4. Image Based Methodologies, Workflows, and Calculation Approaches for Tortuosity

In order to understand the trends in X-ray tomography one has to consider the underlying principles at first. Figure 4.3 represents a schematic illustration of a modern CT-system. The sample is placed between the X-ray source and the detector. The X-rays penetrate the rotating sample so that a multitude of 2D projections are detected under different angles. Algorithms for 3D-reconstruction (e.g., so-called filtered back projection algorithms) are then used to reveal the internal 3D microstructure from the numerous 2D-projections. In absorption/attenuation tomography, the beam—sample interaction and associated beam intensity (I) after travelling through the sample is described by the Beer-Lambert equation

A schematic diagram exhibits the setup for an x ray tomography system. The system is composed of an x ray source, a capillary condenser, a sample on the rotation axis, an objective zone plate, a phase ring, and an x ray detector.

where I₀ is the initial beam intensity, μ is the local attenuation coefficient and s is the beam path vector. Thereby, the attenuation coefficient (μ) is a function of electron density and atomic number (Z). Local variations of this attenuation coefficient are directly related to the materials microstructure, which can be reconstructed in 3D (e.g., by means of filtered back projection). Performance limitations such as the acquisition rates, maximum sample size or inherent noise, can be understood from these basic methodological principles.

As indicated in Fig. 4.4, spatial resolution is only one of the performance relevant characteristics. Other important performance characteristics are for example time resolution/acquisition time and contrast/detection modes. The type of the X-ray source and the optical system also has a strong impact on the performance. Tremendous progress was achieved in all these technological fields. However, there also exists a multitude of interdependencies between the characteristics and parameters mentioned above. For example, faster acquisition usually leads to higher noise and therefore also to weaker contrast and lower effective spatial resolution. In the following, we briefly discuss the most important performance parameters and associated interdependencies.

A diagram illustrates imaging performance parameters. It includes spatial resolution, contrast, time resolution, an x ray source, and optics.

The interaction of X-rays with the material depends on the complex refractive index (n) with a real part (1−δ) and an imaginary part (β_x-ray), i.e.,

The attenuation coefficient (μ) itself depends on the imaginary part (β_x-ray) of the complex refractive index and on the wavelength (λ), where

In materials with a high imaginary part (β_x-ray), the X-rays amplitude is damped, which leads to a lower intensity (i.e., stronger absorption). The attenuation strongly decreases with the beam energy E. More precisely, it holds that β_x-ray = 1/E⁴. Soft X-rays thus provide a better attenuation contrast and also a higher resolution can be achieved. However, stronger absorption at lower beam energy limits the size of samples that can be transmitted and at the same time increases the problem of beam damage with soft X-rays. The contrast between different material constituents can also be improved significantly by using dual-energy X-ray tomography (see e.g., Gondzio et al. [39]).

Phase contrast (PC) imaging is an interesting alternative for materials with a weak attenuation contrast. Local variations of the real part of the refractive index (1–δ, see Eq. 4.2) induce changes of the wavelengths, which lead to beam deflections. These refractive beam-material interactions can be indirectly detected as a phase shift (\(\phi\)). The 3D reconstruction of the local δ-value reveals the materials phase contrast, which, in opposite to the attenuation contrast, increases with beam energy (∆δ/∆β_x-ray = E²). For materials with weak absorption contrast (e.g., in battery electrodes: graphite versus lithium), phase contrast imaging with hard X-rays usually gives better results.

There are different methods for phase contrast imaging (propagation PC, grating-based PC, Zernike PC). These contrast modes require more complex optics, a coherent beam source and more sophisticated 3D reconstructions. However, nowadays even lab-based tomography systems offer the option of phase contrast imaging. For further explanations on attenuation and phase contrast as well as on the principles of 3D reconstruction we refer to Pietsch and Wood [37] and the references therein.

In principle, X-ray microscopy offers three types of magnifications, which are based on X-ray optics, light optics or geometric set-up.

In modern X-ray tomography systems these three magnifying methods are combined, so that a high magnification can be achieved together with a relatively high efficiency and a relatively fast acquisition time (see e.g., [38]). The evolution of 'best' spatial resolution in X-ray tomography over the last 50 years is illustrated in Fig. 4.5 a (adapted from Maire and Withers [36]). This figure shows that it is now possible to reach voxel resolutions in the 10 nm-range by soft X-ray tomography at synchrotron beam lines equipped with Fresnel Zone Plate optics. The resolution power for hard X-rays is weaker. However, soft X-ray tomography suffers from the limited sample thickness, especially for materials with strong absorption (e.g., metals, heavy elements, high density).

2 multi-line graphs. a plots maximum resolution versus year. 3 lines for lens-less, hard, and soft x ray follow a decreasing trend between 0.01 and 10. b plots spatial resolution versus acquisition time. The lines for several light sources follow a decreasing trend between 0.01 and 100.

In general, a fast acquisition time comes at the expense of increased noise. Synchrotron beams with a high brilliance can reduce the noise and are thus particularly well suited for fast X-ray imaging. Nowadays, 4D imaging at 20 to 50 Hz has become possible with white beam synchrotron tomography (see e.g., Maire et al. [40]). As shown in Fig. 4.5b, time resolution and spatial resolution are constraining each other. Both of them also strongly depend on various other aspects such as the facility type (synchrotron versus laboratory systems), beam energy and beam intensity (monochromatic versus white versus mixed beams). Also, the attenuation contrast of the material under investigation and the size of the sample are representing constraining factors.

The best time resolution can be achieved with a white beam (WB) of synchrotron sources due to the relatively high beam flux, however with certain limitations in magnification. Monochromatic beams are better suited to exploit the power of magnifying systems such as FZP, which opens new capabilities for nano-CT [34]. Even lab-based systems with X-ray optics are nowadays capable of providing 50 nm resolutions. However, lab-based systems exhibit a relatively long acquisition time, which is typically several hours per 3D image. A good compromise between short acquisition time, high spatial resolution and reasonable signal to noise ratio can be reached by mixing monochromatic and white beams (e.g., 50% WB). In this way, sub-second tomography with sub-μm resolution has become possible, as reported by F. Büchi (pers. comm., 2021) for the Swiss Light Source (SLS) at Paul Scherrer Institute (violet data point in Fig. 4.5b). For white beam synchrotron tomography, 30 nm spatial resolutions at 1 min acquisition time are reported as current state of the art (Yan et al. [34]).

Many parameters must be considered when optimizing temporal vs. spatial resolutions. The acquisition time can be estimated based on the required number of projections (N), which increases with the image window size and associated number of pixels (q) in horizontal direction, i.e., For example, for q = 1024 pixels, the required number of 2D projections (N) is equal to 1570. With a total rotation of 180° this results in 8.7 projections per 1°. For these settings, 3D imaging at 1 Hz (i.e., 1570 projections per second) requires an acquisition time of 0.64 ms for a single 2D projection. By pushing the limits of fast tomography towards 100 Hz (e.g., 157,000 projections per second), new technological solutions needed to be developed such as fast signal processing (e.g., read out and storage of up to 100 GB per second, see Mokso et al. [41]), highly efficient optics and detector systems (Bührer et al. [19]), more brilliant sources, intelligent 3D reconstruction strategies (e.g., based on a smaller number of projections or segmentation oriented reconstructions using a priori knowledge about the phases, see [42‐45]). Finally, fast data acquisition also calls for dedicated software that enables efficient analysis of the huge 4D image data volumes (e.g., digital image and volume correlation DIC/DVC or 4D particle tracking, see [46‐48]). In all these fields, fast progress and innovation is currently ongoing.

Fast X-ray tomography opens new possibilities for in-situ and in-operando studies of dynamic processes at pore-scale and even below. For lab-based X-ray tomography, the possibilities of high-speed 3D imaging were reviewed in a recent article by Zwanenburg et al. [49]. With synchrotron-tomography, ultrafast 3D imaging can now be performed at 20 Hz and even faster (Maire et al. [40]). In-situ mechanical testing is one prominent example for the application of fast 4D tomography. Nano-mechanical devices for compression, tension, and indentation tests, which are specially designed for dynamic tomography investigations, are nowadays commercially available. Also, the preparation of small samples (e.g., pillars with sizes in the range of mm down to a few μm) suitable for nano-mechanical testing is now relatively straightforward due to the availability of focused ion beam (e.g., Xe + plasma FIB) and laser technologies. In-situ tomography during mechanical testing is thus evolving rapidly, for example in the field of alloys [50] and batteries [15].

Fast tomography is also used in high temperature studies. Villanova et al. [51] reported the 4D-evolution of microstructures and nucleation of nano-droplets upon sintering of alloys at 700 °C.

So-called in-operando studies enable capturing dynamic processes under real life conditions. In-operando studies of electrochemical cells (batteries, fuel cells) are very challenging, because they usually require some in-house development of dedicated experimental equipment, including miniaturization of the cells and of the electrochemical test setup. A tomography system with an efficient optical microscope was designed at the Swiss Light Source (SLS). It was recently used for in-operando studies of water clusters and two-phase flow in PEM fuel cells at temporal and spatial resolutions of < 1–10 Hz and < 1–10 μm, respectively [18, 19, 52, 53].

In battery research, synchrotron based in-operando studies have been used to reveal the 4D microstructure evolution upon degradation and/or (de-)lithiation [54‐56].

With laboratory-based systems the acquisition time is generally longer and rather suitable for the characterization of relatively slow processes in time-lapse mode. Nevertheless, two-phase flow in geological samples was characterized with lab-based tomography at temporal and spatial resolutions of less than 1 min and 15 μm, respectively (Bultreys et al. [27]). The lack of attenuation contrast in two-phase flow can be approached with time-resolved phase contrast imaging as reported by Ohser et al. [57].

4D tomography is thus applicable for dynamic studies in various fields of materials and engineering sciences and also in geology [58‐60] as well as in life sciences [33]. The exciting possibilities of modern high-end X-ray tomography open new possibilities for the investigation of tortuosity effects. The exploitation of these opportunities requires dedicated and efficient solutions in image processing, which will be discussed below in Sects. 4.3–4.5.

The FIB-SEM geometry for serial sectioning is illustrated in Fig. 4.6. In a first step, a cube representing the region of interest is exposed with a high beam current for rapid ion milling. The x–y imaging plane, also called ‘block-face’, is then polished with a lower ion beam current. Subsequently, an SE- or BSE-image is acquired by scanning the block face with the electron beam. A stack of 2D images (i.e., a 3D image volume) is then produced in a fully automated serial sectioning procedure, which consists of two alternating steps: (1) Thin layers of e.g., 10 nm thickness are sequentially removed from the block face with the ion beam and (2) SEM images with a pixel resolution of e.g., 10 nm are acquired from the freshly exposed block face. In this procedure, fiducial markers are used for automated correction of mechanical, magnetic, and electronic drifts. Ideally the milling step size in z-direction is identical to the pixel resolution of the SEM images (x–y plane), which results in isometric voxels.

A 3 D model of the F I B SEM dual beam geometry for serial sectioning system. The region if interest is denoted as a cube with an x y z imaging plane. The ion and electron beams hit the cube, which results in the slicing in the z direction. The angle of stage tilt is 52 degrees.

In most cases, a large number of ca. 1000 or more images would be ideal in order to acquire a representative 3D image volume. The acquisition time for a single slice-and-view cycle includes the following components:

The milling time depends on the beam current (milling rate), on the size of the imaging plane (x–y) and on the thickness of the milled layer (z-direction).

The SEM imaging time depends on the number of pixels (i.e., area of block-face and resolution) and on the dwell time for scanning with the electron beam. Thereby, fast scanning negatively affects the signal-to-noise ratio.

Drift correction is based on images that are taken from specific sample locations, which contain fiducial markers (i.e., reference positions). The time for drift correction(s) thus depends on the imaging conditions (i.e., scan rate, resolution, size of image). In advanced serial sectioning procedures, the drift correction is performed in the x–y plane with SEM- and in the x–z plane with FIB-images.

The beam requires some time for stabilization after switching from electron to ion beam and back.

The total acquisition time thus depends on various parameters such as the size of the 3D image window, the ion beam-current, the electron scan rate, the detector efficiency. Also contrast (or noise) and sputter rates that are characteristic for the material under investigation have an influence on the acquisition time. Depending on the chosen parameters for serial sectioning, the total acquisition time can thus vary significantly. Typically, in a relatively fast setup with small cubes of a few μm edge lengths, an entire slice-and-view cycle takes ca. 30 s. For a stack with 1000 images, this cycling rate results in a total acquisition time of 8 h and 20 min. The acquisition time can easily increase by a factor of 3 to 4, e.g., for larger cubes (tens of μm), for more precise ion milling and/or slower electron scanning (higher signal-to-noise ratio). Often the number of images in the stack is then reduced to only a few 100 images, in order to shorten the total acquisition time. This leads to 3D image volumes with non-isometric dimensions (e.g., 20 × 20 × 5 μm, whereby the 5 μm direction corresponds to the slicing direction).

Over the last years the milling capabilities for serial sectioning have considerably improved due to the appearance of new ion sources and new milling techniques (for details see Bassim et al. [112] and Echlin et al. [109]):

The liquid metal ion source (LMIS) is the basis for conventional Ga FIB. With beam currents between pA and 100 nA, the sputter rates of Ga FIB machines are relatively low,—especially for organic matter and ceramics. In the meanwhile, LMIS works with many different metals (Ga, Al, In, Au, Bi) and alloys. Nevertheless, the milling capabilities of LMIS are still rather limiting. Some improvements of the sputter rates could be achieved with a new ‘rocking milling procedure’.

Magnetically enhanced inductively coupled plasma FIB sources are capable of significantly higher sputter rates. In addition, at beam currents above 10–50 nA, the Xe PFIB provides a much smaller beam diameter than conventional Ga FIB (at similar beam currents). The PFIB thus opens new possibilities for large area serial sectioning, whereby the edge length of the image window can reach dimensions in an order of magnitude of 100 μm. Compared to Ga-FIB, the milling with Xe PFIB is ca. 60 times faster, but at the same time PFIB is capable to reveal small slicing thicknesses (and voxel resolutions) of ca. 10 nm. In addition, Xe PFIB produces less beam damage (i.e., the amorphous surface layer is relatively thin) and it is therefore better suited for 3D EBSD compared to conventional FIB. Applications of large area serial sectioning with Xe PFIB are discussed by Burnett et al. [110] and Zhang et al. [111, 114].

The hollow anode discharge (HAD) Ar source represents the basis for broad ion beam (BIB) machines, which can be used for sequential milling and polishing of large areas up to the mm²-range. For in-plane milling and polishing with a broad ion beam, a metal blend is used with a high milling resistance (e.g., W). HAD Ar ion sources reveal high beam currents up to the μA-range at low beam energies (≤ 5 kV). Milling at low kV induces relatively low beam damage, which makes BIB particularly suitable for 3D EBSD. Generally, the z-resolution (thickness of removed layer) for BIB serial sectioning is in the 100 nm to μm range, but recently more precise BIB-serial sectioning with z-distances as thin as 10 nm were reported [115, 116].

Laser-based systems combined with various microscopy platforms (light microscopy, SEM, FIB-SEM) have been available for many years. Due to limited resolution of the laser, these systems were rather used in the past for targeted feature extraction and micromachining. Thereby, the milling precision of traditional pulsed laser-systems was not suitable for serial sectioning applications. However, modern femtosecond pulsed lasers nowadays provide much higher milling precisions and, at the same time, they cause less beam damage. Recently, a femtosecond laser was integrated into a dual beam PFIB-SEM, which results in a tri-beam system. This device enables precise serial sectioning of large areas in the mm² range. With the tri-beam system, the PFIB can be used for fine polishing after efficient milling with the laser. Typically, the step size of laser milling in z-direction is 0.5–1.5 μm [113, 117, 118].

For comparison, the typical performances of the discussed serial sectioning techniques are shown in Table 4.1. This table also includes ultra-micro tomography serial block face SEM (UMT SBFSEM) [119], which uses a diamond knife for mechanical sectioning. Furthermore, robotic serial sectioning by mechanical polishing [120] is also included for comparison.

Most serial sectioning techniques have a certain tendency towards anisometric voxel resolution. The pixel resolution of the SEM images (i.e., imaging resolution of the x–y plane) is typically in the range of 10 nm. Even large areas up to the mm² range that are produced for example with BIB, laser-PFIB tri-beam or ultra-micro tomography can be efficiently scanned at high resolution by using a stitching approach for the SEM imaging. In contrast, for these serial sectioning methods the step size in z-direction is typically limited to ca. 1 μm, which is 20–100 times larger than the SEM pixel resolution (see the column aspect ratio, resolution in Table 4.1).

The dimensions of the 3D image window (i.e., CEL, cube edge lengths of analyzed volumes) also tend to be anisometric. For example, with PFIB and BIB, the total thickness of the image stack (z-direction) that can be acquired at high slicing resolution within reasonable acquisition time is often 10–50 times smaller than the size of the 2D image window in x–y directions (see Table 4.1, column aspect ratio, CEL). These anisometric properties of serial sectioning are also visualized in Fig. 4.2 by elongated ellipses. The long axes of the ellipses indicate different dimensions of voxels and image windows in x–y—(top left part of ellipses) compared to z-directions (bottom right part of ellipses). Large area serial sectioning is thus particularly well suited for the analysis of anisometric samples such as the thin layers of SOFC electrodes (see e.g., Mahbub et al. [121]).

A significant advantage of destructive serial sectioning compared to X-ray tomography comes from the fact that the exposed surfaces (block-faces) can be probed with many different imaging and surface characterization techniques. Thereby, SEM based serial sectioning benefits considerably from the progress in low-voltage SE- and BSE-imaging, which provide high contrast at high resolution. This progress is mainly due to the innovative improvement of in-lens or through-the-lens detectors [107]. In addition, fast spectral and elemental mappings with silicon drift EDS detectors open-up new possibilities in 3D chemical mapping. Furthermore, new EBSD cameras enable grain orientation mappings with significantly shorter acquisition time, higher spatial resolution, and larger image window size. As discussed by Echlin et al. [109], there is a clear trend in serial sectioning tomography towards larger size of the image windows (e.g., with PFIB, Laser-tribeam or BIB) due to the possibility of combining high milling rates with a high resolution. Another important trend is the evolution towards simultaneous acquisition of multiple signals, which is also called multi-modal tomography (i.e., serial sectioning with simultaneous acquisition of EDX or EBSD together with SE, BSE and even with SIMS), see e.g., the 3D FIB EBSD image data considered in [99, 100].

Modules for qualitative image processing (IP) provide solutions for 3D reconstruction, filtering of image defects, segmentation, and visualization. Some SW packages also provide an IP module for mesh generation. The meshed 3D data is often used as a basis for numerical simulations, e.g., with finite element models.

Such SW modules enable the generation of virtual 3D microstructures with stochastic modeling or with discrete element modeling (DEM), which is an important option for data driven, statistical investigations of micro–macro relationships (see Chap. 5).

Modules for quantitative image processing are used for the determination of morphological microstructure characteristics. In particular, dedicated SW modules are used for characterization of direct geometric tortuosities (τ_{dir_geodesic}, τ_{dir_medial_axis,} τ_{dir_skeleton}, τ_{dir_PTM,} τ_{dir_percolation}, τ_{dir_FMM}, τ_{dir_pore_centroid}) and also for mixed tortuosities.

It must be emphasized that the determination of indirect-tortuosities (τ_{indir_phys_sim}) and mixed tortuosities (τ_{mixed_phys}) cannot be performed with geometric image analysis alone, because the determination of these tortuosities requires modules for 3D numerical simulation of the underlying transport process and extraction of effective or relative transport properties.

In addition to tortuosity, several other morphological characteristics are important in the context with effective transport properties, which are the following:

Further morphological microstructure descriptors for microstructure characterization can be found in [144‐146].

SW packages that are powerful in micro-scale simulations use voxel-based images to capture the structural input and they typically solve one specific transport equation at a time (e.g., Navier–Stokes solver for viscous flow).

SW-packages that are strong in solving coupled processes (e.g., by coupling transport with electrochemistry, with thermal behavior and/or with mechanics) typically use a mesh-based representation of the structural input. The mesh-based representation reduces the data volume significantly. But this benefit comes at the cost of lesser morphological details and precision.

SW packages for multi-physics simulations operating with mesh-based input are thus rather suitable for macro-homogeneous modeling, whereas SW packages operating with voxel-based input are better suited for microstructure simulations.

The SW tools for pore scale modeling are of particular importance for the computation of the two following tortuosity categories:

Indirect tortuosities (τ_{indir_phys} with variations τ_{indir_ele}, τ_{indir_therm}, τ_{indir_diff}, τ_{indir_Kn}, τ_{indir_hydr}) are computed from effective transport properties (conductivity σ_ele, σ_therm, diffusivity D_eff, D_Kn or permeability к), which can be obtained from numerical transport simulations.

Mixed tortuosities (τ_{mixed_phys_streamline}, τ_{mixed_phys_Vav}, with phys = ele, therm, diff or hydr) are computed by image analysis from 3D vector fields, which represent the local flux within the pore structure. SW packages that enable to calculate the mixed tortuosities must be capable of performing both, numerical transport simulations (preferably voxel-based) and quantitative image analysis of 3D vector fields.

Multi-modular SW packages for 3D microstructure analysis and microstructure modeling provide combined solutions for image processing, quantitative microstructure analysis and numerical simulation. An important characteristic is the capability to perform transport simulations with voxel-based structural input. This option is available, for example, in the SW packages GeoDict (Math2Market), PuMa (NASA), Avizo/Amira (ThermoScientific), PerGeos (ThermoScientific) and Pore3D (Elettra Scientific). Voxel-based simulations are capable to capture the microstructure input from tomography with higher precision and accuracy compared to meshed-based simulations. Another important characteristic is the ability to determine and compare different types of tortuosities based on quantitative image processing and numerical modeling. In this context, GeoDict is currently the only SW package that enables characterizing all three tortuosity categories (direct geometric, indirect physics-based and mixed tortuosities) via the included Compute Tortuosity App. The third important characteristic refers to the option of stochastic modeling, which is used to generate virtual 3D microstructures, so-called digital twins. This option is provided, e.g., by GeoDict, Digimat (eXtreme engineering), PuMA and Micress (RWTH-Aachen). These multi-modular SW packages also provide the exciting opportunities to perform digital materials design (DMD). Thereby numerous 3D microstructures can be created, and their performances can be characterized by virtual testing (using voxel-based numerical simulations). Based on the combination of stochastic microstructure modeling, virtual testing and quantitative image analysis, these SW packages also provide outstanding opportunities for statistical investigations of microstructure-property relationships (see Sect. 4.​7 and Chap. 5).

Some SW packages are specifically developed for tortuosity analysis (IP II). A prominent example is TauFactor from Imperial College London (Cooper et al. [147]), which is a Matlab code for voxel-based simulations of diffusive transport using the finite difference method. It provides physics-based indirect tortuosity (τ_{indir_diff}) and the associated tortuosity factor (see Chap. 2, Eq. 2.​15: T = τ²), respectively. Furthermore, TauFactor is also capable to compute various other microstructure characteristics such as porosity, surface area and three phase boundary (TPB) length.

The BruggemanEstimator is a Mathematica code developed at ETH Zurich (Ebner and Wood [148]), which uses the Bruggeman relation (see Chap. 3, Eq. 3.​3: τ = ε^α) for estimation of indirect tortuosity in granular materials. It is primarily designed for the characterization of battery electrodes. Two orthogonal 2D images are used as input for the statistical analysis of particle shapes and particle orientations. Differential effective medium theory is then applied as a tool to predict the Bruggeman exponent (α) and the associated indirect tortuosity.

Fiji plugins for skeletonization (imagej.net/Fiji: skeletonize3D, AnalyzeSkeleton) can be used for the determination of geometric tortuosity (τ_{dir_skeleton}). Moreover, various other microstructure characteristics such as cPSD, MIP-PSD and constrictivity can be determined with the XLib plugin in Fiji (imagej.net/Fiji: XLib [66, 149]).

The SW package MIST [150] for image processing also provides tools for the computation of the geometric tortuosity defined in [151].

Dedicated SW for the computation of mixed tortuosities is rare. Matyka and Koza [152] describe how to implement an in-house code for analysis of volume-averaged tortuosity (τ_{mixed_phys_Vav}).

For completion it must be emphasized that many of the SW packages in 4.3.2.1 also provide interesting options for tortuosity characterization.

Numerous SW packages are available for qualitative image processing (IP I) and visualization. They provide various options for raw data import from tomography, 3D reconstruction and visualization, filtering of noise and correction of image defects (e.g., background correction), segmentation and mesh generation. Some of these image-processing modules are embedded within a larger commercial SW package (e.g., image-processing toolkit in Matlab and in Mathematica). ImageJ/Fiji [153] is an important freeware for image processing. Moreover, numerous SW packages for image processing are developed for medical and life science applications (e.g., ITK, VTK, TTK etc.). However, in many cases, the life science-oriented SW packages rarely consider morphological characteristics that are frequently used in physical and engineering sciences (tortuosity, constrictivity, pore size distributions).

Some SW packages provide dedicated image-processing solutions such as 3D reconstruction, which are specially designed for a certain type of tomography (i.e., processing of raw data from a specific tomography methodology). For example, the ASTRA toolbox [154] is designed for processing of raw data from synchrotron Xray CT. TomViz is dedicated to data processing from electron tomography (TEM, STEM), and Dream3D for processing and analysis of EBSD data from FIB-SEM tomography.

Numerous SW packages are available for so-called multi-physics simulations. They are usually based either on finite element (FEM), finite difference (FDM), finite volume (FVM) or lattice Boltzmann methods (LBM). The SW packages for numerical modeling are particularly strong in simulating coupled processes (i.e., combinations of CFD, transport, electrochemistry, structural mechanics etc.) at different lengths-scales. Well-known commercial SW packages of this type are for example Comsol, Simcenter Star CCM + (Siemens), Ansys/Fluent and Abacus (Dassault). There are also powerful freeware packages and libraries available like OpenFOAM, FreeFEM, FEniCS or SESES.

In most cases the multi-physics modeling approach makes use of a mesh-based structural input. However, as mentioned above, precise descriptions of complex 3D microstructure information from tomography are difficult to achieve with mesh-based representations. Therefore, most SW-packages for multi-physics simulation are better suited for simulations at macro-homogeneous scales and/or scenarios with relatively simple morphologies. Mesh-based simulations are thus not recommended for tortuosity analysis of complex microstructures. Note that GeoDict offers packages for voxel-based multi-physics simulation on the microstructure scale for specific applications (i.e., electrochemistry, structural mechanics, digital rock physics and filtration).

Besides the examples mentioned in part a), only a few additional SW packages are available, which can be used for the generation of virtual 3D microstructures. Freeware packages like ESyS, GenGeo, Yade and Mote3D are based on purely geometric packing of particles using the discrete element method (DEM).

The particle flow code (PFC, Itasca Consulting Group) enables virtual particle packing based on physical interactions (i.e., simulating mechanical densification and/or particle growth and crystallization). As mentioned above, also some multi-modular SW packages (e.g., GeoDict and PuMa) offer the option to generate virtual 3D microstructures. In particular, GeoDict offers specific modules for virtual design of granular and fibrous 3D microstructures. A short review of methods and models from stochastic geometry for the creation of virtual 3D microstructures is given in Sect. 4.7.

In the following Sects. (4.4–4.7), the workflow from 3D image acquisition to quantitative analysis of tortuosity is discussed in more detail.

The calculation approach for geodesic tortuosity (τ_{dir_geodesic}) is very simple and fast. The shortest pathways are defined in terms of the geodesic distance within the voxel space that represents the transporting phase [164]. Sometimes, this approach is also called the direct shortest path searching method (DSPSM) [163]. In the past, most authors dealing with geodesic tortuosity worked with in-house SW. Recently, an option for the computation of geodesic tortuosity was implemented in the commercial GeoDict software. The particular type of geodesic tortuosity introduced in [151] is implemented in the software package MIST [150]. Geodesic tortuosity currently takes a special role among the different geometric tortuosity types because it is used as a basis for empirical relationships between microstructure characteristics (porosity, tortuosity, constrictivity, hydraulic radius) and effective transport properties (see e.g., Stenzel et al. [164], Neumann et al. [165], and the discussion of empirical micro–macro relationships in Chap. 5). It was found in [164] that the geodesic tortuosity has a higher prediction power for estimating effective transport properties compared to medial axis tortuosity (τ_{dir_medial_axis}).

The computation of the medial axis tortuosity (τ_{dir_medial_axis}) is more complicated. It first requires the extraction of a medial axis skeleton [166]. Tortuosity is then computed from a set of shortest pathways along the medial axis skeleton, which are connecting couples of inlet- and outlet-points [64, 65]. Note that there exist many different skeletonization algorithms, which are for example implemented in dedicated modules of the software packages from Avizo/Amira (XPore Network) and/or Fiji (Skeletonize3D). Thereby, the resulting skeletons do not necessarily represent the medial axes. In this case, skeleton tortuosity (τ_{dir_skeleton}) is used as a more general term. Hence, to some degree, the resulting tortuosity values depend on the algorithm used for skeletonization, which leads to an additional complexity and uncertainty.

The different geometries of pathways for medial axis/skeleton tortuosity and geodesic tortuosity are illustrated and compared in Chap. 2 (see Figs. 2.​2, 2.​3 and 2.​7). Empirical data from literature shows that τ_{dir_medial_axis} and τ_{dir_skeleton} are usually not too different from each other, but they are consistently higher than τ_{dir_geodesic} and τ_{dir_FMM}. This consistent order among the different tortuosity types was documented in Chap. 3 (see Figs. 3.​6, 3.​7 and 3.​9).

The fast-marching method tortuosity (τ_{dir_FMM}) is based on the simulation of a propagating front, which reveals the shortest geodesic pathways within the transporting phase [167‐169]. This relatively simple calculation approach is thus very similar to the one used for the computation of geodesic tortuosity. Usually, in-house SW is used in order to determine τ_{dir_FMM}.

The path tracking method tortuosity (τ_{dir_PTM}) can be considered as a fast and simple skeletonization approach, which however is only applicable for structures consisting of packed spheres. The algorithm identifies tetragons consisting of neighboring spheres. The pathways through the interstitial pores are found by connecting the gravity centers of adjacent tetragons in a predefined transport direction [170‐172].

The percolation path tortuosity (τ_{dir_percolation}) is determined with an algorithm that allows the largest possible sphere(s) to travel along the shortest possible path from inlet- to outlet-plane. It should be noted that for a hypothetical case with very small ‘spheres’ (i.e., 1 pixel), the results for τ_{dir_percolation} are identical to those for τ_{dir_geodesic}. However, when the largest possible sphere is considered, the narrow bottlenecks in the pore network hinder the direct passage of the sphere, which leads to longer pathways and higher tortuosity values compared to geodesic tortuosity. These different pathway-geometries are illustrated and compared in Fig. 2.​7 (Chap. 2). The percolation path method is implemented for example in GeoDict, which allows the user to vary the range of sphere radii as well as the number of ‘largest spheres pathways’ to be analyzed (as optional input parameters). The larger the number of pathways, the smaller is the limiting sphere radius, and consequently, the smaller will be the corresponding tortuosity value. Hence, the method can be criticized since the results depend on the chosen parameters. Nevertheless, the percolation path method can be used for identification and visualization of transport pathways with a given, transport-limiting bottleneck size.

The pore-centroid tortuosity (τ_{dir_pore_centroid}) is a quantity, which can be computed by a quick and simple method that is based on determining the center of mass of the transporting phase (e.g., pores) in single 2D slices. The tortuous pathway is then tracked by connecting the mass centers of adjacent 2D slices in transport direction. This method is for example implemented in Avizo. It turns out that the obtained values decrease towards 1 when the volume fraction of the transporting phase increases, but also when the image window size increases (i.e., the center of gravity tends to be identical with the image center). Therefore, the relevance of the pore-centroid tortuosity is questionable.

Materials characteristics and physical laws that are relevant for the simulation of different transport experiments are summarized in Table 4.4. We first consider the case of an electrical conduction experiment and its simulation, respectively. The liquid electrolyte in the pores acts as the transporting phase. The relevant intrinsic property is the electrical conductivity of the electrolyte (σ₀). In the simulation experiment a voltage difference between inlet and outlet planes is applied as driving force (∆U/L). The Laplace equation is solved under the assumption of charge conservation. At steady state conditions, the simulation reveals a constant electrical flux (J_ele). The effective conductivity (σ_eff) can then be calculated by substituting the simulated flux (J_ele) and associated voltage drop divided by the length of the simulation domain (∆U/L) in Ohm’s law (Eq. 2.​21, see Chap. 2). The resulting effective conductivity (σ_eff) is always smaller than the intrinsic conductivity (σ₀) due to the retarding effects from the materials microstructure. These retarding effects are generally attributed to the reduced pore volume fraction (ε < 1) and to the indirect electrical tortuosity (τ_{indir_ele}). Then, Eq. 2.​24b (σ_eff = σ₀ ε/τ_ele²) is usually taken as a quantitative description of the involved micro–macro relationship. Hence, knowing the porosity (ε) from image analysis and the effective conductivity (σ_eff) from simulation, the electrical tortuosity can be computed indirectly according to Eq. 2.​25 (τ_{indir_ele} = √(σ₀ ε/σ_eff)).

As shown in Table 4.4, the material laws and the physical laws for thermal conduction and for bulk diffusion are very similar to those for the electrical conduction. Therefore, the simulation of these transport processes can be performed in a very similar way. From a mathematical point of view, Ohm’s law, Fick’s law, and Fourier’s law reveal exactly the same relationship between the steady state fluxes (electric, diffusive or thermal fluxes), the effective properties (electric conductivity, diffusivity, thermal conductivity) and the applied driving forces (gradients of electric potential, of concentration, and of temperature).

Several authors [8, 81, 162, 163, 173, 174] performed comparative modeling studies using identical 3D microstructures as input for simulations of different transport processes (i.e., bulk diffusion as well as electrical and thermal conduction). These studies document that the three simulation approaches reveal exactly the same results for the relative properties (i.e., X_rel = σ_{eff_ele}/σ_{0_ele} or D_eff/D₀ or K_{eff_thermal}/K_{0_thermal}) and consequently also for the corresponding indirect tortuosities (τ_{indir_ele} = τ_{indir_diff} = τ_{indir_thermal}). In these studies, the consistency of results could be demonstrated even for cases where different numerical methods were used (i.e., FVM, FDM, FVM, LBM and random walk). These findings indicate that, in principle, different numerical simulation approaches for diffusion and conduction are highly reproducible and thus, the corresponding indirect tortuosities for electric and thermal conduction as well as Fick's diffusion can be used interchangeably.

As an exception, it must be emphasized that diffusion in nano-porous media requires a different treatment of the microstructure effects. The so-called Knudsen diffusion, which was discussed earlier (see Eqs. 2.​34–2.​36), is then often simulated with a random walk approach.

In the context of indirect tortuosity, a critical point and a source of uncertainty is the underlying assumption of a known quantitative micro–macro relationship. It is often postulated that the micro–macro relationship in porous media can be described with simple expressions such as Eq. 2.​24b for electrical conduction (σ_eff = σ₀ ε/τ_{indir_ele}²) and with analogous relationships for diffusivity and thermal conduction (e.g., Eq. 2.​31: D_eff = D₀ ε/σ_{indir_ele}²). This is, however, a very simplified assumption, whereby all resistive effects induced by the morphology of the microstructure are lumped together in the indirect tortuosity, except for the volume effect that is accounted for by ε (see Eq. 2.​25: τ_{indir_phys} = √(ε/σ_rel)). For the same 3D microstructures, this calculation approach typically results in indirect tortuosities that are much higher than the direct geometric and the mixed tortuosities (see Chap. 3, Fig. 3.​9, relative order of tortuosity types).

It must be noted that various authors came up with alternative descriptions for the underlying micro–macro relationships. Some authors postulate a more exclusive approach, whereby the bottleneck effect is removed from indirect tortuosity [62, 63, 175‐177]. This exclusion is achieved by adding a distinct constrictivity parameter (β) into the micro–macro relationships (i.e., τ_{indir_phys} = √(ε β/σ_rel), see also Eqs. 2.​26, 2.​27 and 2.​33). As discussed in [63], the exclusive approach with separate treatment of constrictivity results in significantly lower values for the indirect tortuosity compared to the standard definition via Eqs. 2.​24b and 2.​25. The values obtained in this way for indirect tortuosity are then more similar to the values obtained for direct geometric tortuosity.

In contrast, in a more inclusive approach, some authors added the pore volume effect to the indirect tortuosity by removing ε from the equation (i.e., τ_{indir_phys} = √(1/σ_rel)). Then, the corresponding property is sometimes called diffusibility instead of indirect tortuosity [178]. This inclusive approach leads to even higher values for the indirect tortuosity compared to the standard definition.

In summary, this discussion illustrates that the indirect tortuosity heavily depends on the specification of the underlying micro–macro relationships. Regardless which definition one choses, in any case the indirect tortuosity does not capture the true geometric paths lengths. The indirect tortuosity describes some kind of a microstructure resistance, which always requires a clear definition of the underlying micro–macro relationship. Most frequently, the indirect tortuosity is calculated with Eq. 2.​25.

A 3D numerical framework can be established for pore-scale simulation of viscous flow in a similar way as discussed above for electrical conductivity. In this framework, a pressure gradient (∆P/L) is externally applied as driving force (instead of a potential gradient). The transported species and the transporting phase are the viscous medium in the pores. The hydraulic flux (J_hydr) can be computed at steady-state conditions by solving the (Navier-) Stokes equation using different numerical approaches (e.g., FVM, FEM, LBM). Permeability (к) can then be calculated by substituting the simulated hydraulic flux and the corresponding pressure gradient into Darcy's equation (see Table 4.4, see also Eq. 2.​2).

Despite the obvious analogies with conduction and diffusion (Ohm’s law, Fourier’s law, Fick's law), there also exist some fundamental differences in the physical and mathematical description of viscous flow (Navier Stokes equations). An important difference concerns the nature of the effective properties (i.e., permeability versus conductivity and diffusivity). Permeability itself is a pure microstructure property. In contrast to conduction and diffusion, there is no analogy for 'intrinsic permeability'. The intrinsic flow property can be ascribed to viscosity. In principle, permeability (к) is comparable with the relative properties of conduction and diffusion (i.e., к ≈ (σ_{eff_ele}/σ_{0_ele}) ≈ (D_eff/D₀)). These relative properties are entirely dependent on the microstructure. Thereby, small values for the relative properties represent high transport resistances. Nevertheless, whereas relative conductivity and relative diffusivity are dimensionless properties, permeability has units of m². This indicates that the microstructure imposes different limitations to viscous flow compared to conduction and diffusion (which is also obvious from the different forms of differential equations that are used to describe these transport processes). For conduction and diffusion, the microstructure limitations associated with volume fraction, paths lengths and bottlenecks are described by the dimensionless characteristics of porosity (ε), tortuosity (τ) and constrictivity (β) (see e.g., Eqs. 2.​24b, 2.​26, 2.​27, 2.​31 and 2.​33). For flow and permeability, there exists an additional microstructure effect, which is caused by viscous drag at the pore walls. As discussed in Chap. 2, this flow specific effect at the pore walls can be expressed with the squared hydraulic radius (r_h²). Permeability is thus described by a combination of dimensionless characteristics (ε, τ) and a length-dependent characteristic (r_h) according to Eq. 2.​9 (к r_h² ε/τ_hydr²). In principle, the hydraulic tortuosity can now be computed indirectly from Eq. 2.​9 (τ_{indir_hydr} = √(r_h²ε/к)). However, to do so it is necessary to also assess the hydraulic radius (r_h), in addition to permeability and porosity. Until recently, suitable 3D image analysis methods for the measurement of hydraulic radius were lacking for complex microstructures. As discussed in Chap. 2, the Carman-Kozeny equations provide solutions that are valid only for simplified microstructures (packed spheres, parallel tubes). Novel methods of 3D image analysis to determine the hydraulic radius, which can be reliably computed for complex, disordered microstructures, will be discussed in Chap. 5. The lack of suitable methods for characterization of the hydraulic radius may be the main reason why indirect hydraulic tortuosity (τ_{indir_hydr}) has not been considered in previous studies of pore-scale flow. As an alternative approach, it is possible to measure hydraulic tortuosity from simulated 3D velocity fields (and associated streamlines). These types of hydraulic tortuosity (i.e., τ_{mixed_hydr_streamline}, τ_{mixed_hydr_Vav}) however belong to the class of mixed tortuosities and they contain completely different information than the indirect tortuosities (see Sect. 4.5.3).

For indirect tortuosities, it can be summarized that the microstructure resistance is different for viscous flow compared to conduction and diffusion. The computation of indirect hydraulic tortuosity related to flow is more complex and therefore it is hardly used. The physics-based indirect tortuosities for electrical and thermal conduction and for bulk diffusion can be used interchangeably (but not for Knudsen diffusion). In several comparative studies [8, 81, 147, 162, 173, 174] it was shown that different simulation approaches (FVM, FDM, random walk) and different voxel-based SW packages (TauFactor, Avizo, GeoDict, PyTrax) provide almost identical results for the indirect tortuosities of conduction and diffusion (τ_{indir_ele}, τ_{indir_diff}, τ_{indir_thermal}). Tjaden et al. [162] concluded that uncertainties and errors from segmentation and meshing are much more important than those from different simulation approaches.

Commercial and open-source SW packages for 3D numerical simulation of different kinds of transport (conduction, diffusion, flow) and for computation of associated indirect tortuosities are summarized in Table 4.3. For the characterization of complex microstructures, it is recommended to use SW packages that enable transport simulations with a precise (i.e., voxel-based) geometric representation of the microstructure, because this approach is usually more reliable than mesh-based approaches with a reduced number of elements. The voxel-based option is available for example in the SW packages GeoDict, PuMA, Avizo, Amira, PerGeos, Pore3D, TauFactor, Pytrax, OpenLB and Palabos.