4.1 Introduction

In Chap. 2, a new classification scheme with three main tortuosity categories was proposed, which includes direct geometric tortuosities, indirect physics-based tortuosities, and mixed tortuosities (see Fig. 2.8). For each of these categories, the characterization procedure has to follow a specific workflow. In our description of the methods for characterization of tortuosity we follow step by step the general workflows, which are illustrated schematically in Fig. 4.1.

Fig. 4.1
A flow diagram exhibits the steps to measure the direct, mixed, and indirect tortuosities. It starts with 3 D topography, followed by image processing, real and virtual 3 D models, statistical analysis for quantitative relationships, and digital material design.

Schematic illustration of methodologies and workflows for measuring direct geometric, indirect physics-based and mixed tortuosities. Round boxes represent methods and processes, which are discussed in Chap. 4. Rectangular boxes represent either data that is used as input for—or results that are obtained as output from the mentioned processes. Two boxes on top indicate the broader scientific context of tortuosity, which aims to establish quantitative micro–macro relationships and/or to perform digital materials design (see Chap. 5). Legend/abbreviations: ε porosity, PSD pore size distribution, rmin mean bottleneck radius, rmax mean radius of pore bulges, β constrictivity, τ tortuosity, SV surface area per volume, rh hydraulic radius, σ conductivity, D diffusivity, к permeability, PTM path tracking method, FMM fast marching method, Vav Volume averaged, Rwalk random walk, el electric, diff diffusion(-al), hydr hydraulic

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For all three tortuosity categories, the modern characterization approach focuses on the collection and quantitative analysis of 3D information. Thereby, real 3D microstructure models typically originate from experimental samples that are investigated by suitable tomography methods, followed by qualitative image processing (i.e., 3D reconstruction and filtering). Alternatively, virtual 3D microstructure models can be created with methods of stochastic geometry. This approach enables the creation of a large number of virtual 3D models in an efficient way. The virtual 3D microstructure realizations are then used as a basis for parametric studies and data driven microstructure investigations.

3D microstructure models (real or virtual) are considered here as the first step in the workflows for all three tortuosity categories. For measuring direct geometric tortuosities (i.e., τdir_geom), morphological analysis of the transporting phase (typically pore phase) is then performed by means of quantitative image processing. Numerous algorithms are nowadays available (in free and in commercial software tools) for measuring of different geometric tortuosity types.

The 3D microstructure models (real or virtual) can also be used as input for numerical transport simulations, from which effective transport properties and indirect physics-based tortuosities can be derived (τindir_phys_sim). Also here, numerous SW codes are nowadays available to simulate different transports within a 3D microstructure model (e.g., electrical or thermal conduction, bulk diffusion, Knudsen diffusion, viscous flow).

As a third option, the 3D volume fields representing the local flux from numerical transport simulations can be used as basis for the computation of mixed type tortuosities (τmixed_phys_streamline, τmixed_phys_Vav).

It must be noted here, that for the indirect physics-based tortuosities, there also exists an alternative way of determination without using 3D information. This alternative and complementary way is based on experimental measurement of effective transport properties (τindir_phys_exp). However, since we consider 3D analysis as the key to a better understanding of tortuosity and associated path length effects, we are focusing in the present chapter not on the experimental approaches, but on the modern image-based methods.

4.2 Tomography and 3D Imaging

4.2.1 Overview and Introduction to 3D Imaging Methods

In the following sections, we consider four main categories of tomography techniques that are relevant for 3D pore-scale characterization:

  1. (a)

    X-ray tomography

    μ-CT, nano-CT (CT = computed tomography), transmission and scanning X-ray microscopy (TXM, SXM).

  1. (b)

    Serial sectioning methods

    focused ion beam - scanning electron microscopy (Dual Beam FIB-SEM), plasma (P)FIB-SEM, broad ion beam (BIB-SEM), pulsed laser, mechanical sectioning (Ultra-Microtom) and mechanical polishing.

  1. (c)

    3D TEM (transmission electron microscopy)

    scanning transmission electron tomography (3D STEM), electron tomography (ET).

  1. (d)

    Atom probe tomography (APT)

At the beginning of a microstructure investigation there is always the question which tomography method should be chosen. To answer this question, first order criteria are the range of resolution and the size of the image window, which can be obtained with the different methods. A suitable tomography method must be capable to resolve the smallest relevant features of the investigated microstructure. At the same time, the 3D image window should also be large enough to capture the largest objects of interest in a representative way. The minimum size of a 3D image window with statistical relevance is called representative elementary volume (REV). For the determination of REV sizes see e.g., [1, 2]. The requirements of high resolution (small voxels) and at the same time, sufficiently large (i.e., representative) image window sizes are contradictory constraints, which must be addressed when choosing a suitable tomography method for materials characterization. Finding a good compromise for conflicting imaging parameters (i.e., resolution vs. REV) is a challenge, which requires a sound knowledge of the limitations and possibilities of the available tomography methods.

Figure 4.2 illustrates the range of resolutions and image window sizes that can be achieved with X-ray tomography, FIB-SEM tomography, electron tomography and atom probe tomography. Thereby the colored rectangles represent the performance fields that were typically achieved 10–15 years ago (taken from Uchic et al. [3]). At that time the different tomography methods occupied distinct performance fields (regarding resolution and image window size) with almost no overlap.

Fig. 4.2
A graph of volume analyzed versus resolution. The characteristic fields for x-ray, F I B SEM, electron tomography, and atom probe tomography are predicted. The performance of sectioning methods is denoted by the ellipsoids, along with the C T timeline. The S B F has the highest performance.

Graphical representation of important tomography methods characterized by their typical voxel resolutions (x-axis) and size of analyzed volume (y-axis, i.e., image window size). Each diagonal line represents a specific size of data cube (i.e., constant number of voxels), if the 3D image window is isometric. The colored rectangles indicate characteristic performance fields for traditional tomography methods, which are redrawn from Uchic et al. [3]. The green arrows indicate recent methodological evolutions from μ-CT to nano-CT and to large-field-of-view (LFOV) nano-CT. Ellipsoids represent the performance of serial sectioning methods. The elliptical shapes of their performance fields result from the fact that the serial sectioning methods tend to provide anisometric data cubes, because they reveal different properties in x-, y- and z-directions. Legend: nCT = nano-CT, μCT = micro-CT, LFOV = large-field-of-view, PFIB = plasma FIB, UMT SBFSEM = ultra-micro-tomography serial block face SEM, BIB = broad ion beam, mech. SBF = mechanical serial block face sectioning

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In the meanwhile, the resolution power of X-ray tomography has tremendously improved. The evolution from μCT to nanoCT is indicated with a green arrow in Fig. 4.2. For FIB-SEM tomography, the evolution went in the opposite direction. Nowadays the improvement of ion milling efficiency enables to capture much larger image windows. The evolution towards larger image windows also took place due to the introduction of new serial sectioning methods with higher milling rates (e.g., with plasma FIB and pulsed laser), whose performances are indicated with ellipses in Fig. 4.2.

In summary, the performance fields of X-ray CT, FIB-SEM tomography and other serial sectioning methods nowadays show a considerable overlap. However, it must be emphasized that the performance of a tomography method does not only depend on resolution and image window size. In particular, contrast and detection modes, acquisition time, but also the required sample properties (e.g., stability of a sample under specific imaging conditions, required sample size and required sample preparation) must be considered when choosing a suitable tomography method.

4.2.2 X-ray Computed Tomography

The resolution of X-ray tomography (XCT) has tremendously improved over the last 10 years from μm-range down to the 10 nm-range. XCT is now capable to resolve the microstructure at pore scale of almost any material in engineering science (e.g., energy materials used for batteries and fuel cells [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], concrete and asphalt [21, 22], polymer composites [23], 3D-printed materials [24]) as well as materials from geo-applications [25,26,27,28,29,30,31,32] and life sciences [33]. However, it must be emphasized that the progress is not restricted to the resolution power alone. Note that 10 to 15 years ago, XCT mainly was a static 3D methodology with micrometer resolution (μ-CT), which typically provided attenuation contrast. Meanwhile, the high-end version of X-ray tomography provides spatial resolutions down to 10 nm (nano-CT). It can be used as a 4D methodology with fast acquisition times in the sub-second range and it provides multi-mode detection capabilities (i.e., attenuation, phase, diffraction, and chemical contrasts). When speaking about X-ray tomography, we have thus to consider a very versatile group of 3D and 4D imaging methodologies, which continue to make fast progress in various directions, as discussed in a recent paper by Yan et al. [34]. For detailed information we also refer to the excellent review and overview articles by Cocco et al. [35], Maire and Withers [36], Pietsch and Wood [37], Brisard et al. [21], Rawson et al. [33] and Zeiss [38].

The following aspects of XCT are important:

4.2.2.1 Basic Principles of XCT

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

$$I = I_{0} e^{ - \smallint \mu \left( s \right)}ds ,$$
(4.1)
Fig. 4.3
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.

Schematic illustration of a modern, laboratory-based X-ray tomography system, redrawn from Zeiss [38]

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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.

Fig. 4.4
A diagram illustrates imaging performance parameters. It includes spatial resolution, contrast, time resolution, an x ray source, and optics.

Schematic illustration of imaging parameters, which need to be considered, when optimizing image acquisition with X-ray computed tomography (XCT) and/or X-ray microscopy (XM)

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4.2.2.2 Attenuation and Phase Contrasts

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 (βx-ray), i.e.,

$$n = \left( {1 - \delta } \right) + i\beta_{x - ray} .$$
(4.2)

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

$$\mu = 4 \pi \beta_{x - ray} /\lambda .$$
(4.3)

In materials with a high imaginary part (βx-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 βx-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/∆βx-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.

4.2.2.3 Spatial Resolution and Magnification

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

  1. (a)

    X-ray optics

    X-ray optics work with reflection, diffraction, or refraction (see e.g., www.x-ray-optics.de). Recent progress in nanofabrication has boosted the technology for X-ray optics (e.g., Fresnel Zone Plates FZP), which now enables voxel resolutions down to 10 nm. A major drawback of X-ray optics is the fact that it generally reduces the beam flux and therefore leads to longer acquisition times. This can be partially compensated by using brilliant synchrotron sources with a high beam flux.

  2. (b)

    Light optics

    Conventional light optics can be introduced in detection systems that consist of a scintillator and a CCD or CMOS camera. The scintillator converts the X-rays into visible light, which can then be magnified with optical lenses similar to conventional light microscopes. In contrast to X-ray optics, the light optics is highly efficient and fast. However, the diffraction limit of visible light constrains the resolution to little less than 1 μm.

  3. (c)

    Geometric set-up

    In systems with divergent, cone-shaped beams, the image can be geometrically magnified by adjusting the relative positions of source, object, and detector (see Fig. 4.3). The resolution in lens-less systems with purely geometric magnification is typically limited to the μm-range.

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).

Fig. 4.5
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.

Improvement of the spatial resolution in X-ray tomography. a Evolution of best resolution over the last 50 years for lens-less systems (blue), for hard X-ray tomography with Fresnel Zone Plate (FZP) (black) and for soft X-ray tomography with FZP (red). b Evolution of X-ray tomography, illustrating the link between spatial and temporal resolutions. Red: Synchrotron, white beam / Blue: Synchrotron, monochromatic / Green: Laboratory systems. The violet point marks the current high-end performance with sub-second and sub-μm resolutions achieved at synchrotron with 50% white + 50% monochromatic light (pers. communication, F. Büchi, PSI, Swiss Light Source SLS, 2021). The curves in a and b are redrawn from Maire and Withers [36] and updated with current trends presented in the literature. For recent progress in nano-CT see Yan et al. [34]

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4.2.2.4 Time Resolution Versus Spatial Resolution

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.,

$$N = q \pi / 2.$$
(4.4)

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 [42,43,44,45]). 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 [46,47,48]). In all these fields, fast progress and innovation is currently ongoing.

4.2.2.5 Experimental 4D Tomography

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 [54,55,56].

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 [58,59,60] 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.

4.2.3 FIB-SEM Tomography and Serial Sectioning

Commercial dual beam machines combining focused ion beam (FIB) with a scanning electron microscope (SEM) became available around the year 2000. Initially, FIB-SEM was used mainly for failure analysis in semiconductor industries. However, very soon it was recognized that FIB-SEM has a great potential for high-resolution 3D imaging by serial sectioning. First FIB-SEM tomography work was based on in-house developments of a machine-controlled procedure for serial sectioning (Holzer et al. [61]). Already in 2004, voxel resolutions of 6 × 7 × 16 nm could be reached based on the fully automated serial sectioning procedure with integrated drift correction. With a voxel resolution of ca. 10 nm, FIB-SEM tomography opened new possibilities to perform microstructure investigations at the sub-μm scale. FIB-SEM tomography thus became the method of choice for 3D investigations of fine-grained porous media [62,63,64,65,66] at the time when nano-CT was not yet available.

Nowadays, 3D acquisition by ‘slice and view’ is possible with any commercial FIB-SEM machine. Examples for applications of FIB-SEM tomography cover the fields of geological materials (sandstone, shale, coal) [64, 65, 67,68,69,70], zeolite [71], graphite [72, 73], polymers [73], thin films used as optical layers [74], catalysts [75], paper [76] and biomaterials [77]. FIB-SEM tomography is also very important for microstructure investigations of energy materials such as fuel cells [78,79,80,81,82,83,84,85,86,87,88] and batteries [89,90,91,92,93,94].

FIB-SEM serial sectioning can be used in combination with different detector systems such as EDS (energy dispersive spectroscopy) for mapping element concentrations and EBSD (electron backscattered diffraction) for mapping grain orientations and crystallographic information (see e.g., Uchic et al. [3]). With these analytical detection modes, FIB-SEM tomography became particularly important for the study of metals, alloys, and corrosion science [95,96,97,98], but also for battery materials [99,100,101]. Furthermore, the combination of FIB-SEM tomography with a cryo-transfer system enables the study of delicate, water-containing samples such as cement suspensions, swelling clay and biomaterials [102,103,104,105]. Reviews on FIB-SEM tomography and related serial sectioning techniques are given by Holzer and Cantoni [106], Cantoni and Holzer [107], Monteiro and Paciornik [108] and Echlin et al. [109].

As indicated in Fig. 4.2, FIB-SEM tomography initially occupied a niche among other 3D imaging techniques due to its high resolution of ca. 10 nm (nowadays even 5 nm are possible). With X-ray CT, it is only in the last few years that resolutions of less than 100 nm can also be reached. The main limitations of FIB-SEM tomography come from the milling capabilities. With a conventional Ga FIB source, a high milling precision that allows slicing in the range of 10 nm is only possible with relatively low beam currents of ca. 1 nA or less. The corresponding low milling rates lead to relatively long acquisition times of ca. 10–24 h for a stack of 500 to 1000 images. Low milling rates also lead to relatively small sizes of the 3D image window with edge lengths that are typically equal to only a few μm to tens of μm. The milling rates can be increased by using higher ion beam currents, but this comes at the expense of larger beam spot sizes and lower milling precision (i.e., with a decrease of spatial resolution in slicing direction). Fortunately, over the last years, the milling capabilities of FIB-sources improved significantly [110,111,112] and also new and more efficient serial sectioning techniques were introduced such as Plasma FIB and broad ion beam (BIB) [109, 113]. Hence, as indicated in Fig. 4.2, the various serial sectioning techniques for 3D image acquisition nowadays cover a wide range of voxel resolutions from ca. 5 nm to several μm and a wide range of image window sizes with edge lengths from μm to mm.

4.2.3.1 Basic Principle of Serial Sectioning with a Ga + FIB-SEM Dual Beam Machine

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.

Fig. 4.6
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.

Illustration of serial sectioning with a FIB-SEM dual beam system (taken from Holzer et al. [61])

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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:

FIB milling time

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).

SEM imaging time

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.

Time for drift correction(s)

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.

Time for beam stabilization

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).

4.2.3.2 Trends in Serial Sectioning I: Improvement of Milling Capabilities

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]):

Conventional FIB

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’.

Plasma (P)FIB

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].

Broad Ion Beam (BIB)

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].

Pulsed laser and combined tri-beam systems

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.

Table 4.1 Comparison of serial sectioning techniques and associated characteristic properties
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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]).

4.2.3.3 Trends in Serial Sectioning II: Imaging Capabilities and Detection Modes

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].

4.2.4 Electron Tomography

Transmission electron microscopy (TEM) enables for microstructure analysis at the nanoscale and even with atomic resolutions. Due to the invention of aberration corrected lenses, probe sizes as small as 0.05 nm can be reached with TEM [122]. In electron tomography (ET) numerous TEM projections are acquired in a tilt series at different angles, from which the corresponding 3D structure can be reconstructed. Current trends in nano-tomography (both, in ET and X-CT) were recently reviewed by Yan et al. [34].

Distinct ET methods have been developed separately for physical and biological sciences in order to overcome the specific sample-based limitations [123]. In materials science, ET is particularly important for the study of functional materials such as nano-porous materials for chemical engineering, nanoparticle agglomerations or nanostructured catalysts in fuel cells [124, 125].

A major strength of ET is obviously its high resolution-power. However, a relatively short mean free path length of electrons puts strong limitations to the maximum sample thickness, which is ca. 100 nm for mid Z-materials at 200 keV. In nano-tomography mode, ET is typically performed with a resolution of ca. 0.5 nm. The maximum sample and image window sizes are then typically not more than 100–300 nm. In atomic scale tomography mode, ET is performed with < 0.1 nm resolution. The corresponding image window size is then typically not more than 10–20 nm.

Moreover, a particular strength of ET is the ability to detect different signals from the same sample. Thereby one has to distinguish between full field transmission (TEM) and scanning transmission modes (STEM). Full field imaging allows for faster acquisition and low dose imaging of delicate samples. High Z-contrast is achieved in STEM with dark field (HAADF) and bright field (BF) modes. STEM also enables spectroscopic tomography whereby chemical maps are collected with EDS (energy dispersive spectroscopy) or EELS (electron energy loss spectroscopy). In addition, new detection modes are currently evolving, which provide interesting information about the spatial distribution of magnetic and electric fields, strain, grain orientation and/or crystallographic defects (see e.g., [34]).

Current improvements aim to push the limits of ET in various directions:

Acquisition time

The acquisition time is typically in the range of several hours, due to time consuming tilt by tilt tracking of objects. In future, automated repositioning can shorten the acquisition time considerably.

3D reconstruction

The precision of the 3D reconstructions is limited due to a relatively small tilt range (missing wedge problem) and due to a relatively low number of projections. New algorithms based on machine learning are capable to reveal much better 3D reconstructions, despite these limitations.

Sample holders and sample fabrication

New sample holders and stages, as well as improved sample fabrication procedures with automated FIB (producing cylindrical instead of lamellar samples) will contribute to better data acquisition and more reliable 3D reconstructions.

Detector technology

Important improvements can also be expected with respect to the detectors, which are capable to capture different signals (as mentioned above) with higher sensitivity, better signal-to-noise ratio, and faster acquisition time.

4.2.5 Atom Probe Tomography

Atom probe tomography (APT) is capable to perform 3D analysis at the atomic scale (around 0.1–0.3 nm resolution in depth and 0.3–0.5 nm laterally). Electrochemical polishing and focused ion beam (FIB) methods are used for sample preparation in the form of a very sharp tip. A very high electrostatic field (in an order of magnitude of 10 V/nm) is induced at the sharp tip, which is slightly below the point of atom evaporation. Laser or HV pulsing is then superimposed, in order to evaporate single atoms from the tip surface by a field effect (near 100% ionization). The atoms or ions are collected very efficiently with a position sensitive detector (PSD). The detector allows measuring simultaneously the mass of the ions (more precisely: the mass-over-charge ratio) by time of flight and at the same time to reconstruct the original position of the atom on the tip surface. The atoms are progressively removed from the tip so that a 3D image of the material can be reconstructed at the atomic scale.

APT has been successfully applied in materials science for many years, in particular for metals, alloys, and semiconductors (e.g., for the study of interfaces and inter-diffusion phenomena). A review on APT investigations of aluminium alloys was recently given by Ceguerra and Marceau, 2019 [126]. Technical advancements such as the introduction of pulsed laser-assisted field evaporation also enable atom probe analysis of oxides, which extends the field of APT applications to geological materials and metal corrosion (see e.g., Eder et al. [127]).

Air- and temperature-sensitive samples require transfer systems between FIB and atom probe under both vacuum and cryogenic conditions [128, 129]. Such a cryo-transfer system was recently used to study corrosion of nuclear glass. The sample consisted of a nano-porous gel filled with liquid electrolyte. It was shown for the first time that APT is capable to describe the 3D distribution of chemical concentrations at solid–liquid interfaces with (near) atomic resolution (Perea et al. [130]). The size of the 3D image window was 20 × 20 × 20 nm3. APT thus enables to detect variations in the chemical composition of the electrolyte and to combine this chemical information with structural information of tortuous pathways in the nano-porous network.

4.2.6 Correlative Tomography

For the investigation of complex microstructures, the application of a single microscopy method with a fixed resolution and/or with a single detection mode is sometimes not suitable for a representative characterization. For example, in materials with a wide pore size distribution, nano-tomography may be capable to capture small pores and bottlenecks, but the image window is then often too small for capturing the larger pores in a representative way. Alternatively, low-resolution tomography that provides a larger and representative image window may not be capable to resolve the smaller pores and bottlenecks. Fortunately, the contradictory requirements of a high resolution and a large representative image volume can be satisfied with the help of correlative tomography, which makes use of two or more tomography methods with different resolutions and image window sizes. Furthermore, in multi-phase materials, correlative tomography can also be used to capture multimodal information. The combined detection of Z-contrast, chemical- and crystallographic information can then be used as a basis for reliable interpretation, segmentation, and phase identification.

The power of correlative microscopy for advanced microstructure characterization has been recognized for many years. Thereby, complementary microscopy methods with different resolutions and detection modes are applied for the same regions of interest (RoI). Image registration can then be used to combine the information of the spatially overlapping data sets [131, 132]. Initially correlative microscopy was mainly based on the combination of 2D microscopy methods such as light and fluorescence microscopy, AFM, SEM, (S)TEM, (S)XTM (e.g., [133, 134]). However, very soon correlative imaging approaches were also combining 2D microscopy with tomography (see e.g., Caplan et al. [135]). Nowadays, due to the progress in 3D imaging and 3D image processing, the number of studies applying correlative tomography is rapidly increasing. Correlative tomography enables characterizing the full complexity of disordered microstructures by combining multi-modal, multi-scale and multi-dimensional information acquired with multiple 3D techniques from the same region of interest (or from overlapping RoIs).

A full review of correlative tomography is beyond the scope of this article. Overviews of correlative tomography are given by Burnett and Withers [136, 137] for materials science applications, as well as by Bradley and Withers et al. [138] for biomaterials.

In correlative tomography various combinations of 3D techniques are possible. Typically, non-destructive methods at lower resolution such as X-ray CT or confocal laser scanning microscopy are used in a first step. Subsequently, destructive 3D methods (e.g., 3D FIB-SEM, APT, ET) in combination with site-specific sampling techniques (e.g., with laser and with FIB lift-out techniques) are used for zoom-in characterization at higher resolutions. To illustrate the evolution of correlative tomography we briefly present some literature examples from the last 10 years, which are also summarized in Table 4.2.

Table 4.2 Examples of correlative tomography studies from the last 10 years, illustrating the methodological evolution and trends
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In Caplan et al. [135], the correlation of various 2D and 3D methods are discussed in context with a thorough characterization of biomaterials.

Tariq et al. [11] used multi-scale tomography (XCT and FIB-SEM) for the characterization of hierarchical pore structures in ceramics. The cumulative pore size distributions (PSD) obtained with multi-scale tomography are different from those obtained with mercury intrusion porosimetry (MIP). The example illustrates that it is difficult to quantify hierarchical pore structures based on information from different methods (experimental vs. imaging) and different length scales. New up-scaling approaches are needed for integration of multi-scale information in hierarchical pore networks.

Shearing et al. [139] investigated the microstructure of lithium-ion battery electrodes with XCT at different length scales. It was possible to obtain consistent results for porosity, tortuosity, and surface area with different CT scans. Apparently, with the chosen resolutions and sizes of data volumes, it was possible with different tomography methods to capture the relevant features in a representative way.

Burnett et al. [136] used correlative microscopy for the study of metal corrosion, combining multi-scale tomography with 2D maps from EBSD and EDS. This approach enabled to distinguish between pitting and inter-granular corrosion phenomena.

Bradley and Withers [138] used correlative tomography for characterization of biological materials with hierarchical microstructures and anisotropic mechanical properties.

Saif et al. [70] applied multi-scale tomography in combination with various 2D methods (MAPS, high resolution SEM, stitching of multiple SEM images) for characterization of oil shale pyrolysis. The multi-scale and multi-modal information enabled a thorough characterization of the heterogeneous clay microstructures, including accurate identification of porosity, organic matter, and mineralogical composition.

Kwiatowski da Silva et al. [140] used correlative TEM (ET) and atom probe tomography (APT) in combination with multi-scale modeling for characterization of Fe–Mn steels. This approach provides unique insight on the mechanism of Mn segregation to edge dislocations.

Fam et al. [141] used several tomography methods (nano-CT, FIB-SEM, ET) at high resolutions (1–15 nm) for the characterization of hierarchical structures in nano-porous gold catalysts. The results for porosity and pore size vary depending on the method, even though the resolutions were not very different. Most probably this puzzling picture arises from different contrast modes, which have a strong impact on the phase segmentations and associated quantitative analyses.

Keller and Holzer [142] and Keller et al. [143] used XCT, FIB-SEM and ET for a thorough characterization of pores in Opalinus clay. A concept for image-based up-scaling from micro- to meso-scale porosity and associated estimation of permeability is presented. This approach is also capable of capturing the anisotropic transport properties of clays across lengths scales from nm to mm.

In a recent study on PEM Fuel cells, Meyer et al. [16] combined multi-scale XCT with high resolution 2D imaging by He-FIB and TEM. The different methods give complementary information, which is important for accurate identification of relevant features in the heterogeneous multi-layer assembly, such as Pt nanoparticles in the micro-porous catalyst layer (MPL) and meso-pores in the gas diffusion layer (GDL).

For some of the mentioned correlative studies, the achieved resolutions and image window sizes of the 3D datasets are plotted in Fig. 4.7. This Figure illustrates that data volumes produced in current correlative tomography studies are usually smaller than 1,0003 voxels. This is particularly true for the nano-tomography methods. The use of relatively small data volumes in correlative tomography contrasts the general trend of ‘non-correlative’ tomography (i.e., using only one single tomography method), whereby the limits are pushed towards larger image windows and larger data volumes (e.g., 10,000^3 voxels). This comparison points to a certain potential for future development of correlative tomography towards larger image windows, which is particularly helpful for the characterization of materials with complex, heterogeneous microstructures.

Fig. 4.7
A graph plots the volume analyzed versus resolution. The characteristic fields for x-ray, F I B SEM tomography, electron tomography, and atom probe tomography are predicted. The lines for the performance of sectioning methods follow a decreasing trend. The A P T has the highest performance.

Plot of 3D image datasets from correlative tomography studies (see Table 4.2). It illustrates that 3D datasets from correlative tomography are similar or even smaller smaller than 1,0003 voxels, which is opposite to the current trend in ‘normal’ tomography (XCT, PFIB-SEM etc.), where the limits are pushed towards larger image windows and larger data volumes (e.g., 10,0003 voxels)

Full size image

4.3 Available Software Packages for 3D Image Processing and Computation of Tortuosity

As discussed in Chap. 2, there exist numerous types of tortuosities. In the following sections it is discussed how these different tortuosities can be extracted from 3D images. In our description we use the classification scheme and tortuosity nomenclature that was introduced in Chap. 2 (see Fig. 2.8).

The processing and analysis of 3D image data is a complex task. Fortunately, there are numerous software (SW) packages available nowadays for 3D image processing and also for pore scale modeling. Table 4.3 represents a list of these software (SW) packages. The SW packages offer a wide range of applications and opportunities, which are typically structured in different modules. In the following section we will discuss the capabilities of the SW packages with their different modules.

Table 4.3 Compilation of available SW packages with various modules for 3D image processing, modeling, and simulation
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4.3.1 Methodological Modules

The different columns in Table 4.3 from left to right represent specific methodological modules, which are used in the workflows for the characterization of different tortuosity types (see the workflow in Fig. 4.1).

4.3.1.1 Image Processing I (Qualitative)

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.

4.3.1.2 Stochastic Microstructure Modeling

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).

4.3.1.3 Image Processing II (Quantitative)

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 (τdir_geodesic, τdir_medial_axis, τdir_skeleton, τdir_PTM, τdir_percolation, τdir_FMM, τdir_pore_centroid) and also for mixed tortuosities.

It must be emphasized that the determination of indirect-tortuosities (τindir_phys_sim) and mixed tortuosities (τmixed_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:

  • solid and pore volume fractions (\(\phi\), ε)

  • continuous pore (or solid phase) size distributions (cPSD)

  • mean radius of pore bulges (rmax = r50_cPSD)

  • simulated mercury intrusion porosimetry (MIP-PSD)

  • mean bottleneck radius (rmin = r50_MIP-PSD)

  • constrictivity (β)

  • hydraulic radius (rh)

  • specific surface and interface areas (SSA, SIA) and surface areas per volume, respectively (SV, IV)

Further morphological microstructure descriptors for microstructure characterization can be found in [144,145,146].

4.3.1.4 Numerical Simulation of Transport/Pore Scale Modeling

We distinguish two main groups of SW packages with modules for 3D-simulation of physical and/or chemical (transport) processes:

Voxel-based simulations

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).

Mesh-based simulations

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 physics-based tortuosities

Indirect tortuosities (τindir_phys with variations τindir_ele, τindir_therm, τindir_diff, τindir_Kn, τindir_hydr) are computed from effective transport properties (conductivity σele, σtherm, diffusivity Deff, DKn or permeability к), which can be obtained from numerical transport simulations.

Mixed tortuosities

Mixed tortuosities (τmixed_phys_streamline, τmixed_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.

4.3.2 Different Types of SW Packages

The available SW packages can be grouped according to the modules that they include. The following six groups are distinguished in Table 4.3 (from top to bottom):

4.3.2.1 Multi-modular SW Packages

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).

4.3.2.2 SW Packages for Tortuosity Analysis (Quantitative Image Processing and Numerical Simulation)

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 (τindir_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 = εα) 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 (τdir_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 (τmixed_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.

4.3.2.3 SW Packages for Qualitative 3D Image Processing and Visualization

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).

4.3.2.4 SW Packages for Specific Tomography Data

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.

4.3.2.5 SW Packages for Numerical (Multi-physics) Modeling

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).

4.3.2.6 SW Packages for 3D Microstructure Modeling

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.

4.4 From Tomography Raw Data to Segmented 3D Microstructures: Step by Step Example of Qualitative Image Processing

After image acquisition with a suitable tomography method, it is necessary to transform the raw data into a segmented 3D microstructure (see workflow in Fig. 4.1). This transformation typically includes the following steps of qualitative image processing (IP I):

  • corrections of image defects (e.g., noise filtering, background removal and contrast leveling),

  • 3D reconstruction (e.g., alignment of FIB-stack or filtered back projection of CT scans) and, finally,

  • segmentation (i.e., phase identification, object recognition, labeling).

Image processing procedures for all these steps are well established and suitable algorithms are implemented either in freeware or in commercial software packages (see Table 4.3, column ‘Image Processing I’). Details on qualitative image processing can also be found in textbooks and review articles (e.g., Russ [155], Schlüter et al. [156]). Note that the use of machine learning leads to significant advances in the field of image processing. For machine learning algorithms, which are powerful for image segmentation, we refer, e.g., to [157, 158]. Examples of hybrid approaches combining classical image analysis with machine learning are presented in [100, 159], while a deep neural network is trained in [160], which allows for a reliable segmentation of FIB-SEM image data even if shine-through artifacts are present.

Nevertheless, it must be emphasized that it is very difficult to establish standardized procedures for 3D reconstruction and segmentation that allow user independent automation, because each raw data set is somehow unique due to the specific underlying settings associated with the tomography method, the used imaging parameters and the specific sample and materials properties. For each dataset, a careful adaptation of the image processing procedure for 3D reconstruction and segmentation is thus very important in order to achieve reliable quantitative results.

It is beyond the scope of this article to describe the various 3D reconstruction and segmentation procedures for different tomography methods and different materials. Instead, for illustration, we discuss the basic principles of ‘qualitative image processing’ (IP I) for a selected example (see Fig. 4.8). In this example, we consider a dataset from Pecho et al. [82, 83], which was acquired with FIB-SEM tomography from a fine-grained Ni-YSZ anode for solid oxide fuel cells (SOFC). Figure 4.8 a shows three orthogonal cross-sections of the original 3D raw data cube, before and after alignment. The raw data cube consists of 678 gray scale images with 2048 × 1768 pixels (i.e., 2.45 * 109 voxels in total). The initial voxel resolution was 19.5 × 19.5 × 20 nm (i.e., 19.5 nm pixel resolution in SEM-images (x–y plane) and 20 nm step size in FIB-sectioning (z-direction)). The size of the initial 3D image window (i.e., raw data volume) is thus 40 × 34.5 × 13.6 μm (1.87*104 μm3 in total).

Fig. 4.8
Four 3 D models of F I B SEM data. a has 3 models of the sample before and after the alignment and drift correction with distortions. b includes the sample with curtaining. c has a three-phase region. d. 2 diagrams of the sample before and after the removal of the rims. e has a reconstructed sample.

Illustration of the workflow for qualitative image processing (IP I) for a FIB-SEM image stack. The processing includes filtering, 3D reconstruction and segmentation. The images represent the microstructure of a porous Ni-YSZ anode for SOFC [82, 83]. a 3D reconstruction of FIB-SEM raw data (stack of 2D images), before and after correction of drift in x, y- and z-directions, b correction of curtaining, c cropping region of interest (RoI) and segmentation into 3 phases: pores = white, nickel = green, YSZ = red, d removal of artificial rims (thin red line) at pore-nickel interface after threshold segmentation, e visualization of the final 3D microstructure model: pores = black, nickel = white, YSZ = gray

Full size image

The raw data contains the following imperfections that need to be corrected before segmentation:

  • noise caused by relatively fast acquisition rates,

  • gray scale gradients typical for FIB-SEM images that are acquired under an angle of 52°,

  • vertical stripes in y-direction (so called curtaining) caused by materials inhomogeneity and associated variation of the local ion milling rates,

  • distortions in the image stack due to drift that could not be fully compensated during serial sectioning,

  • brightness-flickering from image to image due to detector instabilities and/or charging.

For filtering, 3D-reconstruction and segmentation standard procedures were used, which are implemented in the commercial software GeoDict. Similar options are also available in other SW packages (see Table 4.3: e.g., Avizo, ImageJ, Fiji). In a first step, curtaining and flickering filters are applied for each 2D image of the stack (Fig. 4.8b). Then the 2D images are realigned so that the distortions caused by drift in x, y and z directions are corrected (as shown in Fig. 4.8a). After that, the reconstructed 3D volume is then resampled in order to obtain cubic voxels with edge lengths of 20 nm. Additional 3D image filters to reduce the noise and to increase the contrast are applied in a careful and conservative manner using a so-called non-local means (NLM) algorithm [161]. A suitable region of interest is then cropped (see the colored regions in Fig. 4.8c with 3D image window size of 17.28 × 20.48 × 12.96 μm = total 0.46 * 104 μm3, consisting of 864 × 1024 × 648 voxels = total 0.73 * 109 voxels). Finally, the gray scale volume is then segmented into the three major phases (nickel, YSZ and pores) by means of suitable threshold values obtained from histogram-analysis. Note that there exists a gray-scale gradient at the solid-pore interface due to the limited spatial resolution, which results from the non-finite beam-sample interaction volume (so-called excitation volume). Upon threshold segmentation, the intermediate gray levels at the interface between pores (black/white in c, d) and nickel (white / green in c, d) lead to artificial rims (gray / red in c, d) that are erroneously attributed to the YSZ phase (see thin red line in Fig. 4.8d top). Such erroneous rims represent a typical segmentation-artifact in three-phase materials, also described in [156]. They can be removed with a morphological opening operation, which consists of an erosion step followed by a dilation step (Fig. 4.8d bottom). The final 3D microstructure after filtering and segmentation is visualized in Fig. 4.8e). It represents a suitable input for quantitative image analysis (IP II) and numerical modeling.

4.5 Calculation Approaches for Tortuosity

This section describes methods for the computation of different tortuosity types. The underlying theories and concepts as well as the classification scheme and nomenclature were discussed in Chap. 2. More specific reviews on tortuosity calculation approaches were recently given by Tjaden et al. [162] and Fu et al. [163].

4.5.1 Calculation Approaches and SW for Direct Geometric Tortuosities (τdir_geom)

In principle, almost all geometric tortuosities are based on the analysis of shortest pathways across the 3D microstructure in a predefined direction from inlet- to outlet-planes (i.e., τdir_geom = Leff/L0). Since, typically, there exist numerous shortest pathways connecting numerous couples of inlet- and outlet-points, the analysis of geometric tortuosity generally results in a histogram of paths lengths, from which a mean (effective) length (Leff) with the corresponding mean tortuosity value can be determined. The crux is that the length of shortest pathways can be defined and measured in many different ways. Consequently, there exist various geometric tortuosities. The underlying principles and definitions for direct geometric tortuosities have already been discussed in Sect. 2.4. Here we only present a short summary of the corresponding calculation approaches and refer to suitable SW packages.

4.5.1.1 Direct Geodesic Tortuosity

The calculation approach for geodesic tortuosity (τdir_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 (τdir_medial_axis).

4.5.1.2 Direct Medial Axis Tortuosity

The computation of the medial axis tortuosity (τdir_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 (τdir_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 τdir_medial_axis and τdir_skeleton are usually not too different from each other, but they are consistently higher than τdir_geodesic and τdir_FMM. This consistent order among the different tortuosity types was documented in Chap. 3 (see Figs. 3.6, 3.7 and 3.9).

4.5.1.3 Direct FMM Tortuosity

The fast-marching method tortuosity (τdir_FMM) is based on the simulation of a propagating front, which reveals the shortest geodesic pathways within the transporting phase [167,168,169]. 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 τdir_FMM.

4.5.1.4 Direct PTM Tortuosity

The path tracking method tortuosity (τdir_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 [170,171,172].

4.5.1.5 Direct Percolation Path Tortuosity

The percolation path tortuosity (τdir_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 τdir_percolation are identical to those for τdir_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.

4.5.1.6 Direct Pore Centroid Tortuosity

The pore-centroid tortuosity (τdir_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.

4.5.2 Calculation Approaches and SW for Indirect Physics-Based Tortuosities (τindir_phys)

Indirect, physics-based tortuosities (sometimes also called ‘flux-based’) are determined from effective transport properties, which are measured through specific transport experiments. These transport experiments can be performed either as a real physical experiment in the laboratory or as a virtual experiment by numerical simulation. A detailed discussion of literature dealing with laboratory experiments for electrochemical cells and for diffusion cells can be found in Tjaden et al. [162]. In the present section, however, we focus on the simulation-based approaches, which use 3D microstructure models from tomography as geometric input.

It is often mentioned that the mathematical treatment for different transport processes in numerical simulations is very similar and that the corresponding physics-based tortuosities can therefore be used interchangeably (i.e., it is assumed that τindir_ele = τindir_diff = τindir_therm). This hypothesis needs to be reevaluated critically. The aim is to understand which indirect tortuosities can or cannot be used interchangeably.

4.5.2.1 Comparison of Indirect Electrical, Diffusive, and Thermal Tortuosities

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 (Jele). The effective conductivity (σeff) can then be calculated by substituting the simulated flux (Jele) 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 (σeff) 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 (τindir_ele). Then, Eq. 2.24b (σeff = σ0 ε/τele2) 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 (σeff) from simulation, the electrical tortuosity can be computed indirectly according to Eq. 2.25 (τindir_ele = √(σ0 ε/σeff)).

Table 4.4 Calculation of effective transport properties: summary of material properties and physical laws relevant for transport experiments and associated transport simulation approaches
Full size table

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., Xrel = σeff_ele/σ0_ele or Deff/D0 or Keff_thermal/K0_thermal) and consequently also for the corresponding indirect tortuosities (τindir_ele = τindir_diff = τindir_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 (σeff = σ0 ε/τindir_ele2) and with analogous relationships for diffusivity and thermal conduction (e.g., Eq. 2.31: Deff = D0 ε/σindir_ele2). 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: τindir_phys = √(ε/σrel)). 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, 175,176,177]. This exclusion is achieved by adding a distinct constrictivity parameter (β) into the micro–macro relationships (i.e., τindir_phys = √(ε β/σrel), 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., τindir_phys = √(1/σrel)). 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.

4.5.2.2 Indirect Hydraulic Tortuosity

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 (Jhydr) 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 ≈ (σeff_ele/σ0_ele) ≈ (Deff/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 (rh2). Permeability is thus described by a combination of dimensionless characteristics (ε, τ) and a length-dependent characteristic (rh) according to Eq. 2.9 (к rh2 ε/τhydr2). In principle, the hydraulic tortuosity can now be computed indirectly from Eq. 2.9 (τindir_hydr = √(rh2ε/к)). However, to do so it is necessary to also assess the hydraulic radius (rh), 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 (τindir_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., τmixed_hydr_streamline, τmixed_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 (τindir_ele, τindir_diff, τindir_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.

4.5.3 Calculation Approaches for Mixed Tortuosities

Hydraulic tortuosity cannot easily be determined indirectly from effective properties as this is the case for electric or diffusive tortuosities (e.g., τindir_ele). An alternative approach to characterize hydraulic tortuosity focuses on streamlines representing the flow paths. This approach was discussed already in 1937 by Carman [179]. According to Eq. 2.17, the effective length of the hydraulic flow path is defined as weighted average of streamline lengths (Leff_weighted), from which the hydraulic tortuosity can then be deduced as follows

$$\tau_{mixed\_hydr\_streamline} = \frac{{L_{eff\_weighted} }}{L} = \frac{1}{L}\frac{{\mathop \sum \nolimits_{i} L_{i} w_{i} }}{{\mathop \sum \nolimits_{i} w_{i} }}.$$
(4.5)

As discussed by various authors [180,181,182,183,184,185], the definition and computation of suitable weighting factors (wi) is a major challenge, which puts strong limitations to the practical use of streamline tortuosities. As an alternative approach, it was shown by Matyka and Koza [152] and Duda et al. [186] that mixed hydraulic tortuosity can be computed in a much simpler way, based on the integration of local vector components from a simulated 3D velocity field. This so-called volume-averaged tortuosity was described in Chap. 2, Eq. 2.13 (τmixed_hydr_Vav =  < vc > / < vx > ) and Eq. 2.18.

Hence, both, volume averaged as well as streamline tortuosities, require a 3D vector field from numerical flow simulation as a basis for the computation of mixed hydraulic tortuosity. The underlying flow simulations can be performed with different numerical methods (FVM, FDM, FEM, LBM). Furthermore, these mixed tortuosities can be determined not only for viscous flow but also for other types of transport (i.e., conduction and diffusion), for which a 3D vector field can be computed. Hence, the streamline and volume-averaged tortuosities are also physics- or flux-based tortuosities. However, in contrast to the indirect physics-based tortuosities, streamline and volume-averaged tortuosities are calculated by a geometric analysis of 3D vector fields (and not from the effective property itself). The mixed tortuosities thus bear a higher level of information since they combine physical and geometric information. The mixed tortuosities are thus of major importance if one wants to understand the true path length effects. As discussed in Chap. 3 (compilation of empirical data), the values obtained for mixed tortuosities are in the same range as those from geometric tortuosities, but consistently lower than indirect tortuosities. This finding supports the interpretation that indirect tortuosities overestimate the limiting effects from tortuous pathways.

In literature, different modeling approaches are reported in order to obtain the required 3D vector fields for the calculation of mixed tortuosities. The numerical approach itself (i.e., differences between FVM, LBM etc.) has a lower impact on the resulting tortuosity than, e.g., inaccuracy from meshing or segmentation. In most studies, in-house solutions are used for the analysis of 3D vector fields and for determination of mixed tortuosities. For a detailed description of volume-averaged tortuosity and its implementation see Matyka and Koza [152]. Recently, the option for characterizing volume-averaged tortuosities (τmixed_hydr_Vav, τmixed_diff_Vav, τmixed_ele_Vav, τmixed_therm_Vav) by combining numerical transport simulations with 3D image analysis of the flow fields was implemented in the SW package GeoDict.

4.6 Pore Scale Modeling for Tortuosity Characterization: Examples from Literature

The analysis of mixed and indirect tortuosities is based on pore scale modeling. However, a review of transport modeling techniques and associated equations is beyond the scope of this publication. Instead, we refer to corresponding textbooks (see e.g., Bird et al. [187], Sahimi [188], Bear [189]). Furthermore, it must be emphasized that the phenomena of transport in porous media, the corresponding pore scale modeling and the associated analysis of tortuosity can be very complex. Depending on the system under consideration, complexity can be introduced for example by coupling of a standard transport process (e.g., diffusion or flow) with additional processes such as electrochemical reactions, physical interactions at pore walls (Knudsen effect, adsorption etc.) and/or reactive transport (chemical interaction with solids). Also, the simulation of transport phenomena at different length scales is often an important issue. It is also beyond the scope of this review to discuss the impact of such complexities on tortuosity. Instead, we present some examples from literature with different modeling approaches for material systems with complex transport phenomena.

4.6.1 Examples of Pore Scale Modeling in Geoscience

Saxena et al. [190] define a benchmark with numerous 3D-microstructure models, which are used for comparison of different flow simulation codes (LBM; mesh-based FEM and openFoam; voxel-based FFT and stokes-LIR).

Su et al. [191] describe various methods for pore-scale simulation (2 phase flow), including the pore network model, LBM, Navier–Stokes equation-based interface tracking methods, and smoothed particle hydrodynamics.

Liu et al. [29] present a critical review on computational challenges in petro-physics using micro-CT and up-scaling.

He et al. [192] perform molecular dynamics (MD) simulations of gas diffusion in nano-porous shale in order to evaluate diffusive tortuosity.

Wang et al. [193] present a review of analytical and semi-analytical fluid flow models for ultra-tight hydrocarbon reservoir rocks (including fracking).

Müter et al. [194] simulate diffusion in nano-scale pore networks based on dissipative particle dynamics (DPD).

Tallarek et al. [195] present a multi-scale simulation approach for diffusion in porous media. It covers interfacial dynamics at molecular scale as well as hierarchical porosity at meso- and macro-scales.

Ghanbarian [196] discussed the problem of scale dependency in rocks and soils, which results in scattered plots of tortuosity and diffusion coefficient versus scales. By applying finite-size scaling analysis the data show a quasi-universal trend.

4.6.2 Examples of Pore Scale Modeling for Energy and Electrochemistry Applications

Ryan and Muckerjee [197] give a critical overview on pore-scale modeling approaches for electrochemical devices (i.e., fuel cells and batteries). Particular focus is given to direct numerical simulation (DNS) techniques, which includes particle-based methods (smoothed particle hydrodynamics, dissipative particle dynamics, LBM) and fine-scale CFD methods (voxel-based vs. mesh-based).

Usseglio-Viretta et al. [198] demonstrated how to resolve the discrepancy in tortuosity factor estimation for Li-ion battery electrodes based on a combination of micro- and macro-modeling with experimental characterization.

Lu et al. [199] discuss the concept of digital microstructure design for lithium-ion battery electrodes based on a combination of nano-CT and multi-physics modeling.

Le Houx and Kramer [200] present a review on physics-based modeling of porous lithium-ion battery electrodes.

Zhang et al. [201] describe an experimentally validated pore-scale Lattice Boltzmann model to simulate the performance of redox flow batteries.

Recent publications dealing with the modeling of porous electrodes, the complex transport phenomena at pore scale (Chen et al. [202]), as well as multi-scale phenomena of ion transport (Tao et al. [203]) are addressed specifically.

Fundamental aspects of SOFC modeling such as coupled electrochemistry and transport at micro- to meso-scales as well as impedance analysis are reviewed by Grew and Chiu [204], Hanna et al. [205], Dierickx et al. [206] and Timurkutluk et al. [207].

Models for PEM fuel cells are discussed by Weber et al. [208] (review of transport models), Zenyuk et al. [209] (coupling of pore- and continuum-scales/up-scaling) and Liu et al. [210] (Liquid water transport in GDL).

4.7 Stochastic Microstructure Modeling

Statistical analysis of microstructure effects, for example in studies that aim for establishing quantitative relationships between tortuosity, porosity, and effective transport properties, is generally limited by the availability of suitable 3D image data. In a conventional approach using experimental materials fabrication followed by tomography and image analysis, the number of 3D analyses that can be performed with reasonable effort (in time and money) is usually quite limited. In this context, stochastic microstructure modeling is a powerful method that offers the possibility to increase the amount of 3D image data efficiently. Microstructure modeling is thus particularly important for data driven, statistical investigations of microstructure effects.

Stochastic geometry (also called mathematical morphology by some authors) represents the mathematical basis for stochastic microstructure modeling. Overviews related to the use of stochastic geometry for microstructure modeling are given by Chiu et al. [144], Matheron [211], Jeulin [212], Lantéjoul [213] and Schmidt [214]. In principle, stochastic geometry provides a mathematical toolbox for the generation of virtual, but realistic microstructures, which consists of many different approaches: random point processes, random closed sets, surface processes, random tessellations as well as random geometrical graphs representing spatial networks. The challenge of microstructure modeling is to use appropriate tools of stochastic geometry and mathematical morphology to develop stochastic 3D microstructure models, which allow for the generation of digital twins of a specific microstructure. In principle, two main quality criteria must be fulfilled:

Prediction power

A suitably chosen stochastic model provides virtual 3D microstructures, based on which the structural properties (e.g., tortuosity) and performances (e.g., transport resistance) of real materials can be predicted with a high precision and reliability.

Efficiency

The generation of virtual 3D microstructures with a stochastic model must be efficient in order to enable extensive parameter sweeps for data driven, statistical investigations.

An extensive literature review of microstructure modeling from a materials science perspective is given by Bargmann et al. [215]. In this review, different approaches for stochastic microstructure modelling are discussed in context with the type of microstructure for which these models are suitable (Table 4.5).

Table 4.5 Classification scheme for different types of microstructures, modified after Bargmann et al. [215]
Full size table

In [215], two main approaches for microstructure modeling are distinguished:

Physics-based methods

The physics-based methods aim for the simulation of physical processes liable for microstructure formation. For example, with the phase-field method, physical processes of grain growth or crystallization and associated microstructure formation are described with so-called transformation rules.

Geometrical methods

The geometrical methods aim for mimicking the material's morphology disregarding the underlying physics of microstructure development. A prominent example for this approach is the random packing of particles (spheres, ellipsoids, polyhedron, cylinders, fibers etc.) by means of discrete element modeling (DEM), see e.g., Sheikh and Pak [216].

For almost any type of microstructure, descriptions of suitable models can be found in literature with both, geometrical as well as physics-based approaches. However, only a few SW packages are available for microstructure modeling by stochastic geometry, as shown in Table 4.3. ESyS, GenGeo, YADE and Mote3D represent a small group of dedicated SW packages for microstructure generation, which are based e.g., on discrete element modeling (DEM). Further SW packages with specific modules for microstructure generation are GeoDict (e.g., GrainGeo, FiberGeo), Digimat, and PuMa. The BruggemanEstimator provides a module for the generation of battery structures and Dream3D for simulation of crystalline grain orientation patterns in the context with 3D EBSD (i.e., FIB-SEM tomography).

In the following sections we present short reviews from two important application fields of stochastic microstructure modeling, which are digital materials design for electrochemical devices and digital rock physics (DRP). For stochastic microstructure modeling of cellular and foam materials, see e.g., [217,218,219,220].

4.7.1 Stochastic Modeling for Digital Materials Design (DMD) of Electrochemical Devices

An overview of microstructure modeling approaches for electrochemical devices can be found in Ryan and Mukherjee [197]. Thereby, different stochastic 3D reconstruction methods are presented, which include Monte Carlo modeling, dynamic particle packing, stochastic grids, simulated annealing and controlled random generation. These stochastic models enable the creation of various 3D microstructures that are important for batteries, PEM fuel cells and SOFCs.

Various stochastic models for electrochemical devices and energy materials have been presented in the literature, for example for the fibrous GDL in PEM fuel cells [221, 222], for granular microstructures of battery electrodes [99, 223,224,225,226,227,228,229,230], and for different types of SOFC electrodes [231,232,233,234,235]. Thereby, a particular challenge for the simulation of cermet anodes is the realistic description of connectivity in all three co-existing phases. This challenge can be solved with specific random geometric graphs,—so-called beta-skeletons [232].

In order to discuss stochastic modeling as a basis for digital materials design (DMD), we consider an example of cermet anodes for SOFC, which is illustrated in Fig. 4.9. The aim of this approach is to optimize the anode microstructure, which consists of three phases, namely nickel, YSZ (zirconium oxide) and pore phases. Tomography data of this example is taken from [82, 83]. In an experimental approach, there are typically 3 to 5 main fabrication parameters, which can be used to vary the microstructure of cermet anodes. These parameters are related to composition (Ni/YSZ-ratio, pore former content), grain size of raw materials (powder fineness of Ni-oxide and YSZ) and sintering conditions (temperature, duration and pO2 of gas environment). In order to find an optimized microstructure, it is necessary to perform systematic parametric sweeps, which results in a rather large test matrix. With a conventional experimental approach, this test matrix cannot be covered with a reasonable number of resources. Thus, in most studies, only a few samples can be investigated for example by means of FIB-SEM tomography and quantitative image analysis.

Fig. 4.9
A flow diagram explains the process of the D M D model. It comprises materials fabrication, tomography, stochastic modeling, statistical analysis, digital materials design, and virtual materials testing V M T.

Schematic illustration of the workflow for digital materials design (DMD) of SOFC anodes based on stochastic microstructure modeling. The stochastic model for the creation of numerous virtual anode microstructures is fitted to experimental tomography data (i.e., digital twins (DT) of real anode microstructures), which is why the virtual microstructures have realistic properties (3D-data taken from Pecho et al. [82, 83])

Full size image

On the other hand, using a modern approach of digital materials design (DMD), the statistical basis is enlarged with the help of stochastic microstructure modeling. Thereby, the limited number of available 3D datasets from FIB-SEM tomography is used for calibration of the stochastic microstructure model (i.e., fitting to reality). As shown in Fig. 4.9, the real 3D microstructures from tomography represent the corner stones of an extended virtual parameter space. For each tomography dataset a digital twin is created, whereby the parameters of an appropriately chosen stochastic 3D microstructure model are fitted to the microstructure resolved by 3D imaging. After a successful fitting procedure, the digital twins are statistically similar to the real microstructures observed by 3D imaging. This means that microstructure characteristics and effective properties coincide nicely. Given that the model type is fixed, the fitted parameters of the stochastic microstructure model can be understood as ‘rules’ by means of which 3D microstructures with predefined properties can be produced in a stochastic process, e.g., by randomly placing particles of a certain size, shape, orientation in a 3D image volume. Moreover, the complex information contained in the 3D image data of microstructures is reduced to a relatively small number of model parameters. The fitting of model parameters calibrates the stochastic model to real tomography data. Doing so, a link is established between experimental fabrication parameters, parameters of the stochastic microstructure model and microstructure properties. The stochastic model can then be used to perform extensive parameter sweeps, which mimic the generation of 3D microstructures in a real fabrication process.

Once relationships between fabrication parameters and parameters of the stochastic microstructure model are established, one could, e.g., perform a parametric sweep with three fabrication parameters (Ni-YSZ ratio, sintering temperature, and particle size of YSZ). In the example shown in Fig. 4.9, it is assumed that for each of these fabrication parameters we define 10 different values (e.g., 1100, 1120, 1140, …. 1280 °C for the sintering temperature). This sweep results in 103 different parameter combinations, for each of which the corresponding virtual 3D microstructure will then be created. This leads to a database of 1000 virtual microstructures mimicking the microstructure of differently manufactured electrodes. Such extensive parameter sweeps open new possibilities for data driven optimization of microstructures. Thereby, each virtual 3D microstructure must be analyzed by means of quantitative image analysis (i.e., determination of tortuosity, constrictivity, three phase boundary length, surface area etc.) and by means of numerical modeling (i.e., simulation of gas flow in pores for estimation of permeability and simulation of electric conduction in Ni for estimation of effective conductivity). The invention of highly efficient computational solutions is thus becoming increasingly important. Massive simultaneous cloud computing (MSCC) and the use of artificial intelligence are promising techniques to solve the future challenges of big data analysis for DMD. Thus, novel concepts combining various techniques for modeling and computing represent the methodological basis at the forefront of innovative digital materials research. Thereby, stochastic microstructure modeling is identified as a key technology for digital materials science.

4.7.2 Stochastic Modeling for Digital Rock Physics and Virtual Materials Testing of Porous Media

The progress of 3D imaging, analysis and modeling also opens new possibilities to establish quantitative relationships between morphological microstructure characteristics (e.g., tortuosity, constrictivity, porosity etc.) and effective properties (permeability, diffusivity, strength, elasticity etc.) by means of virtual materials testing (VMT). Thereby, stochastic microstructure modeling provides the statistical basis for such data-driven investigations of microstructure effects. In geoscience this approach is often called digital rocks physics (DRP [236]).

For example, Berg 2012 and 2014 [176, 177] applied DRP to investigate the impact of tortuosity and other microstructure effects on conductivity and permeability of porous rocks, where the investigations are based on 5 micro-CT scans from Bentheimer (1 scan) and Fontainebleau (4 scans) sandstones. A numerical rock model, called e-Core, was used to create additional virtual sandstone microstructures (12 for Bentheimer and 7 for Fontainebleau sandstone) with different porosities, but with realistic properties. In this way, the micro–macro relationships in those sandstones could be described for materials with varying porosities, which is based on a set of 24 microstructures from micro-CT scans and virtual 3D-models.

In the meanwhile, a large amount of 3D image data including CT scans from real samples as well as virtual microstructures from stochastic modeling is available for free download from the ‘Digital Rocks Portal’ [237]. The latter is a public repository focusing on 3D microstructure data of porous media in geoscience. Such data can be used to perform data-driven investigations of micro–macro relationships with a broader data-basis.

Saxena et al. [190] generated a reference dataset consisting of a large variety of 3D microstructures ranging from idealized pipes to realistic digital rocks. The 3D models are used as a benchmark for DRP and for the comparison of different numerical solvers that are used in pore scale simulations (LBM, CFD, voxel based FDM, mesh-based FEM).

Nowadays, also commercial software for digital rocks physics has become available, such as PerGeos or GeoDict (see Table 4.3). These SW packages provide integrated solutions for the entire DRP-workflow, including 3D reconstruction, image analysis and numerical simulation. All these modern tools (including 3D data from repositories, stochastic models, SW packages for image analysis and numerical pore scale modeling) are increasingly used in different combinations for statistical analysis of microstructure effects (see e.g., Fu et al. [163]). The availability of the above-mentioned SW packages and their application to 3D image data representing a vast range of different microstructures will significantly contribute to a better understanding of the different types of tortuosities and of their relationship with effective transport properties in porous media.

In this context, we also refer to the recent work of Prifling et al. [238], where relationships between descriptors of two-phase microstructures, consisting of solid and pores, and their mass transport properties, have been investigated. To that end, a vast database has been generated comprising 90,000 porous microstructures drawn from nine different stochastic models, and their effective diffusivity and permeability as well as various microstructural descriptors have been computed. To the best of our knowledge, this is the largest and most diverse dataset created so far for studying the influence of 3D microstructure on mass transport in porous materials. The microstructures, descriptors, and the code used to study microstructure-property relationships are available open access via the following Zenodo repository: https://zenodo.org/record/4047774.

4.8 Summary

3D microstructure data is the basis for characterization of all three tortuosity categories, i.e., direct geometric, indirect physics-based and mixed tortuosities. The workflows for these 3 categories are illustrated in Fig. 4.1. For each step in these workflows, the underlying principles as well as current trends of methodologies are reviewed.

3D imaging

Nowadays, there are numerous 3D imaging methods available. We consider the four most important techniques, which are:

  • X-ray computed tomography (including micro-CT, nano-CT, transmission and scanning X-ray microscopy (TXM, SXM)),

  • Serial sectioning techniques (including FIB-SEM, PFIB-SEM, BIB-SEM, pulsed laser, Ultra-Microtom and mechanical polishing),

  • Electron tomography (including 3D TEM and 3D STEM) and,

  • Atom probe tomography (APT).

All these imaging techniques are rapidly evolving and improving. Today, the resolutions of these methods cover the lengths-scales from macroscopic scale down to atomic resolution. Depending on the imaging method there are also numerous detection and contrast modes available, which provide microstructure, chemical, crystallographic information and more. Of particular importance is the impressive improvement of time resolution in X-ray imaging, which opens new possibilities for 4D tomography at sub-μm resolution in combination with in-situ and in-operando experiments. In summary, when speaking about 3D imaging and tomography, it must be realized that we are dealing with a very versatile group of methodologies, which continues to make fast progress in various directions including improvement of resolution, acquisition time, detection mode, user friendliness etc. For materials scientist, the question is thus no longer, ‘is there a suitable method available for characterization of my material?’, but rather ‘which method is suitable for characterization of my material?’

SW packages available for image processing and determination of tortuosity

After acquisition of 3D images, the raw data must be processed qualitatively and quantitatively. An extensive list with 75 SW packages is presented in Table 4.3.

The SW packages are grouped according to their modules and to the options that they offer for

  • qualitative image processing (IP I: 3D-reconstruction, filtering, segmentation),

  • quantitative image processing (IP II: various tortuosity types and other morphological characteristics),

  • stochastic microstructure modeling and,

  • numerical simulations (voxel-based vs mesh-based, transport vs multi-physics).

An example is presented, which illustrates the workflow for tortuosity characterization in SOFC electrodes based on FIB-SEM tomography and using the GeoDict SW.

Calculation approaches for tortuosity

For the direct geometric tortuosities, various types (i.e., geodesic, medial axis, FMM, PTM, percolation path, pore centroid) can be determined directly from the segmented 3D image data. For most types, suitable SW codes are available.

The indirect physics-based tortuosities are calculated from effective transport properties (i.e., electric or thermal conductivity, diffusivity or permeability). These effective properties can be determined either by transport simulation or with dedicated experiments. The comparison of involved mathematical and physical laws shows that electric and thermal conduction as well as bulk bulk diffusion are basically identical with each other, and therefore the corresponding indirect electrical, thermal and diffusional tortuosities can be used interchangeably. In contrast, the underlying physical and mathematical laws for viscous flow and associated permeability are different. The indirect hydraulic tortuosity can therefore not be used interchangeably with the other indirect tortuosities related to conduction and diffusion. Due to the indirect calculation approach, this tortuosity category must be interpreted as a bulk measure for the transport resistance, which includes various microstructure limitations (not only the path length effect). This explains, why the estimated values for indirect tortuosities are consistently higher than those for direct geometric and mixed tortuosities.

The procedure to calculate mixed tortuosities (e.g., streamline and volume averaged tortuosities) includes two main steps: simulation of transport (i.e., conduction, diffusion, or flow) and determination of mean path length based on geometric analysis of the simulated flow fields. Hence, mixed tortuosities include both, physics-based information, as well as geometric information on the path lengths. It must be emphasized that the mixed volume averaged tortuosity can be computed simply by integration of the local vector components, which is an elegant, efficient and reliable method. The volume averaged tortuosity is thus considered as the most accurate approach to determine the true path lengths effect.

Pore scale modeling

Transport simulations are the basis for calculation of indirect and mixed tortuosities. For a review of modeling techniques and mathematical equations describing transport in porous media we refer to existing textbooks (see e.g., Bird et al. [187], Sahimi [188], Bear [189]). For many applications with porous media, the transport phenomena can become rather complicated, e.g. due to coupled processes (e.g. reactive transport or poromechanics). Examples from literature dealing with such complex transport phenomena are presented for applications in geoscience and in electrochemistry. However, for standard cases the transport can be simply simulated with one of the above mentioned SW packages (Table 4.3).

Stochastic microstructure modeling

Microstructure modeling by means of stochastic geometry or discrete element modeling (DEM) is a powerful method that offers the possibility to increase the amount of 3D image data efficiently. Microstructure modeling is thus particularly important for data driven, statistical investigations of microstructure effects.

Two main approaches must be distinguished. With the physics-based approach, microstructures are created based on the simulation of the involved physical processes (e.g., crystallization, grain growth or mechanical deformation). With the geometric approach the microstructure is created so that it matches the morphological properties of a real microstructure independent from the physical process and associated history of the material.

Numerous strategies and codes for microstructure modeling are described in literature. The crux is to find a suitable method, which allows for an efficient microstructure realization that matches with the real microstructure properties of the investigated material. Examples are presented, where microstructure modeling is applied in the framework of digital materials design (DMD) for materials in electrochemical devices, as well as in digital rock physics (DRP).