Tortuosity and Microstructure Effects in Porous Media

Tortuosity is a major source of transport resistance from microstructure. The scientific debates on tortuosity and associated microstructure effects are often affected by misunderstandings and misconceptions. The sources of these misunderstandings are briefly described in the introduction chapter. Futhermore, the introduction also gives a short overview of the main book chapters, which are dealing with theory, empirical data, methodologies and quantitative microstructure-property relationships.

Many different definitions of tortuosity can be found in literature. In addition, also many different methodologies are nowadays available to measure or to calculate tortuosity. This leads to confusion and misunderstanding in scientific discussions of the topic. In this chapter, a thorough review of all relevant tortuosity types is presented. Thereby, the underlying concepts, definitions and associated theories are discussed in detail and for each tortuosity type separately. In total, more than 20 different tortuosity types are distinguished in this chapter. In order to avoid misinterpretation of scientific data and misunderstandings in scientific discussions, we introduce a new classification scheme for tortuosity, as well as a systematic nomenclature, which helps to address the inherent differences in a clear and efficient way. Basically, all relevant tortuosity types can be grouped into three main categories, which are (a) the indirect physics-based tortuosities, (b) the direct geometric tortuosities and (c) the mixed tortuosities. Significant differences among these tortuosity types are detected, when applying the different methods and concepts to the same material or microstructure. The present review of the involved tortuosity concepts shall serve as a basis for a better understanding of the inherent differences. The proposed classification and nomenclature shall contribute to more precise and unequivocal descriptions of tortuosity.

It is generally assumed that transport resistance in porous media, which can also be expressed as tortuosity, correlates somehow with the pore volume fraction. Hence, mathematical expressions such as the Bruggeman relation (i.e., τ² = ε^−1/2) are often used to describe tortuosity (τ)—porosity (ε) relationships in porous materials. In this chapter, the validity of such mathematical expressions is critically evaluated based on empirical data from literature. More than 2200 datapoints (i.e., τ – ε couples) are collected from 69 studies on porous media transport. When the empirical data is analysed separately for different material types (e.g., for battery electrodes, SOFC electrodes, sandstones, packed spheres etc.), the resulting τ versus ε—plots do not show clear trend lines, that could be expressed with a mathematical expression. Instead, the datapoints for different materials show strongly scattered distributions in rather ill-defined ‘characteristic’ fields. Overall, those characteristic fields are strongly overlapping, which means that the τ – ε characteristics of different materials cannot be separated clearly. When the empirical data is analysed for different tortuosity types, a much more consistent pattern becomes apparent. Hence, the observed τ − ε pattern indicates that the measured tortuosity values strongly depend on the involved type of tortuosity. A relative order of measured tortuosity values then becomes apparent. For example, the values observed for direct geometric and mixed tortuosities are concentrated in a relatively narrow band close to the Bruggeman trend line, with values that are typically < 2. In contrast, indirect tortuosities show higher values, and they scatter over a much larger range. Based on the analysis of empirical data, a detailed pattern with a very consistent relative order among the different tortuosity types can be established. The main conclusion from this chapter is thus that the tortuosity value that is measured for a specific material, is much more dependent on the type of tortuosity than it is dependent on the material and its microstructure. The empirical data also illustrates that tortuosity is not strictly bound to porosity. As the pore volume decreases, the more scattering of tortuosity values can be observed. Consequently, any mathematical expression that aims to provide a generalized description of τ − ε relationships in porous media must be questioned. A short section is thus provided with a discussion of the limitations of such mathematical expressions for τ − ε relationships. This discussion also includes a description of the rare and special cases, for which the use of such mathematical expressions can be justified.

In this chapter, modern methodologies for characterization of tortuosity are thoroughly reviewed. Thereby, 3D microstructure data is considered as the most relevant basis for characterization of all three tortuosity categories, i.e., direct geometric, indirect physics-based and mixed tortuosities. The workflows for tortuosity characterization consists of the following methodological steps, which are discussed in great detail: (a) 3D imaging (X-ray tomography, FIB-SEM tomography and serial sectioning, Electron tomography and atom probe tomography), (b) qualitative image processing (3D reconstruction, filtering, segmentation) and (c) quantitative image processing (e.g., morphological analysis for determination of direct geometric tortuosity). (d) Numerical simulations are used for the estimation of effective transport properties and associated indirect physics-based tortuosities. Mixed tortuosities are determined by geometrical analysis of flow fields from numerical transport simulation. (e) Microstructure simulation by means of stochastic geometry or discrete element modeling enables the efficient creation of numerous virtual 3D microstructure models, which can be used for parametric studies of micro–macro relationships (e.g., in context with digital materials design or with digital rock physics). For each of these methodologies, the underlying principles as well as the current trends in technical evolution and associated applications are reviewed. In addition, a list with 75 software packages is presented, and the corresponding options for image processing, numerical simulation and stochastic modeling are discussed. Overall, the information provided in this chapter shall help the reader to find suitable methodologies and tools that are necessary for efficient and reliable characterization of specific tortuosity types.

100 years ago, the concept of tortuosity was introduced by Kozeny in order to express the limiting influence of the microstructure on porous media flow. It was also recognized that transport is hindered by other microstructure features such as pore volume fraction, narrow bottlenecks, and viscous drag at the pore surface. The ground-breaking work of Kozeny and Carman makes it possible to predict the macroscopic flow properties (i.e., permeability) based on the knowledge of the relevant microstructure characteristics. However, Kozeny and Carman did not have access to tomography and 3D image analysis techniques, as it is the case nowadays. So, their descriptions were developed by considering simplified models of porous media such as parallel tubes and sphere packings. This simplified setting clearly limits the prediction power of the Carman-Kozeny equations, especially for materials with complex microstructures. Since the ground-breaking work of Kozeny and Carman many attempts were undertaken to improve the prediction power of quantitative expressions that describe the relationship between microstructure characteristics (i.e., tortuosity τ, constrictivity β, porosity ε, hydraulic radius r_h) and effective transport properties (i.e., conductivity σ_eff, diffusivity D_eff, permeability к,). Due to the ongoing progress in tomography, 3D image-processing, stochastic geometry and numerical simulation, new possibilities arise for better descriptions of the relevant microstructure characteristics, which also leads to mathematical expressions with higher prediction power. In this chapter, the 100-years evolution of quantitative expressions describing the micro–macro relationships in porous media is carefully reviewed,—first, for the case of conduction and diffusion,—and second, for flow and permeability.

The following expressions are the once with the highest prediction power:

for conduction and diffusion, and

both, for permeability in porous media.

A short summary is provided for each chapter of the book, which includes a review of theories and a new classification of tortuosity types (chapter 2), a review of empirical data (tortuosity-porosity relationships) from literature (chapter 3), a review of image based methodologies, workflows, and calculation approaches for tortuosity (chapter 4), and a review of quantitative expressions for microstructure—property relationships considering transport in porous media (chapter 5). As a final conclusion, interpretations are given for the three main tortuosity categories: direct geometric, indirect physics-based and mixed tortuosities.