2. Review of Theories and a New Classification of Tortuosity Types

For a given porous medium, tortuosity (τ) is basically defined as the ratio of effective path length (L_eff) over direct path length (L₀) through the considered porous medium ([2‐5]), i.e.

Figure 2.1 schematically illustrates L_eff and L₀. Thereby, the direct path length (L₀) is easily captured, since it is the sample dimension in transport direction. For theoretical treatment of tortuosity, one then simply has to find a suitable definition of the effective path length (L_eff), and for practical application one simply has to find a suitable method to measure effective path length.

A diagram exhibits 3 path lengths of hydraulic flow velocity. It has 3 cuboid microstructures with flow directions. The first structure has effective path length, the second has direct path length, and the third has area and flux. The capillary, axial, and superficial formulas are also given.

Unfortunately, definition and measurement of the effective path length (L_eff) are not as simple as it appears at a first glance, which explains the emergence of numerous different tortuosity concepts over time. As will be shown in Chap. 3, the different types of tortuosities often reveal significantly different values when applied to the same microstructure. These differences and the underlying proliferation of concepts are usually not properly accounted for in the description of tortuosity, e.g., in appropriate papers and conference presentations. This frequently deficient description of tortuosity is partly caused by the fact that there exists no suitable nomenclature for the different types of tortuosities. In addition, very often, researchers in this field are not aware of the inherent differences between the various tortuosity types. This lack of awareness often becomes the source of confusion in scientific debate, and it can also be the source of misinterpretation of acquired data. The basic challenges in this field are thus to foster the awareness in the scientific community for the systematic differences between tortuosity types, and to introduce a useful classification scheme and nomenclature that can then serve as a basis for more precise descriptions and for clearer scientific discussions of the topic.

Two main criteria will be used to classify the different tortuosity types:

In this chapter, we describe the underlying concepts, definitions, and theories for all relevant tortuosity types. Thereby, the evolution of emerging definitions and concepts is presented in a chronological (historical) order. The concept of tortuosity was initially introduced in context with the Carman-Kozeny equations for flow in porous media. All other tortuosity types then evolved and diverged from there. Different branches of physics-based tortuosities (i.e., hydraulic, electric, and diffusional tortuosity types) unfolded in parallel over a long period. These physics-based branches are described in separate subsections. The geometric tortuosity types, which appeared more recently, are then presented in the following subsections. The classification of tortuosity is made even more complicated because there exist also mixed tortuosity types. They are mixed in the sense that they fulfil both criteria, for classification as physics-based, and also for classification as geometric tortuosity.

To establish the basis for more precise descriptions of tortuosity, we thus present a new classification scheme that uses the two mentioned criteria (i.e., method of determination and concept of definition) for a meaningful distinction of all existing tortuosity types. This results in no more than three main categories: (a) direct geometric, (b) indirect physics-based and (c) mixed tortuosity types. This classification scheme also serves as a basis for a systematic and specific nomenclature, which aims to provide all relevant information that is necessary for a scientifically correct treatment of the different tortuosity types.

The concept of electrical tortuosity (τ_ele) was developed since ca. 1940 (see Archie [47]) in parallel to the hydraulic tortuosity concept. The electrical tortuosity describes resistive effects of the microstructure, which limit the effective electrical conductivity (σ_eff) and, equivalently, increase the effective electrical resistance (R_eff) in porous media.

It must be emphasized that electrical conduction in porous media can take place, either, through the pore phase, which is saturated with a liquid electrolyte (whereas the solid phase is insulating). Or, alternatively, the electrical conduction can take place through the solid phase, which is illustrated in Fig. 2.6 for an SOFC-electrode with conductive LSC-phase. In the following section, we consider the case of a porous material saturated with a liquid electrolyte. Ohm's law can be used to describe the electrical flux (J_ele) in the saturated porous media, i.e.,with the potential gradient (ΔU/L₀) as driving force. The resistive formation factor (F_R) was then defined as the ratio of effective electrical resistance of the porous media (R_eff) over the intrinsic resistance of the electrolyte (R₀):

The inverse of the formation factor is the relative conductivity (σ_rel), which is also called microstructure (M)-factor. The difference between effective and intrinsic properties is due to the mentioned resistive effects of the underlying microstructure. According to Archie's law [47], which reads asthe formation factor can also be described as a power law of porosity (ε) with a so-called empirical cementation exponent (m).

Archie's law is widely used in geo- and soil-science. However, it has limited validity because it relates effective transport properties and associated formation factor to a single microstructure characteristic (i.e., porosity) and it ignores all other morphological effects. In an alternative approach by Wyllie and Rose, 1950 [48] the formation factor was described with an additional microstructure characteristic, namely the so-called structural factor (X_ele), defined by

In analogy with Carman's formulation for flow, the structural factor (X_ele) was also considered as an equivalent of the (electrical) tortuosity factor (T_ele = τ_ele²). This relationship is nowadays more commonly formulated as

Hence, the electrical tortuosity (τ_{indir_ele}) can be obtained indirectly by plugging experimental results for the formation factor (or relative conductivity) and porosity into Eq. 2.24, which leads to

This kind of indirect tortuosity (sometimes also called formation tortuosity) is nowadays very prominent because it can be obtained easily from numerical simulations of electric conduction, e.g., with commercial software like GeoDict from Math2Market [49] or with open-source software such as TauFactor from Imperial College London [50]. It must be emphasized that the determination of this indirect electrical tortuosity does not consider any geometric information. It is therefore by no means a measure for the true length of transport pathways. Whenever the tortuosity is afterwards used to determine effective conductivities and/or the resistive effects of microstructure on effective conductivity, respectively, this is a good way to define tortuosity, though.

Katsube et al. [15] performed an extensive investigation on shales that were saturated with electrolyte. The measured values of F_R were in the range from 140 to > 17,000 and porosities were in the range from < 0.01 to 0.1. The corresponding indirect electrical tortuosities took values in the range from 3.4 to 12. Katsube et al. [15] interpreted these values for τ_{indir_ele} as unrealistically high based on geometric considerations. This pessimistic interpretation is in accordance with Carman's 45° argument for the streamlines and associated estimation of \(\sqrt 2\) for streamline tortuosity. Katsube et al. concluded that the empirical results for the indirect electrical tortuosities are unrealistically high because other important microstructure effects (in addition to ε, τ) are not yet included in Eqs. 2.24 and 2.25. Owen [51] and Dullien [52] argued that the influence from narrow bottlenecks needs to be considered as an additional resistive effect. The formation factor was thus redefined by adding constrictivity (β) from Eq. 2.20 (see also [36]), which results in

It must be emphasized that the method for direct measurement of constrictivity from porous media, as presented above in Sect. 2.2.2.4, was only introduced in 2013 (Holzer et al. [38]). Because of a lack of suitable methods, constrictivity was thus not considered—until recently—as a separate microstructure characteristic in the calculation of indirect electrical tortuosity. Consequently, unrealistically high values (> 3) are often reported in literature for the indirect electrical tortuosity. These high tortuosity values must be interpreted as mixed information that includes resistive effects not only from tortuous path lengths but also from narrow bottlenecks.

Nevertheless, nowadays it is more and more accepted that the bottleneck effect and constrictivity should be considered separately from the path length effect. Hence, indirect tortuosity is now sometimes also calculated based on a separate treatment of constrictivity (see e.g. He et al. [53]), i.e.,

Today, the recent progress in numerical simulation and 3D image processing opens new possibilities for the computation of other (mixed) types of electrical tortuosity. In analogy to the hydraulic tortuosity discussed above in Sect. 2.2.2.3, also the electrical tortuosity can be extracted from simulated 3D fields of electrical flux. This approach provides either streamline (τ_{mixed_ele_Streamline}) or volume averaged tortuosities (τ_{mixed_ele_Vav}) (see Matyka and Koza [8] and Duda et al. [7]). For the volume averaged tortuosity, the equation can be rewritten as follows:

Furthermore, it should be noted, that electrical conduction and associated electrical tortuosity are not limited to porous media saturated with electrolyte. As mentioned earlier, the same principles can be used to describe electrical conduction in solid phases of a porous medium or a composite and thereby analyzing the electrical tortuosity of the conducting phase (e.g., electrical conduction in cermet electrodes of solid oxide fuel cells).

This section mainly deals with tortuosity in context with molecular diffusion (also called bulk diffusion), which is the dominant process in systems where chemical interactions between particles (ions, molecules) are negligible. Typically, this is the case for electrolytes with a high level of dilution and/or with an ideal, inert tracer. Also, gas transport at low pressure in macro- and mesoporous media is often dominated by molecular diffusion. Contrariwise, in systems where advection or surface effects (adsorption or dispersion) are important, molecular diffusion may not be the dominant transport process anymore.

The Knudsen number (K_n) is used to distinguish between different diffusion regimes in porous media. K_n is defined by the ratio of mean free path length (λ) over the characteristic length (L_c), i.e.,

The mean free path length (λ) depends on pressure, temperature and on the effective cross-sectional area of the gas species. Typically, it is in the range of 30–200 nm. For example, for air at room temperature and ambient pressure λ is 68 nm [54].

The characteristic length (L_c) is an ill-defined property, but it is usually considered as being equivalent to the characteristic pore radius. Hence, for gas transport in nanoporous media with r₅₀ < 10 nm, the Knudsen number is much larger than 1. In this case, we say that we are in the Knudsen diffusion regime, which is controlled by molecule-wall collisions [55].

For gas transport in macro-porous media, K_n is < 1, meaning that we are in the regime of bulk molecular diffusion, which is controlled by molecule–molecule collisions. For all liquid electrolytes, λ is very small (nm or smaller) so that diffusion in porous media is usually controlled by bulk diffusion.

For Knudsen numbers close to 1, both transport phenomena must be considered. Bosanquet's approximation [55] or the Dusty Gas Model [56] are then often used to model transport in this mixed regime.

In nanoporous materials with K_n \(\gg\) 1 (i.e., in the Knudsen regime), gas diffusion is controlled by collisions with the pore walls. Numerically, this process can be simulated with the random walk method (see e.g., Babovsky [69]). For each particle the corresponding diffusivity can be calculated from displacement length and travel time. For simulations that are based on a large number of particles and sufficiently long travelling time these calculations result in a homogenized effective Knudsen diffusivity (D_{eff_Kn}). The relative Knudsen diffusivity (D_{rel_Kn}) is then again defined as the ratio of effective over intrinsic Knudsen diffusivity, i.e.,

Similar as the relative diffusivity in the bulk diffusion regime, also the relative Knudsen diffusivity is a dimensionless property, which describes the resistive impact of pore microstructure against transport. The corresponding indirect Knudsen tortuosity (τ_{indir_Kn}) is obtained by

Note that such computed values for relative Knudsen diffusivity (D_{rel_Kn}) and for the indirect Knudsen tortuosity (τ_{indir_Kn}) strongly depend on the intrinsic Knudsen diffusivity (D_{0_Kn}, sometimes also called apparent diffusivity (D_a)). D_{0_Kn} itself can be computed using characteristic length (L_c) and thermal velocity (v_th = k_b T/m), i.e.,

The characteristic length (L_c) is a rather ill-defined property, which is somehow related to pore size distribution and to the average pore size, respectively. The uncertainty associated with L_c propagates into the relative Knudsen diffusivity and into the associated Knudsen tortuosity (τ_{indir_Kn}), which is critically discussed by Zalc et al. [70].

Unfortunately, to the best of our knowledge, there is currently no method available for a more direct analysis of Knudsen tortuosity based on effective path lengths (L_eff/L₀), as it is the case for bulk molecular diffusion, for electric conduction and also for viscous flow (see the discussion of mixed-streamline and -volume averaged tortuosities). Knudsen tortuosity is thus generally determined indirectly and therefore it is difficult to understand Knudsen tortuosity as a resistance that is related to distinct morphological features of the pore structure and to the corresponding length of transport pathways.

In literature, the interpretation of Knudsen tortuosity in context with gas diffusion in nanoporous media is highly controversial. For example, Ferguson et al. [71] used different methods (Random walk, FVM) for modeling transport at continuum scale and in the Knudsen regime, which enabled to compute both tortuosities (i.e., for bulk diffusion and for Knudsen diffusion). Also, Gao et al. [72] compared different modeling approaches (Knudsen, Dusty Gas and Oscillator models) that were used to characterize diffusivity and tortuosity in nanoporous media. Gao et al. emphasizes that the resulting tortuosity is highly dependent on the definition of the characteristic pore size (i.e., characteristic length) and on other experimental parameters (chemical species, temperature, pressure). Therefore, Gao et al. [72] suggested using the coordination number (i.e., average number of vertices that are connected to the nodes) instead of the Knudsen tortuosity to describe the impact of nanoscale microstructure on diffusive transport. This approach leads to more stable results and the physical and geometric interpretations of the coordination number are clearer than the indirect Knudsen tortuosity.

In nanoporous materials, transport mechanisms very often consist of a superposition of several processes such as bulk molecular diffusion, Knudsen diffusion, viscous flow, adsorption, and surface diffusion (see the examples in [53, 72]). For such cases with a mixed transport mechanism, it seems no longer possible to maintain the initial tortuosity concept as proposed by Kozeny and Carman, which is based either on the ratio of path lengths (L_eff/L₀) or on the ratio of velocity components (<v_c>/<v_x>), see Eqs. 2.13 and 2.18. It is obvious that the indirect tortuosity, which is then usually applied also for mixed transport mechanisms, has limited value in explaining distinct (geometric) microstructure effects. Also, in this case the indirect tortuosity must be rather understood as a fudge factor that describes the bulk resistive effects from microstructure and should not be interpreted as a measure for effective path lengths.

In very fine-grained, nanoporous media the molecular radii of gas species and the thickness of the surface adsorption layer can be in a similar range as the pore radii. In this case, variations of molecular radii and thickness of adsorption layers can become equally important for effective transport as the pore size and pore structure. Tortuosity in such systems is nowadays often determined with dedicated methods of numerical modeling (e.g., molecular dynamics [53]) or experimental characterization (e.g., NMR [58]), but the link with the initial geometrical tortuosity concept (i.e., with path lengths) is often not clear, since other physical effects (e.g., adsorption) may become dominant for diffusion.

The medial axis tortuosity (τ_{dir_medial_axis}) can be considered as a prototype for the family of geometric tortuosities. Therefore, in literature it is often just called 'geometric tortuosity' without further specification (see e.g., Stenzel et al. [41]). In our nomenclature, we allocate it to the group of ‘direct’ tortuosities since it is derived directly from the 3D microstructure by image analysis, in contrast to the indirect tortuosity types. The computation of medial axis tortuosity is based on several rather complex image-processing steps, which are illustrated in Fig. 2.6: The raw data from tomography is first segmented into its constituent phases. The example in Fig. 2.6a shows a fuel cell anode with the phases nickel, Gd-doped ceria, and pores [73]. For each phase a medial axis skeleton (MAS, see Chap. 5 of Soille [74] for details) is then produced (Fig. 2.6b, c). The shortest pathways through the MAS network, which connect couples of inlet and outlet points, are found e.g., with the help of the Dijkstra algorithm. For propagation algorithms on graphs to determine shortest path lengths, the reader is referred to Jeulin et al. [75]. The voxel-based skeleton is then transformed into a 3D graph, i.e., into a network representation consisting of vertices (nodes) and edges (branches) between them. A small portion of a 3D graph is shown in Fig. 2.6d. Note that further information can be attached to edges and vertices in form of additional characteristics of the local microstructure (e.g., local pore size and bottleneck size of each segment, coordinates, and coordination nr of each node etc.). The analysis of the 3D graph is computationally cheap. It reveals a distribution of medial axis tortuosities for each space direction (Fig. 2.6e). Based on graph analysis, tortuosity information of each pathway can be combined with local pore characteristics, such as the paths orientation. The example from Keller et al. [76, 77] in Fig. 2.6f shows a stereographic projection of pore path orientations in an anisotropic clay rock. The path orientation information is combined with the medial axis tortuosities of each path. Tortuosity values are indicated with a color code. In this example, the pathways parallel to the bedding plane (yz-direction) have lower medial axis tortuosities (2.7, dark blue) compared to the pathways perpendicular to the bedding plane (x-direction: up to 13, yellow), which of course has a strong impact on the anisotropy of the macroscopic permeability.

A diagram exhibits the steps involved in an image processing unit. It includes the construction of a solid oxide fuel cell anode, incorporating C G O and N i with pores, the generation of medial axis skeletons, graph analysis, a plot of tortuosity, and a stereoplot of path orientations.

Reproducibility of medial axis tortuosity among different research groups may be a challenge, since there exist many different methods for skeleton extraction from 3D voxel data (see Soille [74] for a detailed description of different skeletonization algorithms). Throughout the present paper, we use the following nomenclature: If the skeleton is not a medial axis representation, then we speak of τ_{dir_skeleton} instead of τ_{dir_medial_axis}. The procedure for generation of a medial axis skeleton (MAS) can be found e.g., in Lindquist et al. [78]. MAS generation is typically based on an iterative erosion process called topological thinning. The resulting skeleton consists of lines or curves with a thickness of one voxel. These curves are always located in the center of the pore bodies. As previously mentioned, the shortest pathways between couples of inlet and outlet points can then be algorithmically computed, e.g., by means of the Dijkstra algorithm [79]. It turns out that for complex pore structures a robust skeletonization procedure is challenging.

Efforts have been made to develop algorithms for an accelerated computation of shortest pathways in skeleton networks. Besides the propagation algorithm in Jeulin et al. [75] mentioned above, TESAR (Tree-structure Extraction algorithm delivering Skeletons that are Accurate and Robust) was introduced by Sato et al. [80]. Thereby the distance field used in the Dijkstra algorithm is modified, such that the shortest pathways in a pore network can be found in a fast and reliable way. This algorithm thus also reveals a medial axis skeleton (MAS). It was later implemented in commercial software, such as Avizo Fire (thermofischer.com), which was used by many authors as a basis for graph analysis to compute medial axis tortuosity (see e.g., [41, 73, 76, 77, 81, 82]; examples are shown in Fig. 2.6). Keller et al. [76] also presented a detailed description of the transformation steps from a voxel-based skeleton, via a 3D graph to a vector-based list of segments and nodes (assigned with local pore characteristics, such as effective lengths, orientation, pore size and bottleneck size).

Open-source software solutions for skeletonization and extraction of medial axes are available e.g., as Matlab code (Tort3D described by Al-Raoush and Madhoun [83]) or in ImageJ/Fiji (see https://​imagej.​net/​Skeletonize3D and imagej.​net/​AnalyzeSkeleton).

Furthermore, Thiedemann et al. [84] also presented methodological details that can be used for an extension of medial axis tortuosity towards weighting of pathways for their fluxes. This approach is based on a detailed analysis of the 3D graph including various local structural characteristics (see also Jungnickel [85] for a general reference to graph theory). Thereby the bottleneck size, which limits the flux of a specific pathway, is used for weighting the segments and local path lengths, which in turn affects the overall statistics of effective path lengths and the associated medial axis tortuosity. The weighting with bottleneck size can be understood intuitively as an analogue of the previously discussed weighting of streamlines by flux. A similar approach was recently also described by Nemati et al. [25].

In summary, the computation of medial axis tortuosity is based on various complex image-processing steps (e.g., skeleton extraction, 3D graph analysis). Therefore, its implementation is complex, and its computation may be time consuming. A major disadvantage is the fact that skeleton extraction can be done in many ways, which may lead to different results for the same type of geometric tortuosity. Moreover, it becomes difficult to interpret the extracted skeletons as transport networks in case of high porosities. Nevertheless, based on fundamental graph theory, this approach allows combining tortuosity with other local characteristics (bottleneck size, pore path orientation, connectivity), which are important for understanding the influence of microstructure on effective transport properties.

The path tracking method (PTM) was introduced by Sobieski et al. [86‐88]. It allows for a fast computation of a geometric tortuosity (i.e., τ_{dir_PTM}) that is very similar to the skeleton approach discussed in the previous section. However, the PTM algorithm is only applicable for microstructures consisting of packed spheres. It identifies tetragonal structures formed by four neighboring spheres. These tetragons represent an approximation of the interstitial pore space. The algorithm then finds the shortest pathways through the material by connecting gravity centers in the base triangles of neighboring tetragons. The resulting pathways from PTM are similar to the shortest pathways in a network skeleton. Despite this similarity, it must be emphasized that the pathways used for PTM differ in general from the medial axis representation.

In [87], a comparison of the PTM tortuosity (τ_{dir_PTM}) with the hydraulic volume averaged tortuosity (τ_{mixed_hydr_Vav}) is presented, which shows almost identical results. However, whereas the geometric PTM approach is fast and easy, the hydraulic volume averaged tortuosity (based on e.g., LBM simulations) is computationally expensive and time consuming. Hence, PTM is a fast method for measuring geometric tortuosity in 3D models of packed spheres or particles.

The concept of geodesic tortuosity (τ_{dir_geod}) relies on the geodesic metric in image data introduced by Lantuéjoul [89] and is used, e.g., in [41, 90, 91] to compute the corresponding geodesic tortuosity. The geodesic tortuosity is based on a statistical analysis of shortest path lengths (L_eff) from inlet- to outlet-planes divided by the sample length (L₀). For this purpose, the shortest path lengths are defined in terms of geodesic distances within the set of those voxels that represent the transporting phase (see e.g., Stenzel et al. [41]). Figure 2.7 illustrates the difference between geodesic (red) and medial axis (green) tortuosity. For the medial axis tortuosity there is only one starting point per pore body in the inlet plane, whereas for the geodesic tortuosity all voxels of the transporting phase in the inlet plane are taken as starting points. With increasing distance, the numerous pathways starting from different seed points concentrate on a few geodesic tracks. In both cases (i.e., geodesic and medial axis tortuosities) the shortest pathways can be calculated using the Dijkstra algorithm [79, 92]. However, whereas in the case of medial axis tortuosity the shortest pathways are restricted to the centerlines, for the case of geodesic tortuosity all voxels of the transporting phase are interpreted as vertices of a graph, which are connected to their neighboring voxels (26-neighborhood). Thus, in average, the geodesic pathways are shorter than the medial axis pathways. Note that—due to discretization errors on the voxel grid—single geodesic pathways might be longer than the corresponding paths along the skeleton. In [41] the following relationship

3 illustrations present the 3 pathways of medial axis, geodesic, and percolation path tortuosities. a and c. The largest spheres migrate from inlet to outlet. b. A set of 2 groups of lines are propagating from inlet to outlet through the medial axis and percolation path.

has been empirically derived by linear regression, using 43 virtual microstructures generated by a specific type of a 3D stochastic microstructure model. The coefficient of determination R² quantifying the goodness of fit for linear regression was equal to 0.81. Note that R² is between 0 and 1, where 1 indicates a perfect fit.

A mathematical formalization of geodesic tortuosity in the framework of random sets, a key object in stochastic geometry and mathematical morphology for microstructure characterization [46, 93], was recently provided by Neumann et al. [44, 94], while a slightly modified version of geodesic tortuosity was presented by Barman et al. [11].

The fast-marching method (FMM) tortuosity is very similar to the geodesic tortuosity, in the sense that it also considers geodesic distances within the voxel space of a given phase. The FMM algorithm is based on the simulation of a propagating front from inlet- to outlet-plane, which is described e.g., by Vicente et al. [95]. In particular, FMM solves the following Eikonal equation (see Sethian et al. [96‐99])on the voxel grid, where T(x) is the arrival time at location x and F(x) is the speed of the front. For each voxel the minimum arrival time is computed and by considering the speed of the front, this results in a distance map representing the shortest path lengths (L_FMM). For each pixel in the outlet plane, the corresponding FMM tortuosity can be calculated by dividing L_FMM through the sample length (L₀).

Jørgensen et al., 2011 [100, 101] describes the FMM method in context with microstructure analysis of SOFC electrodes. Thereby, FMM is also used to extract additional (local) information of the transporting phase network, such as the distribution of interface distances, distribution of characteristic path diameters and identification of dead-end pores. A further application of FMM was presented by Taiwo et al. [102] for battery electrodes. A recipe for the implementation of FMM can be found in Appendix E (supplementary info) of Hamann et al. [103].

In summary, the FMM tortuosity is very similar to the geodesic tortuosity, in the sense that it also finds the shortest (geodesic) pathways within the voxel space representing the transporting phase. In addition, it is computationally cheap and relatively fast.

Percolation path tortuosity is based on an algorithm that finds the pathway(s) with the least constricting bottleneck(s) (i.e., with the largest minimum bottleneck size). Hence, this algorithm allows the largest possible sphere(s) to travel from inlet- to outlet-plane and, at the same time, it finds the shortest path through the network for this sphere. Tortuosity is then defined as ratio of percolation path length over direct length (τ_{dir_percolation} = L_percolation/L₀). This method is, for example, implemented in the GeoDict Software (www.​math2market.​com). Thereby it is possible not only to calculate the percolation path for a single largest sphere but also for a defined number (n) of largest spheres. Hence, it enables us to find the n least constricting pathways and it calculates the corresponding mean tortuosity.

As shown in Fig. 2.7c, the blue sphere (with radius corresponding to the least constricting bottleneck) cannot pass through the narrow bottlenecks of the direct pathway (left) and hence it must take a deviation (right pathway). Percolation tortuosity is thus often larger than medial axis tortuosity, which takes more direct pathways through narrow bottlenecks.

The percolation pathways capture the maximum possible opening, which can be intuitively associated with a pathway of high flux. In contrast, the pathways for medial axis and geodesic tortuosities capture their shortest pathways regardless of the bottleneck radius, and therefore medial axis and geodesic tortuosities do not represent characteristics that can be related to pathways of high flux. Considering percolation path tortuosity for varying radii reveals interesting insights on porous microstructures going beyond the information gained by geodesic tortuosity. This is demonstrated using an example of paper-based materials in [46, 104].

For the pore centroid method (see e.g., [81, 101]), the 3D image volume is processed as a stack of 2D images. In each 2D section the position of the center of mass is determined for the transporting phase (e.g., pores). These centers are then tracked in transporting direction perpendicular to the 2D images, which results in one single tortuous centroid path. The centroid tortuosity is then calculated as the ratio of the effective centroid path length (L_eff) over the sample thickness (L₀). The pore centroid method is a quick and simple method, which is, e.g., implemented in the Avizo Software (www.​thermofischer.​com). For increasing volume fractions, the mass center approaches the image center and thus, pore centroid tortuosity goes to one. One can think of simple examples for microstructures with low values of centroid tortuosity (close to one), where the actual transportation paths are highly tortuous. Hence, the relevance of the centroid tortuosity in context with microstructure—property relationships is highly uncertain.

Finally, further approaches for the extraction of geometric tortuosity from 3D images can be found in literature. They are usually based on distance propagation and/or shortest path algorithms (see examples in [105‐107]).

The above-presented review reveals a multitude of different tortuosity types. However, in literature, in conference presentations and in associated scientific discussions dealing with tortuosity, the type of tortuosity under consideration is very often not properly defined. This lack of information often becomes the source of confusion and misunderstanding. For clarification, we propose to use a rough classification scheme with only three main categories of tortuosities. For a more precise specification, we introduce a systematic nomenclature that builds on the simple classification scheme. The nomenclature aims to provide all relevant details that are necessary for proper interpretation of the specific tortuosity under consideration. Both, the classification scheme and the detailed nomenclature approach are illustrated in Fig. 2.8.

A chart describes the tortuosity classification scheme and nomenclature. The scheme includes direct, mixed, and indirect. The nomenclature includes the specify method of determination, specify underlying concept of definition, and indirect tortuosity only.

The classification of tortuosity (Fig. 2.8, top) is based on two main criteria:

The method of determination is in most cases either based on a direct geometric analysis of the microstructure using tomography and 3D image analysis (called direct τ). Alternatively, tortuosity can be deduced indirectly from effective transport properties (called indirect τ).

The concept of definition is in most cases either based on the assumption that tortuosity and associated path lengths (L_eff) are geometric properties of the microstructure (called geometric τ) or, alternatively, the definition emphasizes that tortuosity is a function of the transport process under investigation, such as viscous flow, diffusion, or electric conduction (called physics-based τ).

It turns out that the method of determination and the concept of definition are generally linked with each other in a specific way, which leads to a reduction to three main categories of tortuosity. The first category consists of direct, geometric tortuosity types. The second category consists of indirect, physics-based tortuosity types. And the third category consists of mixed types, including streamline and volume averaged tortuosities (i.e., τ_{mixed_phys_streamline}, τ_{mixed_phys_Vav}). The definition of mixed types emphasizes both, the dependency on the transport process (i.e., physics-based) as well as the geometric aspect of path lengths and associated tortuosity. Characterization of mixed types is challenging, because the required information cannot be obtained directly from 3D image analysis of the microstructure. First, it requires some numerical 3D simulation to compute a transport-specific volume field of velocities or fluxes. In a second step, 3D image analysis is then used to extract the lengths of streamlines or the volume-averaged ratio of vector components from these volume fields (see Sect. 2.2.2.3).

Based on our classification scheme we introduce a new tortuosity nomenclature (Fig. 2.8, bottom), which consists of 2 or 3 terms. With the first term we describe the method of determination, which is one of three main categories (i.e., direct, mixed, or indirect determination of tortuosity).

The nomenclature then contains additional information on the underlying concept of definition. With the second (and third) term we specify details related to the underlying concept of definition. This can be either, a specification of the geometric analysis (for direction tortuosities), or a specification of the involved transport process (for indirect tortuosities). For mixed tortuosities, it is necessary to specify both, the geometric analysis, and the involved transport process. In the following section, the nomenclature is rigorously applied to all relevant tortuosity types, which are treated in this book.