations are complemented by preliminary numerical simulations for finite sample sizes with soft spheres. A similar analytic approach could be used to study packings of spheres inside cylinders of fixed dimensions, but with a variation in the number of spheres.

Diverse chiral assemblies of nanoparticles directed by achiral block copolymers via nanochannel confinement

It is a challenging task to realize large-area manufacture of chiral geometries of nanoparticles in solid-state materials, which exhibit strongly chiroptical responses in the visible and near-infrared ranges. Herein, novel nanocomposites, made from mixtures of achiral block copolymers and nanoparticles in a geometrically confined environment, are conceptually proposed to construct the chiral assemblies of nanoparticles through a joint theoretical-calculation framework and experimental discussion. It is found that the nanochannel-confined block copolymers self-assemble into a family of intrinsically chiral architectures, which serve as structural scaffolds to direct the chiral arrangement of nanoparticles. Through calculations of chiral order parameters and simulations of discrete dipole approximation, it is further demonstrated that certain members of this family of nanoparticle assemblies exhibit intense chiroptical activity, which can be tailored by the nanochannel radius and the nanoparticle loading. These findings highlight the multiple levels of structural control over a class of chiral assemblies of nanoparticles and the functionalities of emerging materials via careful design and selection of nanocomposites.

Tissue self-organization underlies morphogenesis of the notochord

The notochord is a conserved axial structure that in vertebrates serves as a hydrostatic scaffold for embryonic axis elongation and, later on, for proper spine assembly. It consists of a core of large fluid-filled vacuolated cells surrounded by an epithelial sheath that is encased in extracellular matrix. During morphogenesis, the vacuolated cells inflate their vacuole and arrange in a stereotypical staircase pattern. We investigated the origin of this pattern and found that it can be achieved purely by simple physical principles. We are able to model the arrangement of vacuolated cells within the zebrafish notochord using a physical model composed of silicone tubes and water-absorbing polymer beads. The biological structure and the physical model can be accurately described by the theory developed for the packing of spheres and foams in cylinders. Our experiments with physical models and numerical simulations generated several predictions on key features of notochord organization that we documented and tested experimentally in zebrafish. Altogether, our data reveal that the organization of the vertebrate notochord is governed by the density of the osmotically swelling vacuolated cells and the aspect ratio of the notochord rod. We therefore conclude that self-organization underlies morphogenesis of the vertebrate notochord.This article is part of the Theo Murphy meeting issue on 'Mechanics of development'.

Liquid crystals of hard rectangles on flat and cylindrical manifolds

The self-assembly of rectangular particles on flat and curved substrates was investigated using density functional theory and simulations.

Non-Equilibrium Self-Assembly of Monocomponent and Multicomponent Tubular Structures in Rotating Fluids

When suspended in a denser rotating fluid, lighter particles experience a cylindrically symmetric confining potential that drives their crystallization into either monocomponent or unprecedented binary tubular packing. These assemblies form around the fluid's axis of rotation, can be dynamically interconverted (upon accelerating or decelerating the fluid), can exhibit preferred chirality, and can be made permanent by solidifying the fluid. The assembly can be extended to fluids forming multiple concentric interfaces or to systems of bubbles forming both ordered and "gradient" structures within curable polymers.

Simulation and observation of line-slip structures in columnar structures of soft spheres

We present the computed phase diagram of columnar structures of soft spheres under pressure, of which the main feature is the appearance and disappearance of line slips, the shearing of adjacent spirals, as pressure is increased. A comparable experimental observation is made on a column of bubbles under forced drainage, clearly exhibiting the expected line slip.

Corrected Article: Simulation and observation of line-slip structures in columnar structures of soft spheres [Phys. Rev. E 96, 012610 (2017)]

This corrects the article DOI: 10.1103/PhysRevE.96.012610.

Phyllotaxis, disk packing, and Fibonacci numbers

We consider the evolution of the packing of disks (representing the position of buds) that are introduced at the top of a surface which has the form of a growing stem. They migrate downwards, while conforming to three principles, applied locally: dense packing, homogeneity, and continuity. We show that spiral structures characterized by the widely observed Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...), as well as related structures, occur naturally under such rules. Typical results are presented in an animation.

Plastic deformation of tubular crystals by dislocation glide

Tubular crystals, two-dimensional lattices wrapped into cylindrical topologies, arise in many contexts, including botany and biofilaments, and in physical systems such as carbon nanotubes. The geometrical principles of botanical phyllotaxis, describing the spiral packings on cylinders commonly found in nature, have found application in all these systems. Several recent studies have examined defects in tubular crystals associated with crystalline packings that must accommodate a fixed tube radius. Here we study the mechanics of tubular crystals with variable tube radius, with dislocations interposed between regions of different phyllotactic packings. Unbinding and separation of dislocation pairs with equal and opposite Burgers vectors allow the growth of one phyllotactic domain at the expense of another. In particular, glide separation of dislocations offers a low-energy mode for plastic deformations of solid tubes in response to external stresses, reconfiguring the lattice step by step. Through theory and simulation, we examine how the tube's radius and helicity affects, and is in turn altered by, the mechanics of dislocation glide. We also discuss how a sufficiently strong bending rigidity can alter or arrest the deformations of tubes with small radii.

Hard sphere packings within cylinders

Arrangements of identical hard spheres confined to a cylinder with hard walls have been used to model experimental systems, such as fullerenes in nanotubes and colloidal wire assembly. Finding the densest configurations, called close packings, of hard spheres of diameter σ in a cylinder of diameter D is a purely geometric problem that grows increasingly complex as D/σ increases, and little is thus known about the regime for D > 2.873σ. In this work, we extend the identification of close packings up to D = 4.00σ by adapting Torquato-Jiao's adaptive-shrinking-cell formulation and sequential-linear-programming (SLP) technique. We identify 17 new structures, almost all of them chiral. Beyond D ≈ 2.85σ, most of the structures consist of an outer shell and an inner core that compete for being close packed. In some cases, the shell adopts its own maximum density configuration, and the stacking of core spheres within it is quasiperiodic. In other cases, an interplay between the two components is observed, which may result in simple periodic structures. In yet other cases, the very distinction between the core and shell vanishes, resulting in more exotic packing geometries, including some that are three-dimensional extensions of structures obtained from packing hard disks in a circle.

Clusters of polyhedra in spherical confinement

Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to [Formula: see text] constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.

Controlling Chirality of Entropic Crystals

Colloidal crystal structures with complexity and diversity rivaling atomic and molecular crystals have been predicted and obtained for hard particles by entropy maximization. However, thus far homochiral colloidal crystals, which are candidates for photonic metamaterials, are absent. Using Monte Carlo simulations we show that chiral polyhedra exhibiting weak directional entropic forces self-assemble either an achiral crystal or a chiral crystal with limited control over the crystal handedness. Building blocks with stronger faceting exhibit higher selectivity and assemble a chiral crystal with handedness uniquely determined by the particle chirality. Tuning the strength of directional entropic forces by means of particle rounding or the use of depletants allows for reconfiguration between achiral and homochiral crystals. We rationalize our findings by quantifying the chirality strength of each particle, both from particle geometry and potential of mean force and torque diagrams.

Coexistence of both gyroid chiralities in individual butterfly wing scales of Callophrys rubi

The wing scales of the Green Hairstreak butterfly Callophrys rubi consist of crystalline domains with sizes of a few micrometers, which exhibit a congenitally handed porous chitin microstructure identified as the chiral triply periodic single-gyroid structure. Here, the chirality and crystallographic texture of these domains are investigated by means of electron tomography. The tomograms unambiguously reveal the coexistence of the two enantiomeric forms of opposite handedness: the left- and right-handed gyroids. These two enantiomers appear with nonequal probabilities, implying that molecularly chiral constituents of the biological formation process presumably invoke a chiral symmetry break, resulting in a preferred enantiomeric form of the gyroid structure. Assuming validity of the formation model proposed by Ghiradella H (1989) J Morphol 202(1):69-88 and Saranathan V, et al. (2010) Proc Natl Acad Sci USA 107(26):11676-11681, where the two enantiomeric labyrinthine domains of the gyroid are connected to the extracellular and intra-SER spaces, our findings imply that the structural chirality of the single gyroid is, however, not caused by the molecular chirality of chitin. Furthermore, the wing scales are found to be highly textured, with a substantial fraction of domains exhibiting the <001> directions of the gyroid crystal aligned parallel to the scale surface normal. Both findings are needed to completely understand the photonic purpose of the single gyroid in gyroid-forming butterflies. More importantly, they show the level of control that morphogenesis exerts over secondary features of biological nanostructures, such as chirality or crystallographic texture, providing inspiration for biomimetic replication strategies for synthetic self-assembly mechanisms.

Confinement without boundaries: anisotropic diffusion on the surface of a cylinder

Densely packed systems of thermal particles in curved geometries are frequently encountered in biological and microfluidic systems. In 2D systems, at sufficiently high surface coverage, diffusive motion is widely known to be strongly affected by physical confinement, e.g., by the walls. In this work, we explore the effects of confinement by shape, not rigid boundaries, on the diffusion of discs by confining them to the surface of a cylinder. We find that both the magnitude and the directionality of lateral diffusion is strongly influenced by the radius of the cylinder. An anisotropy between diffusion in the longitudinal and circumferential direction of the cylinder develops. We demonstrate that the origin of this effect lies in the fact that screw-like packings of mono- and oligodisperse discs on the surface of a cylinder induce preferential collective motions in the circumferential direction, but also show that even in polydisperse systems lacking such order an intrinsic finite size confinement effect increases diffusivity in the circumferential direction.

Flow and jamming of granular suspensions in foams

The drainage of particulate foams is studied under conditions where the particles are not trapped individually by constrictions of the interstitial pore space. The drainage velocity decreases continuously as the particle volume fraction φ(p) increases. The suspensions jam--and therefore drainage stops--for values φ*(p) which reveal a strong effect of the particle size. In accounting for the particular geometry of the foam, we show that φ*(p) accounts for unusual confinement effects when the particles pack into the foam network. We model quantitatively the overall behavior of the suspension--from flow to jamming--by taking into account explicitly the divergence of its effective viscosity at φ*(p). Beyond the scope of drainage, the reported jamming transition is expected to have a deep significance for all aspects related to particulate foams, from aging to mechanical properties.

Theory of cylindrical dense packings of disks

We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analyzing the rules for phyllotactic indices of related structures and the variation of the density for line-slip structures, close to the symmetric ones. We show that rhombic structures, which are of a lower density, are always unstable, i.e., can be increased in density by small perturbations.

Hard rods in a cylindrical pore: the nematic-to-smectic phase transition

The effect of cylindrical confinement on the phase behaviour of a system of parallel hard rods is studied using Onsager's second-virial theory. The hard rods are represented as hard cylinders of diameter D and length L, while the cylindrical pore is infinite with diameter W. The interaction between the wall and the rods is hard repulsive, and it is assumed that molecules are parallel to the surface of the pore (planar anchoring). In very narrow pores (D < W < 2D), the structure is homogeneous and the system behaves as a one-dimensional Tonks gas. For wider pores, inhomogeneous fluid structures emerge because of the lowering of the average excluded volume due to the wall-particle interaction. The bulk nematic-smectic A phase transition is replaced by a transition between inhomogeneous nematic and smectic A phases. The smectic is destabilized with respect to the nematic for decreasing pore width; this effect becomes substantial for W < 10D. For W > 100D, results for bulk and confined fluids agree well due to the short range effect of the wall (∼ 3-4D).

Dimensional crossover of hard parallel cylinders confined on cylindrical surfaces

We derive, from the dimensional-crossover criterion, a fundamental-measure density functional for parallel hard curved rectangles moving on a cylindrical surface. We derive it from the density functional of circular arcs of length σ with centers of mass located on an external circumference of radius R(0). The latter functional in turn is obtained from the corresponding two-dimensional functional for a fluid of hard disks of radius R on a flat surface with centers of mass confined onto a circumference of radius R(0). Thus the curved length of closest approach between the two centers of mass of hard disks on this circumference is σ=2R(0)sin(-1)(R/R(0)), the length of the circular arcs. From the density functional of circular arcs, and by applying a dimensional expansion procedure to the spatial dimension orthogonal to the plane of the circumference, we finally obtain the density functional of curved rectangles of edge lengths σ and L. Along with the derivation, we show that, when the centers of mass of the disks are confined to the exterior circumference of a circle of radius R(0),(i) for R(0)>R, the exact Percus one-dimensional (1D) density functional of circular arcs of length 2R(0)sin(-1)(R/R(0)) is obtained, and (ii) for R(0)<R, the zero-dimensional limit (a cavity that can hold one particle at most) is recovered. We also show that, for R(0)>R, the obtained functional is equivalent to that of parallel hard rectangles on a flat surface of the same lengths, except that now the density profile of curved rectangles is a periodic function of the azimuthal angle, ρ(φ,z)=ρ(φ+2π,z). The phase behavior of a fluid of aligned curved rectangles is obtained by calculating the free-energy branches of smectic, columnar, and crystalline phases for different values of the ratio R(0)/R in the range 1<R(0)/R≤4; the smectic phase turns out to be the most stable except for R(0)/R=4, where the crystalline phase becomes reentrant in a small range of packing fractions. When R(0)/R<1 the transition is absent, since the density functional of curved rectangles reduces to the 1D Percus functional.

Theory of interacting dislocations on cylinders

We study the mechanics and statistical physics of dislocations interacting on cylinders, motivated by the elongation of rod-shaped bacterial cell walls and cylindrical assemblies of colloidal particles subject to external stresses. The interaction energy and forces between dislocations are solved analytically, and analyzed asymptotically. The results of continuum elastic theory agree well with numerical simulations on finite lattices even for relatively small systems. Isolated dislocations on a cylinder act like grain boundaries. With colloidal crystals in mind, we show that saddle points are created by a Peach-Koehler force on the dislocations in the circumferential direction, causing dislocation pairs to unbind. The thermal nucleation rate of dislocation unbinding is calculated, for an arbitrary mobility tensor and external stress, including the case of a twist-induced Peach-Koehler force along the cylinder axis. Surprisingly rich phenomena arise for dislocations on cylinders, despite their vanishing Gaussian curvature.

Packing confined hard spheres denser with adaptive prism phases

We show that hard spheres confined between two parallel hard plates pack denser with periodic adaptive prismatic structures which are composed of alternating prisms of spheres. The internal structure of the prisms adapts to the slit height which results in close packings for a range of plate separations, just above the distance where three intersecting square layers fit exactly between the plates. The adaptive prism phases are also observed in real-space experiments on confined sterically stabilized colloids and in Monte Carlo simulations at finite pressure.

Self-assembly of colloidal particles into strings in a homogeneous external electric or magnetic field

Colloidal particles with a dielectric constant (magnetic susceptibility) mismatch with the surrounding solvent acquire a dipole moment in a homogeneous external electric (magnetic) field. The resulting dipolar interactions can lead to aggregation of the particles into string-like clusters. Recently, several methods have been developed to make these structures permanent. However, especially when multiple particle sizes and/or more complex shapes than single spheres are used, the parameter space for these experiments is enormous. We therefore use Monte Carlo simulations to investigate the structure of the self-assembled string-like aggregates in binary mixtures of dipolar hard and charged spheres, as well as dipolar hard asymmetric dumbbells. Binary mixtures of spheres aggregate in different types of clusters depending on the size ratio of the spheres. For highly asymmetric systems, the small spheres form ring-like and flame-like clusters around strings of large spheres, while for size ratios closer to 1, alternating strings of both large and small spheres are observed. For asymmetric dumbbells, we investigate both the effect of size ratio and dipole moment ratio, leading to a large variety of cluster shapes, including chiral clusters.

Optimal filling of shapes

We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In n-dimensional space, if the objects are polydisperse n-balls, we show that solutions correspond to sets of maximal n-balls. For polygons, we provide a heuristic for finding solutions of maximal disks. We consider the properties of ideal distributions of N disks as N→∞. We note an analogy with energy landscapes.

Dense packings of spheres in cylinders: simulations

We study the optimal packing of hard spheres in an infinitely long cylinder, using simulated annealing, and compare our results with the analogous problem of packing disks on the unrolled surface of a cylinder. The densest structures are described and tabulated in detail up to D/d=2.873 (ratio of cylinder and sphere diameters). This extends previous computations into the range of structures which include internal spheres that are not in contact with the cylinder.

Densest columnar structures of hard spheres from sequential deposition

The rich variety of densest columnar structures of identical hard spheres inside a cylinder can surprisingly be constructed from a simple and computationally fast sequential deposition of cylinder-touching spheres, if the cylinder-to-sphere diameter ratio is D is an element of [1,2.7013]. This provides a direction for theoretically deriving all these densest structures and for constructing such densest packings with nano-, micro-, colloidal, or charged particles, which all self-assemble like hard spheres.