Geodesic Tortuosity Research Articles

Optimizing the Pore Structure and Geometry of Polycaprolactone/Graphene Scaffold to Promote Osteogenesis

Introduction: Pore structure and geometry are crucial in scaffold development for tissue engineering. From this viewpoint, pore geometry characterization methods will aid in understanding the influence of pore structure (pore size and diffusivity) on its properties. These properties determine how oxygen and water enter the scaffold, forming adsorption states. Previous studies showed that the addition of graphene (G) to polycaprolactone (PCL) increased the pore size, and played a significant role in tissue regeneration. Therefore, this study aimed to evaluate pore structure, geometry, and viability that are suitable for osteogenesis. Materials and methods: Morphology, size, and size distribution of PCL and PCL/G scaffolds were measured at concentrations of 1, 2, and 3 wt% G for the pore structure, while the connectivity of the scaffold's pore was analyzed using the geodesic tortuosity. This is essential because these factors promote the viability of osteogenesis-supporting cells. Results and discussion: The results showed that 3 wt% G had a good water diffusivity rate, larger pores, better connectivity, and viability than other concentrations. Conclusion: In conclusion, the presence of G in scaffolds affects the pore geometry and structure. This results in several advantageous effects that support osteogenesis, such as increased water and nutrient uptake, enhanced waste metabolism transport, and increased viability of the cells. HIGHLIGHTS Pore structure and geometry are essential for tissue engineering scaffold development. 3 wt% G concentration exhibited better interconnectivity compared to PCL. Larger pore sizes improve fluid flow because of the higher diffusivity of the scaffold. MG-63 cells, resembling osteoblasts, exhibited enhanced proliferation at 3 wt% G. GRAPHICAL ABSTRACT

Mesh Density and Geodesic Tortuosity in Planar Triangular Tesselations Devoted to Fracture Mechanics

Mesh Density and Geodesic Tortuosity in Planar Triangular Tesselations Devoted to Fracture Mechanics

Changes in filtration and capacitance properties of highly porous reservoir in underground gas storage: CT-based and geomechanical modeling

The paper presents the results of comprehensive studies of filtration and capacitance properties of highly porous reservoir rocks of the aquifer of an underground gas storage facility. The geomechanical part of the research included studying the dependence of rock permeability on the stress-strain state in the vicinity of the wells, and physical modeling of the implementation of the method of increasing the permeability of the wellbore zone - the method of directional unloading of the reservoir. The digital part of the research included computed tomography (CT)-based computer analysis of the internal structure, pore space characteristics, and filtration properties before and after the tests. According to the results of physical modeling of deformation and filtration processes, it is found that the permeability of rocks before fracture depends on the stress-strain state insignificantly, and this influence is reversible. However, when downhole pressure reaches 7-8 MPa, macrocracks in the rock begin to grow, accompanied by irreversible permeability increase. Porosity, geodesic tortuosity and permeability values were obtained based on digital studies and numerical modeling. A weak degree of transversal anisotropy of the filtration properties of rocks was detected. Based on the analysis of pore size distribution, pressure field and flow velocities, high homogeneity and connectivity of the rock pore space is shown. The absence of pronounced changes in pore space characteristics and pore permeability after non-uniform triaxial loading rocks was shown. On the basis of geometrical analysis of pore space, the reasons for weak permeability anisotropy were identified. The filtration-capacitance properties obtained from the digital analysis showed very good agreement with the results of field and laboratory measurements. The physical modeling has confirmed the efficiency of application of the directional unloading method for the reservoir under study. The necessary parameters of its application were calculated: bottomhole geometry, stage of operation, stresses and pressure drawdown value.

Three-dimensional reconstruction and computational analysis of a structural battery composite electrolyte

Structural batteries are multifunctional composite materials that can carry mechanical load and store electrical energy. Their multifunctionality requires an ionically conductive and stiff electrolyte matrix material. For this purpose, a bi-continuous polymer electrolyte is used where a porous solid phase holds the structural integrity of the system, and a liquid phase, which occupies the pores, conducts lithium ions. To assess the porous structure, three-dimensional topology information is needed. Here we study the three-dimensional structure of the porous battery electrolyte material using combined focused ion beam and scanning electron microscopy and transfer into finite element models. Numerical analyses provide predictions of elastic modulus and ionic conductivity of the bi-continuous electrolyte material. Characterization of the three-dimensional structure also provides information on the diameter and volume distributions of the polymer and pores, as well as geodesic tortuosity.

Quantifying the Impact of 3D Pore Space Morphology on Soil Gas Diffusion in Loam and Sand

Effective diffusion is an important macroscopic property for assessing transport in porous media. Numerical computations on segmented 3D CT images yield precise estimates for diffusive properties. On the other hand, geometrical descriptors of pore space such as porosity, specific surface area and further transport-related descriptors can be easily computed from 3D CT images and are closely linked to diffusion processes. However, the investigation of quantitative relationships between these descriptors and diffusive properties for a diverse range of porous structures is still ongoing. In the present paper, we consider three different soil samples of each loam and sand for a total of six samples, whose 3D microstructure is quantitatively investigated using univariate as well as bivariate probability distributions of geometrical pore space descriptors. This information is used for investigating microstructure–property relationships by means of empirically derived regression formulas, where a particular focus is put on the differences between loam and sand samples. Due to the analytical nature of these formulas, it is possible to obtain a deeper understanding for the relationship between the 3D pore space morphology and the resulting diffusive properties. In particular, it is shown that formulas existing so far in the literature for predicting soil gas diffusion can be significantly improved by incorporating further geometrical descriptors such as geodesic tortuosity, chord lengths, or constrictivity of the pore space. The robustness of these formulas is investigated by fitting the regression parameters on different data sets as well as by applying the empirically derived regression formulas to data that is not used for model fitting. Among others, it turns out that a formula based on porosity as well as mean and standard deviation of geodesic tortuosity performs best with regard to the coefficient of determination and the mean absolute percentage error. Moreover, it is shown that regarding the prediction of diffusive properties the concept of geodesic tortuosity is superior to geometric tortuosity, where the latter is based on the creation of a skeleton of the pore space.

Investigating the influence of the calendering process on the 3D microstructure of single-layer and two-layer cathodes in lithium-ion batteries using synchrotron tomography

Further improvement of the performance of lithium-ion batteries is a central goal in state-of-the-art battery research to satisfy the permanently growing demands. This can be achieved by optimizing the 3D microstructure of the electrodes, which in turn depends on the material composition used as well as on the corresponding production parameters. Since the calendering process of electrodes has a significant impact on their microstructure, the goal of this paper is to further deepen the understanding of how varying the compaction load changes the microstructure of lithium-ion battery cathodes. For this purpose, three different sample sets are analyzed and compared to each other, using image data gained by synchrotron tomography. More precisely, we consider differently compacted thin and thick cathodes as well as a set of porosity-graded two-layer cathode samples. A phase-based as well as a particle-based segmentation of the tomographic image data is performed to allow for an extensive morphological analysis of the cathode samples, where, among others, mean geodesic tortuosity of the pore space and particle connectivity are considered to quantify changes of the 3D microstructure. Furthermore, for the two-layer sample set, a special focus is put on microstructural changes of both layers and the interface between them.

Generating digital twins of mesoporous silica by graph-based stochastic microstructure modeling

Silica monoliths are hierarchically structured, versatile materials that are widely used in analytical separation science, e.g., liquid chromatography. Their functionality strongly depends on the 3D morphology of their macropore and mesopore spaces. In the present paper, we consider three differently manufactured silica monolith samples, where the process conditions of their hydrothermal treatment (affecting, e.g., mesopore size) have been varied, and present a parametric stochastic 3D microstructure model that is able to generate digital twins of the resulting mesopore spaces. The model, which is based on random point processes and relative neighborhood graphs, theoretically guarantees the complete connectivity of both, the silica phase and the mesopores. The parametric model is fitted to electron tomographic image data. For this purpose, we optimize a cost function that is based on empirically derived relationships between model parameters and volume fraction, mean geodesic tortuosity and constrictivity. Validation is performed regarding further microstructure characteristics, which are not used for model fitting, and regarding effective diffusivity, which is numerically simulated by a particle-tracking algorithm based on random walks.

Predicting permeability via statistical learning on higher-order microstructural information

Quantitative structure–property relationships are crucial for the understanding and prediction of the physical properties of complex materials. For fluid flow in porous materials, characterizing the geometry of the pore microstructure facilitates prediction of permeability, a key property that has been extensively studied in material science, geophysics and chemical engineering. In this work, we study the predictability of different structural descriptors via both linear regressions and neural networks. A large data set of 30,000 virtual, porous microstructures of different types, including both granular and continuous solid phases, is created for this end. We compute permeabilities of these structures using the lattice Boltzmann method, and characterize the pore space geometry using one-point correlation functions (porosity, specific surface), two-point surface-surface, surface-void, and void-void correlation functions, as well as the geodesic tortuosity as an implicit descriptor. Then, we study the prediction of the permeability using different combinations of these descriptors. We obtain significant improvements of performance when compared to a Kozeny-Carman regression with only lowest-order descriptors (porosity and specific surface). We find that combining all three two-point correlation functions and tortuosity provides the best prediction of permeability, with the void-void correlation function being the most informative individual descriptor. Moreover, the combination of porosity, specific surface, and geodesic tortuosity provides very good predictive performance. This shows that higher-order correlation functions are extremely useful for forming a general model for predicting physical properties of complex materials. Additionally, our results suggest that artificial neural networks are superior to the more conventional regression methods for establishing quantitative structure–property relationships. We make the data and code used publicly available to facilitate further development of permeability prediction methods.

On Variability and Interdependence of Local Porosity and Local Tortuosity in Porous Materials: a Case Study for Sack Paper

The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.

Quantifying the influence of microstructure on effective conductivity and permeability: Virtual materials testing

Effective conductivity and permeability of a versatile, graph-based model of random structures are investigated numerically. This model, originally introduced in Gaiselmann et al. (2014) allows one to simulate a wide class of realistic materials. In the present work, an extensive dataset of two-phase microstructures with wide-ranging morphological features is used to assess the relationship between microstructure and effective transport properties, which are computed using Fourier-based methods on digital images. Our main morphological descriptors are phase volume fractions, mean geodesic tortuosity, two “hydraulic radii” for characterizing the length scales of heterogeneities, and a “constrictivity” parameter that describes bottleneck effects. This additional parameter, usually not considered in homogenization theories, is an essential ingredient for predicting transport properties, as observed in Gaiselmann et al. (2014). We modify the formula originally developed in Stenzel et al. (2016) for predicting the effective conductivity and propose a formula for permeability. For the latter one, different geometrical definitions of the hydraulic radius are compared. Our predictions are validated using tomographic image data of fuel cells.

Stochastic 3D Modeling of Three-Phase Microstructures for Predicting Transport Properties: A Case Study

We compare two conceptually different stochastic microstructure models, i.e., a graph-based model and a pluri-Gaussian model, that have been introduced to model the transport properties of three-phase microstructures occurring, e.g., in solid oxide fuel cell electrodes. Besides comparing both models, we present new results regarding the relationship between model parameters and certain microstructure characteristics. In particular, an analytical expression is obtained for the expected length of triple phase boundary per unit volume in the pluri-Gaussian model. As a case study, we consider 3D image data which show a representative cutout of a solid oxide fuel cell anode obtained by FIB-SEM tomography. The two models are fitted to image data and compared in terms of morphological characteristics (like mean geodesic tortuosity and constrictivity) as well as in terms of effective transport properties. The Stokes flow in the pore phase and effective conductivities in the solid phases are computed numerically for realizations of the two models as well as for the 3D image data using Fourier methods. The local and effective physical responses of the model realizations are compared to those obtained from 3D image data. Finally, we assess the accuracy of the two methods to predict permeability as well as electronic and ionic conductivities of the anode.

Estimation of geodesic tortuosity and constrictivity in stationary random closed sets

AbstractWe investigate the problem of estimating geodesic tortuosity and constrictivity as two structural characteristics of stationary random closed sets. They are of central importance for the analysis of effective transport properties in porous or composite materials. Loosely speaking, geodesic tortuosity measures the windedness of paths, whereas the notion of constrictivity captures the appearance of bottlenecks resulting from narrow passages within a given materials phase. We first provide mathematically precise definitions of these quantities and introduce appropriate estimators. Then, we show strong consistency of these estimators for unboundedly growing sampling windows. In order to apply our estimators to real data sets, the extent of edge effects needs to be controlled. This is illustrated using a model for a multiphase material that is incorporated in solid oxide fuel cells.

Prediction of diffusive transport through polymer films from characteristics of the pore geometry

Diffusive transport through porous materials is to a large extent determined by the microstructure of the material. To design materials with controlled transport properties, it is hence important to connect properties of the pore geometry to diffusive transport rates. Different kinds of microstructures from a stochastic model are generated and multiplicative regression is used to find relationships between geometric measures of the microstructures and numerically simulated diffusive transport. The main results are that the geodesic tortuosity explains a large part of the transport variation, and that the standard deviations we introduce further improves prediction. It was found that it is best to compute the tortuosity using the whole pore space, instead of using only the inlet, as is commonly done. The effects of calculating the measures using small samples of the pore structure were investigated, and a method for minimizing errors resulting from boundary effects was proposed. © 2018 American Institute of Chemical Engineers AIChE J, 65: 446–457, 2019

Quantification of the microstructure, effective hydraulic radius and effective transport properties changed by the coke deposition during the crude oil in-situ combustion

Coke deposition during crude oil in-situ combustion (ISC) is an important phenomenon that significantly impacts the pore topology and permeability. In this study, X-ray computed microtomography and a specific image processing procedure were used to reconstruct the micro-tomographic images of packed beds with coke deposition. From the reconstructed images, the microstructural parameters related to the transport were analyzed, such as the effective porosity, the constrictivity and the geodesic tortuosity. The Lattice Boltzmann method was used to simulate the species diffusion and fluid flow through the microstructures to quantify the mass diffusivity and permeability. The experimental work was performed to validate the digital microstructures and the simulated permeability. The effects of the coke deposition on the pore topology and the permeability were analyzed. The coke deposition pattern showed significant deposition in the pore throat. Analyses of the pore size distribution lead to a more reasonable geometric approach to measure the effective hydraulic radius for the better permeability prediction. Based on the effective transport properties and the microstructural parameters, a developed permeability relation was introduced by factorizing the permeability into two distinct contributions from the characteristic length and the microstructural effect. The permeability model describes the main influences of all the relevant geometric parameters on the permeability reduction by the coke deposition.

Big data for microstructure‐property relationships: A case study of predicting effective conductivities

The analysis of big data is changing industries, businesses and research as large amounts of data are available nowadays. In the area of microstructures, acquisition of (3‐D tomographic image) data is difficult and time‐consuming. It is shown that large amounts of data representing the geometry of virtual, but realistic 3‐D microstructures can be generated using stochastic microstructure modeling. Combining the model output with physical simulations and data mining techniques, microstructure‐property relationships can be quantitatively characterized. Exemplarily, we aim to predict effective conductivities given the microstructure characteristics volume fraction, mean geodesic tortuosity, and constrictivity. Therefore, we analyze 8119 microstructures generated by two different stochastic 3‐D microstructure models. This is—to the best of our knowledge—by far the largest set of microstructures that has ever been analyzed. Fitting artificial neural networks, random forests and classical equations, the prediction of effective conductivities based on geometric microstructure characteristics is possible. © 2017 American Institute of Chemical Engineers AIChE J, 63: 4224–4232, 2017

Microstructure-property relationships in a gas diffusion layer (GDL) for Polymer Electrolyte Fuel Cells, Part II: pressure-induced water injection and liquid permeability

The performance of polymer electrolyte fuel cells (PEFC) strongly depends on a controlled water management within the porous layers. For this purpose we investigate liquid water transport in a commercial gas diffusion layer (SGL 25BA) on the pore scale. X-ray tomography experiments combined with pressure-induced water injection provide 3D images of the liquid water distribution inside the GDL at incremental pressure steps between 0 and 100mbar.The breakthrough behavior of the liquid phase is highly anisotropic. In through-plane (tp) direction first bubble points appear at the outlet plane already at 5mbar and the ‘breakthrough' then evolves continuously over an extended pressure range up to >30mbar. For in-plane (ip) direction the breakthrough is discontinuous and takes place at 27mbar. Simulations of the intrusion process reveal that the different breakthrough behaviors are mainly triggered by different ip- and tp-transport distances. Short tp-transport distances through the thin gas diffusion layer (ca. 100μm) are responsible for the characteristic continuous tp-breakthrough behavior, which is thus attributed to a so-called short-range effect.Dedicated methods for 3D-image analysis adapted to fibrous GDL microstructures were presented in part I. With these methods we quantify all microstructure characteristics that are relevant for liquid permeability. These characteristics of pore and liquid phases include size distributions of bulges and bottlenecks, connectivity, effective volume fractions, geodesic tortuosity, constrictivity and hydraulic radius. Quantitative relationships are established between these microstructure characteristics and the liquid permeability, which provide a better understanding of the underlying microstructure limitations for injection and liquid transport.For the in-plane direction the liquid permeability is limited to roughly a similar extent by tortuosity, constrictivity and effective volume fraction. In contrast, for through-plane direction relatively low volume fractions of the liquid phase put stronger limitations to the liquid permeability than tortuosity, constrictivity and hydraulic radius.The curves for relative permeability vs. saturation (and vs. capillary pressure, respectively) achieved from 3D-analysis reveal complex but characteristic (reproducible) shapes with concave, linear and convex segments. The shape of these segments can be attributed to distinct microstructure effects. In contrast, the conventional macroscopic descriptions from literature cannot capture these complex shapes and the underlying microstructure effects. Future investigations with different GDL materials are necessary in order to understand whether these complex shapes for the relative permeability represent a general feature of gas diffusion layers or if they are specific to the investigated SGL material.

Microstructure-property relationships in a gas diffusion layer (GDL) for Polymer Electrolyte Fuel Cells, Part I: effect of compression and anisotropy of dry GDL

New quantitative relationships are established between effective properties (gas diffusivity, permeability and electrical conductivity) for a dry GDL (25 BA) from SGL Carbon with the corresponding microstructure characteristics from 3D analysis. These microstructure characteristics include phase volume fractions, geodesic tortuosity, constrictivity and hydraulic radius. The latter two parameters include information from two different size distribution curves for bulges (continuous PSD) and for bottlenecks (MIP-PSD). X-ray tomographic microscopy is performed for GDL at different compression levels and the micro-macro-relationships are then established for the in-plane and through-plane directions. The predicted properties based on these relationships are compared with numerical transport simulations, which give very similar results and which can be summarized as follows:Gas diffusivity is higher in the in-plane than in the through-plane direction. Its variation with compression is mainly related to changes of porosity and geodesic tortuosity. Permeability is dominated by variations in hydraulic radius. Through-plane permeability is slightly higher than in-plane. Anisotropy of electrical conductivity is controlled by tortuosity, which is higher for the through-plane direction. A table with new quantitative relationships is provided, which are considered to be more accurate and precise than older descriptions (e.g. Carman-Kozeny, Bruggeman), because they are based on detailed topological information from 3D analysis. Furthermore, when using these relationships as input for macro-homogenous modeling, this enables to simulate microstructure effects of real GDL (SGL 25 BA) more accurately. In future, the same methodology can be used to study micro-macro relationships in wet GDL and to predict relative liquid permeability and relative gas diffusivity.

Predicting effective conductivities based on geometric microstructure characteristics

Empirical relationships between effective conductivities in porous and composite materials and their geometric characteristics such as volume fraction , tortuosity τ and constrictivity β are established. For this purpose, 43 virtually generated 3D microstructures with varying geometric characteristics are considered. Effective conductivities are determined by numerical transport simulations. Using error‐minimization the following relationships have been established: and (simplified formula) with intrinsic conductivity σ0, geodesic tortuosity and relative prediction errors of 19% and 18%, respectively. We critically analyze the methodologies used to determine tortuosity and constrictivity. Comparing geometric tortuosity and geodesic tortuosity, our results indicate that geometric tortuosity has a tendency to overestimate the windedness of transport paths. Analyzing various definitions of constrictivity, we find that the established definition describes the effect of bottlenecks well. In summary, the established relationships are important for a purposeful optimization of materials with specific transport properties, such as porous electrodes in fuel cells and batteries. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1834–1843, 2016