Wave propagation and band tails of two-dimensional disordered systems in the thermodynamic limit.

Understanding the nature and formation of band gaps associated with the propagation of electromagnetic, electronic, or elastic waves in disordered materials as a function of system size presents fundamental and technological challenges. In particular, a basic question is whether band gaps in disordered systems exist in the thermodynamic limit. To explore this issue, we use a two-stage ensemble approach to study the formation of complete photonic band gaps (PBGs) for a sequence of increasingly large systems spanning a broad range of two-dimensional photonic network solids with varying degrees of local and global order, including hyperuniform and nonhyperuniform types. We discover that the gap in the density of states exhibits exponential tails and the apparent PBGs rapidly close as the system size increases for nearly all disordered networks considered. The only exceptions are sufficiently stealthy hyperuniform cases for which the band gaps remain open and the band tails exhibit a desirable power-law scaling reminiscent of the PBG behavior of photonic crystals in the thermodynamic limit.

Designer pair statistics of disordered many-particle systems with novel properties.

The knowledge of exact analytical functional forms for the pair correlation function g2(r) and its corresponding structure factor S(k) of disordered many-particle systems is limited. For fundamental and practical reasons, it is highly desirable to add to the existing database of analytical functional forms for such pair statistics. Here, we design a plethora of such pair functions in direct and Fourier spaces across the first three Euclidean space dimensions that are realizable by diverse many-particle systems with varying degrees of correlated disorder across length scales, spanning a wide spectrum of hyperuniform, typical nonhyperuniform, and antihyperuniform ones. This is accomplished by utilizing an efficient inverse algorithm that determines equilibrium states with up to pair interactions at positive temperatures that precisely match targeted forms for both g2(r) and S(k). Among other results, we realize an example with the strongest hyperuniform property among known positive-temperature equilibrium states, critical-point systems (implying unusual 1D systems with phase transitions) that are not in the Ising universality class, systems that attain self-similar pair statistics under Fourier transformation, and an experimentally feasible polymer model. We show that our pair functions enable one to achieve many-particle systems with a wide range of translational order and self-diffusion coefficients D, which are inversely related to one another. One can design other realizable pair statistics via linear combinations of our functions or by applying our inverse procedure to other desirable functional forms. Our approach facilitates the inverse design of materials with desirable physical and chemical properties by tuning their pair statistics.

Local number fluctuations in ordered and disordered phases of water across temperatures: Higher-order moments and degrees of tetrahedrality.

The isothermal compressibility (i.e., related to the asymptotic number variance) of equilibrium liquid water as a function of temperature is minimal under near-ambient conditions. This anomalous non-monotonic temperature dependence is due to a balance between thermal fluctuations and the formation of tetrahedral hydrogen-bond networks. Since tetrahedrality is a many-body property, it will also influence the higher-order moments of density fluctuations, including the skewness and kurtosis. To gain a more complete picture, we examine these higher-order moments that encapsulate many-body correlations using a recently developed, advanced platform for local density fluctuations. We study an extensive set of simulated phases of water across a range of temperatures (80-1600 K) with various degrees of tetrahedrality, including ice phases, equilibrium liquid water, supercritical water, and disordered nonequilibrium quenches. We find clear signatures of tetrahedrality in the higher-order moments, including the skewness and excess kurtosis, which scale for all cases with the degree of tetrahedrality. More importantly, this scaling behavior leads to non-monotonic temperature dependencies in the higher-order moments for both equilibrium and non-equilibrium phases. Specifically, under near-ambient conditions, the higher-order moments vanish most rapidly for large length scales, and the distribution quickly converges to a Gaussian in our metric. However, under non-ambient conditions, higher-order moments vanish more slowly and hence become more relevant, especially for improving information-theoretic approximations of hydrophobic solubility. The temperature non-monotonicity that we observe in the full distribution across length scales could shed light on water's nested anomalies, i.e., reveal new links between structural, dynamic, and thermodynamic anomalies.

Equilibrium states corresponding to targeted hyperuniform nonequilibrium pair statistics.

The Zhang-Torquato conjecture [G. Zhang and S. Torquato, Phys. Rev. E, 2020, 101, 032124.] states that any realizable pair correlation function g2(r) or structure factor S(k) of a translationally invariant nonequilibrium system can be attained by an equilibrium ensemble involving only (up to) effective two-body interactions. To further test and study this conjecture, we consider two singular nonequilibrium models of recent interest that also have the exotic hyperuniformity property: a 2D "perfect glass" and a 3D critical absorbing-state model. We find that each nonequilibrium target can be achieved accurately by equilibrium states with effective one- and two-body potentials, lending further support to the conjecture. To characterize the structural degeneracy of such a nonequilibrium-equilibrium correspondence, we compute higher-order statistics for both models, as well as those for a hyperuniform 3D uniformly randomized lattice (URL), whose higher-order statistics can be very precisely ascertained. Interestingly, we find that the differences in the higher-order statistics between nonequilibrium and equilibrium systems with matching pair statistics, as measured by the "hole" probability distribution, provide measures of the degree to which a system is out of equilibrium. We show that all three systems studied possess the bounded-hole property and that holes near the maximum hole size in the URL are much rarer than those in the underlying simple cubic lattice. Remarkably, upon quenching, the effective potentials for all three systems possess local energy minima (i.e., inherent structures) with stronger forms of hyperuniformity compared to their target counterparts. Our methods are expected to facilitate the self-assembly of tunable hyperuniform soft-matter systems.

Multifunctional composites for elastic and electromagnetic wave propagation.

Composites are ideally suited to achieve desirable multifunctional effective properties since the best properties of different materials can be judiciously combined with designed microstructures. Here, we establish cross-property relations for two-phase composite media that link effective elastic and electromagnetic wave characteristics to one another, including the respective effective wave speeds and attenuation coefficients, which facilitate multifunctional material design. This is achieved by deriving accurate formulas for the effective electromagnetic and elastodynamic properties that depend on the wavelengths of the incident waves and the microstructure via the spectral density. Our formulas enable us to explore the wave characteristics of a broad class of disordered microstructures because they apply, unlike conventional formulas, to a wide range of incident wavelengths (i.e., well beyond the long-wavelength regime). This capability enables us to study the dynamic properties of exotic disordered "hyperuniform" composites that can have advantages over crystalline ones, such as nearly optimal, direction-independent properties and robustness against defects. We specifically show that disordered "stealthy" hyperuniform microstructures exhibit novel wave characteristics (e.g., low-pass filters that transmit waves "isotropically" up to a finite wavenumber). Our cross-property relations for the effective wave characteristics can be applied to design multifunctional composites via inverse techniques. Design examples include structural components that require high stiffness and electromagnetic absorption; heat sinks for central processing units and sound-absorbing housings for motors that have to efficiently emit thermal radiation and suppress mechanical vibrations; and nondestructive evaluation of the elastic moduli of materials from the effective dielectric response.

Realizability of iso-g2 processes via effective pair interactions.

An outstanding problem in statistical mechanics is the determination of whether prescribed functional forms of the pair correlation function g2(r) [or equivalently, structure factor S(k)] at some number density ρ can be achieved by many-body systems in d-dimensional Euclidean space. The Zhang-Torquato conjecture states that any realizable set of pair statistics, whether from a nonequilibrium or equilibrium system, can be achieved by equilibrium systems involving up to two-body interactions. To further test this conjecture, we study the realizability problem of the nonequilibrium iso-g2 process, i.e., the determination of density-dependent effective potentials that yield equilibrium states in which g2 remains invariant for a positive range of densities. Using a precise inverse algorithm that determines effective potentials that match hypothesized functional forms of g2(r) for all r and S(k) for all k, we show that the unit-step function g2, which is the zero-density limit of the hard-sphere potential, is remarkably realizable up to the packing fraction ϕ = 0.49 for d = 1. For d = 2 and 3, it is realizable up to the maximum "terminal" packing fraction ϕc = 1/2d, at which the systems are hyperuniform, implying that the explicitly known necessary conditions for realizability are sufficient up through ϕc. For ϕ near but below ϕc, the large-r behaviors of the effective potentials are given exactly by the functional forms exp[ - κ(ϕ)r] for d = 1, r-1/2 exp[ - κ(ϕ)r] for d = 2, and r-1 exp[ - κ(ϕ)r] (Yukawa form) for d = 3, where κ-1(ϕ) is a screening length, and for ϕ = ϕc, the potentials at large r are given by the pure Coulomb forms in the respective dimensions as predicted by Torquato and Stillinger [Phys. Rev. E 68, 041113 (2003)]. We also find that the effective potential for the pair statistics of the 3D "ghost" random sequential addition at the maximum packing fraction ϕc = 1/8 is much shorter ranged than that for the 3D unit-step function g2 at ϕc; thus, it does not constrain the realizability of the unit-step function g2. Our inverse methodology yields effective potentials for realizable targets, and, as expected, it does not reach convergence for a target that is known to be non-realizable, despite the fact that it satisfies all known explicit necessary conditions. Our findings demonstrate that exploring the iso-g2 process via our inverse methodology is an effective and robust means to tackle the realizability problem and is expected to facilitate the design of novel nanoparticle systems with density-dependent effective potentials, including exotic hyperuniform states of matter.

Phoamtonic designs yield sizeable 3D photonic band gaps.

We show that it is possible to construct foam-based heterostructures with complete photonic band gaps. Three-dimensional foams are promising candidates for the self-organization of large photonic networks with combinations of physical characteristics that may be useful for applications. The largest band gap found is based on 3D Weaire-Phelan foam, a structure that was originally introduced as a solution to the Kelvin problem of finding the 3D tessellation composed of equal-volume cells that has the least surface area. The photonic band gap has a maximal size of 16.9% (at a volume fraction of 21.6% for a dielectric contrast [Formula: see text]) and a high degree of isotropy, properties that are advantageous in designing photonic waveguides and circuits. We also present results for 2 other foam-based heterostructures based on Kelvin and C15 foams that have somewhat smaller but still significant band gaps.

Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks.

Through an extensive series of high-precision numerical computations of the optimal complete photonic band gap (PBG) as a function of dielectric contrast α for a variety of crystal and disordered heterostructures, we reveal striking universal behaviors of the gap sensitivity S(α)≡dΔ(α)/dα, the first derivative of the optimal gap-to-midgap ratio Δ(α). In particular, for all our crystal networks, S(α) takes a universal form that is well approximated by the analytic formula for a 1D quarter-wave stack, S_{QWS}(α). Even more surprisingly, the values of S(α) for our disordered networks converge to S_{QWS}(α) for sufficiently large α. A deeper understanding of the simplicity of this universal behavior may provide fundamental insights about PBG formation and guidance in the design of novel photonic heterostructures.

Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions.

The creation of disordered hyperuniform materials with extraordinary optical properties (e.g., large complete photonic band gaps) requires a capacity to synthesize large samples that are effectively hyperuniform down to the nanoscale. Motivated by this challenge, we propose a feasible equilibrium fabrication protocol using binary paramagnetic colloidal particles confined in a 2D plane. The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system effectively hyperuniform. We show that hyperuniformity is a general consequence of low isothermal compressibilities, which makes our protocol suitable to treat more general systems with other long-ranged interactions, dimensionalities, and/or polydispersity. Our methodology paves the way to synthesize large photonic hyperuniform materials that function in the visible to infrared range and hence may accelerate the discovery of novel photonic materials.

Sensitivity of pair statistics on pair potentials in many-body systems.

We study the sensitivity and practicality of Henderson's theorem in classical statistical mechanics, which states that the pair potential v(r) that gives rise to a given pair correlation function g2(r) [or equivalently, the structure factor S(k)] in a classical many-body system at number density ρ and temperature T is unique up to an additive constant. While widely invoked in inverse-problem studies, the utility of the theorem has not been quantitatively scrutinized to any large degree. We show that Henderson's theorem has practical shortcomings for disordered and ordered phases for certain densities and temperatures. Using proposed sensitivity metrics, we identify illustrative cases in which distinctly different potential functions give very similar pair correlation functions and/or structure factors up to their corresponding correlation lengths. Our results reveal that due to a limited range and precision of pair information in either direct or reciprocal space, there is effective ambiguity of solutions to inverse problems that utilize pair information only, and more caution must be exercised when one claims the uniqueness of any resulting effective pair potential found in practice. We have also identified systems that possess virtually identical pair statistics but have distinctly different higher-order correlations. Such differences should be reflected in their individually distinct dynamics (e.g., glassy behaviors). Finally, we prove a more general version of Henderson's theorem that extends the uniqueness statement to include potentials that involve two- and higher-body interactions.

Self-Similar Dynamics of Nuclear Packing in the Early Drosophila Embryo.

Embryonic development starts with cleavages, a rapid sequence of reductive divisions that result in an exponential increase of cell number without changing the overall size of the embryo. In Drosophila, the final four rounds of cleavages occur at the surface of the embryo and give rise to ∼6000 nuclei under a common plasma membrane. We use live imaging to study the dynamics of this process and to characterize the emergent nuclear packing in this system. We show that the characteristic length scale of the internuclear interaction scales with the density, which allows the densifying embryo to sustain the level of structural order at progressively smaller length scales. This is different from nonliving materials, which typically undergo disorder-order transition upon compression. To explain this dynamics, we use a particle-based model that accounts for density-dependent nuclear interactions and synchronous divisions. We reproduce the pair statistics of the disordered packings observed in embryos and recover the scaling relation between the characteristic length scale and the density both in real and reciprocal space. This result reveals how the embryo can robustly preserve the nuclear-packing structure while being densified. In addition to providing quantitative description of self-similar dynamics of nuclear packings, this model generates dynamic meshes for the computational analysis of pattern formation and tissue morphogenesis.

Structural degeneracy in pair distance distributions.

A traditional basic descriptor of many-particle systems has been the distribution of interparticle pair distances. In the case of structureless particles at thermal equilibrium, with just additive pair interactions, this suffices to determine pressure and mean energy. However, it is usually the case that a given set of pair distances can emerge from a multiplicity of distinguishable many-particle configurations. This paper focuses on the ways in which such a configuration detail can be overlooked. After providing some elementary small-system examples in which full pair distance specification still permits distinct configurational pattern ambiguity, subsequent analysis concentrates on large-system classical canonical ensembles. In that context, configurational degeneracy is analyzed in two-dimensional systems for the shape distribution of triangles, whose chirality occurrence can be controlled by suitable three-particle interactions. For many-particle systems in three dimensions, the possibility is explored that a set of three-particle "pair-invisible" interactions can exist which modify the three-particle distribution function, but which have no effect on the pair distribution function, and thus remain undetected by conventional diffraction experiments. For illustration, a specific mathematical example is presented, applicable to the case where two-particle interactions vanish.

Light Localization in Local Isomorphism Classes of Quasicrystals.

We study a continuum of photonic quasicrystal heterostructures derived from local isomorphism (LI) classes of pentagonal quasicrystal tilings. These tilings are obtained by direct projection from a five-dimensional hypercubic lattice. We demonstrate that, with the sole exception of the Penrose LI class, all other LI classes result in degenerate, effectively localized states, with precisely predictable and tunable properties (frequencies, frequency splittings, and densities). We show that localization and tunability are related to a mathematical property of the pattern known as "restorability," i.e., whether the tiling can be uniquely specified given only a set of rules that fix all allowed clusters smaller than some bound.

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.

Binary mixtures of charged colloids: a potential route to synthesize disordered hyperuniform materials.

Disordered hyperuniform materials are a new, exotic class of amorphous matter that exhibits crystal-like behavior, in the sense that volume-fraction fluctuations are suppressed at large length scales, and yet they are isotropic and do not display diffraction Bragg peaks. These materials are endowed with novel photonic, phononic, transport and mechanical properties, which are useful for a wide range of applications. Motivated by the need to fabricate large samples of disordered hyperuniform systems at the nanoscale, we study the small-wavenumber behavior of the spectral density of binary mixtures of charged colloids in suspension. The interaction between the colloids is approximated by a repulsive hard-core Yukawa potential. We find that at dimensionless temperatures below 0.05 and dimensionless inverse screening lengths below 1.0, which are experimentally accessible, the disordered systems become effectively hyperuniform. Moreover, as the temperature and inverse screening length decrease, the level of hyperuniformity increases, as quantified by the "hyperuniformity index". Our results suggest an alternative approach to synthesize large samples of effectively disordered hyperuniform materials at the nanoscale under standard laboratory conditions. In contrast with the usual route to synthesize disordered hyperuniform materials by jamming particles, this approach is free from the burden of applying high pressure to compress the systems.

Large-Scale Structure and Hyperuniformity of Amorphous Ices.

We investigate the large-scale structure of amorphous ices and transitions between their different forms by quantifying their large-scale density fluctuations. Specifically, we simulate the isothermal compression of low-density amorphous ice (LDA) and hexagonal ice to produce high-density amorphous ice (HDA). Both HDA and LDA are nearly hyperuniform; i.e., they are characterized by an anomalous suppression of large-scale density fluctuations. By contrast, in correspondence with the nonequilibrium phase transitions to HDA, the presence of structural heterogeneities strongly suppresses the hyperuniformity and the system becomes hyposurficial (devoid of "surface-area fluctuations"). Our investigation challenges the largely accepted "frozen-liquid" picture, which views glasses as structurally arrested liquids. Beyond implications for water, our findings enrich our understanding of pressure-induced structural transformations in glasses.

Existence of isostatic, maximally random jammed monodisperse hard-disk packings.

We generate jammed packings of monodisperse circular hard-disks in two dimensions using the Torquato-Jiao sequential linear programming algorithm. The packings display a wide diversity of packing fractions, average coordination numbers, and order as measured by standard scalar order metrics. This geometric-structure approach enables us to show the existence of relatively large maximally random jammed (MRJ) packings with exactly isostatic jammed backbones and a packing fraction (including rattlers) of [Formula: see text]. By contrast, the concept of random close packing (RCP) that identifies the most probable packings as the most disordered misleadingly identifies highly ordered disk packings as RCP in 2D. Fundamental structural descriptors such as the pair correlation function, structure factor, and Voronoi statistics show a strong contrast between the MRJ state and the typical hyperstatic, polycrystalline packings with [Formula: see text] that are more commonly obtained using standard packing protocols. Establishing that the MRJ state for monodisperse hard disks is isostatic and qualitatively distinct from commonly observed polycrystalline packings contradicts conventional wisdom that such a disordered, isostatic packing does not exist due to a lack of geometrical frustration and sheds light on the nature of disorder. This prompts the question of whether an algorithm may be designed that is strongly biased toward generating the monodisperse disk MRJ state.

Structural Characterization and Statistical-Mechanical Model of Epidermal Patterns.

In proliferating epithelia of mammalian skin, cells of irregular polygon-like shapes pack into complex, nearly flat two-dimensional structures that are pliable to deformations. In this work, we employ various sensitive correlation functions to quantitatively characterize structural features of evolving packings of epithelial cells across length scales in mouse skin. We find that the pair statistics in direct space (correlation function) and Fourier space (structure factor) of the cell centroids in the early stages of embryonic development show structural directional dependence (statistical anisotropy), which is a reflection of the fact that cells are stretched, which promotes uniaxial growth along the epithelial plane. In the late stages, the patterns tend toward statistically isotropic states, as cells attain global polarization and epidermal growth shifts to produce the skin's outer stratified layers. We construct a minimalist four-component statistical-mechanical model involving effective isotropic pair interactions consisting of hard-core repulsion and extra short-range soft-core repulsion beyond the hard core, whose length scale is roughly the same as the hard core. The model parameters are optimized to match the sample pair statistics in both direct and Fourier spaces. By doing this, the parameters are biologically constrained. In contrast with many vertex-based models, our statistical-mechanical model does not explicitly incorporate information about the cell shapes and interfacial energy between cells; nonetheless, our model predicts essentially the same polygonal shape distribution and size disparity of cells found in experiments, as measured by Voronoi statistics. Moreover, our simulated equilibrium liquid-like configurations are able to match other nontrivial unconstrained statistics, which is a testament to the power and novelty of the model. The array of structural descriptors that we deploy enable us to distinguish between normal, mechanically deformed, and pathological skin tissues. Our statistical-mechanical model enables one to generate tissue microstructure at will for further analysis. We also discuss ways in which our model might be extended to better understand morphogenesis (in particular the emergence of planar cell polarity), wound healing, and disease-progression processes in skin, and how it could be applied to the design of synthetic tissues.