Perspective: Basic understanding of condensed phases of matter via packing models

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Photonic amorphous topological insulator

The current understanding of topological insulators and their classical wave analogs, such as photonic topological insulators, is mainly based on topological band theory. However, standard band theory does not apply to amorphous phases of matter, which are formed by non-crystalline lattices with no long-range positional order but only short-range order, exhibiting unique phenomena such as the glass-to-liquid transition. Here, we experimentally investigate amorphous variants of a Chern number-based photonic topological insulator. By tuning the disorder strength in the lattice, we demonstrate that photonic topological edge states can persist into the amorphous regime prior to the glass-to-liquid transition. After the transition to a liquid-like lattice configuration, the signatures of topological edge states disappear. This interplay between topology and short-range order in amorphous lattices paves the way for new classes of non-crystalline topological photonic bandgap materials.

Phase Diagram and Structure Map of Binary Nanoparticle Superlattices from a Lennard-Jones Model

A first-principles prediction of the binary nanoparticle phase diagram assembled by solvent evaporation has eluded theoretical approaches. In this paper, we show that a binary system interacting through the Lennard-Jones (LJ) potential contains all experimental phases in which nanoparticles are effectively described as quasi hard spheres. We report a phase diagram consisting of 53 equilibrium phases, whose stability is quite insensitive to the microscopic details of the potentials, thus giving rise to some type of universality. Furthermore, we show that binary lattices may be understood as consisting of certain particle clusters, i.e., motifs, that provide a generalization of the four conventional Frank-Kasper polyhedral units. Our results show that metastable phases share the very same motifs as equilibrium phases. We discuss the connection with packing models, phase diagrams with repulsive potentials, and the prediction of likely experimental superlattices.

Hard convex lens-shaped particles: Characterization of dense disordered packings

Among the family of hard convex lens-shaped particles (lenses), the one with aspect ratio equal to 2/3 is "optimal" in the sense that the maximally random jammed (MRJ) packings of such lenses achieve the highest packing fraction ϕ_{MRJ}≃0.73 [G. Cinacchi and S. Torquato, Soft Matter 14, 8205 (2018)1744-683X10.1039/C8SM01519H]. This value is only a few percent lower than ϕ_{DKP}=0.76210⋯, the packing fraction of the corresponding densest-known crystalline (degenerate) packings [G. Cinacchi and S. Torquato, J. Chem. Phys. 143, 224506 (2015)JCPSA60021-960610.1063/1.4936938]. By exploiting the appreciably reduced propensity that a system of such optimal lenses has to positionally and orientationally order, disordered packings of them are progressively generated by a Monte Carlo method-based procedure from the dilute equilibrium isotropic fluid phase to the dense nonequilibrium MRJ state. This allows us to closely monitor how the (micro)structure of these packings changes in the process of formation of the MRJ state. The gradual changes undergone by the many structural descriptors calculated here can coherently and consistently be traced back to the gradual increase in contacts between the hard particles until the isostatic mean value of ten contact neighbors per lens is reached at the effectively hyperuniform MRJ state. Compared to the MRJ state of hard spheres, the MRJ state of such optimal lenses is denser (less porous), more disordered, and rattler-free. This set of characteristics makes them good glass formers. It is possible that this conclusion may also hold for other hard convex uniaxial particles with a correspondingly similar aspect ratio, be they oblate or prolate, and that, by using suitable biaxial variants of them, that set of characteristics might further improve.

Predicting maximally random jammed packing density of non-spherical hard particles via analytical continuation of fluid equation of state

We developed a formalism for accurately predicting the density of MRJ packing state of a wide spectrum of congruent non-spherical hard particles in 3D via analytical fluid EOS.

Influence of Softness on the Stability of Binary Colloidal Crystals

Mixtures of two types of nanoparticles can self-assemble into a wide variety of binary colloidal crystals (also called binary nanoparticle superlattices), which are interesting for their structural diversity and potential applications. Although so-called packing models-which usually treat the particles as "hard" with only excluded volume interactions-seem to explain many reported dense crystalline phases, these models often fail to predict the right structure. Here, we examine the role of soft repulsive interparticle interactions on binary colloidal crystals comprising two sizes of spherical particles; such "softness" can arise due to ligand shells or screened electrostatics. We determine the ground state phase diagram of binary systems of particles interacting with an additive inverse power law potential using a basin hopping algorithm to calculate the enthalpy of an extremely large pool of candidate structures. We find that a surprisingly small amount of softness can destabilize dense packings in favor of less densely packed structures, which provides further evidence that considerations beyond packing are necessary for describing many of the observed phases of binary colloidal crystals. Importantly, we find that several of the phases stabilized by softness, which are characterized by relatively few interparticle contacts and a tendency for local icosahedral order, are more likely to be observed experimentally than those predicted by packing models. We also report a previously unknown dense AB₄ phase and conduct free energy calculations to examine how the stability of several crystals will vary with temperature. Our results further our understanding of why particular binary colloidal crystals form and will be useful as a reference for experimentalists working with softly repulsive colloids.

Tuning the Porosity of Supraparticles

Supraparticles consisting of nano- or microparticles have potential applications as, for example, photonic crystals, drug carriers, or heterogeneous catalysts. To avoid the use of solvent or processing liquid, one can make supraparticles by evaporating droplets of aqueous suspensions from super-liquid-repellent surfaces. Herein, a method to adjust the porosity of supraparticles is described; a high porosity is desired, for example, in catalysis. To prepare highly porous TiO₂ supraparticles, polymer nanoparticles are co-dispersed in the suspension. Supraparticles are formed through evaporation of aqueous suspension droplets on superamphiphobic surfaces followed by calcination of the sacrificial polymer particles. The increase of porosity of up to 92% resulted in enhanced photocatalytic activity while maintaining sufficient mechanical stability.

Argon-neon binary diagram and ArNe2 Laves phase

Mixtures of argon and neon have been experimentally studied under high pressure. One stoichiometric compound, with ArNe₂ composition, is observed in this system. It is a Laves phase with a hexagonal MgZn₂ structure, stable up to at least 65 GPa, the highest pressure reached in the experiments. Its equation of state follows closely the one of an ideal Ar+2Ne mixture. The binary phase diagram of the Ar-Ne system resembles the diagram predicted for hard sphere mixtures with a similar atomic radius ratio, suggesting that no electronic interactions appear in this system in this pressure range. ArNe₂ can be a convenient quasihydrostatic pressure transmitting medium under moderate pressure.

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.

Methodology to construct large realizations of perfectly hyperuniform disordered packings

Disordered hyperuniform packings (or dispersions) are unusual amorphous two-phase materials that are endowed with exotic physical properties. Such hyperuniform systems are characterized by an anomalous suppression of volume-fraction fluctuations at infinitely long-wavelengths, compared to ordinary disordered materials. While there has been growing interest in such singular states of amorphous matter, a major obstacle has been an inability to produce large samples that are perfectly hyperuniform due to practical limitations of conventional numerical and experimental methods. To overcome these limitations, we introduce a general theoretical methodology to construct perfectly hyperuniform packings in d-dimensional Euclidean space R^{d}. Specifically, beginning with an initial general tessellation of space by disjoint cells that meets a "bounded-cell" condition, hard particles of general shape are placed inside each cell such that the local-cell particle packing fractions are identical to the global packing fraction. We prove that the constructed packings with a polydispersity in size are perfectly hyperuniform in the infinite-sample-size limit, regardless of particle shapes, positions, and numbers per cell. We use this theoretical formulation to devise an efficient and tunable algorithm to generate extremely large realizations of such packings. We employ two distinct initial tessellations: Voronoi as well as sphere tessellations. Beginning with Voronoi tessellations, we show that our algorithm can remarkably convert extremely large nonhyperuniform packings into hyperuniform ones in R^{2} and R^{3}. Implementing our theoretical methodology on sphere tessellations, we establish the hyperuniformity of the classical Hashin-Shtrikman multiscale coated-spheres structures, which are known to be two-phase media microstructures that possess optimal effective transport and elastic properties. A consequence of our work is a rigorous demonstration that packings that have identical tessellations can either be nonhyperuniform or hyperuniform by simply tuning local characteristics. It is noteworthy that our computationally designed hyperuniform two-phase systems can easily be fabricated via state-of-the-art methods, such as 2D photolithographic and 3D printing technologies. In addition, the tunability of our methodology offers a route for the discovery of novel disordered hyperuniform two-phase materials.

From jamming to fast compaction dynamics in granular binary mixtures

Binary granular mixtures are known to show various packing arrangements depending on both fractions and size ratios of their components. While the final packing fraction can be estimated by geometrical arguments, the dynamics of the pile submitted to gentle vibrations towards a dense state is seen to be highly size ratio dependent. We observe experimentally a diverging compaction characteristic time close to a critical size ratio, such that the grain mobility in the packing is the lowest close to the percolation threshold, when small particles can pass through the voids left by the large ones. Moreover, we evidence a fast compaction dynamics regime when the grain size ratio is large enough.

Hyperuniformity order metric of Barlow packings

The concept of hyperuniformity has been a useful tool in the study of density fluctuations at large length scales in systems ranging across the natural and mathematical sciences. One can rank a large class of hyperuniform systems by their ability to suppress long-range density fluctuations through the use of a hyperuniformity order metric Λ[over ¯]. We apply this order metric to the Barlow packings, which are the infinitely degenerate densest packings of identical rigid spheres that are distinguished by their stacking geometries and include the commonly known fcc lattice and hcp crystal. The "stealthy stacking" theorem implies that these packings are all stealthy hyperuniform, a strong type of hyperuniformity, which involves the suppression of scattering up to a wave vector K. We describe the geometry of three classes of Barlow packings, two disordered classes and small-period packings. In addition, we compute a lower bound on K for all Barlow packings. We compute Λ[over ¯] for the aforementioned three classes of Barlow packings and find that, to a very good approximation, it is linear in the fraction of fcc-like clusters, taking values between those of least-ordered hcp and most-ordered fcc. This implies that the value of Λ[over ¯] of all Barlow packings is primarily controlled by the local cluster geometry. These results highlight the special nature of anisotropic stacking disorder, which provides impetus for future research on the development of anisotropic order metrics and hyperuniformity properties.

Random sequential adsorption of cuboids

The subject of this study was random sequential adsorption of cuboids of axes length ratio of a : 1 : b for a ∈ [0.3, 1.0] and b ∈ [1.0, 2.0], and the aim of this study was to find a shape that provides the highest packing fraction. The obtained results show that the densest packing fraction is 0.401 87 ± 0.000 97 and is reached for axes ratios near cuboids of 0.75:1:1.30. Kinetics of packing growth was also studied, and it was observed that its power-law character seems not to be governed by the number of cuboid degrees of freedom. The microstructural properties of obtained packings were studied in terms of density correlation function and propagation of orientational ordering.

Massive replica exchange Monte Carlo algorithm: a tool to access high pressure thermodynamics of hard systems

We have explored the idea of producing the equilibrium equation of state, i.e. the pressure as a function of packing fraction, βP(φ), of a confined system up to very high pressures to yield the configuration that leads to the maximum packing fraction. For this purpose we have massively implemented the replica exchange Monte Carlo algorithm in graphics processing units (GPUs), in such a way that each GPU core handles a single simulation cell. This yields a very easy scheme to implement parallelization for a very large amount of replicas (thousands), which densely sample configuration space. We have tested this idea with a very well studied system, i.e. discs confined in a circular cavity, for a number of particles N ≤ 125. In all cases, our outcomes for configurations having maximum packing fractions are in perfect agreement with those already reported and conjectured optimal in the literature, for which there is no formal mathematical proof, strongly suggesting that they are indeed optimal configurations. Furthermore, in most cases, we have obtained the same function βP(φ), by compressing loose random configurations and by decompressing copies of the configuration having the largest packing fraction. This reveals numerically that the so obtained maximum packing configurations are the correct answer.

Hard convex lens-shaped particles: metastable, glassy and jammed states

We generate and study dense positionally and/or orientationally disordered, including jammed, monodisperse packings of hard convex lens-shaped particles (lenses). Relatively dense isotropic fluid configurations of lenses of various aspect ratios are slowly compressed via a Monte Carlo method based procedure. Under this compression protocol, while 'flat' lenses form a nematic fluid phase (where particles are positionally disordered but orientationally ordered) and 'globular' lenses form a plastic solid phase (where particles are positionally ordered but orientationally disordered), 'intermediate', neither 'flat' nor 'globular', lenses do not form either mesophase. In general, a crystal solid phase (where particles are both positionally and orientationally ordered) does not spontaneously form during lengthy numerical simulation runs. In correspondence to those volume fractions at which a transition to the crystal solid phase would occur in equilibrium, a 'downturn' is observed in the inverse compressibility factor versus volume fraction curve beyond which this curve behaves essentially linearly. This allows us to estimate the volume fraction at jamming of the dense non-crystalline packings so generated. These packings are nematic for 'flat' lenses and plastic for 'globular' lenses, while they are robustly isotropic for 'intermediate' lenses, as confirmed by the calculation of the τ order metric, among other quantities. The structure factors S(k) of the corresponding jammed states tend to zero as the wavenumber k goes to zero, indicating they are effectively hyperuniform (i.e., their infinite-wavelength density fluctuations are anomalously suppressed). Among all possible lens shapes, 'intermediate' lenses with aspect ratio around 2/3 are special because they are those that reach the highest volume fractions at jamming while being positionally and orientationally disordered and these volume fractions are as high as those reached by nematic jammed states of 'flat' lenses and plastic jammed states of 'globular' lenses. All of their attributes, taken together, make such 'intermediate' lens packings particularly good glass-forming materials.