Modeling heterogeneous materials via two-point correlation functions: basic principles

Deep-subwavelength engineering of stealthy hyperuniformity

Light behaviours in disordered materials have been of research interest primarily at length scales beyond or comparable to the wavelength of light, because order and disorder are often believed to be almost indistinguishable in the subwavelength regime according to effective medium theory (EMT). However, it was demonstrated that the breakdown of EMT occurs even at deep-subwavelength scales when interface phenomena, such as the Goos-Hänchen effect, dominate light flows. Here we develop the engineering of disordered multilayers at deep-subwavelength scales to achieve angle-selective manipulation of wave localization. To examine the disorder-dependent EMT breakdown, we classify the intermediate regime of microstructural phases between deep-subwavelength crystals and uncorrelated disorder through the concept of stealthy hyperuniformity (SHU). We devise material phase transitions from SHU to uncorrelated disorder for distinct angular responses of wave localization by tailoring the short-range and long-range order in SHU multilayers. The result paves the way to the realization of deep-subwavelength disordered metamaterials, bridging the fields of disordered photonics and metamaterials.

Three-dimensional construction of hyperuniform, nonhyperuniform, and antihyperuniform disordered heterogeneous materials and their transport properties via spectral density functions

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function χ[over ̃]_{_{V}}(k), which can be obtained experimentally from scattering data, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability S(t) and effective dynamic dielectric constant ε_{e} for electromagnetic wave propagation. Moreover, χ[over ̃]_{_{V}}(k) determines rigorous upper bounds on the fluid permeability K. Given the importance of χ[over ̃]_{_{V}}(k), we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical functional forms for χ[over ̃]_{_{V}}(k) that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by autocovariance function χ_{_{V}}(r) with a power-law tail, resulting in microstructures that contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability S(t) and estimate the fluid permeability K associated with all of the constructed materials directly from the corresponding spectral densities. Although it is well established that the long-time asymptotic scaling behavior of S(t) only depends on the functional form of χ[over ̃]_{_{V}}(k), with the stealthy hyperuniform and antihyperuniform media, respectively, achieving the most and least efficient transport, we find that varying the length-scale parameter within each class of χ[over ̃]_{_{V}}(k) functions can also lead to orders of magnitude variation of S(t) at intermediate and long time scales. Moreover, we find that increasing the solid volume fraction ϕ_{1} and correlation length a in the constructed media generally leads to a decrease in the dimensionless fluid permeability K/a^{2}, while the antihyperuniform media possess the largest K/a^{2} among the four classes of materials with the same ϕ_{1} and a. These results indicate the feasibility of employing parameterized spectral densities for designing composites with targeted transport properties.

Machine learning-enabled multiscale modeling platform for damage sensing digital twin in piezoelectric composite structures

Nondestructive evaluation (NDE) of aerospace structures plays a crucial role in their successful operation under harsh environments. Most NDE methods, however, lack real-time in-situ predictive capabilities of evolving damage and are conducted in a post-mortem manner. This paper proposes a damage-sensing digital twin (DT) for piezoelectric composite structures incorporating microscale morphology and mechanisms into structural damage to address this shortcoming. The DT framework consists of a two-step computational process integrating multiscale-multiphysics modeling with machine learning (ML) tools to detect damage progression in the piezoelectric composite structure using electrical signal measurements at a few surface points. A parametrically upscaled coupled constitutive damage mechanics (PUCCDM) model is developed for this DT with the representation of microstructural morphology and mechanisms in macroscopic constitutive relations. Artificial neural network operates on micro-electro-mechanical data to derive PUCCDM constitutive coefficients as functions of underlying representative aggregated microstructural parameters (RAMPs). The PUCCDM model-simulated macroscopic electromechanical and damage fields, in conjunction with RAMPs, provide a comprehensive time-dependent dataset for a convolutional long-short-term memory (ConvLSTM) network to learn microstructure-dependent electrical and damage field correlations. The DT is consequently used to successfully predict location-specific damage from electrical signals at a limited number of sensors.

Bioinspired Hierarchical Porous Architecture for Enhanced Kinetics and Mechanical Integrity in Thick Cathode

Increasing the thickness of the electrodes is considered the primary strategy to elevate battery energy density. However, as the thickness increases, rate performance, cycling performance, and mechanical stability are affected due to the sluggish ion transfer kinetics and compromised structural integrity. Inspired by the natural hierarchical porous structure of trees, electrodes with bioinspired architecture are fabricated to address these challenges. Specifically, electrodes with aligned columns consist of tree-inspired vertical channels, and hierarchical pores are constructed by screen printing and ice-templating, imparting enhanced electrochemical and mechanical performance. Employing an aqueous-based binder, the LiNi0.8Mn0.1Co0.1O2 cathode achieves a high areal energy density of 15.1 mWh cm-2 at a rate of 1C at mass loading of 26.0 mg cm-2, benefitting from the multiscale pores that elevated charge transfer kinetics in the thick electrode. The electrodes demonstrate capacity retention of 90% at the 100th cycle at a high current density of 5.2 mA cm-2. To understand the mechanisms that promote electrode performance, simplified electro-chemo-mechanical models are developed, the drying process and the charge-discharge process are simulated. The simulation results suggested that the improved performance of the designed electrode benefits from the lower ohmic overpotential and less strain gradient and stress concentration due to the hierarchical porous architecture.

Microstructural characterization of DEM-based random packings of monodisperse and polydisperse non-convex particles

Understanding of hard particles in morphologies and sizes on microstructures of particle random packings is of significance to evaluate physical and mechanical properties of many discrete media, such as granular materials, colloids, porous ceramics, active cells, and concrete. The majority of previous lines of research mainly dedicated microstructure analysis of convex particles, such as spheres, ellipsoids, spherocylinders, cylinders, and convex-polyhedra, whereas little is known about non-convex particles that are more close to practical discrete objects in nature. In this study, the non-convex morphology of a three-dimensional particle is devised by using a mathematical-controllable parameterized method, which contains two construction modes, namely, the uniformly distributed contraction centers and the randomly distributed contraction centers. Accordingly, three shape parameters are conceived to regulate the particle geometrical morphology from a perfect sphere to arbitrary non-convexities. Random packing models of hard non-convex particles with mono-/poly-dispersity in sizes are then established using the discrete element modeling Diverse microstructural indicators are utilized to characterize configurations of non-convex particle random packings. The compactness of non-convex particles in packings is characterized by the random close packing fraction fd and the corresponding average coordination number Z. In addition, four statistical descriptors, encompassing the radial distribution function g(r), two-point probability function S2(i)(r), lineal-path function L(i)(r), and cumulative pore size distribution function F(δ), are exploited to demonstrate the high-order microstructure information of non-convex particle random packings. The results demonstrate that the particle shape and size distribution have significant effects on Z and fd; the construction mode of the randomly distributed contraction centers can yield higher fd than that of the uniformly distributed contraction centers, in which the upper limit of fd approaches to 0.632 for monodisperse sphere packings. Moreover, non-convex particles of sizes following the famous Fuller distribution of the power-law distribution of the exponent q = 2.5, have the highest fd (≈0.761) with respect to other q. In contrast, the particle shapes have an almost negligible effect on the four statistical descriptors, but they are remarkably sensitive to particle packing fraction fp and size distribution. The results can provide sound guidance for custom-design of granular media by tailoring specific microstructures of particles.

Modeling complex polycrystalline alloys using a Generative Adversarial Network enabled computational platform

Creating statistically equivalent virtual microstructures (SEVM) for polycrystalline materials with complex microstructures that encompass multi-modal morphological and crystallographic distributions is a challenging enterprise. Cold spray-formed (CSF) AA7050 alloy containing coarse-grained prior particles and ultra-fine grains (UFG) and additively manufactured (AM) Ti64 alloys with alpha laths in beta substrates. The paper introduces an approach strategically integrating a Generative Adversarial Network (GAN) for multi-modal microstructures with a synthetic microstructure builder DREAM.3D for packing grains conforming to statistics in electron backscatter diffraction (EBSD) maps for generating SEVMs of CSF and AM alloy microstructures. A robust multiscale model is subsequently developed for self-consistent coupling of crystal plasticity finite element model (CPFEM) for coarse-grained crystals with an upscaled constitutive model for UFGs. Sub-volume elements are simulated for efficient computations and their responses are averaged for overall stress-strain response. The methods developed are important for image-based micromechanical modeling that is necessary for microstructure-property relations.

Topological Data Analysis for Particulate Gels

Soft gels, formed via the self-assembly of particulate materials, exhibit intricate multiscale structures that provide them with flexibility and resilience when subjected to external stresses. This work combines particle simulations and topological data analysis (TDA) to characterize the complex multiscale structure of soft gels. Our TDA analysis focuses on the use of the Euler characteristic, which is an interpretable and computationally scalable topological descriptor that is combined with filtration operations to obtain information on the geometric (local) and topological (global) structure of soft gels. We reduce the topological information obtained with TDA using principal component analysis (PCA) and show that this provides an informative low-dimensional representation of the gel structure. We use the proposed computational framework to investigate the influence of gel preparation (e.g., quench rate, volume fraction) on soft gel structure and to explore dynamic deformations that emerge under oscillatory shear in various response regimes (linear, nonlinear, and flow). Our analysis provides evidence of the existence of hierarchical structures in soft gels, which are not easily identifiable otherwise. Moreover, our analysis reveals direct correlations between topological changes of the gel structure under deformation and mechanical phenomena distinctive of gel materials, such as stiffening and yielding. In summary, we show that TDA facilitates the mathematical representation, quantification, and analysis of soft gel structures, extending traditional network analysis methods to capture both local and global organization.

Evaluation of three-point correlation functions from structural images on CPU and GPU architectures: Accounting for anisotropy effects

Structures, or spatial arrangements of matter and energy, including some fields (e.g., velocity or pressure) are ubiquitous in research applications and frequently require description for subsequent analysis, or stochastic reconstruction from limited data. The classical descriptors are two-point correlation functions (CFs), but the computation of three-point statistics is known to be advantageous in some cases as they can probe non-Gaussian signatures, not captured by their two-point counterparts. Moreover, n-point CFs with n≥3 are believed to possess larger information content and provide more information about studied structures. In this paper, we have developed algorithms and code to compute S_{3},C_{3},F_{sss},F_{ssv}, and F_{svv} with a right-angle and arbitrary triangle pattern. The former was believed to be faster to compute, but with the help of precomputed regular positions we achieved the same speed for arbitrary pattern. In this work we also implement and demonstrate computations of directional three-point CFs-for this purpose right-triangular pattern seems to be superior due to explicit orientation and high coverage. Moreover, we assess the errors in CFs' evaluation due to image or pattern rotations and show that they have minor effect on accuracy of computations. The execution times of our algorithms for the same number of samples are orders of magnitude lower than in existing published counterparts. We show that the volume of data produced gets unwieldy very easily, especially if computations are performed in frequency domain. For these reasons until information content of different sets of correlation functions with different "n-pointness" is known, advantages of CFs with n>3 are not clear. Nonetheless, developed algorithms and code are universal enough to be easily extendable to any n with increasing computational and random access memory (RAM) burden. All results are available as part of open-source package correlationfunctions.jl [V. Postnicov et al., Comput. Phys. Commun. 299, 109134 (2024)10.1016/j.cpc.2024.109134.]. As described in this paper, three-point CFs computations can be immediately applied in a great number of research applications, for example: (1) flow and transport velocity fields analysis or any data with non-Gaussian signatures, (2) deep learning for structural and physical properties, and (3) structure taxonomy and categorization. In all these and numerous other potential cases the ability to compute directional three-point functions may be crucial. Notably, the organization of the code functions allows computation of cross correlation, i.e., one can compute three-point CFs for multiphase images (while binary structures were used in this paper for simplicity of explanations).

Randomizing the growth of silica nanofibers for whiteness

In colloids, the shape influences the function. In silica, straight nanorods have already been synthesized from water-in-oil emulsions. By contrast, curly silica nanofibers have been less reported because the underlying growth mechanism remains unexplored, hindering further morphology control for applications. Herein, we describe the synthetic protocol for silica nanofibers with a tunable curliness based on the control of the water-in-oil emulsion droplets. Systematically decreasing the droplet size and increasing their contact angle, the Brownian motion of the droplets intensifies during the silica growth, thus increasing the random curliness of the nanofibers. This finding is supported by simplistic theoretical arguments and experimentally verified by varying the temperature to finely tune the curliness. Assembling these nanofibers toward porous disordered films enhances multiple scattering in the visible range, resulting in increased whiteness in contrast to films constructed by spherical and rod-like building units, which can be useful for, e.g., coatings and pigments.

Unlocking the potential: analyzing 3D microstructure of small-scale cement samples from space using deep learning

Due to the prohibitive cost of transporting raw materials into Space, in-situ materials along with cement-like binders are poised to be employed for extraterrestrial construction. A unique methodology for obtaining microstructural topology of cement samples hydrated in microgravity environment at the International Space Station (ISS) is presented here. Distinctive Scanning Electron Microscopy (SEM) micrographs of hardened tri-calcium silicate (C3S) samples were used as exemplars in a deep learning-based microstructure reconstruction framework. The proposed method aids in generation of an ensemble of microstructures that is inherently statistical in nature, by utilizing sparse experimental data such as the C3S samples hydrated in microgravity. The hydrated space-returned samples had exhibited higher porosity content (~70 %) with the portlandite phase assuming an elongated plate-like morphology. Qualitative assessment of the volumetric slices from the reconstructed volumes showcased similar visual characteristics to that of the target 2D exemplar. Detailed assessment of the reconstructed volumes was carried out using statistical descriptors, and was further compared against micro-CT virtual data. The reconstructed volumes captured the unique microstructural morphology of the hardened C3S samples of both space-returned and ground-based samples, and can be directly employed as Representative Volume Element (RVE) to characterize mechanical/transport properties.

Measuring structural nonstationarity: The use of imaging information to quantify homogeneity and inhomogeneity

Heterogeneity is the concept we encounter in numerous research areas and everyday life. While "not mixing apples and oranges" is easy to grasp, a more quantitative approach to such segregation is not always readily available. Consider the problem from a different angle: To what extent does one have to make apples more orange and oranges more "apple-shaped" to put them into the same basket (according to their appearance alone)? This question highlights the central problem of the blurred interface between heterogeneous and homogeneous, which also depends on the metrics used for its identification. This work uncovers the physics of structural stationarity quantification, based on correlation functions (CFs) and clustering based on CFs different between image subregions. By applying the methodology to a wide variety of synthetic and real images of binary porous media, we confirmed computationally that only periodically unit-celled structures and images produced by stationary processes with resolutions close to infinity are strictly stationary. Natural structures without recurring unit cells are only weakly stationary. We established a physically meaningful definition for these stationarity types and their distinction from nonstationarity. In addition, the importance of information content of the chosen metrics is highlighted and discussed. We believe the methodology as proposed in this contribution will find its way into numerous research areas dealing with materials, structures, and measurements and modeling based on structural imaging information.

Computational design of anisotropic stealthy hyperuniform composites with engineered directional scattering properties

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here we generalize the Fourier-space-based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)1359-645410.1016/j.actamat.2017.09.053] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ[over ̃]_{_{V}}(k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform microstructures with χ[over ̃]_{_{V}}(k)=0 for k∈Ω, i.e., complete suppression of scattering in an "exclusion" region Ω around the origin in Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped, and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions and are nonhyperuniform along others. We find that while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions impose strong constraints on the global symmetry of the resulting media, they can still possess structures at a local level that are nearly isotropic. Both the isotropic and anisotropic hyperuniform microstructures associated with the elliptical-disk, square, and rectangular Ω possess phase-inversion symmetry over certain range of volume fractions and a percolation threshold ϕ_{c}≈0.5. On the other hand, the directionally hyperuniform microstructures associated with the butterfly-shaped and lemniscate-shaped Ω do not possess phase-inversion symmetry and percolate along certain directions at much lower volume fractions. We also apply our general procedure to construct stealthy nonhyperuniform systems. Our construction algorithm enables one to control the statistical anisotropy of composite microstructures via the shape, size, and symmetries of Ω, which is crucial to engineering directional optical, transport, and mechanical properties of two-phase composite media.

A novel machine learning approach for surface roughness quantification and optimization of cast-on-strap lead-antimony alloy via two-point correlation function

Surface roughness has a negative impact on the materials' lifetime. It accelerates pitting corrosion, increases effective heat transfer, and increases the rate of effective charge loss. However, controlled surface roughness is desirable in many applications. The automotive lead-acid battery is very sensitive to such effects. In our case study, the cast-on-strap machine has the largest effect on the surface roughness of the lead-antimony alloy. In this regard, statistical correlation functions are commonly used as statistical morphological descriptors for heterogeneous correlation functions. Two-point correlation functions are fruitful tools to quantify the microstructure of two-phase material structures. Herein, we demonstrate the use of the two-point correlation function to quantify surface roughness and optimize lead-antimony poles and straps used in the lead-acid battery as a solution to reduce their electrochemical corrosion when used in highly corrosive media. However, we infer that this method can be used in surface roughness mapping in a wide range of applications, such as pipes submerged in seawater as well as laser cutting. The possibility of using information obtained from the two-point correlation function and applying the simulated annealing procedure to optimize the surface micro-irregularities is investigated. The results showed successful surface representation and optimization that agree with the initially proposed hypothesis.

Topological invariance in whiteness optimisation

Maximizing the scattering of visible light within disordered nano-structured materials is essential for commercial applications such as brighteners, while also testing our fundamental understanding of light-matter interactions. The progress in the research field has been hindered by the lack of understanding how different structural features contribute to the scattering properties. Here we undertake a systematic investigation of light scattering in correlated disordered structures. We demonstrate that the scattering efficiency of disordered systems is mainly determined by topologically invariant features, such as the filling fraction and correlation length, and residual variations are largely accounted by the surface-averaged mean curvature of the systems. Optimal scattering efficiency can thus be obtained from a broad range of disordered structures, especially when structural anisotropy is included as a parameter. These results suggest that any disordered system can be optimised for whiteness and give comparable performance, which has far-reaching consequences for the industrial use of low-index materials for optical scattering.

Quantifying microstructures of earth materials using higher-order spatial correlations and deep generative adversarial networks

The key to most subsurface processes is to determine how structural and topological features at small length scales, i.e., the microstructure, control the effective and macroscopic properties of earth materials. Recent progress in imaging technology has enabled us to visualise and characterise microstructures at different length scales and dimensions. However, one limitation of these technologies is the trade-off between resolution and sample size (or representativeness). A promising approach to this problem is image reconstruction which aims to generate statistically equivalent microstructures but at a larger scale and/or additional dimension. In this work, a stochastic method and three generative adversarial networks (GANs), namely deep convolutional GAN (DCGAN), Wasserstein GAN with gradient penalty (WGAN-GP), and StyleGAN2 with adaptive discriminator augmentation (ADA), are used to reconstruct two-dimensional images of two hydrothermally rocks with varying degrees of complexity. For the first time, we evaluate and compare the performance of these methods using multi-point spatial correlation functions-known as statistical microstructural descriptors (SMDs)-ultimately used as external tools to the loss functions. Our findings suggest that a well-trained GAN can reconstruct higher-order, spatially-correlated patterns of complex earth materials, capturing underlying structural and morphological properties. Comparing our results with a stochastic reconstruction method based on a two-point correlation function, we show the importance of coupling training/assessment of GANs with higher-order SMDs, especially in the case of complex microstructures. More importantly, by quantifying original and reconstructed microstructures via different GANs, we highlight the interpretability of these SMDs and show how they can provide valuable insights into the spatial patterns in the synthetic images, allowing us to detect common artefacts and failure cases in training GANs.

Enhancing Mechanical Metamodels with a Generative Model-Based Augmented Training Dataset

Modeling biological soft tissue is complex in part due to material heterogeneity. Microstructural patterns, which play a major role in defining the mechanical behavior of these tissues, are both challenging to characterize, and difficult to simulate. Recently, machine learning (ML)-based methods to predict the mechanical behavior of heterogeneous materials have made it possible to more thoroughly explore the massive input parameter space associated with heterogeneous blocks of material. Specifically, we can train ML models to closely approximate computationally expensive heterogeneous material simulations where the ML model is trained on datasets of simulations with relevant spatial heterogeneity. However, when it comes to applying these techniques to tissue, there is a major limitation: the number of useful examples available to characterize the input domain under study is often limited. In this work, we investigate the efficacy of both ML-based generative models and procedural methods as tools for augmenting limited input pattern datasets. We find that a Style-based Generative Adversarial Network with an adaptive discriminator augmentation mechanism is able to successfully leverage just 1,000 example patterns to create authentic generated patterns. And, we find that diverse generated patterns with adequate resemblance to real patterns can be used as inputs to finite element simulations to meaningfully augment the training dataset. To enable this methodological contribution, we have created an open access Finite Element Analysis simulation dataset based on Cahn-Hilliard patterns. We anticipate that future researchers will be able to leverage this dataset and build on the work presented here.

Synthesizing controlled microstructures of porous media using generative adversarial networks and reinforcement learning

For material modeling and discovery, synthetic microstructures play a critical role as digital twins. They provide stochastic samples upon which direct numerical simulations can be conducted to populate material databases. A large ensemble of simulation data on synthetic microstructures may provide supplemental data to inform and refine macroscopic material models, which might not be feasible from physical experiments alone. However, synthesizing realistic microstructures with realistic microstructural attributes is highly challenging. Thus, it is often oversimplified via rough approximations that may yield an inaccurate representation of the physical world. Here, we propose a novel deep learning method that can synthesize realistic three-dimensional microstructures with controlled structural properties using the combination of generative adversarial networks (GAN) and actor-critic (AC) reinforcement learning. The GAN-AC combination enables the generation of microstructures that not only resemble the appearances of real specimens but also yield user-defined physical quantities of interest (QoI). Our validation experiments confirm that the properties of synthetic microstructures generated by the GAN-AC framework are within a 5% error margin with respect to the target values. The scientific contribution of this paper resides in the novel design of the GAN-AC microstructure generator and the mathematical and algorithmic foundations therein. The proposed method will have a broad and substantive impact on the materials community by providing lenses for analyzing structure-property-performance linkages and for implementing the notion of 'materials-by-design'.

Ultraefficient reconstruction of effectively hyperuniform disordered biphase materials via non-Gaussian random fields

Disordered hyperuniform systems are statistically isotropic and possess no Bragg peaks like liquids and glasses, yet they suppress large-scale density fluctuations in a similar manner as in perfect crystals. The unique hyperuniform long-range order in these systems endow them with nearly optimal transport, electronic, and mechanical properties. The concept of hyperuniformity was originally introduced for many-particle systems and has subsequently been generalized to biphase heterogeneous materials such as porous media, composites, polymers, and biological tissues for unconventional property discovery. Existing methods for rendering realizations of disordered hyperuniform biphase materials reconstruction typically employ stochastic optimization such as the simulated annealing approach, which requires many iterations. Here, we propose an explicit ultraefficient method for reconstructing effectively hyperuniform biphase materials, based on the second-order non-Gaussian random fields where no additional tuning step or iteration is needed. Both the effectively hyperuniform microstructure and the latent material property field can be simultaneously generated in a single reconstruction. Moreover, our method can also incorporate hierarchical uncertainties in the heterogeneous materials, including both uncertainties in the disordered material microstructure and material property variation within each phase. The efficiency and feasibility of the proposed reconstruction method are demonstrated via a wide spectrum of examples spanning from isotropic to anisotropic, effectively hyperuniform to nonhyperuniform, and antihyperuniform systems. Our ultraefficient reconstruction method can be readily incorporated into material design, probabilistic analysis, optimization, and discovery of novel disordered hyperuniform heterogeneous materials.

Generative adversarial networks for reconstruction of three-dimensional porous media from two-dimensional slices

In many branches of earth sciences, the problem of rock study on the microlevel arises. However, a significant number of representative samples is not always feasible. Thus the problem of the generation of samples with similar properties becomes actual. In this paper we propose a deep learning architecture for three-dimensional porous medium reconstruction from two-dimensional slices. We fit a distribution on all possible three-dimensional structures of a specific type based on the given data set of samples. Then, given partial information (central slices), we recover the three-dimensional structure around such slices as the most probable one according to that constructed distribution. Technically, we implement this in the form of a deep neural network with encoder, generator, and discriminator modules. Numerical experiments show that this method provides a good reconstruction in terms of Minkowski functionals.

Quantifying microstructural evolution via time-dependent reduced-dimension metrics based on hierarchical n-point polytope functions

We devise reduced-dimension metrics for effectively measuring the distance between two points (i.e., microstructures) in the microstructure space and quantifying the pathway associated with microstructural evolution, based on a recently introduced set of hierarchical n-point polytope functions P_{n}. The P_{n} functions provide the probability of finding particular n-point configurations associated with regular n polytopes in the material system, and are a special subset of the standard n-point correlation functions S_{n} that effectively decompose the structural features in the system into regular polyhedral basis with different symmetries. The nth order metric Ω_{n} is defined as the L_{1} norm associated with the P_{n} functions of two distinct microstructures. By choosing a reference initial state (i.e., a microstructure associated with t_{0}=0), the Ω_{n}(t) metrics quantify the evolution of distinct polyhedral symmetries and can in principle capture emerging polyhedral symmetries that are not apparent in the initial state. To demonstrate their utility, we apply the Ω_{n} metrics to a two-dimensional binary system undergoing spinodal decomposition to extract the phase separation dynamics via the temporal scaling behavior of the corresponding Ω_{n}(t), which reveals mechanisms governing the evolution. Moreover, we employ Ω_{n}(t) to analyze pattern evolution during vapor deposition of phase-separating alloy films with different surface contact angles, which exhibit rich evolution dynamics including both unstable and oscillating patterns. The Ω_{n} metrics have potential applications in establishing quantitative processing-structure-property relationships, as well as real-time processing control and optimization of complex heterogeneous material systems.

Deep-learning-based porous media microstructure quantitative characterization and reconstruction method

Microstructure characterization and reconstruction (MCR) is one of the most important components of discovering processing-structure-property relations of porous media behavior and inverse porous media design in computational materials science. Since the algorithms for describing and controlling the geometric configuration of microstructures need to solve a large number of variables and involve multiobjective conditions, the existing MCR methods have difficulty in gaining a perfect trade-off among the quantitative generation and characterization capability and the reconstruction quality. In this work, an improved 3D Porous Media Microstructure (3DPmmGAN) generative adversarial network based on deep-learning algorithm is demonstrated for high-quality microstructures generation with high controllability and high prediction accuracy. The proposed 3DPmmGAN allows the model to utilize unlabeled data for complex high-randomness microstructures end-to-end training within an acceptable time consumption. Further analysis shows that the trained network has good adaptivity for microstructures with different random geometric configurations, and can quantitatively control the generated structure according to semantic conditions, and can also quantitatively predict complex microstructure features. The key results suggest the proposed 3DPmmGAN is a powerful tool to accelerate the preparation and the initial characterization of 3D porous media, and potentially maximize the design efficiency for porous media.

Abnormal Aggregation of Invasive Cancer Cells Induced by Collective Polarization and ECM-Mediated Mechanical Coupling in Coculture Systems

Studies on pattern formation in coculture cell systems can provide insights into many physiological and pathological processes. Here, we investigate how the extracellular matrix (ECM) may influence the patterning in coculture systems. The model coculture system we use is composed of highly motile invasive breast cancer cells, initially mixed with inert nonmetastatic cells on a 2D substrate and covered with a Matrigel layer introduced to mimic ECM. We observe that the invasive cells exhibit persistent centripetal motion and yield abnormal aggregation, rather than random spreading, due to a “collective pulling” effect resulting from ECM-mediated transmission of active contractile forces generated by the polarized migration of the invasive cells along the vertical direction. The mechanism we report may open a new window for the understanding of biological processes that involve multiple types of cells.

Diffusion spreadability as a probe of the microstructure of complex media across length scales

Understanding time-dependent diffusion processes in multiphase media is of great importance in physics, chemistry, materials science, petroleum engineering, and biology. Consider the time-dependent problem of mass transfer of a solute between two phases and assume that the solute is initially distributed in one phase (phase 2) and absent from the other (phase 1). We desire the fraction of total solute present in phase 1 as a function of time, S(t), which we call the spreadability, since it is a measure of the spreadability of diffusion information as a function of time. We derive exact direct-space formulas for S(t) in any Euclidean space dimension d in terms of the autocovariance function as well as corresponding Fourier representations of S(t) in terms of the spectral density, which are especially useful when scattering information is available experimentally or theoretically. These are singular results because they are rare examples of mass transport problems where exact solutions are possible. We derive closed-form general formulas for the short- and long-time behaviors of the spreadability in terms of crucial small- and large-scale microstructural information, respectively. The long-time behavior of S(t) enables one to distinguish the entire spectrum of microstructures that span from hyperuniform to nonhyperuniform media. For hyperuniform media, disordered or not, we show that the "excess" spreadability, S(∞)-S(t), decays to its long-time behavior exponentially faster than that of any nonhyperuniform two-phase medium, the "slowest" being antihyperuniform media. The stealthy hyperuniform class is characterized by an excess spreadability with the fastest decay rate among all translationally invariant microstructures. We obtain exact results for S(t) for a variety of specific ordered and disordered model microstructures across dimensions that span from hyperuniform to antihyperuniform media. Moreover, we establish a remarkable connection between the spreadability and an outstanding problem in discrete geometry, namely, microstructures with "fast" spreadabilities are also those that can be derived from efficient "coverings" of space. We also identify heretofore unnoticed, to our best knowledge, remarkable links between the spreadability S(t) and NMR pulsed field gradient spin-echo amplitude as well as diffusion MRI measurements. This investigation reveals that the time-dependent spreadability is a powerful, dynamic-based figure of merit to probe and classify the spectrum of possible microstructures of two-phase media across length scales.

Understanding degeneracy of two-point correlation functions via Debye random media

It is well known that the degeneracy of two-phase microstructures with the same volume fraction and two-point correlation function S_{2}(r) is generally infinite. To elucidate the degeneracy problem explicitly, we examine Debye random media, which are entirely defined by a purely exponentially decaying two-point correlation function S_{2}(r). In this work, we consider three different classes of Debye random media. First, we generate the "most probable" class using the Yeong-Torquato construction algorithm [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)1063-651X10.1103/PhysRevE.57.495]. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function. We structurally discriminate these three classes of Debye random media from one another by ascertaining their other statistical descriptors, including the pore-size, surface correlation, chord-length probability density, and lineal-path functions. We also compare and contrast the percolation thresholds as well as the diffusion and fluid transport properties of these degenerate Debye random media. We find that these three classes of Debye random media are generally distinguished by the aforementioned descriptors, and their microstructures are also visually distinct from one another. Our work further confirms the well-known fact that scattering information is insufficient to determine the effective physical properties of two-phase media. Additionally, our findings demonstrate the importance of the other two-point descriptors considered here in the design of materials with a spectrum of physical properties.

Three-dimensional reconstruction of porous media using super-dimension-based adjacent block-matching algorithm

As porous media play an essential role in a variety of industrial applications, it is essential to understand their physical properties. Nowadays, the super-dimensional (SD) reconstruction algorithm is used to stochastically reconstruct a three-dimensional (3D) structure of porous media from a given two-dimensional image. This algorithm exhibits superiority in accuracy compared with classical algorithms because it learns information from the real 3D structure. However, owing to the short development time of the SD algorithm, it also has some limitations, such as inexact porosity characterization, long run time, blocking artifacts, and suboptimal accuracy that may be improved. To mitigate these limitations, this study presents the design of a special template that contains two parts of data (i.e., adjacent blocks and a central block); the proposed method matches adjacent blocks during reconstruction and assigns the matched central block to the area to be reconstructed. Furthermore, we design two important mechanisms during reconstruction: one for block matching and the other for porosity control. To verify the effectiveness of the proposed method compared with an existing SD method, both methods were tested on silica particle material and three homogeneous sandstones with different porosities; meanwhile, we compared the proposed method with a multipoint statistics method and a simulated annealing method. The reconstructed results were then compared with the target both visually and quantitatively. The experimental results indicate that the proposed method can overcome the aforementioned limitations and further improve the accuracy of existing methods. This method achieved 4-6 speedup factor compared with the traditional SD method.

Critical pore radius and transport properties of disordered hard- and overlapping-sphere models

Transport properties of porous media are intimately linked to their pore-space microstructures. We quantify geometrical and topological descriptors of the pore space of certain disordered and ordered distributions of spheres, including pore-size functions and the critical pore radius δ_{c}. We focus on models of porous media derived from maximally random jammed sphere packings, overlapping spheres, equilibrium hard spheres, quantizer sphere packings, and crystalline sphere packings. For precise estimates of the percolation thresholds, we use a strict relation of the void percolation around sphere configurations to weighted bond percolation on the corresponding Voronoi networks. We use the Newman-Ziff algorithm to determine the percolation threshold using universal properties of the cluster size distribution. The critical pore radius δ_{c} is often used as the key characteristic length scale that determines the fluid permeability k. A recent study [Torquato, Adv. Wat. Resour. 140, 103565 (2020)10.1016/j.advwatres.2020.103565] suggested for porous media with a well-connected pore space an alternative estimate of k based on the second moment of the pore size 〈δ^{2}〉, which is easier to determine than δ_{c}. Here, we compare δ_{c} to the second moment of the pore size 〈δ^{2}〉, and indeed confirm that, for all porosities and all models considered, δ_{c}^{2} is to a good approximation proportional to 〈δ^{2}〉. However, unlike 〈δ^{2}〉, the permeability estimate based on δ_{c}^{2} does not predict the correct ranking of k for our models. Thus, we confirm 〈δ^{2}〉 to be a promising candidate for convenient and reliable estimates of the fluid permeability for porous media with a well-connected pore space. Moreover, we compare the fluid permeability of our models with varying degrees of order, as measured by the τ order metric. We find that (effectively) hyperuniform models tend to have lower values of k than their nonhyperuniform counterparts. Our findings could facilitate the design of porous media with desirable transport properties via targeted pore statistics.

Impact of Geometrical Disorder on Phase Equilibria of Fluids and Solids Confined in Mesoporous Materials

Porous solids used in practical applications often possess structural disorder over broad length scales. This disorder strongly affects different properties of the substances confined in their pore spaces. Quantifying structural disorder and correlating it with the physical properties of confined matter is thus a necessary step toward the rational use of porous solids in practical applications and process optimization. The present work focuses on recent advances made in the understanding of correlations between the phase state and geometric disorder in nanoporous solids. We overview the recently developed statistical theory for phase transitions in a minimalistic model of disordered pore networks: linear chains of pores with statistical disorder. By correlating its predictions with various experimental observations, we show that this model gives notable insight into collective phenomena in phase-transition processes in disordered materials and is capable of explaining self-consistently the majority of the experimental results obtained for gas-liquid and solid-liquid equilibria in mesoporous solids. The potentials of the theory for improving the gas sorption and thermoporometry characterization of porous materials are discussed.

Generation and structural characterization of Debye random media

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semianalytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks) and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here.

Probing information content of hierarchical n-point polytope functions for quantifying and reconstructing disordered systems

Disordered systems are ubiquitous in physical, biological, and material sciences. Examples include liquid and glassy states of condensed matter, colloids, granular materials, porous media, composites, alloys, packings of cells in avian retina, and tumor spheroids, to name but a few. A comprehensive understanding of such disordered systems requires, as the first step, systematic quantification, modeling, and representation of the underlying complex configurations and microstructure, which is generally very challenging to achieve. Recently, we introduced a set of hierarchical statistical microstructural descriptors, i.e., the "n-point polytope functions" P_{n}, which are derived from the standard n-point correlation functions S_{n}, and successively included higher-order n-point statistics of the morphological features of interest in a concise, explainable, and expressive manner. Here we investigate the information content of the P_{n} functions via optimization-based realization rendering. This is achieved by successively incorporating higher-order P_{n} functions up to n=8 and quantitatively assessing the accuracy of the reconstructed systems via unconstrained statistical morphological descriptors (e.g., the lineal-path function). We examine a wide spectrum of representative random systems with distinct geometrical and topological features. We find that, generally, successively incorporating higher-order P_{n} functions and, thus, the higher-order morphological information encoded in these descriptors leads to superior accuracy of the reconstructions. However, incorporating more P_{n} functions into the reconstruction also significantly increases the complexity and roughness of the associated energy landscape for the underlying stochastic optimization, making it difficult to convergence numerically.

Three-dimensional multiscale fusion for porous media on microtomography images of different resolutions

Accurately acquiring the three-dimensional (3D) image of a porous medium is an imperative issue for the prediction of multiple physical properties. Considering the inherent nature of the multiscale pores contained in porous media such as tight sandstones, to completely characterize the pore structure, one needs to scan the microstructure at different resolutions. Specifically, low-resolution (LR) images cover a larger field of view (FOV) of the sample, but are lacking small-scale features, whereas high-resolution (HR) images contain ample information, but sometimes only cover a limited FOV. To address this issue, we propose a method for fusing the spatial information from a two-dimensional (2D) HR image into a 3D LR image, and finally reconstructing an integrated 3D structure with added fine-scale features. In the fusion process, the large-scale structure depicted by the 3D LR image is fixed as background and the 2D image is utilized as training image to reconstruct a small-scale structure based on the background. To assess the performance of our method, we test it on a sandstone scanned with low and high resolutions. Statistical properties between the reconstructed image and the target are quantitatively compared. The comparison indicates that the proposed method enables an accurate fusion of the LR and HR images because the small-scale information is precisely reproduced within the large one.

Microstructure synthesis using style-based generative adversarial networks

This work considers the usage of StyleGAN architecture for the task of microstructure synthesis. The task is the following: Given number of samples of structure we try to generate similar samples at the same time preserving its properties. Since the considered architecture is not able to produce samples of sizes larger than the training images, we propose to use image quilting to merge fixed-sized samples. One of the key features of the considered architecture is that it uses multiple image resolutions. We also investigate the necessity of such an approach.

Enhanced Oxygen Electrocatalysis in Heterostructured Ceria Electrolytes for Intermediate-Temperature Solid Oxide Fuel Cells

Heterostructured composite ceria electrolytes have been shown to accelerate the oxygen reduction activity and provide a new approach to improve solid oxide fuel cell (SOFC) performance. In this study, barium carbonate was added to gadolinium-doped ceria, Gd_0.2Ce_0.8O_2-δ (GDC) electrolyte to improve the electrochemical performance of intermediate-temperature SOFCs. The heterostructured electrolyte was formed by the addition of 5 wt % BaCO₃ to a GDC electrolyte, resulting in a reaction during sintering that formed well-dispersed BaCe_0.8Gd_0.2O_3-δ (BCG) throughout the electrolyte. The resulting material was tested as an electrolyte using La_0.6Sr_0.4Co_0.2Fe_0.8O_3-δ as a cathode, resulting in a dramatic reduction to the polarization resistance of more than half the value (600 and 700 °C, the resistance was reduced from 2.49 and 0.23 Ω cm² to 1.21 and 0.12 Ω cm²) obtained by using pure GDC as an electrolyte. Furthermore, full cell SOFC tests employing the heterostructured electrolyte conducted during overextended durations indicated that the BCG phase in the 5BCG-GDC electrolyte was stable in an air atmosphere with no observed reactions with residual CO₂. This approach of tailoring surface reactivity by tailoring the composition and structure of the electrolyte as opposed to electrode materials provides an alternative method to improve fuel cell performance.

Realizations of highly heterogeneous collagen networks via stochastic reconstruction for micromechanical analysis of tumor cell invasion

Three-dimensional (3D) collective cell migration in a collagen-based extracellular matrix (ECM) is among one of the most significant topics in developmental biology, cancer progression, tissue regeneration, and immune response. Recent studies have suggested that collagen-fiber mediated force transmission in cellularized ECM plays an important role in stress homeostasis and regulation of collective cellular behaviors. Motivated by the recent in vitro observation that oriented collagen can significantly enhance the penetration of migrating breast cancer cells into dense Matrigel which mimics the intravasation process in vivo [Han et al. Proc. Natl. Acad. Sci. USA 113, 11208 (2016)PNASA60027-842410.1073/pnas.1610347113], we devise a procedure for generating realizations of highly heterogeneous 3D collagen networks with prescribed microstructural statistics via stochastic optimization. Specifically, a collagen network is represented via the graph (node-bond) model and the microstructural statistics considered include the cross-link (node) density, valence distribution, fiber (bond) length distribution, as well as fiber orientation distribution. An optimization problem is formulated in which the objective function is defined as the squared difference between a set of target microstructural statistics and the corresponding statistics for the simulated network. Simulated annealing is employed to solve the optimization problem by evolving an initial network via random perturbations to generate realizations of homogeneous networks with randomly oriented fibers, homogeneous networks with aligned fibers, heterogeneous networks with a continuous variation of fiber orientation along a prescribed direction, as well as a binary system containing a collagen region with aligned fibers and a dense Matrigel region with randomly oriented fibers. The generation and propagation of active forces in the simulated networks due to polarized contraction of an embedded ellipsoidal cell and a small group of cells are analyzed by considering a nonlinear fiber model incorporating strain hardening upon large stretching and buckling upon compression. Our analysis shows that oriented fibers can significantly enhance long-range force transmission in the network. Moreover, in the oriented-collagen-Matrigel system, the forces generated by a polarized cell in collagen can penetrate deeply into the Matrigel region. The stressed Matrigel fibers could provide contact guidance for the migrating cell cells, and thus enhance their penetration into Matrigel. This suggests a possible mechanism for the observed enhanced intravasation by oriented collagen.

Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

The quantitative characterization of the microstructure of random heterogeneous media in d-dimensional Euclidean space R^{d} via a variety of n-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface F_{ss} and surface-void F_{sv} correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-r behaviors of these functions, which are linked to the mean curvature of the system. Then we demonstrate that one can reduce the computational complexity of the problem, without sacrificing accuracy, by extracting the necessary interfacial information from a cut of the d-dimensional statistically homogeneous and isotropic system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated "stealthy" patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures.

Pore-Scale Simulations of Porous Electrodes of Li-O2 Batteries at Different Saturation Levels

This study reconstructs pore-scale structures of battery electrodes from scanning electron microscopy images, quantitatively studies the distribution of the electrolyte at various saturations, and simulates the discharge performance of Li-O₂ batteries. This research sheds lights on the critical role of liquid-gas two-phase mass transfer within the porous electrode on the electrochemical performance of batteries. It is found that fully saturated electrodes (100% saturation) have high oxygen-transfer resistance, which will impede the battery performance at typical electrode thickness (∼200 μm). On the contrary, overdried battery (with <50% saturations) electrodes have poor electrochemical performance because dry pores are inactive for electrochemical reactions. In addition, the low electrolyte saturation level leads to low ionic conductivity and high mass transfer resistance of the lithium ion. Carefully designed electrodes with the mixture of lyophilic and lyophobic pores could achieve similar discharge capacity (>7 A h/g) at high current (20 A/m²) with lyophilic electrodes that are fully saturated by the electrolyte at low current (1 A/m²). The findings from this study enable further research to significantly increase (by orders of magnitude) the operating current and power of the Li-O₂ battery and accelerate its deployment to transport and stationary applications.

Improved multipoint statistics method for reconstructing three-dimensional porous media from a two-dimensional image via porosity matching

Reconstructing a three-dimensional (3D) structure from a single two-dimensional training image (TI) is a challenging issue. Multiple-point statistics (MPS) is an effective method to solve this problem. However, in the traditional MPS method, errors occur while statistical features of reconstruction, such as porosity, connectivity, and structural properties, deviate from those of TI. Due to the MPS reconstruction mechanism that the voxel being reconstructed is dependent on the reconstructed voxel, it may cause error accumulation during simulations, which can easily lead to a significant difference between the real 3D structure and the reconstructed result. To reduce error accumulation and improve morphological similarity, an improved MPS method based on porosity matching is proposed. In the reconstruction, we search the matching pattern in the TI directly. Meanwhile, a multigrid approach is also applied to capture the large-scale structures of the TI. To demonstrate its superiority over the traditional MPS method, our method is tested on different sandstone samples from many aspects, including accuracy, stability, generalization, and flow characteristics. Experimental results show that the reconstruction results by the improved MPS method effectively match the CT sandstone samples in correlation functions, local porosity distribution, morphological parameters, and permeability.

Pore-scale statistics of flow and transport through porous media

Flow in porous media is known to be largely affected by pore morphology. In this work, we investigate the effects of pore geometry on the transport and spatial correlations of flow through porous media in two distinct pore structures arising from three-dimensional assemblies of overlapping and nonoverlapping spheres. Using high-resolution direct numerical simulations (DNS), we perform Eulerian and Lagrangian analysis of the flow and transport characteristics in porous media. We show that the Eulerian velocity distributions change from nearly exponential to Gaussian distributions as porosity increases. A stretched exponential distribution can be used to represent this behavior for a wide range of porosities. Evolution of Lagrangian velocities is studied for the uniform injection rule. Evaluation of tortuosity and trajectory length distributions of each porous medium shows that the model of overlapping spheres results in higher tortuosity and more skewed trajectory length distributions compared to the model of nonoverlapping spheres. Wider velocity distribution and higher tortuosity for overlapping spheres model give rise to non-Fickian transport while transport in nonoverlapping spheres model is found to be Fickian. Particularly, for overlapping spheres model our analysis of first-passage time distribution shows that the transport is very similar to those observed for sandstone. Finally, using three-dimensional (3D) velocity field obtained by DNS at the pore-scale, we quantitatively show that despite the randomness of pore-space, the spatially fluctuating velocity field and the 3D pore-space distribution are strongly correlated for a range of porous media from relatively homogeneous monodisperse sphere packs to Castlegate sandstone.

Markov prior-based block-matching algorithm for superdimension reconstruction of porous media

A superdimension reconstruction algorithm is used for the reconstruction of three-dimensional (3D) structures of a porous medium based on a single two-dimensional image. The algorithm borrows the concepts of "blocks," "learning," and "dictionary" from learning-based superresolution reconstruction and applies them to the 3D reconstruction of a porous medium. In the neighborhood-matching process of the conventional superdimension reconstruction algorithm, the Euclidean distance is used as a criterion, although it may not really reflect the structural correlation between adjacent blocks in an actual situation. Hence, in this study, regular items are adopted as prior knowledge in the reconstruction process, and a Markov prior-based block-matching algorithm for superdimension reconstruction is developed for more accurate reconstruction. The algorithm simultaneously takes into consideration the probabilistic relationship between the already reconstructed blocks in three different perpendicular directions (x, y, and z) and the block to be reconstructed, and the maximum value of the probability product of the blocks to be reconstructed (as found in the dictionary for the three directions) is adopted as the basis for the final block selection. Using this approach, the problem of an imprecise spatial structure caused by a point simulation can be overcome. The problem of artifacts in the reconstructed structure is also addressed through the addition of hard data and by neighborhood matching. To verify the improved reconstruction accuracy of the proposed method, the statistical and morphological features of the results from the proposed method and traditional superdimension reconstruction method are compared with those of the target system. The proposed superdimension reconstruction algorithm is confirmed to enable a more accurate reconstruction of the target system while also eliminating artifacts.

Effect of the image resolution on the statistical descriptors of heterogeneous media

The characterization and reconstruction of heterogeneous materials, such as porous media and electrode materials, involve the application of image processing methods to data acquired by scanning electron microscopy or other microscopy techniques. Among them, binarization and decimation are critical in order to compute the correlation functions that characterize the microstructure of the above-mentioned materials. In this study, we present a theoretical analysis of the effects of the image-size reduction, due to the progressive and sequential decimation of the original image. Three different decimation procedures (random, bilinear, and bicubic) were implemented and their consequences on the discrete correlation functions (two-point, line-path, and pore-size distribution) and the coarseness (derived from the local volume fraction) are reported and analyzed. The chosen statistical descriptors (correlation functions and coarseness) are typically employed to characterize and reconstruct heterogeneous materials. A normalization for each of the correlation functions has been performed. When the loss of statistical information has not been significant for a decimated image, its normalized correlation function is forecast by the trend of the original image (reference function). In contrast, when the decimated image does not hold statistical evidence of the original one, the normalized correlation function diverts from the reference function. Moreover, the equally weighted sum of the average of the squared difference, between the discrete correlation functions of the decimated images and the reference functions, leads to a definition of an overall error. During the first stages of the gradual decimation, the error remains relatively small and independent of the decimation procedure. Above a threshold defined by the correlation length of the reference function, the error becomes a function of the number of decimation steps. At this stage, some statistical information is lost and the error becomes dependent on the decimation procedure. These results may help us to restrict the amount of information that one can afford to lose during a decimation process, in order to reduce the computational and memory cost, when one aims to diminish the time consumed by a characterization or reconstruction technique, yet maintaining the statistical quality of the digitized sample.

Rational design of stealthy hyperuniform two-phase media with tunable order

Disordered stealthy hyperuniform materials are exotic amorphous states of matter that have attracted recent attention because of their novel structural characteristics (hidden order at large length scales) and physical properties, including desirable photonic and transport properties. It is therefore useful to devise algorithms that enable one to design a wide class of such amorphous configurations at will. In this paper, we present several algorithms enabling the systematic identification and generation of discrete (digitized) stealthy hyperuniform patterns with a tunable degree of order, paving the way towards the rational design of disordered materials endowed with novel thermodynamic and physical properties. To quantify the degree of order or disorder of the stealthy systems, we utilize the discrete version of the τ order metric, which accounts for the underlying spatial correlations that exist across all relevant length scales in a given digitized two-phase (or, equivalently, a two-spin state) system of interest. Our results impinge on a myriad of fields, ranging from physics, materials science and engineering, visual perception, and information theory to modern data science.

Collagen Organization in Facet Capsular Ligaments Varies With Spinal Region and With Ligament Deformation

The spinal facet capsular ligament (FCL) is primarily comprised of heterogeneous arrangements of collagen fibers. This complex fibrous structure and its evolution under loading play a critical role in determining the mechanical behavior of the FCL. A lack of analytical tools to characterize the spatial anisotropy and heterogeneity of the FCL's microstructure has limited the current understanding of its structure-function relationships. Here, the collagen organization was characterized using spatial correlation analysis of the FCL's optically obtained fiber orientation field. FCLs from the cervical and lumbar spinal regions were characterized in terms of their structure, as was the reorganization of collagen in stretched cervical FCLs. Higher degrees of intra- and intersample heterogeneity were found in cervical FCLs than in lumbar specimens. In the cervical FCLs, heterogeneity was manifested in the form of curvy patterns formed by collections of collagen fibers or fiber bundles. Tensile stretch, a common injury mechanism for the cervical FCL, significantly increased the spatial correlation length in the stretch direction, indicating an elongation of the observed structural features. Finally, an affine estimation for the change of correlation length under loading was performed which gave predictions very similar to the actual values. These findings provide structural insights for multiscale mechanical analyses of the FCLs from various spinal regions and also suggest methods for quantitative characterization of complex tissue patterns.

Statistical methods for mechanical characterization of randomly reinforced media

Advanced materials with heterogeneous microstructure attract extensive interest of researchers and engineers due to combination of unique properties and ability to create materials that are most suitable for each specific application. One of the challenging tasks is development of models of mechanical behavior for such materials since precision of the obtained numerical results highly depends on level of consideration of features of their heterogeneous microstructure. In most cases, numerical modeling of composite structures is based on multiscale approaches that require special techniques for establishing connection between parameters at different scales. This work offers a review of instruments of the statistics and the probability theory that are used for mechanical characterization of heterogeneous media with random positions of reinforcements. Such statistical descriptors are involved in assessment of correlations between the microstructural components and are parts of mechanical theories which require formalization of the information about microstructural morphology. Particularly, the paper addresses application of the instruments of statistics for geometry description and media reconstruction as well as their utilization in homogenization methods and local stochastic stress and strain field analysis.

Inferring low-dimensional microstructure representations using convolutional neural networks

We apply recent advances in machine learning and computer vision to a central problem in materials informatics: the statistical representation of microstructural images. We use activations in a pretrained convolutional neural network to provide a high-dimensional characterization of a set of synthetic microstructural images. Next, we use manifold learning to obtain a low-dimensional embedding of this statistical characterization. We show that the low-dimensional embedding extracts the parameters used to generate the images. According to a variety of metrics, the convolutional neural network method yields dramatically better embeddings than the analogous method derived from two-point correlations alone.

Microstructure and mechanical properties of hyperuniform heterogeneous materials

A hyperuniform random heterogeneous material is one in which the local volume fraction fluctuations in an observation window decay faster than the reciprocal window volume as the window size increases. Recent studies show that this class of materials are endowed with superior physical properties such as large isotropic photonic band gaps and optimal transport properties. Here we employ a stochastic optimization procedure to systematically generate realizations of hyperuniform heterogeneous materials with controllable short-range order, which is partially quantified using the two-point correlation function S_{2}(r) associated with the phase of interest. Specifically, our procedure generalizes the widely used Yeong-Torquato reconstruction procedure by including an additional constraint for hyperuniformity, i.e., the volume integral of the autocovariance function χ(r)=S_{2}(r)-ϕ^{2} over the whole space is zero. In addition, we only require the reconstructed S_{2} to match the target function up to a certain cutoff distance γ, in order to give the system sufficient degrees of freedom to satisfy the hyperuniform condition. By systematically increasing the γ value for a given S_{2}, one can produce a spectrum of hyperuniform heterogeneous materials with varying degrees of partial short-range order compatible with the specified S_{2}. The mechanical performance including both elastic and brittle fracture behaviors of the generated hyperuniform materials is analyzed using a volume-compensated lattice-particle method. For the purpose of comparison, the corresponding nonhyperuniform materials with the same short-range order (i.e., with S_{2} constrained up to the same γ value) are also constructed and their mechanical performance is analyzed. Here we consider two specific S_{2} including the positive exponential decay function and the correlation function associated with an equilibrium hard-sphere system. For the constructed systems associated with these two specific functions, we find that although the hyperuniform materials are softer than their nonhyperuniform counterparts, the former generally possess a significantly higher brittle fracture strength than the latter. This superior mechanical behavior is attributed to the lower degree of stress concentration in the material resulting from the hyperuniform microstructure, which is crucial to crack initiation and propagation.