3.1 Introduction

The review of concepts and theories (see Chap. 2) reveals a multitude of different approaches, how tortuosity can be defined and measured nowadays. This diversity raises the question, whether the different tortuosity types can be used interchangeably, which is, however, only the case when they reveal identical or very similar results when applied to the same sample and microstructure. If it turns out, that different tortuosity types reveal different results, then the question arises, whether these differences are systematic, predictable, and understandable. A deeper understanding of inherent differences will improve our interpretation of measured tortuosity data and sharpen the scientific discussion of the topic.

To address these issues, we compile and analyse empirical data from the literature in this chapter. Table 3.1 represents a list of data sets with tortuosity-porosity (τ − ε) couples collected in almost 70 different studies. Each line in this table represents a set of data points for a specific class of porous material, which is characterized with a specific tortuosity type. More than 2200 data points were collected semi-automatically from literature (using the webplotdigitizer [70]), in order to investigate and identify specific τ–ε patterns.

τ–ε relationships are collected for the following 7 important classes of porous materials:

  1. (a)

    Solid oxide fuel cell (SOFC) electrodes and sintered ceramics,

  2. (b)

    Gas diffusion layers (GDL) of polymer electrolyte membrane fuel cells (PEM-FC) and other fibrous materials (e.g., paper, particle filters),

  3. (c)

    Battery electrodes, mostly from Li-ion batteries (LIB),

  4. (d)

    Geological materials (sandstones, clays, soils),

  5. (e)

    2D-models of granular materials (packed circles, ellipses, squares, or rectangles)

  6. (f)

    3D-models of granular materials (packed spheres and ellipsoids, mono-sized and poly-dispersed), and experimental model materials (e.g., glass beads)

  7. (g)

    3D-models of networked pore structures (from stochastic simulation) and foams.

In Sect. 3.2 we analyse this data to see, whether different materials classes reveal specific τ–ε patterns. In our comparison of empirical data, we also consider the different tortuosity types as a criterion for discrimination. In Sect. 3.3, the data from the seven material classes is thus also used to investigate the impact of different tortuosity types on the observed τ–ε patterns. Thereby, we use the classification scheme from Chap. 2 with the main categories: (I) direct geometric τ, (II) mixed τ and (III) indirect physics-based tortuosities (see also Fig. 2.8). In Sect. 3.4, we consider some specific examples with datasets, where different tortuosity types are applied to the very same microstructure. From these investigations, a consistent pattern can be deduced, which shows how various tortuosity types differ from each other (Sect. 3.5). Finally, in Sect. 3.6, the mathematical descriptions of τ–ε relationships in literature (e.g., Bruggeman, Archie, Maxwell etc.) are reviewed in the light of the findings in previous sections.

3.2 Empirical Data for Different Materials and Microstructure Types

The empirical data in tables 3.1 covers a large variety of materials with a wide range of porosities and with different microstructure characteristics. The data includes, for example, simple structures consisting of mono-sized spheres, but also more complex structures of packed fibers, foams, and composite materials (e.g., SOFC electrodes).

The empirical tortuosity-porosity (τ–ε) data is plotted in Fig. 3.1a–g for each specific material type separately. Overall, these plots show a relatively large scatter, so that no specific τ–ε relationship that could be described with a simple trend line and/or a mathematical expression (such as τ = εx) can be attributed to the specific material types. This finding indicates that even within a single material type the structural variety is large. However, when taking a closer look at Fig. 3.1a–g, for each material type a characteristic field (i.e., a dense cloud of data points) can be observed in the τ–ε plots.

Figure 3.1h illustrates that the characteristic fields of the different material types tend to have a strong overlap. Hence, it seems that τ–ε relationships are not suitable to capture the strong differences between the involved types of materials and structures. Nevertheless, for simple microstructures, such as packed spheres and ellipsoids, the scatter of characteristic fields is relatively small (see Fig. 3.2 e, f). For these rather simple microstructures, the corresponding data points are in a narrow band close to and often slightly above the Bruggeman trend line (i.e., T = τ2 = 1/ε0.5). For all other types of materials and microstructures, the tortuosity-porosity (τ–ε) data shows as strong scattering.

It must be emphasized that the plots in Fig. 3.1 do not distinguish between different types of tortuosities, which may be one reason for the observed scatter of τ–ε datapoints.

The underlying source file for Fig. 3.1, with detailed information to 69 references, can be downloaded from the electronic appendix (Supplementary File 3.1).

Table 3.1 a–g contains list with empirical tortuosity-porosity (τ–ε)-data from literature 2204 tortuosity-porosity (τ–ε) data points are collected from 69 references in literature. The corresponding source file for Table 3.1, which contains additional detailed information, can be downloaded from the electronic appendix (Supplementary File 3.1). It must be emphasized that ‘porosity’ in this table always means the effective porosity, where this information is available. The effective porosity represents the fraction of pore space, which forms a contiguous network, and which excludes trapped pores. In subsequent figures derived from this table the effective porosity is plotted.
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Table 3.1b PEM FC (GDL) and fibrous materials. Legend: (Nr of dataset) Reference [Ref Nr], τ type, (τ methodical details) investigated material
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Table 3.1c Battery electrodes. Legend: (Nr of dataset) Reference [Ref Nr], τ type, (τ methodical details) investigated material
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Table 3.1d Earth Science and Geo-engineering (sandstones, clays/soils/mortar). Legend: (Nr of dataset) Reference [Ref Nr], τ type, (τ methodical details) investigated material
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Table 3.1e Granular porous media I: 2D models (circles, ellipsoids, rectangles). Legend: (Nr of dataset) Reference [Ref Nr], τ type, (τ methodical details) investigated material
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Table 3.1f Granular porous media II: 3D models (packed spheres, ellipsoids, sands) and model materials (glass beads, sand). Legend: (Nr of dataset) Reference [Ref Nr], τ type, (τ methodical details) investigated material
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Table 3.1g Foams and stochastic models of networked structures with continuous phases. Legend: (Nr of dataset) Reference [Ref Nr], τ type, (τ methodical details) investigated material
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Fig. 3.1
4 scatter plots of tortuosity versus porosity. The fit line begins along the y axis and follows a decreasing trend. Several plots for different materials are scattered between 0 and 11. The scattered regions are shaded with different color codes. Plot g for open cell foams has the highest value.figure 1figure 1figure 1

Plots of tortuosity versus effective porosity for different materials and microstructures, as presented in Tables 3.1. a SOFC electrodes and porous ceramics from Table 3.1a, b Battery electrodes from Table 3.1c, c PEM GDL and fibrous materials from Table 3.1b, d Rocks and geological materials from Table 3.1d, e 2D granular materials from Table 3.1e, f 3D granular materials from Table 3.1f, g Stochastic models of networked structures and open foams from Table 3.1g, h For each material type a characteristic field is indicated with a specific color. The underlying source file for Fig. 3.1, with detailed information to 69 references, can be downloaded from the electronic appendix (Supplementary file 3.1)

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3.3 Empirical Data for Different Tortuosity Types

The same empirical data from Tables 3.1, which in Fig. 3.1 is grouped into different material types, is now replotted in Fig. 3.2, and thereby regrouped into different tortuosity types. The results in Fig. 3.2 illustrate a pronounced pattern, whereby the indirect tortuosities (Fig. 3.2a) scatter over a much larger range than the mixed (Fig. 3.2b) and geometric (Fig. 3.2c, d) tortuosity types.

In Fig. 3.2e geometric and mixed tortuosities are plotted together. But compared to Fig. 3.2a–d, they are plotted with a different scale on the y-axis. The characteristic field in red, representing geodesic and FMM tortuosities, shows the lowest tortuosity values. The measured values for geodesic and FMM tortuosities are often slightly below the Bruggeman trend line and only rarely they take values larger than 2. In contrast, the Bruggeman trend line typically defines the lower bound for mixed tortuosities (green characteristic field) and for those geometric tortuosities, which are derived from a skeleton (medial axis and PTM, blue characteristic field). For porosities below 0.3, the medial axis/PTM tortuosities often also reach values greater than 2. It must be mentioned here that for mixed tortuosities, the empirical data at low porosities is missing, but it is expected that the mixed tortuosity values also show a strong increase with decreasing porosity, in a similar way as it is observed, e.g., for medial axis and PTM tortuosities.

The comparison of empirical results in Figs. 3.1 and 3.2, indicate that the large scatter observed for most materials and microstructure types can be mainly attributed to data points containing indirect tortuosity types. A surprising insight is the fact that the different tortuosity types (Fig. 3.2) give a more characteristic pattern than the different material types (Fig. 3.1). The latter were discussed in the previous section. This indicates that in general the value of tortuosity is more strongly influenced by the type of tortuosity than by the type of material or microstructure. This finding strongly emphasizes the necessity of clearly defining (and choosing carefully) the type of tortuosity, which is used for the characterization of porous media.

Fig. 3.2
5 scatter plots of tortuosity versus porosity. The fit line begins along the y axis and follows a decreasing trend. Several plots for various tortuosity types are scattered between 0 and 11. Plot a for indirect tortuosities has the highest value at (10, 10.5). Values are estimated.figure 2figure 2

Plots of tortuosity versus effective porosity with color codes for different tortuosity types (data from Tables 3.1). a Indirect electrical and diffusional tortuosities, b Mixed tortuosities, c Direct geometric tortuosities (medial axis, skeleton and PTM), d Direct geodesic tortuosities, e Comparison of mixed and direct tortuosity types (characteristic fields are highlighted with different colors). Note that in Fig. 3.2e, the y-axis has a different scale compared to Fig. 3.2a–d

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3.4 Direct Comparison of Tortuosity Types Based on Selected Data Sets

The data reviewed in the previous sections documents a kind of universal pattern with systematic differences between the various tortuosity types. In this section, we focus on examples that illustrate the difference between specific tortuosity types for a given material and microstructure. This investigation is based on selected data sets from Tables 3.1, where more than one tortuosity type is characterized for the same material or microstructure. For each example, we refer to the corresponding (Nr of data set) in Tables 3.1, which correspond to the [Ref Nr].

3.4.1 Example 1: Indirect Versus Direct Pore Centroid Tortuosity

τindir_diff versus τdir_pore_centroid from SOFC and battery electrodes.

Cooper et al. [5, 6] (dataset Nr 6a-e) used 5 different methods to extract tortuosity from an SOFC cathode material (LSCF). The data set includes four indirect tortuosities that were determined with different simulation approaches (6b: τindir_therm StarCCM+, mesh-based, 6c: τindir_diff, AvizoXlab, voxel-based, 6d: τindir_diff, TauFactor, voxel-based, 6e: τindir_diff, random walk algorithm, voxel-based). As shown in Fig. 3.3, all four approaches give very similar results for the indirect tortuosities (red circles). In comparison, the direct geometric tortuosity (6a: τdir_pore_centroid) for the same sample is significantly lower (red squares).

Fig. 3.3
A scatter plot of tortuosity versus porosity. A fit is plotted through (30, 1.35) and (70, 1.1). 4 lines for indirect tortuosity follow a decreasing trend above the fit line. The line for direct tortuosities follows a decreasing trend near the fit line. The plots are distributed along the lines.

Comparison of indirect (circles) versus direct tortuosities (squares) for SOFC cathodes (red symbols indicating values from Cooper et al. [5, 6]) and for battery electrodes (blue symbols indicating values from Cooper et al. [32]). Both materials show the same pattern, where the values for indirect tortuosity (τindir_diff_sim) are consistently higher than those for geometric tortuosity (τdir_pore_centroid). Note that the values for geometric tortuosity are very close to the Bruggeman trend line

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Cooper et al. [32] (dataset Nr 32a-b, see Fig. 3.3, blue symbols) also presented a comparison of direct geometric tortuosities (32a: τdir_pore_centroid) with indirect diffusional tortuosities (32b: τindir_therm) for a Li-ion battery electrode (LiFePO4). The overall pattern and even the specific tortuosity-porosity values are very similar when comparing the battery electrode (blue) with the SOFC electrode (red). For both materials the values for indirect tortuosity (τindir_diff/therm) are consistently higher than those for geometric tortuosity (τdir_pore_centroid). From this data, Cooper et al. [5] deduced the following relationship between indirect and direct tortuosities:

$$\tau_{dir\_pore\_centroid} = 0.5\ln \tau_{diff\_indir} + 1$$
(3.1)

3.4.2 Example 2: Indirect Versus Direct Medial Axis Tortuosity

τindir_ele_exp versus τdir_medial_axis from porous ceramic membranes.

Wiedenmann et al. [14] (Dataset Nr. 14 in Table 3.1) presented a comparison of direct medial axis tortuosities (14a: τdir_medial_axis) versus indirect electrical tortuosities (14b: τindir_ele, from EIS experiments) for two different separation membranes in alkaline electrolysis cells, consisting of sintered olivine and wollastonite. As shown in Fig. 3.4 the microstructures vary from fine grained and dense (ε = 0.27) to coarse-grained and open porous (ε = 0.80). Despite the large variation of porosity, the values for medial axis tortuosity (filled squares) are all in a very narrow range (1.62–1.84). The 3D visualizations in Fig. 3.4 [14] document nicely, that the basic geometry of all pore networks remains very similar for all samples, except for the coarseness, which increases with porosity (i.e., scaling). When changing porosity, the obtained values for geometric tortuosity thus remain almost constant (1.73 ± 0.11).

Fig. 3.4
A graph of tortuosity versus porosity. A fit is plotted through (20, 1.5) and (90, 1.02). 3 lines for direct and indirect tortuosities follow a decreasing trend above the fit line, and 1 direct tortuosities line follows an increasing trend between 1.4 and 2.4. The 3 D microstructures are provided.

Comparison of indirect (τindir_ele_exp) versus direct tortuosities (τdir_medial_axis) for porous membranes consisting of wollastonite (red) and olivine (blue) from Wiedenmann et al. [14]. The geometric tortuosities (filled squares) vary in a narrow range (1.6–1.8), whereas the indirect tortuosities (open circles) show significant variation (1.4–2.4). The variation of effective properties and associated indirect tortuosities is attributed to variations of the size of characteristic bottlenecks and corresponding constrictivities

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In contrast, the indirect tortuosities (open circles) increase significantly with decreasing porosity from 1.4 to 2.2 for olivine (blue) and from 1.6 to 2.4 for wollastonite (red). Wiedenmann et al. [14] and Holzer et al. [13] documented that the effective properties in these samples scale with constrictivity (i.e., bottlenecks), but not with geometric tortuosity. Therefore, they concluded that the variation of indirect tortuosities in this example rather represents the resistive effects arising from variations in the size of characteristic bottlenecks, rather than effects from path length variations.

3.4.3 Example 3: Indirect Versus Direct Geodesic Tortuosity

τindir_ele/therm_sim versus τindir_diff_exp versus τdir_geodesic/FMM from porous Zr-oxide and PEM GDL.

Figure 3.5 represents a comparison of direct geometric tortuosities (geodesic/FMM) with indirect tortuosities (diffusive/electric/thermal—from simulation and experiment) based on data from Holzer et al. [19], Tjaden et al. [8] and Holzer et al. [29, 30]. As shown in Fig. 3.5, all four studies show that the values for geometric tortuosities (τdir_geodesic and τdir_FMM, closed squares) are systematically lower than the results for indirect tortuosity (open circles and crosses). The samples of the three studies cover a wide range of effective porosities from 4 to 76 vol-%. In addition, the involved microstructures in sintered ceramics (Zr-oxide, YSZ) are very different from those in fibrous PEM GDL. Despite this obvious microstructural variation, all samples show the same consistent pattern. The direct geometric tortuosities (geodesic/FMM) vary hardly and are always below the Bruggeman trend line. Only for very low porosities (ε < 0.2) the geodesic tortuosities start to increase moderately. Apparently, the geodesic path lengths are not very sensitive to variation of the pore volume fraction. In contrast the indirect tortuosities (open symbols and crosses) are always significantly higher than the Bruggeman trend line and they also show much stronger variation. For each series of material/microstructure, a trend of increasing indirect tortuosities is observed when lowering the porosity.

Fig. 3.5
A scatter plot of tortuosity versus porosity. A fit is plotted through (0, 6) and (80, 1.1). The plots for direct and indirect tortuosities are distributed on either side of the fit. Two 3 D models of porous material are given with microscopic image. The indirect tortuosities have the highest value.

Comparison of indirect (τindir_ele_exp/sim) versus direct geometric tortuosities (τdir_geodesic, τdir_FMM) for various porous materials from Holzer et al. [19] (black symbols), Tjaden et al. [8] (blue symbols), and Holzer et al. [29, 30] (red and green symbols). The direct geometric tortuosities (closed symbols) vary only in a narrow range below the Bruggeman trend line. In contrast, for the same samples, the indirect tortuosities (open symbols) show much higher values

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3.4.4 Example 4: Indirect Versus Medial Axis Versus Geodesic Tortuosity

τindir_ele_sim versus τdir_medial_axis versus τdir_geodesic from stochastic 3D structures.

Gaiselmann et al. [67] and Stenzel et al. [66] performed in-depth investigations on the relationship between microstructure characteristics and effective transport properties based on virtual 3D structures generated by stochastic modelling. The so-called stochastic spatial graph model (SSGM) provides 3D structures with a connected transporting phase even at low volume fractions. A large number of microstructures covering a wide range of microstructure characteristics (i.e., volume fractions, phase size distributions, constrictivity and path lengths) were created and analysed. Three different tortuosities (medial axis, geodesic, and indirect tortuosities) can be compared based on datasets Nr. 66 and Nr. 67. It should be noted, that in these studies, electric conduction in the solid phase and associated solid phase microstructure was investigated. However, the effect of microstructure (e.g., tortuosity) on transport in solid phases is basically the same as in the pore phase.

The results in Fig. 3.6 document that the indirect tortuosities (τindir_ele_sim) are consistently higher than the direct geometric tortuosities. Figure 3.6 also reveals slight differences between direct medial axis and direct geodesic tortuosities. The medial axis tortuosity (τdir_medial_axis, red symbols, Nr 67) is slightly higher than the Bruggeman trend line. In contrast, the geodesic tortuosity (τdir_geodesic, black, Nr 66b) is usually below the Bruggeman trend line. However, for small volume fractions of the transporting phase (i.e., ε < 20 vol-%), both tortuosities (i.e., geodesic, and medial axis) show similar values.

Fig. 3.6
A scatter plot of tortuosity versus porosity. A fit is plotted through (10, 0) and (80, 1). The tortuosity plots are distributed on either side of the fit. The indirect tortuosities have the highest value. 3 microstructures of tubular network, high anisotropy, and narrow bottlenecks are depicted.

Comparison of indirect (τindir_ele_sim) versus direct geometric tortuosities (τdir_geodesic, τdir_medial_axis) for virtual 3D structures created with a spatial stochastic graph model (SSGM) from Gaiselmann et al. [67] and Stenzel et al. [66]. The direct geometric tortuosities (black squares, red triangles) vary in a narrow range, which is close to the Bruggeman trend line. For the same samples, the indirect tortuosities (blue open circles) show much higher values and a stronger variation

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3.4.5 Example 5: Direct Medial Axis Versus Direct Geodesic Tortuosity

τdir_medial_axis versus τdir_geodesic from SOFC anodes and silica monoliths.

The relationship between two direct geometric tortuosities (i.e., geodesic, and medial axis tortuosities) is also investigated by Holzer et al. [11] and Pecho et al. [12] for SOFC anodes and by Hormann et al. [64] for silica monoliths. Figure 3.7 clearly shows that geodesic tortuosities (squares) are systematically lower than the medial axis tortuosities (triangles), even though they are measured for the same samples. Furthermore, the Bruggeman trend line can be roughly considered as the boundary between the characteristic fields for these two geometric tortuosity types. These findings are compatible with the results from Gaiselmann et al. [67] and Stenzel et al. [66] in Example 4 (Fig. 3.6).

Fig. 3.7
A scatter plot of tortuosity versus porosity. A fit is plotted through (2, 3) and (80, 1.05). The tortuosity plots are distributed on either side of the fit between 1 and 2.3. The direct medial axis tortuosities have the highest value.

Comparison of geodesic (τdir_geodesic) versus medial axis tortuosities (τdir_medial_axis) for SOFC anodes (Holzer et al. [11] and Pecho et al. [12]) and for silica monoliths (Hormann et al. [64]). The geodesic tortuosity (τdir_geodesic) typically shows values close to or below the Bruggeman trend line. In contrast, the medial axis tortuosities (τdir_medial_axis) show consistently higher values

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3.4.6 Example 6: Mixed Streamline Versus Mixed Volume Averaged Tortuosity

τmixed_ele/hydr_streamline versus τmixed_ele/hydr_Vav (mixed tortuosity types) for simulated particle packing (2D and 3D).

This example uses datasets with mixed tortuosity types, including the streamline tortuosity (τmixed_phys_streamline) and the volume-averaged tortuosity (τmixed_phys_Vav) for flow, conduction, and diffusion.

Saomoto et al. [55] simulated hydraulic flow and electrical conduction in simple 2D structures consisting of mono-sized circles and/or ellipsoids. Both mixed tortuosities (i.e., streamline and area-averaged) are extracted from the electric and hydraulic flow fields and their values are plotted in Fig. 3.8. Surprisingly the results for both mixed tortuosity types are almost identical (if considering results for one specific type of transport). For example, the electric streamline tortuosity (red square with crosses) is nearly identical to the electric volume-averaged tortuosity (blue square with crosses) (i.e., τmixed_ele_streamline ≅ τmixed_ele_Vav). The same holds for hydraulic tortuosities (i.e., τmixed_hydr_streamline ≅ τmixed_hydr_Vav). However, a significant difference is observed between electric and hydraulic tortuosities. The characteristic field for mixed electric tortuosities (i.e., τmixed_ele_streamline, τmixed_ele_Vav), which is highlighted in red color, is lower than the characteristic field for mixed hydraulic tortuosities (i.e., τmixed_hydr_streamline, τmixed_hydr_Vav), which is highlighted in blue. The boundary between these characteristic fields is roughly identical with the Bruggeman trend line.

Fig. 3.8
A scatter plot of tortuosity versus porosity. A fit is plotted through (7, 2) and (100, 1). The tortuosity plots are distributed on either side of the fit between 1 and 2. The plots for mixed hydraulic tortuosities are higher than the others. The characteristic field is shaded with various colors.

Comparison of different mixed tortuosities (τmixed_ele/diff/hydr_streamline, τmixed_ele/diff/hydr_Vav). In principle, the datasets for mono-sized spheres (and ellipsoids) from Saomoto et al. [55] (dataset 55, 2D) and from Sheikh and Pak [61] (dataset 61, 3D) show hardly a difference between streamline and volume averaged tortuosities. However, a systematic difference is caused by the underlying transport mechanism, in the sense that mixed hydraulic tortuosities (characteristic field in blue, and yellow crosses) are higher than the corresponding mixed electric tortuosities (characteristic field in red)

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Sheikh and Pak [61] (green crosses) reported diffusive volume-averaged tortuosities (τmixed_diff_ Vav) from 3D poly-dispersed spheres, which are close to the values of mixed electric tortuosities in Saomoto et al. [55], but lower than the mixed hydraulic tortuosities in [55]. This finding is compatible with the general expectation that diffusive and electrical transport properties and associated tortuosities are almost identical to each other and that the hydraulic tortuosities are generally somewhat higher, see e.g., Clennell [71].

Finally, results of volume averaged hydraulic tortuosities (τmixed_hydr_Vav) for gas diffusion layers (GDL) in PEM fuel cells are presented by Froning et al. [28] (crosses highlighted in yellow). Note that the microstructures of fibrous GDL considered by Froning et al. [28] are significantly different from those of poly-dispersed sphere packing in Sheikh and Pack [61]. Nevertheless, the characteristic values of τmixed_hydr_Vav for these two materials (GDL in [28] and packed spheres in [61]) are rather similar. This shows again that the tortuosity values are usually more strongly depending on the type of tortuosity and less strongly on the type of material and/or microstructure.

Overall, the results presented in this example show that the values of mixed tortuosities are relatively low (i.e., close to the Bruggeman trend line). Values larger than 2 are rarely observed and can only be expected for structures with either low porosities (ε < 0.2) or with strong anisotropy effects. Similar as observed previously for the various geometric tortuosities, also the mixed tortuosities show a relatively small scatter, and the values are usually in the range between 1 and 2 (i.e., roughly compatible with Carman’s estimation of √2). This is in remarkable contrast to the indirect tortuosities (not analysed in this example), which scatter over a much larger range and often take values much larger than 2.

3.5 Relative Order of Tortuosity Types

Fig. 3.9
A profile exhibits an overview of tortuosity-porosity data. It has the patterns for direct, mixed, and indirect tortuosities with their respective parameters and characteristic ranges. The direct geometric and mixed types have the lowest value, and the indirect hydraulic type has the highest value.

The review of tortuosity-porosity data from literature reveals a universal pattern that is characterized by a consistent relative order among the different tortuosity types. This pattern is observed for many different materials and microstructures. Geometric and mixed tortuosity types typically show relatively low values close to the Bruggeman trend line. In contrast, the indirect tortuosities show higher values with a larger variability

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3.5.1 Summary of Empirical Data: Global Pattern of Tortuosity Types

The empirical data from literature reveals a consistent pattern among the tortuosity types, in the sense that certain tortuosity types give consistently higher values than others. This relative order of tortuosity types, which is schematically illustrated in Fig. 3.9, is valid for a wide range of materials with very different microstructures.

Basically, the indirect tortuosities scatter over a much wider range than the direct geometric and the mixed tortuosity types. The direct and mixed types rarely take values greater than 2, whereas for the indirect tortuosities much larger values are measured,—in some cases even higher than 20.

For the geometric tortuosities, two subgroups can be distinguished. For medial axis and path tracking method (PTM) tortuosities, the Bruggeman trend line represents the lower bound. In contrast, geodesic and fast marching method (FMM) tortuosities typically show values that are equal or even lower than the Bruggeman trend line.

The values of mixed tortuosity types (streamline and volume averaged tortuosities) roughly overlap with the values for the direct geometric tortuosities (i.e., usually the mixed tortuosities are also close to the Bruggeman trend line—see e.g., Fu et al. [72]). Example 6 indicates that the mixed streamline tortuosities are identical with the mixed volume-averaged tortuosities, provided that the same transport mechanism is considered (see e.g., Saomoto et al. [55]). However, the mixed electrical, diffusional, or thermal tortuosities are consistently lower than the mixed hydraulic tortuosities. The Bruggeman trend line separates the two characteristic fields for mixed electrical/diffusional and for mixed hydraulic tortuosities (see Fig. 3.8).

3.5.2 Interpretation of Different Tortuosity Categories

3.5.2.1 Direct and Mixed Tortuosities

The direct and mixed tortuosities are based on geometric analyses and therefore they can be considered as true or realistic measures for the path lengths through the porous medium under investigation. Consequently, in order to predict the impact of microstructure on effective transport properties, it is not sufficient to merely consider the geometric or mixed tortuosities, since other morphological effects (e.g., bottlenecks/constrictivity) in addition to path length variation also have an influence on the effective transport properties. Predictions of effective transport properties based on distinct estimations of the path length effect (tortuosity), bottleneck effect (constrictivity) and viscous drag at pore walls (hydraulic radius) are extensively discussed in Chap. 5. Using tortuosity types that give a realistic estimation of the true path length is the key to understanding and differentiating different microstructure effects that limit the transport in porous media.

3.5.2.2 Indirect Tortuosities

The indirect tortuosities are derived from effective (or relative) properties that are known from experiment or simulation. The indirect tortuosities can thus be considered as a measure for the bulk microstructure resistance against transport. The large values and the large variability observed for indirect tortuosities are due to the fact, that they capture all different kinds of microstructure limitations, including resistive effects from narrow bottlenecks. Indirect tortuosities are thus not a realistic measure for the length of transport paths since they tend to overestimate the effect of pure path lengths significantly. Indirect tortuosities can also be interpreted as fudge factor that describes the ratio of relative transport property over porosity (i.e., √(σrel/ε) or √(Drel/ε)). For viscous flow, the indirect hydraulic tortuosity is rarely calculated, since viscous drag expressed by hydraulic radius makes calculation more complicated (see discussion in Chap. 2).

3.6 Tortuosity–Porosity Relationships in Literature

3.6.1 Mathematical Expressions for τ–ε Relationships and Their Limitations

Numerous mathematical expressions describing tortuosity–porosity (τ–ε) relationships can be found in literature. The different τ–ε relationships are reviewed by Shen and Chen [73], Ghanbarian [74], Tjaden et al. [75] and Idris et al. [76]. Table 3.2 represents a selection of mathematical τ − ε relationships from literature.

Table 3.2 Mathematical tortuosity—porosity (τ–ε) relationships from literature
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Note that very different mathematical expressions are proposed—usually logarithmic and power-law functions. As shown in Fig. 3.10a, the resulting τ–ε curves diverge greatly from each other. It may be argued that the observed variety results from the fact, that these τ–ε relationships are derived for different material types and different microstructures (e.g., battery electrodes, clays). However, most of these relationships are derived for packed spheres.

Fig. 3.10
A multi-line graph and 4 scatter plots of tortuosity versus porosity. a. 25 lines follow a decreasing trend between 1 and 10. b. The plots are distributed in a decreasing manner. c. The plots are distributed in an increasing manner. d and e. The plots are distributed on either side of the slope.

a Mathematical tortuosity–porosity (τ–ε) relationships from literature. The numbers refer to the equations given in Table 3.2. Colors indicate which type of ‘physics-based’ tortuosity is considered. It must be emphasized that in the source papers it is usually not indicated, whether these are indirect, mixed, or direct tortuosities, b, c, d and e plots of tortuosity (τdir_geodesic,), constrictivity (β), and relative electric conductivity (σrel) versus porosity (ε, on the x-axis). Red, blue, and black data points represent three distinct types of 3D microstructures, which are generated with different stochastic models. More than 2000 different 3D structures are investigated, which cover a wide range of microstructure characteristics (Figs. 3.10b, c, e are taken from Neumann et al. [77], and Fig. 3.10d is from Stenzel et al. [78])

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The observed chaotic picture of mathematical expressions (Fig. 3.10a) contrasts with the empirical data, which shows a clear pattern when plotted separately for different tortuosity types, as summarized in Fig. 3.9. Moreover, the empirical data typically results in characteristic τ–ε fields (as shown in Fig. 3.2 for the different tortuosity types), but usually it does not result in clearly defined τ–ε curves. This is particularly not the case when plotted for specific material types (see Fig. 3.1) and thereby not distinguishing the involved tortuosity types.

The large scatter for the mathematical expressions in Fig. 3.10a thus illustrates that there is no consensus how tortuosity varies with porosity. In fact, this finding is questioning the underlying assumption that variations of tortuosity and porosity are strictly related, so that the τ–ε relationship could be described with one universal mathematical formula.

The behaviour of microstructure characteristics with varying porosity can be investigated in a more general way, when 3D image analysis is applied to a large number of 3D microstructure models. Results from such studies are shown in Fig. 3.10b–e (taken from Neumann et al. [77] and Stenzel et al. [78]). It must be emphasized that these investigations are using only one specific tortuosity type, which is geodesic tortuosity. These data sets indicate that the microstructure characteristics (ε, β, τdir_geodesic) can vary in a large range within the theoretically possible constellations, except for high porosities, where tortuosity asymptotically tends to 1 (see Fig. 3.10d). For small porosities the variation of geodesic tortuosity is particularly large, but also at very high porosities (ε > 0.8) there is still significant variation in the tortuosity values (Fig. 3.10d). A similar behaviour is expected for all the other direct geometric and mixed tortuosities. These examples thus indicate that there exists no clear correlation between porosity and tortuosity (and associated effective path length, respectively), which could be described with a universal mathematical τ–ε expression. As shown in Fig. Fig. 3.10e, there exists also no strict correlation between relative conductivity and porosity. Hence, it is not justified to expect a strict correlation between indirect tortuosity and porosity, since the first one is derived from relative conductivity.

3.6.2 Mathematical Expressions for τ–ε Relationships and Their Justification

Nevertheless, mathematical expressions for tortuosity–porosity (τ–ε) relationships may have their justification in context with specific cases of controlled microstructure variation. Such a case was described by Archie [79], who presented experimental data for porous sediments saturated with an electrolyte. For the special case, where all samples originate from the same sedimentary unit, the experimental results show a correlation between electric resistance and porosity. Archie’s law (see Eqs. 2.22 and 2.23: FR = ε−m = 1/σrel) describes this correlation using a so-called cementation factor (m) as exponent. Wyllie and Rose [80] redefined the relationship between electric resistance (formation factor, FR, respectively) and porosity by introducing the so-called structural factor (Eq. 2.24: FR = Xele/ε = 1/σrel). Thereby, Xele can be interpreted as being equivalent with Carman’s tortuosity factor (Xele = T = τele2). This leads to Eq. 2.25 (i.e., τindir_ele2 = ε/σrel), which is widely used to compute indirect tortuosity. Thereby, indirect tortuosity can be considered as some kind of proportionality between porosity and effective properties (e.g., transport resistance or conductivity). For the special case where effective or relative properties (FR, σrel) correlate with porosity, it follows that there must be also a strict correlation with indirect tortuosity, such that Archie’s law can be reformulated as follows:

$$\tau_{indir\_ele}^{2} = \varepsilon^{1 - m}$$
(3.2)

It must be emphasized that Archie’s cementation factor (m > 1) is a fit parameter, which is valid only for a special case of microstructure variation. The cementation factor takes a specific value for a series of rocks, which all have undergone the same diagenetic process (i.e., solidification process transforming loose sediment into a solid rock). The common history of these sediments in the same geo-environment led to a characteristic variation of the microstructure, so that sizes of pores and bottlenecks, pore connectivity and transport path lengths vary in a characteristic way with porosity. Hence, in this case the correlation of microstructure with porosity is controlled by the diagenetic conditions and by the associated cementation process. Due to this controlled correlation, it is possible to find a suitable mathematical expression for the τ − ε relationship of rocks in a single sedimentary unit. Thereby the indirect tortuosity represents the lumped sum of all microstructure effects. With respect to Archie’s law, it must be realized that the fitting of the cementation factor (m) has to be repeated if the rocks originate from a different sedimentary unit, because then these rocks were exposed to different diagenetic conditions, and therefore they are characterized by different τ − ε relationships.

Such special cases are also known from materials engineering when microstructure variations are performed in a controlled way. An example is described by Holzer et al. [19], where sintering temperatures are changed systematically, but all the other parameters (e.g., composition and sinter time) for the fabrication of porous ceramic membranes are kept constant. This results in a systematic variation of microstructure characteristics (τdir_geodesic, β) and effective properties (σeff, σrel) with porosity (ε). Hence, for such controlled microstructure variation a suitable mathematical expression can be found for the observed τ − ε relationship. However, this mathematical expression may no longer be valid, when one of the other fabrication parameters (e.g., composition or sinter time, grinding and particle size distribution of ceramic powders) is changed.

The Bruggeman equation is a special case of Archie’s τ − ε relationship (see Eq. 3.2), where the ‘cementation factor’ (m) is fitted for a special microstructure type. For example, it was found that m = 1.5 for packed spheres (and m = 2 for cylindrical particles), which leads to

$$\tau_{indir\_ele}^{2} = \varepsilon^{ - 0.5} .$$
(3.3)

In battery research, a modified version of the Bruggeman equation, given by

$$\tau_{indir\_ele}^{2} = \gamma \varepsilon^{1 - \alpha }$$
(3.4)

is nowadays often used (see Thorat et al. [96]), where γ is an additional fit parameter. Various authors [96, 105,106,107,108,109,110] presented experimental and numerical fits of Eq. 3.4 (e.g., Nrs 27–30 in Table 3.2) for different battery electrodes, which was critically reviewed by Tjaden et al. [75]. Thereby it is well illustrated that the variations of γ (0.1–2.6) and α (1.27–5.2) are very large and the resulting τ − ε curves differ significantly from each other, depending on the type of battery material. In addition, Tjaden et al. [75] also report examples from literature, which show that for many battery materials the Bruggeman equation and its modifications are simply not applicable. Tjaden et al. [75] thus concluded that tortuosity-porosity relationships such as the Bruggeman equation are only applicable to microstructures ‘which are similar to the microstructure used to derive the respective relationship’ (i.e., for special cases).

These selected examples related to Archie’s and Bruggeman’s equations illustrate that τ − ε relationships should not be mistaken as universal laws. In general, when varying materials microstructures are considered, the different microstructure characteristics (ε, β, τ, rh, rmin, rmax) can vary independently from each other,—at least to some degree. Therefore, empirical data shows significant scatter of τ − ε datapoints for different material types (Fig. 3.1) and also for different tortuosity types (Fig. 3.2). A similar scatter of datapoints is also observed for other morphological characteristics (e.g., constrictivity) and effective properties (see Fig. 3.10b–e). The empirical data illustrate that, in general, there exists no simple and clear correlation between porosity and the other relevant microstructure characteristics (tortuosity, constrictivity, hydraulic radius).

Hence, we conclude that mathematical formulas for tortuosity-porosity relationships are valid only when the following special conditions are fulfilled:

  1. (a)

    τ − ε relationship is defined for a specific type of tortuosity.

A suitable classification scheme and associated nomenclature are given in Sect. 2.6 (see Fig. 2.8). The empirical data show that the scatter of data points is generally much smaller for direct geometric and mixed tortuosity types (Fig. 3.2b–e), compared to the indirect tortuosities (Fig. 3.2a).

  1. (b)

    τ − ε relationship is defined for a specific type of material and microstructure.

Thereby, the materials under consideration fulfil a systematic microstructure variation (e.g., rocks from the same sedimentary unit, which all had similar conditions during sedimentation and diagenesis). For simple microstructures such as packed spheres it is more probable that microstructure variation (i.e., densification) results in a systematic correlation between tortuosity and porosity (Fig. 3.1e, f), compared to more complex microstructures (e.g., SOFC electrodes, fibrous materials, foams), which tend to show a larger scatter (Figs. 3.1g, h) of τ-ε couples.

3.7 Summary

Nowadays, there are numerous methods available for the characterization of tortuosity in porous media. Despite the progress in characterization, there still exist many controversies, misconceptions, and confusions about tortuosity, which mainly come from the fact that the awareness for systematic differences between tortuosity types is missing. Hence, in many studies and discussions, the different tortuosity types are not clearly distinguished and addressed. As a first step toward solving this problem, we proposed to use a clear terminology. For this purpose, a new classification scheme and a new tortuosity-nomenclature were introduced in Chap. 2 (see e.g., Fig. 2.8). As a second step, the nature, and the extent of the systematic differences between the various tortuosity types need to be documented and illustrated. In order to investigate these differences, a large collection of empirical data from 69 different references was presented and analysed in this chapter (see Table 3.1).

Based on the collection of empirical data from literature, plots of tortuosity (τ) versus porosity (ε) are presented for different classes of materials and microstructures. For simple microstructures such as monosized sphere packings, the τ − ε plot shows a characteristic field that defines a relatively narrow band close to the Bruggeman trend line. However, for most material classes, which have more complicated, disordered, and stochastic microstructures (such as fuel cell electrodes, foams, rocks, and soils), the corresponding characteristic fields in the τ − ε plots are expanded over much wider domains. In addition, the characteristic fields for these materials classes are strongly overlapping. This overlap is observed even for material classes with significantly different microstructural architectures (e.g., cellular foams, fibrous textiles or sintered ceramics fabricated from powders). Obviously, the different materials and microstructures cannot be distinguished easily based on their τ − ε characteristics.

For the same collection of empirical data, τ − ε characteristics are replotted, but this time separately for the different types of tortuosities. Thereby, systematic differences can be observed among the main tortuosity categories: The indirect tortuosities show relatively high tortuosity values (typically higher than 2 and sometimes up to 20) and their τ − ε characteristics scatter over a wide range. In contrast, τ − ε characteristics of the direct geometric tortuosities and of the mixed tortuosities are typically concentrated in a narrow band close to the Bruggeman trend line (i.e., τ < 1.5 for ε > 0.3).

Systematic differences are also observed between tortuosity types within the same tortuosity category. For example, within the mixed category, the hydraulic tortuosity is consistently higher than the electrical and diffusional tortuosities. Similarly, within the category of direct geometric tortuosities, the medial axis tortuosity is consistently higher than the geodesic type. The τ − ε plots of empirical data thus document a relative order among the different tortuosity types. This consistent pattern is graphically illustrated in Fig. 3.9.

Hence, a consistent τ − ε pattern is observed when the empirical data is plotted for different tortuosity types. However, when the same data is plotted for different materials and microstructures, the corresponding pattern of the characteristic fields is less clear. It follows that the tortuosity value that is measured for a specific material, is much more dependent on the type of tortuosity than it is dependent on the material and its microstructure. This illustrates the need for a clear distinction between the different tortuosity types, including the need for a careful selection of a suitable method and the use of a clear terminology (i.e., nomenclature).

Based on the detailed description of the underlying definitions (see Chap. 2) and based on the documentation of the characteristic τ − ε patterns (present chapter), the main characteristics of the three main tortuosity categories can be summarized as follows:

The direct tortuosities (geodesic, FMM, medial axis, PTM, etc.) as well as the mixed tortuosities (streamline and volume-averaged) are based on geometric analyses and therefore, they provide realistic estimations of the true path lengths. The differences among these tortuosity types are relatively small, and they reflect the existing variations of the underlying geometric and methodological concepts.

The indirect tortuosities are derived from effective (or relative) properties that are known from experiment or simulation. The indirect tortuosities can thus be considered as a measure for the bulk microstructure resistance against transport, which includes also other resistive effects such as the bottleneck effect. Indirect tortuosities should therefore not be misinterpreted as a realistic measure for the length of transport pathways, but they should be rather considered as a bulk factor or fudge factor, which describes the ratio of relative transport property over porosity (e.g., τindir_ele = √(σrel/ε)).

Finally, the empirical data also illustrates that tortuosity is not strictly bound to porosity. As the pore volume decreases, the more scattering of tortuosity values can be observed. Consequently, any mathematical expression that aims to provide a generalized description of τ − ε relationships in porous media must be treated with caution (especially in cases without specification of the corresponding type of tortuosity).