Minkowski tensor shape analysis of cellular, granular and porous structures

Data-Driven Based Characterization of Anisotropic Mechanical Properties for Cancellous Bone From Low-Resolution CT Images

OBJECTIVES

Exploring the anisotropic mechanical behavior of cancellous bone is crucial for in-vivo bone biomechanical analysis. However, it is challenging to characterize anisotropic mechanical behaviors under low-resolution (LR) clinical CT images due to a lack of microstructural information. The data-driven method proposed in this article accurately characterizes the anisotropic mechanical properties of cancellous bone from LR clinical CT images.

METHODS

The trabecular bone cubes of sheep are used to obtain a high-resolution (HR) micro-CT and an LR clinical CT image dataset. First, an auto-encoder model is trained using HR image data. Microstructural features are extracted by the encoder. A fast super-resolution (FSR) model is trained to map LR bone cubes to the features extracted from corresponding HR samples. The pretrained FSR model is used to convert LR clinical CT images to encoded microstructural features. The features are later used to predict target histomorphological parameters, anisotropic elastic tensors, and fabric tensors based on a fully connected neural network.

RESULTS

The data-driven model accurately predicts the elastic tensor and fabric tensor of trabecular bones with LR CT images with 0.6 mm/pixel spatial resolution. It was verified that LR clinical CT images could generate microstructural information using a generative deep-learning model and an up-sampling operation.

SIGNIFICANCE

This study proves that clinical medical images of cancellous bone can be used for analysis of complex mechanical properties using a data-driven method, which is useful for real-time bone defect diagnosis and personalized bone prosthesis design in clinical application.

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.

Experimentally characterizing the spatially varying anisotropic mechanical property of cancellous bone via a Bayesian calibration method

Traditional experimental tests for characterizing bone's mechanical properties usually hypothesize a uniaxial stress condition without quantitatively evaluating the influence of spatially varying principal material orientations, which cannot accurately predict the mechanical properties distribution of bones in vivo environment. In this study, a Bayesian calibrating procedure was developed using quantified multiaxial stress to investigate cancellous bone's local anisotropic elastic performance around joints as the spatial variation of main bearing orientations. First, the bone cube specimens from the distal femur of sheep are prepared using traditional anatomical axes. The multiaxial stress state of each bone specimen is calibrated using the actual principal material orientations derived from fabric tensor at different anatomical locations. Based on the calibrated multiaxial stress state, the process of identifying mechanical properties is described as an inverse problem. Then, a Bayesian calibration procedure based on a surrogate constitutive model was developed via multiaxial stress correction to identify the anisotropic material parameters. Finally, a comparison between the experiment and simulation results is discussed by applying the optimal model parameters obtained from the Bayesian probability distribution. Compared to traditional uniaxial methods, our results prove that the calibration based on the spatial variation of the main bearing orientations can significantly improve the accuracy of characterizing regional anisotropic mechanical responses. Moreover, we determine that the actual mechanical property distribution is influenced by complicated mechanical stimulation. This study provides a novel method to evaluate the spatially varying mechanical properties of bone tissues enduring complex mechanical loading accurately and effectively. It is expected to provide more realistic mechanical design targets in vivo for a personalized artificial bone prosthesis in clinical treatment.

Approximate Packing of Binary Mixtures of Cylindrical Particles

Particle packing plays an essential role in industry and chemical engineering. In this work, the discrete element method is used to generate the cylindrical particles and densify the binary cylindrical particle mixtures under the poured packing conditions. The influences of the aspect ratio and volume fraction of particles on the packing structure are measured by planar packing fraction. The Voronoi tessellation is used to quantify the porous structure of packing. The cumulative distribution functions of local packing fractions and the probability distributions of the reduced free volume of Voronoi cells are calculated to describe the local packing characteristics of binary mixtures with different volume fractions. As a result, it is observed that particles with larger aspect ratios in the binary mixture tend to orient randomly, and the particles with smaller aspect ratios have a preferentially horizontal orientation. Results also show that the less dense packings are obtained for mixtures with particles of higher aspect ratios and mixtures with a larger fraction of elongated cylindrical particles.

A Dataset of 3D Structural and Simulated Transport Properties of Complex Porous Media

Physical processes that occur within porous materials have wide-ranging applications including - but not limited to - carbon sequestration, battery technology, membranes, oil and gas, geothermal energy, nuclear waste disposal, water resource management. The equations that describe these physical processes have been studied extensively; however, approximating them numerically requires immense computational resources due to the complex behavior that arises from the geometrically-intricate solid boundary conditions in porous materials. Here, we introduce a new dataset of unprecedented scale and breadth, DRP-372: a catalog of 3D geometries, simulation results, and structural properties of samples hosted on the Digital Rocks Portal. The dataset includes 1736 flow and electrical simulation results on 217 samples, which required more than 500 core years of computation. This data can be used for many purposes, such as constructing empirical models, validating new simulation codes, and developing machine learning algorithms that closely match the extensive purely-physical simulation. This article offers a detailed description of the contents of the dataset including the data collection, simulation schemes, and data validation.

AI-driven quantification of ground glass opacities in lungs of COVID-19 patients using 3D computed tomography imaging

Objectives

Ground-glass opacity (GGO)—a hazy, gray appearing density on computed tomography (CT) of lungs—is one of the hallmark features of SARS-CoV-2 in COVID-19 patients. This AI-driven study is focused on segmentation, morphology, and distribution patterns of GGOs.

Method

We use an AI-driven unsupervised machine learning approach called PointNet++ to detect and quantify GGOs in CT scans of COVID-19 patients and to assess the severity of the disease. We have conducted our study on the “MosMedData”, which contains CT lung scans of 1110 patients with or without COVID-19 infections. We quantify the morphologies of GGOs using Minkowski tensors and compute the abnormality score of individual regions of segmented lung and GGOs.

Results

PointNet++ detects GGOs with the highest evaluation accuracy (98%), average class accuracy (95%), and intersection over union (92%) using only a fraction of 3D data. On average, the shapes of GGOs in the COVID-19 datasets deviate from sphericity by 15% and anisotropies in GGOs are dominated by dipole and hexapole components. These anisotropies may help to quantitatively delineate GGOs of COVID-19 from other lung diseases.

Conclusion

The PointNet++ and the Minkowski tensor based morphological approach together with abnormality analysis will provide radiologists and clinicians with a valuable set of tools when interpreting CT lung scans of COVID-19 patients. Implementation would be particularly useful in countries severely devastated by COVID-19 such as India, where the number of cases has outstripped available resources creating delays or even breakdowns in patient care. This AI-driven approach synthesizes both the unique GGO distribution pattern and severity of the disease to allow for more efficient diagnosis, triaging and conservation of limited resources.

Structural evolution of granular cubes packing during shear-induced ordering

Packings of granular particles may transform into ordered structures under external agitation, which is a special type of out-of-equilibrium self-assembly. Here, evolution of the internal packing structures of granular cubes under cyclic rotating shearing has been analyzed using magnetic resonance imaging techniques. Various order parameters, different types of contacts and clusters composed of face-contacting cubes, as well as the free volume regions in which each cube can move freely have been analyzed systematically to quantify the ordering process and the underlying mechanism of this granular self-assembly. The compaction process is featured by a first rapid formation of orientationally ordered local structures with faceted contacts, followed by further densification driven by free-volume maximization with an almost saturated degree of order. The ordered structures are strongly anisotropic with contacting ordered layers in the vertical direction while remaining liquid-like in the horizontal directions. Therefore, the constraint of mechanical stability for granular packings and the thermodynamic principle of entropy maximization are both effective in this system, which we propose can be reconciled by considering different depths of supercooling associated with various degrees of freedom.

The local and global geometry of trabecular bone

The organization and shape of the microstructural elements of trabecular bone govern its physical properties, are implicated in bone disease, and serve as blueprints for biomaterial design. To devise fundamental structure-property relationships and design truly bone-mimicking biomaterials, it is essential to characterize trabecular bone structure from the perspective of geometry, the mathematical study of shape. Using micro-CT images from 70 donors at five different sites, we analyze the local and global geometry of human trabecular bone in detail, respectively by quantifying surface curvatures and Minkowski functionals. We find that curvature density maps provide distinct and sensitive shape fingerprints for bone from different sites. Contrary to a common assumption, these curvature maps also show that bone morphology does not approximate a minimal surface but exhibits a much more intricate curvature landscape. At the global (or integral) perspective, our Minkowski analysis illustrates that trabecular bone exhibits other types of anisotropy/ellipticity beyond interfacial orientation, and that anisotropy varies substantially within the trabecular structure. Moreover, we show that the Minkowski functionals unify several traditional morphometric indices. Our geometric approach to trabecular morphometry provides a fundamental language of shape that could be useful for bone failure prediction, understanding geometry-driven tissue growth, and the design of bone-mimicking tissue scaffolds. STATEMENT OF SIGNIFICANCE: The architecture of trabecular bone is key in determining bone properties, and is often a starting point for the design of bone-substitutes. Despite the substantial history of bone morphometry, a fundamental characterization of trabecular bone geometry is still lacking. Therefore, we introduce a robust framework to quantify local and global trabecular bone geometry, which we apply to hundreds of micro-CT scans. Our approach relies on quantifying surface curvatures and Minkowski functionals, which are the most fundamental local and global shape quantifiers. Our results show that these shape metrics are sensitive to differences between bone types and unify traditional metrics within a single mathematical framework. This geometrical framework could also be useful to design bone-mimicking scaffolds and understand geometry-driven tissue growth.

AI-DRIVEN QUANTIFICATION OF GROUND GLASS OPACITIES IN LUNGS OF COVID-19 PATIENTS USING 3D COMPUTED TOMOGRAPHY IMAGING

Objectives

Ground-glass opacity (GGO) - a hazy, gray appearing density on computed tomography (CT) of lungs - is one of the hallmark features of SARS-CoV-2 in COVID-19 patients. This AI-driven study is focused on segmentation, morphology, and distribution patterns of GGOs.

Method

We use an AI-driven unsupervised machine learning approach called PointNet++ to detect and quantify GGOs in CT scans of COVID-19 patients and to assess the severity of the disease. We have conducted our study on the "MosMedData", which contains CT lung scans of 1110 patients with or without COVID-19 infections. We quantify the morphologies of GGOs using Minkowski tensors and compute the abnormality score of individual regions of segmented lung and GGOs.

Results

PointNet++ detects GGOs with the highest evaluation accuracy (98%), average class accuracy (95%), and intersection over union (92%) using only a fraction of 3D data. On average, the shapes of GGOs in the COVID-19 datasets deviate from sphericity by 15% and anisotropies in GGOs are dominated by dipole and hexapole components. These anisotropies may help to quantitatively delineate GGOs of COVID-19 from other lung diseases.

Conclusion

The PointNet++ and the Minkowski tensor based morphological approach together with abnormality analysis will provide radiologists and clinicians with a valuable set of tools when interpreting CT lung scans of COVID-19 patients. Implementation would be particularly useful in countries severely devastated by COVID-19 such as India, where the number of cases has outstripped available resources creating delays or even breakdowns in patient care. This AI-driven approach synthesizes both the unique GGO distribution pattern and severity of the disease to allow for more efficient diagnosis, triaging and conservation of limited resources.

Key Points

Our approach to GGO analysis has four distinguishing features:We combine an unsupervised computer vision approach with convex hull and convex points algorithms to segment and preserve the actual structure of the lung.To the best of our knowledge, we are the first group to use PointNet++ architecture for 3D visualization, segmentation, classification, and pattern analysis of GGOs.We make abnormality predictions using a deep network and Cox proportional hazards model using lung CT images of COVID-19 patients.We quantify the shapes and sizes of GGOs using Minkowski tensors to understand the morphological variations of GGOs within the COVID-19 cohort.

Characterizing the hyperuniformity of ordered and disordered two-phase media

The hyperuniformity concept provides a unified means to classify all perfect crystals, perfect quasicrystals, and exotic amorphous states of matter according to their capacity to suppress large-scale density fluctuations. While the classification of hyperuniform point configurations has received considerable attention, much less is known about the classification of hyperuniform two-phase heterogeneous media, which include composites, porous media, foams, cellular solids, colloidal suspensions, and polymer blends. The purpose of this article is to begin such a program for certain two-dimensional models of hyperuniform two-phase media by ascertaining their local volume-fraction variances σ_{_{V}}^{2}(R) and the associated hyperuniformity order metrics B[over ¯]_{V}. This is a highly challenging task because the geometries and topologies of the phases are generally much richer and more complex than point-configuration arrangements, and one must ascertain a broadly applicable length scale to make key quantities dimensionless. Therefore, we purposely restrict ourselves to a certain class of two-dimensional periodic cellular networks as well as periodic and disordered or irregular packings of circular disks, some of which maximize their effective transport and elastic properties. Among the cellular networks considered, the honeycomb networks have minimal values of the hyperuniformity order metrics B[over ¯]_{V} across all volume fractions. On the other hand, for all packings of circular disks examined, the triangular-lattice packings have the smallest values of B[over ¯]_{V} for the possible range of volume fractions. Among all structures studied here, the triangular-lattice packing of circular disks have the minimal values of the order metric for almost all volume fractions. Our study provides a theoretical foundation for the establishment of hyperuniformity order metrics for general two-phase media and a basis to discover new hyperuniform two-phase systems with desirable bulk physical properties by inverse design procedures.

Dynamical arrest of topological defects in 2D hyperuniform disk packings

We investigate collective motions of points in 2D systems, orchestrated by Lloyd algorithm. The algorithm iteratively updates a system by minimising the total quantizer energy of the Voronoi landscape of the system. As a result of a tradeoff between energy minimisation and geometric frustration, we find that optimised systems exhibit a defective landscape along the process, where strands of 5- and 7-coordinated dislocations are embedded in the hexatic phase. In particular, dipole defects, each of which is the simplest possible pair of a pentagon and a heptagon, come into the picture of dynamical arrest, as the system freezes down to a disordered hyperuniform state. Moreover, we explore the packing fractions of 2D disk packings associated to the obtained hyperuniform systems by considering the maximum inscribed disks in their Voronoi cells.

Pattern detection in colloidal assembly: A mosaic of analysis techniques

Characterization of the morphology, identification of patterns and quantification of order encountered in colloidal assemblies is essential for several reasons. First of all, it is useful to compare different self-assembly methods and assess the influence of different process parameters on the final colloidal pattern. In addition, casting light on the structures formed by colloidal particles can help to get better insight into colloidal interactions and understand phase transitions. Finally, the growing interest in colloidal assemblies in materials science for practical applications going from optoelectronics to biosensing imposes a thorough characterization of the morphology of colloidal assemblies because of the intimate relationship between morphology and physical properties (e.g. optical and mechanical) of a material. Several image analysis techniques developed to investigate images (acquired via scanning electron microscopy, digital video microscopy and other imaging methods) provide variegated and complementary information on the colloidal structures under scrutiny. However, understanding how to use such image analysis tools to get information on the characteristics of the colloidal assemblies may represent a non-trivial task, because it requires the combination of approaches drawn from diverse disciplines such as image processing, computational geometry and computational topology and their application to a primarily physico-chemical process. Moreover, the lack of a systematic description of such analysis tools makes it difficult to select the ones more suitable for the features of the colloidal assembly under examination. In this review we provide a methodical and extensive description of real-space image analysis tools by explaining their principles and their application to the investigation of two-dimensional colloidal assemblies with different morphological characteristics.

Universality of internal structure characteristics in granular media under shear

We examine the signatures of internal structure emerged from quasistatic shear responses of granular materials based on three-dimensional discrete element simulations. Granular assemblies consisting of spheres or nonspherical particles of different polydispersity are sheared from different initial densities and under different loading conditions (drained or undrained) steadily to reach the critical state (a state featured by constant stress and constant volume). The radial distribution function used to measure the packing structure is found to remain almost unchanged during the shearing process, regardless of the initial states or loading conditions of an assembly. Its specific form, however, varies with polydispersities in both grain size and grain shape. Set Voronoi tessellation is employed to examine the characteristics of local volume and anisotropy, and deformation. The local inverse solid fraction and anisotropy index are found following inverse Weibull and log-normal distributions, respectively, which are unique at the critical states. With further normalization, an invariant distribution for local volume and anisotropy is observed whose function can be determined by the polydispersities in both particle size and grain shape but bears no relevance to initial densities or loading conditions (or paths). An invariant Gaussian distribution is found for the local deformation for spherical packings, but no invariant distribution can be found for nonspherical packings where the distribution of normalized local volumetric strain is dependent on initial states.

Strength of bacterial adhesion on nanostructured surfaces quantified by substrate morphometry

Microbial adhesion and the subsequent formation of resilient biofilms at surfaces are decisively influenced by substrate properties, such as the topography. To date, studies that quantitatively link surface topography and bacterial adhesion are scarce, as both are not straightforward to quantify. To fill this gap, surface morphometry combined with single-cell force spectroscopy was performed on surfaces with irregular topographies on the nano-scale. As surfaces, hydrophobized silicon wafers were used that were etched to exhibit surface structures in the same size range as the bacterial cell wall molecules. The surface structures were characterized by a detailed morphometric analysis based on Minkowski functionals revealing both qualitatively similar features and quantitatively different extensions. We find that as the size of the nanostructures increases, the adhesion forces decrease in a way that can be quantified by the area of the surface that is available for the tethering of cell wall molecules. In addition, we observe a bactericidal effect, which is more pronounced on substrates with taller structures but does not influence adhesion. Our results can be used for a targeted development of 3D-structured materials for/against bio-adhesion. Moreover, the morphometric analysis can serve as a future gold standard for characterizing a broad spectrum of material structures.

Novel morphometric analysis of higher order structure of human radial peri-papillary capillaries: relevance to retinal perfusion efficiency and age

We apply novel analyses to images of superficial capillaries that are located near and around the optic disc of the human retina: the radial peri-papillary capillaries (RPCs). Due to their unique perfusion of the nerve fibre layer the RPCs are particularly significant for optic-neuropathies. The inputs to the analysis were z-stacks from 3D confocal fluorescence microscopy from 62 human retinas aged 9 to 84 years. Our aim was to find morphometric correlates of age. The retinas had no ophthalmic history. The analysis was undertaken in two stages: (1) converting the z-stacks to 3D tubular networks of vessels, and (2) characterizing the tubular networks using features derived from the Minkowski functionals (MFs). The MFs measure: the capillary volume, surface area, mean breadth, and Euler number. The mean breadth is related to tortuosity, wall shear stress and resistance to flow, and the Euler number is related to the density of loops (collaterals). Features derived from the surface area, mean breadth and Euler number were most related to age (all p ≤ 0.006). The results indicate the importance of pressure-equalizing loops and tortuosity as quantitative measures related to perfusion efficiency. The novel morphometric analysis could quantify disease-related accelerated aging and vessel malformation.

Universal hidden order in amorphous cellular geometries

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

Disordered hyperuniformity implies a hidden order on length scales that can be found in various amorphous materials. Klatt et al. analyse the evolution of random point patterns using Llyod’s algorithm and show that they converge to an effectively hyperuniform state regardless of the initial conditions.

Skinny emulsions take on granular matter

Our understanding of the structural features of foams and emulsions has advanced significantly over the last 20 years. However, with a search for "super-stable" liquid dispersions, foam and emulsion science employs increasingly complex formulations which create solid-like visco-elastic layers at the bubble/drop surfaces. These lead to elastic, adhesive and frictional forces between bubbles/drops, impacting strongly how they pack and deform against each other, asking for an adaptation of the currently available structural description. The possibility to modify systematically the interfacial properties makes these dispersions ideal systems for the exploration of soft granular materials with complex interactions. We present here a first systematic analysis of the structural features of such a system using a model silicone emulsion containing millimetre-sized polyethylene glycol drops (PEG). Solid-like drop surfaces are obtained by polymeric cross-linking reactions at the PEG-silicone interface. Using a novel droplet-micromanipulator, we highlight the presence of elastic, adhesive and frictional interactions between two drops. We then provide for the first time a full tomographic analysis of the structural features of these emulsions. An in-depth analysis of the angle of repose, local volume fraction distributions, pair correlation functions and the drop deformations for different skin formulations allow us to put in evidence the striking difference with "ordinary" emulsions having fluid-like drop surfaces. While strong analogies with frictional hard-sphere systems can be drawn, these systems display a set of unique features due to the high deformability of the drops which await systematic exploration.

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.

Scale-free crystallization of two-dimensional complex plasmas: Domain analysis using Minkowski tensors

Experiments of the recrystallization processes in two-dimensional complex plasmas are analyzed to rigorously test a recently developed scale-free phase transition theory. The "fractal-domain-structure" (FDS) theory is based on the kinetic theory of Frenkel. It assumes the formation of homogeneous domains, separated by defect lines, during crystallization and a fractal relationship between domain area and boundary length. For the defect number fraction and system energy a scale-free power-law relation is predicted. The long-range scaling behavior of the bond-order correlation function shows clearly that the complex plasma phase transitions are not of the Kosterlitz, Thouless, Halperin, Nelson, and Young type. Previous preliminary results obtained by counting the number of dislocations and applying a bond-order metric for structural analysis are reproduced. These findings are supplemented by extending the use of the bond-order metric to measure the defect number fraction and furthermore applying state-of-the-art analysis methods, allowing a systematic testing of the FDS theory with unprecedented scrutiny: A morphological analysis of lattice structure is performed via Minkowski tensor methods. Minkowski tensors form a complete family of additive, motion covariant and continuous morphological measures that are sensitive to nonlinear properties. The FDS theory is rigorously confirmed and predictions of the theory are reproduced extremely well. The predicted scale-free power-law relation between defect fraction number and system energy is verified for one more order of magnitude at high energies compared to the inherently discontinuous bond-order metric. It is found that the fractal relation between crystalline domain area and circumference is independent of the experiment, the particular Minkowski tensor method, and the particular choice of parameters. Thus, the fractal relationship seems to be inherent to two-dimensional phase transitions in complex plasmas. Minkowski tensor analysis turns out to be a powerful tool for investigations of crystallization processes. It is capable of revealing nonlinear local topological properties, however, still provides easily interpretable results founded on a solid mathematical framework.

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

Can exotic disordered "stealthy" particle configurations tolerate arbitrarily large holes?

The probability of finding a spherical cavity or "hole" of arbitrarily large size in typical disordered many-particle systems in the infinite-system-size limit (e.g., equilibrium liquid states) is non-zero. Such "hole" statistics are intimately linked to the thermodynamic and nonequilibrium physical properties of the system. Disordered "stealthy" many-particle configurations in d-dimensional Euclidean space [Doublestruck R]^d are exotic amorphous states of matter that lie between a liquid and crystal that prohibit single-scattering events for a range of wave vectors and possess no Bragg peaks [Torquato et al., Phys. Rev. X, 2015, 5, 021020]. In this paper, we provide strong numerical evidence that disordered stealthy configurations across the first three space dimensions cannot tolerate arbitrarily large holes in the infinite-system-size limit, i.e., the hole probability has compact support. This structural "rigidity" property apparently endows disordered stealthy systems with novel thermodynamic and physical properties, including desirable band-gap, optical and transport characteristics. We also determine the maximum hole size that any stealthy system can possess across the first three space dimensions.

Curvature driven motion of a bubble in a toroidal Hele-Shaw cell

We investigate the equilibrium properties of a single area-minimizing bubble trapped between two narrowly separated parallel curved plates. We begin with the case of a bubble trapped between concentric spherical plates. We develop a model which shows that the surface energy of the bubble is lower when confined between spherical plates than between flat plates. We confirm our findings by comparing against Surface Evolver simulations. We then derive a simple model for a bubble between arbitrarily curved parallel plates. The energy is found to be higher when the local Gaussian curvature of the plates is negative and lower when the curvature is positive. To check the validity of the model, we consider a bubble trapped between concentric tori. In the toroidal case, we find that the sensitivity of the bubble's energy to the local curvature acts as a geometric potential capable of driving bubbles from regions with negative to positive curvature.

The Quantified Characterization Method of the Micro-Macro Contacts of Three-Dimensional Granular Materials on the Basis of Graph Theory

We have attempted a multiscale and quantified characterization method of the contact in three-dimensional granular material made of spherical particles, particularly in cemented granular material. Particle contact is defined as a type of surface contact with voids in its surroundings, rather than a point contact. Macro contact is a particle contact set satisfying the restrictive condition of a two-dimensional manifold with a boundary. On the basis of graph theory, two dual geometrical systems are abstracted from the granular pack. The face and the face set, which satisfies the two-dimensional manifold with a boundary in the solid cell system, are extracted to characterize the particle contact and the macro contact, respectively. This characterization method is utilized to improve the post-processing in DEM (Discrete Element Method) from a micro perspective to describe the macro effect of the cemented granular material made of spherical particles. Since the crack has the same shape as its corresponding contact, this method is adopted to characterize the crack and realize its visualization. The integral failure route of the sample can be determined by a graph theory algorithm. The contact force is assigned to the weight value of the face characterizing the particle contact. Since the force vectors can be added, the macro contact force can be solved by adding the weight of its corresponding faces.

Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development

The formation of the biophotonic gyroid material in butterfly wing scales is an exceptional feat of evolutionary engineering of functional nanostructures. It is hypothesized that this nanostructure forms by chitin polymerization inside a convoluted membrane of corresponding shape in the endoplasmic reticulum. However, this dynamic formation process, including whether membrane folding and chitin expression are simultaneous or sequential processes, cannot yet be elucidated by in vivo imaging. We report an unusual hierarchical ultrastructure in the butterfly Thecla opisena that, as a solid material, allows high-resolution three-dimensional microscopy. Rather than the conventional polycrystalline space-filling arrangement, a gyroid occurs in isolated facetted crystallites with a pronounced size gradient. When interpreted as a sequence of time-frozen snapshots of the morphogenesis, this arrangement provides insight into the formation mechanisms of the nanoporous gyroid material as well as of the intracellular organelle membrane that acts as the template.

A geometric exploration of stress in deformed liquid foams

We explore an alternate way of looking at the rheological response of a yield stress fluid: using discrete geometry to probe the heterogeneous distribution of stress in soap froth. We present quasi-static, uniaxial, isochoric compression and extension of three-dimensional random monodisperse soap froth in periodic boundary conditions and examine the stress and geometry that result. The stress and shape anisotropy of individual cells is quantified by Q, a scalar measure derived from the interface tensor that gauges each cell's contribution to the global stress. Cumulatively, the spatial distribution of highly deformed cells allows us to examine how stress is internally distributed. The topology of highly deformed cells, how they arrange relative to one another in space, gives insight into the heterogeneous distribution of stress.

Hyperuniformity and its generalizations

Disordered many-particle hyperuniform systems are exotic amorphous states of matter that lie between crystal and liquid: They are like perfect crystals in the way they suppress large-scale density fluctuations and yet are like liquids or glasses in that they are statistically isotropic with no Bragg peaks. These exotic states of matter play a vital role in a number of problems across the physical, mathematical as well as biological sciences and, because they are endowed with novel physical properties, have technological importance. Given the fundamental as well as practical importance of disordered hyperuniform systems elucidated thus far, it is natural to explore the generalizations of the hyperuniformity notion and its consequences. In this paper, we substantially broaden the hyperuniformity concept along four different directions. This includes generalizations to treat fluctuations in the interfacial area (one of the Minkowski functionals) in heterogeneous media and surface-area driven evolving microstructures, random scalar fields, divergence-free random vector fields, and statistically anisotropic many-particle systems and two-phase media. In all cases, the relevant mathematical underpinnings are formulated and illustrative calculations are provided. Interfacial-area fluctuations play a major role in characterizing the microstructure of two-phase systems (e.g., fluid-saturated porous media), physical properties that intimately depend on the geometry of the interface, and evolving two-phase microstructures that depend on interfacial energies (e.g., spinodal decomposition). In the instances of random vector fields and statistically anisotropic structures, we show that the standard definition of hyperuniformity must be generalized such that it accounts for the dependence of the relevant spectral functions on the direction in which the origin in Fourier space is approached (nonanalyticities at the origin). Using this analysis, we place some well-known energy spectra from the theory of isotropic turbulence in the context of this generalization of hyperuniformity. Among other results, we show that there exist many-particle ground-state configurations in which directional hyperuniformity imparts exotic anisotropic physical properties (e.g., elastic, optical, and acoustic characteristics) to these states of matter. Such tunability could have technological relevance for manipulating light and sound waves in ways heretofore not thought possible. We show that disordered many-particle systems that respond to external fields (e.g., magnetic and electric fields) are a natural class of materials to look for directional hyperuniformity. The generalizations of hyperuniformity introduced here provide theoreticians and experimentalists new avenues to understand a very broad range of phenomena across a variety of fields through the hyperuniformity "lens."

The microscopic structure of mono-disperse granular heaps and sediments of particles on inclined surfaces

Granular heaps of particles created by deposition of mono-disperse particles raining from an extended source of finite size are characterized by a non-homogeneous field of density. It was speculated that this inhomogeneity is due to the transient shape of the sediment during the process of construction of the heap, thus reflecting the history of the creation of the heap. By comparison of structural characteristics of the heap with sediments created on top of inclined planes exploiting the method of Minkowski tensors, we provide further evidence to support this hypothesis. Moreover, for the case of sediments generated by homogeneous rain on surfaces, we provide relationships between the inclination of the surface and the Minkowski measures characterizing the isotropy of local particle environments.

Stereological estimation of particle shape and orientation from volume tensors

In the present paper, we describe new robust methods of estimating cell shape and orientation in 3D from sections. The descriptors of 3D cell shape and orientation are based on volume tensors which are used to construct an ellipsoid, the Miles ellipsoid, approximating the average cell shape and orientation in 3D. The estimators of volume tensors are based on observations in several optical planes through sampled cells. This type of geometric sampling design is known as the optical rotator. The statistical behaviour of the estimator of the Miles ellipsoid is studied under a flexible model for 3D cell shape and orientation. In a simulation study, the lengths of the axes of the Miles ellipsoid can be estimated with coefficients of variation of about 2% if 100 cells are sampled. Finally, we illustrate the use of the developed methods in an example, involving neurons in the medial prefrontal cortex of rat.

Assessing structural correlations and heterogeneity length scales in functional porous polymers from physical reconstructions

A general, model-free, quantitative approach to the key morphological properties of a porous polymer monolith is presented. After 3D reconstruction, image-based analysis delivers detailed spatial and spatially correlated information on the structural heterogeneities in the void space and the polymer skeleton. Identified heterogeneities, which limit the monolith's performance in targeted applications, are traced back to the preparation process.

Direct relations between morphology and transport in Boolean models

We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allow us to evaluate a power-law relation between the Euler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.

Pattern-fluid interpretation of chemical turbulence

The spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems, from biological tissue to evolutionary population dynamics. The standard model for pattern formation in general, and for Turing patterns in chemical reaction-diffusion systems in particular, are deterministic nonlinear partial differential equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern. However, these models fail to fully explain the experimental observation of turbulent patterns with spatio-temporal disorder in chemical systems. Here we introduce a pattern-fluid model as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction-diffusion system. The transition from turbulent to stationary patterns is then interpreted as a condensation phenomenon, where the nonlinearity forces one single mode to dominate the ensemble. This model leads to better reproduction of the experimental concentration profiles for the "stationary phases" and reproduces the turbulent chemical patterns observed by Q. Ouyang and H. L. Swinney [Chaos 1, 411 (1991)].

A lower bound on the number of mechanisms for discriminating fourth and higher order spatial correlations

Research on single striate cortical neurons has often concentrated on their responses to stimuli defined by two-point correlations. Texture discrimination studies using a relatively small palette of isotrigon textures have indicated that we are sensitive to third and higher-order spatial correlations. To further evaluate the underlying mechanisms of texture discrimination subjects discriminated random binary noise patterns from ten new isotrigon texture types. Factor analysis revealed that as few as three mechanisms may govern the detection of fourth and higher order image structure. This supports the findings of previous studies using different isotrigon textures. The computation of higher-order correlations by the brain is neurophysiologically plausible. The mechanisms identified in this study may represent some short range nonlinear combination of recursive and/or rectifying processes.

Angularly anisotropic correlation in granular packings

We present an x-ray microtomography study of the three-dimensional structural correlations in monodisperse granular packings. By measuring an orientation-dependent pair correlation function, we find that the local structure shows an angularly anisotropic orientation correlation. The correlation is strongest along the major axis of the local Minkowski tensor of the Voronoi cell. It turns out that this anisotropic correlation is consistent with the existence of some locally favored structures. The study suggests the importance of high-order structural correlations in random granular packings.

Characterization of maximally random jammed sphere packings: Voronoi correlation functions

We characterize the structure of maximally random jammed (MRJ) sphere packings by computing the Minkowski functionals (volume, surface area, and integrated mean curvature) of their associated Voronoi cells. The probability distribution functions of these functionals of Voronoi cells in MRJ sphere packings are qualitatively similar to those of an equilibrium hard-sphere liquid and partly even to the uncorrelated Poisson point process, implying that such local statistics are relatively structurally insensitive. This is not surprising because the Minkowski functionals of a single Voronoi cell incorporate only local information and are insensitive to global structural information. To improve upon this, we introduce descriptors that incorporate nonlocal information via the correlation functions of the Minkowski functionals of two cells at a given distance as well as certain cell-cell probability density functions. We evaluate these higher-order functions for our MRJ packings as well as equilibrium hard spheres and the Poisson point process. It is shown that these Minkowski correlation and density functions contain visibly more information than the corresponding standard pair-correlation functions. We find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ packings, consistent with previous findings for other pair correlations [A. Donev et al., Phys. Rev. Lett. 95, 090604 (2005)PRLTAO0031-900710.1103/PhysRevLett.95.090604], indicating that large-scale volume fluctuations are suppressed by accompanying large Voronoi cells with small cells, and vice versa. In contrast to the aforementioned local Voronoi statistics, the correlation functions of the Voronoi cells qualitatively distinguish the structure of MRJ sphere packings (prototypical glasses) from that of not only the Poisson point process but also the correlated equilibrium hard-sphere liquids. Moreover, while we did not find any perfect icosahedra (the locally densest possible structure in which a central sphere contacts 12 neighbors) in the MRJ packings, a preliminary Voronoi topology analysis indicates the presence of strongly distorted icosahedra.

Integral-geometry characterization of photobiomodulation effects on retinal vessel morphology

The morphological characterization of quasi-planar structures represented by gray-scale images is challenging when object identification is sub-optimal due to registration artifacts. We propose two alternative procedures that enhances object identification in the integral-geometry morphological image analysis (MIA) framework. The first variant streamlines the framework by introducing an active contours segmentation process whose time step is recycled as a multi-scale parameter. In the second variant, we used the refined object identification produced in the first variant to perform the standard MIA with exact dilation radius as multi-scale parameter. Using this enhanced MIA we quantify the extent of vaso-obliteration in oxygen-induced retinopathic vascular growth, the preventative effect (by photobiomodulation) of exposure during tissue development to near-infrared light (NIR, 670 nm), and the lack of adverse effects due to exposure to NIR light.

Local fluctuations and spatial correlations in granular flows under constant-volume quasistatic shear

We investigate the local fluctuations in dense granular media subjected to athermal, quasistatic shearing, based on three-dimensional discrete element method simulations. By shearing granular assemblies of different polydispersities under constant-volume constraint, we quantify the characteristics of local structures (in terms of local volume and local anisotropy) and local deformation (using local shear strain and nonaffine displacement). The distribution of the local volume in a granular medium is found unchanged during the entire shearing process, which indicates a constant temperaturelike compactivity for the material. The compactivity is not, however, equilibrated among different particle groups in a polydisperse assembly. The local structures of a disordered granular assembly are inherently anisotropic. The fluctuations in local anisotropy can be well captured by a gamma or mixed-gamma distribution function, which is also unchanged during the shear. The local anisotropic orientation evolves towards the coaxial direction of the stress anisotropy with shear. The deformation characteristics of a jammed granular medium have their origins in the structural amorphousness. The local shear strain field depicts clear shear transformation zones which act as plasticity carriers. The spatial correlation of the local shear strains exhibits a fourfold pattern which is stronger in the stress deviatoric planes than in the stress isotropic plane. The fluctuations of nonaffine displacement suggest an isotropic granular temperature and an isotropic spatial correlation independent of the stress state. Both the local strain and the nonaffine displacement exhibit a power-law decayed distribution with a long-range correlation. We further modify the shear-transformation-zone theory to predict the pressure-dependent constitutive behavior of a sheared granular material and compare its prediction with our simulation data.

Solvation of a sponge-like geometry

Periodic entanglements of filaments and networks, which resemble sponge-like materials, are often found as self-assembled materials. The interaction between the geometry of the assembly and a solvent in its interstices can dictate the geometric configuration of the structure as well as influence macroscopic properties such as swelling and mechanics. In this paper, we show the calculation of the solvation free energy as a function of the solute–solvent interaction from hydrophilic to hydrophobic, for a candidate entanglement of filaments. We do this using the morphometric approach to solvation free energy, a method that disentangles geometric properties from thermodynamic coefficients, which we compute via density functional theory.

Locally countable properties and the perceptual salience of textures

The human ability to discriminate structured from uniformly random binary textures has been shown to exploit third- and higher-order pixel correlations. We examine this ability in an experiment using a large number of texture families that can only be distinguished on the basis of these higher-order correlations. This study investigates statistical models based on possible explanatory variables involving spatial interactions of up to four pixels. Some of these explanatory variables have been recently associated with natural images, and others are somewhat less intuitive and are used here for the first time, to our knowledge. Our models are constructed using intraclass and cross-class feature selection by means of lasso/elastic net optimization and extensive cross-validation. We focus on a special set of locally countable image measures that seem to parsimoniously capture the observed discrimination performance. Among the measures underpinning the best models, we highlight a concept that can only exist in nine-pixel or larger image patches, but nonetheless is calculable based on the multiplicity of specific four-pixel patches in a texture. We show that this single geometric concept provides significant clues to explain texture discrimination.