Phase behavior of colloidal superballs: shape interpolation from spheres to cubes

Quasi-2d fluids of dipolar superballs in an external field

The structure of quasi-2d solutions of dipolar superballs in the fluid state has been determined by Metropolis Monte Carlos simulations in the absence and the presence of an external field. Superballs are 3d objects characterized by a one shape parameter. Here, superballs resembling cubes, but possessing rounded edges, have been used. Examination has been made for several magnitudes of the dipole moment in three different dipole directions. In the limit of a cube, the directions become (i) the center of mass - the center of a face (001) direction, (ii) the center of mass - the center of an edge (011) direction, and (iii) the center of mass - the corner (111) direction. At a small dipole moment, the superballs are translationally and orientationally disordered, and the dipoles become partially orientationally ordered in the presence of the field parallel to the plane of the superballs. At a large dipole moment, chains of superballs are formed, and the chains become parallel in the presence of the field. The chains remain separated for the dipole in the 001-direction and form bundles for the 011- and 111-directions. The different structures obtained for the different dipole directions are interpreted in terms of how compatible the dipole-dipole interaction is with the cube-cube interaction at short separation for the different directions of the dipole moment. Hence, the structural richness arises from an interplay of the different symmetries of a cube and of the field of a dipole.

Interplay of particle shape and suspension properties: a study of cube-like particles

With advances in anisotropic particle synthesis, particle shape is now a feasible parameter for tuning suspension properties. However, there is a need to determine how these newly synthesized particles affect suspension properties and a need to solve the inverse problem of inferring the particle shape from property measurements. Either way, accurate suspension property predictions are required. Towards this end, we calculated a set of dilute suspension properties for a family of cube-like particles that smoothly interpolate between spheres and cubes. Using three conceptually different methods, we numerically computed the electrical properties of particle suspensions, including the intrinsic conductivity of perfect conductors and insulators. We also considered hydrodynamic properties relevant to particle solutions including the hydrodynamic radius, the intrinsic viscosity and the intrinsic solvent diffusivity. Additionally, we determined the second osmotic virial coefficient using analytic expressions along with numerical integration. As the particles became more cube-like, we found that all of the properties investigated become more sensitive to particle shape.

Shape-sensitive crystallization in colloidal superball fluids

Guiding the self-assembly of materials by controlling the shape of the individual particle constituents is a powerful approach to material design. We show that colloidal silica superballs crystallize into canted phases in the presence of depletants. Some of these phases are consistent with the so-called "Λ1" lattice that was recently predicted as the densest packing of superdisks. As the size of the depletant is reduced, however, we observe a transition to a square phase. The differences in these entropically stabilized phases result from an interplay between the size of the depletants and the fine structure of the superball shape. We find qualitative agreement of our experimental results both with a phase diagram computed on the basis of the volume accessible to the depletants and with simulations. By using a mixture of depletants, one of which is thermosensitive, we induce solid-to-solid phase transitions between square and canted structures. The use of depletant size to leverage fine features of the shape of particles in driving their self-assembly demonstrates a general and powerful mechanism for engineering novel materials.

Directional self-assembly of permanently magnetised nanocubes in quasi two dimensional layers

To design modern materials with a specific response, the consequences of directionally dependent interactions on the self-assembly of constituent nanoparticles need to be properly understood. Directionality arises in the study of anisometric nanoparticles, where geometry has a drastic effect on the properties observed. Given the fact that magnetic interactions are inherently anisotropic, if one constructs these particles from a magnetic medium, an interesting interplay between the two sources of directionality will occur. We have investigated this scenario by exploring systems of dipolar nanocube monolayers. Using an applied analytical approach, in combination with molecular dynamics simulations, we have determined the ground state structures of individual monolayer clusters. Taking inspiration from experiments, two different fixed dipole orientations for the permanent magnetisation of the nanocubes were considered: the first aligned along the [001] crystallographic axis of each cube, and the second along the [111] axis. We discovered that the structure of the ground state is distinctly different for the two systems of permanently magnetised nanocubes; [001] cubes form dipolar chains in the ground state, whereas those with [111] orientation adopt square lattice structures. The discovered configurations in the ground state represent two different structural motifs, as yet unobserved in the ground state of other magnetic nanoparticle systems.

Formation and liquid permeability of dense colloidal cube packings

The liquid permeability of dense random packings of cubic colloids with rounded corners is studied for solid hematite cubes and hollow microporous silica cubes. The permeabilities of these two types of packings are similar, confirming that the micropores in the silica shell of the hollow cubes do not contribute to the permeability. From the Brinkman screening length √k of ∼16 nm, we infer that the relevant pores are indeed intercube pores. Furthermore, we relate the permeability to the volume fraction and specific solid volume of the cubes using the Kozeny-Carman relation. The Kozeny-Carman relation contains a constant that accounts for the topology and size distribution of the pores in the medium. The constant obtained from our study with aspherical particles is of the same order of magnitude as those from studies with spherical and ellipsoidal particles, which supports the notion that the Kozeny-Carman relation is applicable for any dense particle packing with (statistically) isotropic microstructures, irrespective of the particle shape.

An experimental and simulation study on the self-assembly of colloidal cubes in external electric fields

When a suspension of colloidal particles is placed in an oscillating electric field, the contrast in dielectric constant between the particles and the solvent induces a dipole moment in each of the colloidal particles. The resulting dipole-dipole interactions can strongly influence the phase behavior of the system. We investigate the phase behavior of cube-shaped colloidal particles in electric fields, using both experiments and Monte Carlo simulations. In addition to a string fluid phase and a body centered tetragonal (BCT) crystal phase, we observe a columnar phase consisting of hexagonally ordered strings of rotationally disordered cubes. By simulating the system for a range of pressures and electric field strengths, we map out the phase diagram, and compare the results to the experimentally observed phases. Additionally, we estimate the accuracy of a point-dipole approximation on the alignment of cubes in string-like clusters.

Disordered strictly jammed binary sphere packings attain an anomalously large range of densities

Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. Here we report that disordered strictly jammed binary packings (packings that remain mechanically stable under general shear deformations and compressions) can be produced with an anomalously large range of average packing fractions 0.634≤φ≤0.829 for small to large sphere radius ratios α restricted to α≥0.100. Surprisingly, this range of average packing fractions is obtained for packings containing a subset of spheres (called the backbone) that are exactly strictly jammed, exactly isostatic, and also generated from random initial conditions. Additionally, the average packing fractions of these packings at certain α and small sphere relative number concentrations x approach those of the corresponding densest known ordered packings. These findings suggest for entropic reasons that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. We also identify an unusual feature of the packing fraction of jammed backbones (packings with rattlers excluded). The backbone packing fraction is about 0.624 over the majority of the α-x plane, even when large numbers of small spheres are present in the backbone. Over the (relatively small) area of the α-x plane where the backbone is not roughly constant, we find that backbone packing fractions range from about 0.606 to 0.829, with the volume of rattler spheres comprising between 1.6% and 26.9% of total sphere volume. To generate isostatic strictly jammed packings, we use an implementation of the Torquato-Jiao sequential linear programming algorithm [Phys. Rev. E 82, 061302 (2010)], which is an efficient producer of inherent structures (mechanically stable configurations at the local maxima in the density landscape). The identification and explicit construction of binary packings with such high packing fractions could have important practical implications for granular composites where density is critical both to material properties and fabrication cost, including for solid propellants, concrete, and ceramics. The densities and structures of jammed binary packings at various α and x are also relevant to the formation of a glass phase in multicomponent metallic systems.

Phase diagram and structural diversity of a family of truncated cubes: degenerate close-packed structures and vacancy-rich states

Using Monte Carlo simulations and free-energy calculations, we determine the phase diagram of a family of truncated hard cubes, where the shape evolves smoothly from a cube via a cuboctahedron to an octahedron. A remarkable diversity in crystal phases and close-packed structures is found, including a fully degenerate crystal structure, several plastic crystals, as well as vacancy-stabilized crystal phases, all depending sensitively on the precise particle shape. Our results illustrate the intricate relation between phase behavior and building-block shape, and can guide future experimental studies on polyhedral-shaped nanoparticles.

Bending and elongation effects on the random packing of curved spherocylinders

Studies on the macroscopic and microscopic packing properties of nonconvex particles are scarce. As a common concave form, the curved spherocylinder is used in the simulations, and its bending and elongation effects on the random packings are investigated numerically with sphere assembly models and a relaxation algorithm. The aspect ratio is demonstrated to be the main factor regarding the packing density. However, at certain aspect ratios of low densities around 0.3-0.4, the density of curved spherocylinders may increase by 15% more than that of the straight ones, indicating that bending is also a contributor to the packing density. The excluded volume of the curved spherocylinder decreases with the increase of the bending angle, indicating that the excluded volume is applicable in explaining the bending effect on the packing density variation of nonconvex particles. The packings are verified to be randomly distributed in orientation with no significant layering or in-plane order. The local arrangements are further analyzed from the radial distribution function and contact results. The results show that the random packings of nonconvex particles have significant differences and richer characteristics on both the macroscopic and microscopic properties compared with convex objects.

Phase transitions and thermodynamic properties of dense assemblies of truncated nanocubes and cuboctahedra

Inspired by recent advances on the self-assembly of non-spherical nanoparticles, Monte Carlo simulations of the packing and thermodynamic properties of truncated nanocubes and cuboctahedra have been performed. The ergodicity problem was overcome by a modified Wang-Landau entropic sampling algorithm and equilibrium structural and thermodynamic properties were computed over a wide density range for both non-interacting and interacting particles. We found a structural transition from a simple cubic to a rhombohedral order when the degree of truncation exceeds a value of 0.9.

Organizing principles for dense packings of nonspherical hard particles: not all shapes are created equal

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids and certain smoothly shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. All of our organizing principles are applied to and tested against the most comprehensive set of both convex and concave particle shapes examined to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres, and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, "lens-shaped" particles, square pyramids, and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles.

Densest binary sphere packings

The densest binary sphere packings in the α-x plane of small to large sphere radius ratio α and small sphere relative concentration x have historically been very difficult to determine. Previous research had led to the prediction that these packings were composed of a few known "alloy" phases including, for example, the AlB(2) (hexagonal ω), HgBr(2), and AuTe(2) structures, and to XY(n) structures composed of close-packed large spheres with small spheres (in a number ratio of n to 1) in the interstices, e.g., the NaCl packing for n=1. However, utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [Torquato and Jiao, Phys. Rev. E 82, 061302 (2010)], we have discovered that many more structures appear in the densest packings. For example, while all previously known densest structures were composed of spheres in small to large number ratios of one to one, two to one, and very recently three to one, we have identified densest structures with number ratios of seven to three and five to two. In a recent work [Hopkins et al., Phys. Rev. Lett. 107, 125501 (2011)], we summarized these findings. In this work, we present the structures of the densest-known packings and provide details about their characteristics. Our findings demonstrate that a broad array of different densest mechanically stable structures consisting of only two types of components can form without any consideration of attractive or anisotropic interactions. In addition, the structures that we have identified may correspond to currently unidentified stable phases of certain binary atomic and molecular systems, particularly at high temperatures and pressures.

Continuous phase transformation in nanocube assemblies

The phase behavior of 3D assemblies of nanocubes in a ligand-rich solution upon solvent evaporation was experimentally investigated using small-angle x-ray scattering and electron microscopy. We observed a continuous transformation of assemblies between simple cubic and rhombohedral phases, where a variable angle of rhombohedral structure is determined by ligand thickness. We established a quantitative relationship between the particle shape evolution from cubes to quasispheres and the lattice distortion during the transformation, with a pathway exhibiting the highest known packing.

Orientationally ordered phase produced by fully antinematic interactions: a simulation study

We consider here a classical model, consisting of D(2h) symmetric particles, whose centers of mass are associated with a three-dimensional simple-cubic lattice; the pair potential is isotropic in orientation space, and restricted to nearest neighbors. Two orthonormal triads define orientations of a pair of interacting particles; the simplest potential models proposed in the literature can be written as a linear combination involving the squares of the scalar products between corresponding unit vectors only, thus depending on three parameters, and making the interaction model rather versatile. A coupling constant with negative sign tends to keep the two interacting unit vectors parallel to each other, whereas a positive sign tends to keep them mutually orthogonal (antinematic coupling). We address here a special, extreme case of the above family, involving only antinematic couplings: more precisely, three antinematic terms whose coefficients are set to a common positive value (hence the name PPP model). The model under investigation produces a doubly degenerate pair ground state; the nearest-neighbor range of the interaction and the bipartite character of the lattice can propagate the pair ground state and increase the overall degeneracy, but without producing frustration. The model was investigated by a simplified molecular field treatment as well as by Monte Carlo simulation, whose results suggested a second-order transition to a low-temperature biaxially ordered phase; ground-state configurations producing orientational order have been selected by thermal fluctuations. The molecular field treatment also predicted a continuous transition, and was found to overestimate the transition temperature by a factor 2.

Thermodynamic stability of the cubatic phase of hard cut spheres evaluated by expanded ensemble simulations

The system of hard cut spheres (disk-shaped particles formed by symmetrically truncating the end caps of a sphere) exhibits an intriguing "cubatic" phase with cubic orientational symmetry. However, it is unclear whether this phase is metastable with respect to the columnar phase. We attempt to provide an answer to this question by carrying out free energy calculations by the expanded ensemble Monte Carlo method. We conclude that there may be a very small region of cubatic stability in the vicinity of the isotropic-cubatic phase transition, but that that transition would need to be determined more accurately to obtain a definitive answer. We also comment on the efficacy of the expanded ensemble method for these kinds of calculations.

Mesophase behaviour of polyhedral particles

Translational and orientational excluded-volume fields encoded in particles with anisotropic shapes can lead to purely entropy-driven assembly of morphologies with specific order and symmetry. To elucidate this complex correlation, we carried out detailed Monte Carlo simulations of six convex space-filling polyhedrons, namely, truncated octahedrons, rhombic dodecahedrons, hexagonal prisms, cubes, gyrobifastigiums and triangular prisms. Simulations predict the formation of various new liquid-crystalline and plastic-crystalline phases at intermediate volume fractions. By correlating these findings with particle anisotropy and rotational symmetry, simple guidelines for predicting phase behaviour of polyhedral particles are proposed: high rotational symmetry is in general conducive to mesophase formation, with low anisotropy favouring plastic-solid behaviour and intermediate anisotropy (or high uniaxial anisotropy) favouring liquid-crystalline behaviour. It is also found that dynamical disorder is crucial in defining mesophase behaviour, and that the apparent kinetic barrier for the liquid-mesophase transition is much lower for liquid crystals (orientational order) than for plastic solids (translational order).

Athermal jamming of soft frictionless Platonic solids

A mechanically based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high-density phases. We confirm that the large system jamming threshold of 0.623 ± 0.003 for tetrahedra is consistent with experiments on tetrahedral dice. Also, the extremely short-ranged translational correlations of packed tetrahedra observed in experiments are confirmed here, in contrast with those of thermally simulated glasses. Although highly ordered phases are observed to form for small numbers of cubes and dodecahedra, the short correlation length scale suppresses ordering in large systems, resulting in packings that are mechanically consistent with "orientationally disordered" contacts (point-face and edge-edge contacts). Mild nematic ordering is observed for large systems of cubes, whereas angular correlations for the remaining shapes are ultrashort ranged. In particular the angular correlation function of tetrahedra agrees with that recently observed experimentally for tetrahedral dice. Power-law scaling exponents for energy with respect to distance from the jamming threshold exhibit a clear dependence on the "highest-order" percolating contact topology. These nominal exponents are 6, 4, and 2 for configurations having percolating point-face (or edge-edge), edge-face, and face-face contacts, respectively. Jamming contact number is approximated for small systems of tetrahedra, icosahedra, dodecahedra, and octahedra with order and packing representative of larger systems. These Platonic solids exhibit hypostatic behavior, with average jamming contact number between the isostatic value for spheres and that of asymmetric particles. These shapes violate the isostatic conjecture, displaying contact number that decreases monotonically with sphericity. The common symmetry of dual polyhedra results in local translational structural similarity. Systems of highly spherical particles possessing icosahedral symmetry, such as icosahedra or dodecahedra, exhibit structural behavior similar to spheres, including jamming contact number and radial distribution function. These results suggest that although continuous rotational symmetry is broken by icosahedra and dodecahedra, the structural features of disordered packings of these particles are well replicated by spheres. Octahedra and cubes, which possess octahedral symmetry, exhibit similar local translational ordering, despite exhibiting strong differences in nematic ordering. In general, the structural features of systems with tetrahedra, octahedra, and cubes differ significantly from those of sphere packings.