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Layered “mosaic” metal-halide perovskite materials display a wide-variety of microstructures that span the order–disorder spectrum and can be tuned via the composition of their constituent B-site octahedral species. Such materials are typically modeled using computationally expensive ab initio methods, but these approaches are greatly limited to small sample sizes. Here, we develop a highly efficient hard-particle packing algorithm to model large samples of these layered complex alloys that enables an accurate determination of the geometrical and topological properties of the B-site arrangements within the plane of the inorganic layers across length scales. Our results are in good agreement with various experiments and, therefore, our algorithm bypasses the need for full-blown ab initio calculations. The accurate predictive power of our algorithm demonstrates how our minimalist hard-particle model effectively captures complex interactions and dynamics like incoherent thermal motion, out-of-plane octahedral tilting, and bond compression/stretching. We specifically show that the composition-dependent miscibility predicted by our algorithm for certain silver–iron and copper–indium layered alloys is consistent with previous experimental observations. We further quantify the degree of mixing in the simulated structures across length scales using our recently developed sensitive “mixing” metric. The large structural snapshots provided by our algorithm also shed light on previous experimentally measured magnetic properties of a copper–indium system. The generalization of our algorithm to model 3D perovskite alloys is also discussed. In summary, our packing model and mixing metric enable one to accurately explore the enormous space of hypothetical layered mosaic alloy compositions and identify materials with potentially desirable optoelectronic and magnetic properties. 

Nonequilibrium hard-sphere hyperuniform glasses (left panel) exist within the stealthy hyperuniform ground-state manifold. Accessible packing fractions for allχwithin the disordered regime (grey region in the left panel). 

Isospectrality is a general fundamental concept often involving whether various operators can have identical spectra, i.e., the same set of eigenvalues. In the context of the Laplacian operator, the famous question “Can one hear the shape of a drum?” concerns whether different shaped drums can have the same vibrational modes. The isospectrality of a lattice in $d$-dimensional Euclidean space ${\mathbb{R}}^d$is a tantamount to whether it is uniquely determined by its theta series, i.e., the radial distribution function $g_2\left(r\right)$. While much is known about the isospectrality of Bravais lattices across dimensions, little is known about this question of more general crystal (periodic) structures with an $n$-particle basis ( $n\ge 2$). Here, we ask what is $n_{\text{min}}\left(d\right)$, the minimum value of $n$for inequivalent (i.e., unrelated by isometric symmetries) crystals with the same theta function in space dimension $d$? To answer these questions, we use rigorous methods as well as a precise numerical algorithm that enables us to determine the minimum multiparticle basis of inequivalent isospectral crystals. Our algorithm identifies isospectral four-, three- and two-particle bases in one, two, and three spatial dimensions, respectively. For many of these isospectral crystals, we rigorously show that they indeed possess identically the same $g_2\left(r\right)$'s for all values of $r$. Based on our analyses, we conjecture that $n_{\text{min}}\left(d\right)=4$, 3, 2 for $d=1$, 2, 3, respectively. The identification of isospectral crystals enables one to study the degeneracy of the ground-state under the action of isotropic pair potentials. Indeed, using inverse statistical-mechanical techniques, we find an isotropic pair potential whose low-temperature configurations in two dimensions obtained via simulated annealing can lead to both of two isospectral crystal structures with $n=3$, the proportion of which can be controlled by the cooling rate. Our findings provide general insights into the structural and ground-state degeneracies of crystal structures as determined by radial pair information. Published by the American Physical Society2024 

Formulating order metrics that sensitively quantify the degree of order/disorder in many-particle systems in $d$-dimensional Euclidean space ${\mathbb{R}}^d$across length scales is an outstanding challenge in physics, chemistry, and materials science. Since an infinite set of $n$-particle correlation functions is required to fully characterize a system, one must settle for a reduced set of structural information, in practice. We initiate a program to use the local number variance ${\sigma }_N^2\left(R\right)$associated with a spherical sampling window of radius $R$(which encodes pair correlations) and an integral measure derived from it ${\mathrm{\Sigma }}_N(R_i,R_j)$that depends on two specified radial distances $R_i$and $R_j$. Across the first three space dimensions ( $d=1,2,3$), we find these metrics can sensitively describe and categorize the degree of order/disorder of 41 different models of antihyperuniform, nonhyperuniform, disordered hyperuniform, and ordered hyperuniform many-particle systems at a specified length scale $R$. Using our local variance metrics, we demonstrate the importance of assessing order/disorder with respect to a specific value of $R$. These local order metrics could also aid in the inverse design of structures with prescribed length-scale-specific degrees of order/disorder that yield desired physical properties. In future work, it would be fruitful to explore the use of higher-order moments of the number of points within a spherical window of radius $R$[S. Torquato , ] to devise even more sensitive order metrics. Published by the American Physical Society2024 

The probability of finding a spherical “hole” of a given radius r contains crucial structural information about many-body systems. Such hole statistics, including the void conditional nearest-neighbor probability functions GV(r), have been well studied for hard-sphere fluids in d-dimensional Euclidean space Rd. However, little is known about these functions for hard-sphere crystals for values of r beyond the hard-sphere diameter, as large holes are extremely rare in crystal phases. To overcome these computational challenges, we introduce a biased-sampling scheme that accurately determines hole statistics for equilibrium hard spheres on ranges of r that far extend those that could be previously explored. We discover that GV(r) in crystal and hexatic states exhibits oscillations whose amplitudes increase rapidly with the packing fraction, which stands in contrast to GV(r) that monotonically increases with r for fluid states. The oscillations in GV(r) for 2D crystals are strongly correlated with the local orientational order metric in the vicinity of the holes, and variations in GV(r) for 3D states indicate a transition between tetrahedral and octahedral holes, demonstrating the power of GV(r) as a probe of local coordination geometry. To further study the statistics of interparticle spacing in hard-sphere systems, we compute the local packing fraction distribution f(ϕl) of Delaunay cells and find that, for d ≤ 3, the excess kurtosis of f(ϕl) switches sign at a certain transitional global packing fraction. Our accurate methods to access hole statistics in hard-sphere crystals at the challenging intermediate length scales reported here can be applied to understand the important problem of solvation and hydrophobicity in water at such length scales. 

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. 

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. 

Abstract In this study, multiscale physics‐informed neural networks (MscalePINNs) are employed for the inverse design of finite‐size photonic materials with stealthy hyperuniform (SHU) disordered geometries. Specifically, MscalePINNs are shown to capture the fast spatial variations of complex fields scattered by arrays of dielectric nanocylinders arranged according to isotropic SHU point patterns, thus enabling a systematic methodology to inversely retrieve their effective dielectric profiles. This approach extends the recently developed high‐frequency homogenization theory of hyperuniform media and retrieves more general permittivity profiles for applications‐relevant finite‐size SHU and optical systems, unveiling unique features related to their isotropic nature. In particular, the existence of a transparency region beyond the long‐wavelength approximation is numerically corroborated, enabling the retrieval of effective and isotropic locally homogeneous media even without disorder‐averaging, in contrast to the case of uncorrelated Poisson random patterns. The flexible multiscale network approach introduced here enables the efficient inverse design of more general effective media and finite‐size optical metamaterials with isotropic electromagnetic responses beyond the limitations of traditional homogenization theories.