Modeling heterogeneous materials via two-point correlation functions. II. Algorithmic details and applications

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.

Modeling flow of Carreau fluids in porous media

Carreau fluids occur routinely in porous medium systems for a range of applications, and the dependence of the viscosity for such fluids on the rate of strain tensor poses challenges to modeling at an averaged macroscale. Traditional approaches for macroscale modeling such flows have relied upon experimental observations of flows for generalized Newtonian fluids (GNFs) and a phenomenological approach referred to herein as the shift factor. A recently developed approach based upon averaging conservation and thermodynamic equations from the microscale for Cross model GNFs is extended to the case of Carreau fluids and shown to predict the flow through both isotropic and anisotropic media accurately without the need for GNF-flow experiments. The model is formulated in terms of rheological properties, a standard Newtonian resistance tensor, and a length-scale tensor, which does require estimation. An approach based upon measures of the morphology and topology of the pore space is developed to approximate this length-scale tensor. Thus, this work provides the missing components needed to predict Carreau GNF macroscale flow with only rheological information for the fluid and analysis of the pore morphology and topology independent of any fluid flow experiments. Accuracy of predictions based upon this approach is quantified, and extension to other GNFs is straightforward.

Three-dimensional reconstruction of granular porous media based on deep generative models

Reconstruction of microstructure in granular porous media, which can be viewed as granular assemblies, is crucial for studying their characteristics and physical properties in various fields concerned with the behavior of such media, including petroleum geology and computational materials science. In spite of the fact that many existing studies have investigated grain reconstruction, most of them treat grains as simplified individuals for discrete reconstruction, which cannot replicate the complex geometrical shapes and natural interactions between grains. In this work, a hybrid generative model based on a deep-learning algorithm is proposed for high-quality three-dimensional (3D) microstructure reconstruction of granular porous media from a single two-dimensional (2D) slice image. The method extracts 2D prior information from the given image and generates the grain set as a whole. Both a self-attention module and effective pattern loss are introduced in a bid to enhance the reconstruction ability of the model. Samples with grains of varied geometrical shapes are utilized for the validation of our method, and experimental results demonstrate that our proposed approach can accurately reproduce the complex morphology and spatial distribution of grains without any artificiality. Furthermore, once the model training is complete, rapid end-to-end generation of diverse 3D realizations from a single 2D image can be achieved.

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.

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.

Fast reconstruction of multiphase microstructures based on statistical descriptors

In this paper, we propose a hierarchical simulated annealing of erosion method (HSAE) to improve the computational efficiency of multiphase microstructure reconstruction, whose computational efficiency can be improved by an order of magnitude. Reconstruction of the two-dimensional (2D) and three-dimensional (3D) multiphase microstructures (pore, grain, and clay) based on simulated annealing (SA) and HSAE are performed. In the reconstruction of multiphase microstructure with HSAE and SA, three independent two-point correlation functions are chosen as the morphological information descriptors. The two-point cluster function which contains significant high-order statistical information is used to verify the reconstruction results. From the analysis of 2D reconstruction, it can find that the proposed HSAE technique not only improves the quality of reconstruction, but also improves the computational efficiency. The reconstructions of our proposed method are still imperfect. This is because the used two-point correlation functions contain insufficient information. For the 3D reconstruction, the two-point correlation functions of the 3D generation are in excellent agreement with those of the original 2D image, which illustrates that our proposed method is effective for the reconstruction of 3D microstructure. The comparison of the energy vs computational time between the SA and HSAE methods shows that our presented method is an order of magnitude faster than the SA method. That is because only some of the pixels in the overall hierarchy need to be considered for sampling.

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.

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.

A machine learning model for predicting the ballistic impact resistance of unidirectional fiber-reinforced composite plate

It has been a vital issue to ensure both the accuracy and efficiency of computational models for analyzing the ballistic impact response of fiber-reinforced composite plates (FRCP). In this paper, a machine learning (ML) model is established in an effort to bridge the ballistic impact protective performance and the characteristics of microstructure for unidirectional FRCP (UD-FRCP), where the microstructure of the UD-FRCP is characterized by the two-point correlation function. The results showed that the ML model, after trained by 175 cases, could reasonably predict the ballistic impact energy absorption of the UD-FRCP with a maximum error of 13%, indicating that the model can ensure both computational accuracy and efficiency. Besides, the model's critical parameter sensitivities are investigated, and three typical ML algorithms are analyzed, showing that the gradient boosting regression algorithm has the highest accuracy among these algorithms for the ballistic impact problem of UD-FRCP. The study proposes an effective solution for the traditional difficulty of the ballistic impact simulation of composites with both high efficiency and accuracy.

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.

A Two-Stage Reconstruction of Microstructures with Arbitrarily Shaped Inclusions

The main goal of our research is to develop an effective method with a wide range of applications for the statistical reconstruction of heterogeneous microstructures with compact inclusions of any shape, such as highly irregular grains. The devised approach uses multi-scale extended entropic descriptors (ED) that quantify the degree of spatial non-uniformity of configurations of finite-sized objects. This technique is an innovative development of previously elaborated entropy methods for statistical reconstruction. Here, we discuss the two-dimensional case, but this method can be generalized into three dimensions. At the first stage, the developed procedure creates a set of black synthetic clusters that serve as surrogate inclusions. The clusters have the same individual areas and interfaces as their target counterparts, but random shapes. Then, from a given number of easy-to-generate synthetic cluster configurations, we choose the one with the lowest value of the cost function defined by us using extended ED. At the second stage, we make a significant change in the standard technique of simulated annealing (SA). Instead of swapping pixels of different phases, we randomly move each of the selected synthetic clusters. To demonstrate the accuracy of the method, we reconstruct and analyze two-phase microstructures with irregular inclusions of silica in rubber matrix as well as stones in cement paste. The results show that the two-stage reconstruction (TSR) method provides convincing realizations for these complex microstructures. The advantages of TSR include the ease of obtaining synthetic microstructures, very low computational costs, and satisfactory mapping in the statistical context of inclusion shapes. Finally, its simplicity should greatly facilitate independent applications.

Quantifying accuracy of stochastic methods of reconstructing complex materials by deep learning

Time and cost are two main hurdles to acquiring a large number of digital image I of the microstructure of materials. Thus, use of stochastic methods for producing plausible realizations of materials' morphology based on one or very few images has become an increasingly common practice in their modeling. The accuracy of the realizations is often evaluated using two-point microstructural descriptors or physics-based modeling of certain phenomena in the materials, such as transport processes or fluid flow. In many cases, however, two-point correlation functions do not provide accurate evaluation of the realizations, as they are usually unable to distinguish between high- and low-quality reconstructed models. Calculating flow and transport properties of the realization is an accurate way of checking the quality of the realizations, but it is computationally expensive. In this paper a method based on machine learning is proposed for evaluating stochastic approaches for reconstruction of materials, which is applicable to any of such methods. The method reduces the dimensionality of the realizations using an unsupervised deep-learning algorithm by compressing images and realizations of materials. Two criteria for evaluating the accuracy of a reconstruction algorithm are then introduced. One, referred to as the internal uncertainty space, is based on the recognition that for a reconstruction method to be effective, the differences between the realizations that it produces must be reasonably wide, so that they faithfully represent all the possible spatial variations in the materials' microstructure. The second criterion recognizes that the realizations must be close to the original I and, thus, it quantifies the similarity based on an external uncertainty space. Finally, the ratio of two uncertainty indices associated with the two criteria is considered as the final score of the accuracy of a stochastic algorithm, which provides a quantitative basis for comparing various realizations and the approaches that produce them. The proposed method is tested with images of three types of heterogeneous materials in order to evaluate four stochastic reconstruction algorithms.

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.

Computed Tomography 3D Super-Resolution with Generative Adversarial Neural Networks: Implications on Unsaturated and Two-Phase Fluid Flow

Fluid flow characteristics are important to assess reservoir performance. Unfortunately, laboratory techniques are inadequate to know these characteristics, which is why numerical methods were developed. Such methods often use computed tomography (CT) scans as input but this technique is plagued by a resolution versus sample size trade-off. Therefore, a super-resolution method using generative adversarial neural networks (GANs) was used to artificially improve the resolution. Firstly, the influence of resolution on pore network properties and single-phase, unsaturated, and two-phase flow was analysed to verify that pores and pore throats become larger on average and surface area decreases with worsening resolution. These observations are reflected in increasingly overestimated single-phase permeability, less moisture uptake at lower capillary pressures, and high residual oil fraction after waterflooding. Therefore, the super-resolution GANs were developed which take low (12 µm) resolution input and increase the resolution to 4 µm, which is compared to the expected high-resolution output. These results better predicted pore network properties and fluid flow properties despite the overestimation of porosity. Relevant small pores and pore surfaces are better resolved thus providing better estimates of unsaturated and two-phase flow which can be heavily influenced by flow along pore boundaries and through smaller pores. This study presents the second case in which GANs were applied to a super-resolution problem on geological materials, but it is the first one to apply it directly on raw CT images and to determine the actual impact of a super-resolution method on fluid predictions.

Effect of An Image Resolution Change on the Effective Transport Coefficient of Heterogeneous Materials

Electrochemical electrodes comprise multiple phenomena at different scales. Several works have tried to model such phenomena using statistical techniques. This paper proposes a novel process to work with reduced size images to reconstruct microstructures with the Simulated Annealing method. Later, using the Finite Volume Method, it is verified the effect of the image resolution on the effective transport coefficient (ETC). The method can be applied to synthetic images or images from the Scanning Electron Microscope. The first stage consists of obtaining the image of minimum size, which contains at least 98% of the statistical information of the original image, allowing an equivalent statistical study. The image size reduction was made by applying an iterative decimation over the image using the normalized coarseness to compare the amount of information contained at each step. Representative improvements, especially in processing time, are achieved by reducing the size of the reconstructed microstructures without affecting their statistical behavior. The process ends computing the conduction efficiency from the microstructures. The simulation results, obtained from two kinds of images from different materials, demonstrate the effectivity of the proposed approach. It is important to remark that the controlled decimation allows a reduction of the processor and memory use during the reconstruction and ETC computation of electrodes.

The relationship between domain-specific subjective cognitive decline and Alzheimer's pathology in normal elderly adults

We evaluated the associations of subjective (self-reported everyday cognition [ECog]) and objective cognitive measures with regional amyloid-β (Aβ) and tau accumulation in 86 clinically normal elderly subjects from the Alzheimer's Disease Neuroimaging Initiative. Regression analyses were conducted to identify whether individual ECog domains (Memory, Language, Organization, Planning, Visuospatial, and Divided Attention) were equally or differentially associated with regional [18F]florbetapir and [18F]flortaucipir uptake and how these associations compared to those obtained with objective cognitive measures. A texture analysis, the weighted 2-point correlation, was used as an additional approach for estimating the whole-brain tau burden without positron emission tomography intensity normalization. Although the strongest models for ECog domains included either tau (planning and visuospatial) or Aβ (memory and organization), the strongest models for all objective measures included Aβ. In Aβ-negative participants, the strongest models for all ECog domains of executive functioning included tau. Our results indicate differential associations of individual subjective cognitive domains with Aβ and tau in clinically normal adults. Detailed characterization of ECog may render a valuable prescreening tool for pathological prediction.

Reconstruction of porous media from extremely limited information using conditional generative adversarial networks

Porous media are ubiquitous in both nature and engineering applications. Therefore, their modeling and understanding is of vital importance. In contrast to direct acquisition of three-dimensional (3D) images of this type of medium, obtaining its subregion (s) such as 2D images or several small areas can be feasible. Therefore, reconstructing whole images from limited information is a primary technique in these types of cases. Given data in practice cannot generally be determined by users and may be incomplete or only partially informed, thus making existing reconstruction methods inaccurate or even ineffective. To overcome this shortcoming, in this study we propose a deep-learning-based framework for reconstructing full images from their much smaller subareas. In particular, conditional generative adversarial network is utilized to learn the mapping between the input (a partial image) and output (a full image). To ensure the reconstruction accuracy, two simple but effective objective functions are proposed and then coupled with the other two functions to jointly constrain the training procedure. Because of the inherent essence of this ill-posed problem, a Gaussian noise is introduced for producing reconstruction diversity, thus enabling the network to provide multiple candidate outputs. Our method is extensively tested on a variety of porous materials and validated by both visual inspection and quantitative comparison. It is shown to be accurate, stable, and even fast (∼0.08 s for a 128×128 image reconstruction). The proposed approach can be readily extended by, for example, incorporating user-defined conditional data and an arbitrary number of object functions into reconstruction, while being coupled with other reconstruction methods.

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.

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.

Accurate modeling and evaluation of microstructures in complex materials

Accurate characterization of heterogeneous materials is of great importance for different fields of science and engineering. Such a goal can be achieved through imaging. Acquiring three- or two-dimensional images under different conditions is not, however, always plausible. On the other hand, accurate characterization of complex and multiphase materials requires various digital images (I) under different conditions. An ensemble method is presented that can take one single (or a set of) I(s) and stochastically produce several similar models of the given disordered material. The method is based on a successive calculating of a conditional probability by which the initial stochastic models are produced. Then, a graph formulation is utilized for removing unrealistic structures. A distance transform function for the Is with highly connected microstructure and long-range features is considered which results in a new I that is more informative. Reproduction of the I is also considered through a histogram matching approach in an iterative framework. Such an iterative algorithm avoids reproduction of unrealistic structures. Furthermore, a multiscale approach, based on pyramid representation of the large Is, is presented that can produce materials with millions of pixels in a matter of seconds. Finally, the nonstationary systems-those for which the distribution of data varies spatially-are studied using two different methods. The method is tested on several complex and large examples of microstructures. The produced results are all in excellent agreement with the utilized Is and the similarities are quantified using various correlation functions.

Reconstruction of a digital core containing clay minerals based on a clustering algorithm

It is difficult to obtain a core sample and information for digital core reconstruction of mature sandstone reservoirs around the world, especially for an unconsolidated sandstone reservoir. Meanwhile, reconstruction and division of clay minerals play a vital role in the reconstruction of the digital cores, although the two-dimensional data-based reconstruction methods are specifically applicable as the microstructure reservoir simulation methods for the sandstone reservoir. However, reconstruction of clay minerals is still challenging from a research viewpoint for the better reconstruction of various clay minerals in the digital cores. In the present work, the content of clay minerals was considered on the basis of two-dimensional information about the reservoir. After application of the hybrid method, and compared with the model reconstructed by the process-based method, the digital core containing clay clusters without the labels of the clusters' number, size, and texture were the output. The statistics and geometry of the reconstruction model were similar to the reference model. In addition, the Hoshen-Kopelman algorithm was used to label various connected unclassified clay clusters in the initial model and then the number and size of clay clusters were recorded. At the same time, the K-means clustering algorithm was applied to divide the labeled, large connecting clusters into smaller clusters on the basis of difference in the clusters' characteristics. According to the clay minerals' characteristics, such as types, textures, and distributions, the digital core containing clay minerals was reconstructed by means of the clustering algorithm and the clay clusters' structure judgment. The distributions and textures of the clay minerals of the digital core were reasonable. The clustering algorithm improved the digital core reconstruction and provided an alternative method for the simulation of different clay minerals in the digital cores.

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.

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.

Reconstructing the microstructure of polyimide-silicalite mixed-matrix membranes and their particle connectivity using FIB-SEM tomography

Properties of a composite material made of a continuous matrix and particles often depend on microscopic details, such as contacts between particles. Focusing on processing raw focused-ion beam scanning electron microscope (FIB-SEM) tomography data, we reconstructed three mixed-matrix membrane samples made of 6FDA-ODA polyimide and silicalite-1 particles. In the first step of image processing, backscattered electron (BSE) and secondary electron (SE) signals were mixed in a ratio that was expected to obtain a segmented 3D image with a realistic volume fraction of silicalite-1. Second, after spatial alignment of the stacked FIB-SEM data, the 3D image was smoothed using adaptive median and anisotropic nonlinear diffusion filters. Third, the image was segmented using the power watershed method coupled with a seeding algorithm based on geodesic reconstruction from the markers. If the resulting volume fraction did not match the target value quantified by chemical analysis of the sample, the BSE and SE signals were mixed in another ratio and the procedure was repeated until the target volume fraction was achieved. Otherwise, the segmented 3D image (replica) was accepted and its microstructure was thoroughly characterized with special attention paid to connectivity of the silicalite phase. In terms of the phase connectivity, Monte Carlo simulations based on the pure-phase permeability values enabled us to calculate the effective permeability tensor, the main diagonal elements of which were compared with the experimental permeability. In line with the hypothesis proposed in our recent paper (Čapek, P. et al. (2014) Comput. Mater. Sci. 89, 142-156), the results confirmed that the existence of particle clusters was a key microstructural feature determining effective permeability.

Evaluating the morphological completeness of a training image

Understanding the three-dimensional (3D) stochastic structure of a porous medium is helpful for studying its physical properties. A 3D stochastic structure can be reconstructed from a two-dimensional (2D) training image (TI) using mathematical modeling. In order to predict what specific morphology belonging to a TI can be reconstructed at the 3D orthogonal slices by the method of 3D reconstruction, this paper begins by introducing the concept of orthogonal chords. After analyzing the relationship among TI morphology, orthogonal chords, and the 3D morphology of orthogonal slices, a theory for evaluating the morphological completeness of a TI is proposed for the cases of three orthogonal slices and of two orthogonal slices. The proposed theory is evaluated using four TIs of porous media that represent typical but distinct morphological types. The significance of this theoretical evaluation lies in two aspects: It allows special morphologies, for which the attributes of a TI can be reconstructed at a special orthogonal slice of a 3D structure, to be located and quantified, and it can guide the selection of an appropriate reconstruction method for a special TI.

Heterogeneous force network in 3D cellularized collagen networks

Collagen networks play an important role in coordinating and regulating collective cellular dynamics via a number of signaling pathways. Here, we investigate the transmission of forces generated by contractile cells in 3D collagen-I networks. Specifically, the graph (bond-node) representations of collagen networks with collagen concentrations of 1, 2 and 4 mg ml^-1 are derived from confocal microscopy data and used to model the network microstructure. Cell contraction is modeled by applying correlated displacements at specific nodes of the network, representing the focal adhesion sites. A nonlinear elastic model is employed to characterize the mechanical behavior of individual fiber bundles including strain hardening during stretching and buckling under compression. A force-based relaxation method is employed to obtain equilibrium network configurations under cell contraction. We find that for all collagen concentrations, the majority of the forces are carried by a small number of heterogeneous force chains emitted from the contracting cells, which is qualitatively consistent with our experimental observations. The force chains consist of fiber segments that either possess a high degree of alignment before cell contraction or are aligned due to fiber reorientation induced by cell contraction. The decay of the forces along the force chains is significantly slower than the decay of radially averaged forces in the system, suggesting that the fibreous nature of biopolymer network structure can support long-range force transmission. The force chains emerge even at very small cell contractions, and the number of force chains increases with increasing cell contraction. At large cell contractions, the fibers close to the cell surface are in the nonlinear regime, and the nonlinear region is localized in a small neighborhood of the cell. In addition, the number of force chains increases with increasing collagen concentration, due to the larger number of focal adhesion sites in collagen networks with high concentrations.

Efficient three-phase reconstruction of heterogeneous material from 2D cross-sections via phase-recovery algorithm

Digital reconstruction of a complex heterogeneous media from the limited statistical information, mostly provided by different imaging techniques, is the key to the successful computational analysis of this important class of materials. In this study, a novel approach is presented for three-dimensional (3D) reconstruction of a three-phase microstructure from its statistical information provided by two-dimensional (2D) cross-sections. In this three-step method, first two-point correlation functions (TPCFs) are extracted from the cross-section(s) using a spectral method suitable for the three-phase media. In the next step, 3D TPCFs are approximated for all vectors in a representative volume element (RVE). Finally, the 3D microstructure is realized from the full-set TPCFs obtained in the previous step, using a modified phase-recovery algorithm. The method is generally applicable to any complex three-phase media, here illustrated for an SOFC anode microstructure. The capabilities and shortcomings of the method are then investigated by performing a qualitative comparison between example cross-sections obtained computationally and their experimental equivalents. Finally, it is shown that the method almost conserves key microstructural properties of the media including tortuosity, percolation and three-phase boundary length (TPBL).

Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media

We present a strong relationship between the microstructural characteristics of, and the fluid velocity fields confined to, three-dimensional random porous materials. The relationship is revealed through simultaneously extracting correlation functions R_{uu}(r) of the spatial (Eulerian) velocity fields and microstructural two-point correlation functions S_{2}(r) of the random porous heterogeneous materials. This demonstrates that the effective physical transport properties depend on the characteristics of complex pore structure owing to the relationship between R_{uu}(r) and S_{2}(r) revealed in this study. Further, the mean excess plot was used to investigate the right tail of the streamwise velocity component that was found to obey light-tail distributions. Based on the mean excess plot, a generalized Pareto distribution can be used to approximate the positive streamwise velocity distribution.

Multigrid hierarchical simulated annealing method for reconstructing heterogeneous media

A reconstruction methodology based on different-phase-neighbor (DPN) pixel swapping and multigrid hierarchical annealing is presented. The method performs reconstructions by starting at a coarse image and successively refining it. The DPN information is used at each refinement stage to freeze interior pixels of preformed structures. This preserves the large-scale structures in refined images and also reduces the number of pixels to be swapped, thereby resulting in a decrease in the necessary computational time to reach a solution. Compared to conventional single-grid simulated annealing, this method was found to reduce the required computation time to achieve a reconstruction by around a factor of 70-90, with the potential of even higher speedups for larger reconstructions. The method is able to perform medium sized (up to 300(3) voxels) three-dimensional reconstructions with multiple correlation functions in 36-47 h.

Dynamic reconstruction of heterogeneous materials and microstructure evolution

Reconstructing heterogeneous materials from limited structural information has been a topic that attracts extensive research efforts and still poses many challenges. The Yeong-Torquato procedure is one of the most popular reconstruction techniques, in which the material reconstruction problem based on a set of spatial correlation functions is formulated as a constrained energy minimization (optimization) problem and solved using simulated annealing [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)]. The standard two-point correlation function S2 has been widely used in reconstructions, but can also lead to large structural degeneracy for certain nearly percolating systems. To improve reconstruction accuracy and reduce structural degeneracy, one can successively incorporate additional morphological information (e.g., nonconventional or higher-order correlation functions), which amounts to reshaping the energy landscape to create a deep (local) energy minimum. In this paper, we present a dynamic reconstruction procedure that allows one to use a series of auxiliary S2 to achieve the same level of accuracy as those incorporating additional nonconventional correlation functions. In particular, instead of randomly sampling the microstructure space as in the simulated annealing scheme, our procedure utilizes a series of auxiliary microstructures that mimic a physical structural evolution process (e.g., grain growth). This amounts to constructing a series auxiliary energy landscapes that bias the convergence of the reconstruction to a favored (local) energy minimum. Moreover, our dynamic procedure can be naturally applied to reconstruct an actual microstructure evolution process. In contrast to commonly used evolution reconstruction approaches that separately generate individual static configurations, our procedure continuously evolves a single microstructure according to a time-dependent correlation function. The utility of our procedure is illustrated by successfully reconstructing nearly percolating hard-sphere packings and particle-reinforced composites as well as the coarsening process in a binary metallic alloy.

Universal spatial correlation functions for describing and reconstructing soil microstructure

Structural features of porous materials such as soil define the majority of its physical properties, including water infiltration and redistribution, multi-phase flow (e.g. simultaneous water/air flow, or gas exchange between biologically active soil root zone and atmosphere) and solute transport. To characterize soil microstructure, conventional soil science uses such metrics as pore size and pore-size distributions and thin section-derived morphological indicators. However, these descriptors provide only limited amount of information about the complex arrangement of soil structure and have limited capability to reconstruct structural features or predict physical properties. We introduce three different spatial correlation functions as a comprehensive tool to characterize soil microstructure: 1) two-point probability functions, 2) linear functions, and 3) two-point cluster functions. This novel approach was tested on thin-sections (2.21×2.21 cm2) representing eight soils with different pore space configurations. The two-point probability and linear correlation functions were subsequently used as a part of simulated annealing optimization procedures to reconstruct soil structure. Comparison of original and reconstructed images was based on morphological characteristics, cluster correlation functions, total number of pores and pore-size distribution. Results showed excellent agreement for soils with isolated pores, but relatively poor correspondence for soils exhibiting dual-porosity features (i.e. superposition of pores and micro-cracks). Insufficient information content in the correlation function sets used for reconstruction may have contributed to the observed discrepancies. Improved reconstructions may be obtained by adding cluster and other correlation functions into reconstruction sets. Correlation functions and the associated stochastic reconstruction algorithms introduced here are universally applicable in soil science, such as for soil classification, pore-scale modelling of soil properties, soil degradation monitoring, and description of spatial dynamics of soil microbial activity.