© The Author(s), published by EDP Sciences, 2024

Notations and vocabulary:

The vocabulary used in the article is explicitly given. Knowing the confusion in the literature about geometric, geodesic or morphological tortuosities, it is necessary to clearly define the terms used in this article. Hereafter, the term geometric is chosen to qualify the tortuosity assessed by means of geodesic and Euclidean distance transforms.

The article defines notations and groups them based on their referents to improve readability. First, the notations associated with microstructures:

X: the porous microstructure,

l, w and h: primary grain parameters defining a platelet, used for simulating microstructures with a Boolean model,

V_v: volume fraction of primary grains of a homogeneous Boolean model,

V_a, R_a, V_i and V_o: heterogeneous Boolean model parameters formed of spheres, named aggregates, with different volume fractions of grains inside and outside,

n_r: number of realization of a model of microstructure,

σ: standard deviation of measurements over the n_r realizations,

l_σ: confidence intervals of measurements over the n_r realizations.

Notations for distance:

D_E: Euclidean distance,

D_G: geodesic distance,

L: Euclidean distance between two opposite planes of digitalized cubes of materials,

D_E(p_m, p_n): value of the Euclidean distance map at point p_m from p_n,

D_G(p_m, p_n; X): value of the geodesic distance map in X at point p_m from p_n,

V_D(n): volume of the direct reconstruction of X at step n,

V_G(n): volume of the geodesic reconstruction of X at step n.

Notations for tortuosity:

τ: geometric tortuosity of a path,

τ_g: geometric tortuosity of X considering the minimal geodesic path between two opposite planes,

τ_m: geometric tortuosity of X considering the mean value of the geodesic paths between two opposite planes,

τ_P: geometric tortuosity of X considering the Peyrega et al. definition,

τ_B: geometric tortuosity of X considering the Berrocal et al. definition,

τ_G: geometric tortuosity of X considering the Gommes et al. first definition,

: geometric tortuosity of X considering the Gommes et al. second definition.

Notations for M-tortuosity:

S₁: a set of points p_n in X,

: a set of points associated to p_n.

τ_n,m: geometric tortuosity between the two points p_n and p_m of X,

L_n: local insight of X from p_n,

T: representative tortuosity or M-tortuosity of X.

1 Introduction

Structural analysis of materials is a common issue in materials science, being of particular interest since connections have been highlighted between the structure and the physicochemical parameters of these materials, especially for porous media [1–3]. Among the usual concepts used for characterization purposes, tortuosity [4] lies on a central position in materials science [5–7]. Tortuosity has been reviewed many times from distinct perspectives. Clennell [5] presents the first exhaustive review focusing on physical meanings. Ghanbarian et al. [7] propose an update while discussing the relationship with percolation theory. Fu et al. [6] describe the various algorithms for physical assessment of tortuosity. Among the various notions extracted from the concept of tortuosity, the geometric tortuosity has a central position, having the potential to be used in numerous applications dealing with transport properties of materials, such as diffusion analysis [8, 9]. The geometric tortuosity is connected to several concepts such as percolation, connectivity, sinuosity and constrictivity [10–12]. In practice, the aim is to extend geometric tortuosity from a single pore to the characterization of the entire microstructure. The flow under consideration is often assumed to propagate in a given direction, simplifying the analysis of the microstructure as a whole. Therefore, such a tortuosity appears to be a well-suited notion for characterizing digitalized cubes of materials (see Fig. 1a). Nevertheless, its implementation required some adaptations in cases as irregular pieces of materials as γ-alumina samples [13] (see Fig. 1b), for which the definitions in the literature are not applicable.

Fig. 1 Microstructures illustrations. (a) Boolean schemes of spheres generated in a cube where straightforward definition of propagation direction for tortuosity calculations is possible. (b) real γ-alumina sample, imaged by electron tomography (resolution around 1 nm·vx−1, vx standing for voxel), where a proper propagation direction is delicate to define. Volume (a) was generated using [35] and volumes (a) and (b) were rendered using [35].

The M-tortuosity was introduced as an efficient and fast way breaking free directional aspects of previous approaches to address the challenge by stochastically probing microstructures [14, 15]. The M-tortuosity answers practically to the requirement of application to any complex microstructures as the microstructure presented in Figure 1b. Nevertheless, a practical need of versatility subsists for this descriptor in reflecting more or less some microstructural properties, especially those related to percolation and sinuosity. Furthermore, no study has discussed the relevance of M-tortuosity for practical purposes, unlike the standard definitions used in various applications [8, 9, 16]. Additionally, there are no articles that compare the chosen classical approaches with each other.

Consequently, our contribution is a novel version of the M-tortuosity definition. We introduce new parameters that promote or inhibit geodesic paths based on their length and eccentricity. We propose an alternative version of the M-tortuosity that uses arithmetic means instead of harmonic means, simplifying the original formulation. These novel definitions ensure proper disconnection handling. Furthermore, our new perspective on microstructure analysis is compared to state-of-the-art definitions, as shown in Figure 2. This positions our descriptor in relation to others while emphasizing its potential for analyzing transport properties in atypical contexts. Finally, to the best of our knowledge, this is the first time that different purely morphological definitions of tortuosity are compared.

Fig. 2 Strategies to extend geometric tortuosity from paths to microstructures. Illustrations of various ways to characterize whole microstructures based on geometric tortuosity definition (Eq. (8)). (a) path of minimal length connecting entry (green) and exit (red) planes (Eq. (2)), (b) average of paths connecting entry and exit planes (Eq. (3)), (c) point tortuosities, i.e., tortuosity map, assessed by forward and backward propagations between two opposite planes (Eq. (4)), (d) M-tortuosity averaging local tortuosities between random locations.

In this article, state-of-the-art definitions of tortuosity are briefly recalled, all based on the definition of the geometric tortuosity making use of well-known concepts as accessibility, connectivity and sinuosity. The original definition of the M-tortuosity is defined and its extension is proposed in the following section, after discussing limitations and challenges related to standard tortuosities. In the results and discussion section, stochastic multi-scale microstructure models of binary materials are considered and analyzed to compare the different tortuosimetric descriptors. The M-tortuosity behaves similarly to the other definitions when faced to the evolution of morphological features as volume fraction, making it a potential equivalent of the latter when they do not fit to the material to be analyzed.

2 Tortuosities

Geometric tortuosity is based on several well-known concepts in numerical characterization of materials. Accessibility is described by the percolation theory, as it assesses the existence of a path, fully included in the microstructure, connecting two regions, usually the entry and the exit. Connectivity follows accessibility and focuses on quantifying the degree of interconnection of the network, i.e. the microstructure. Usually, the Euler number, or the Euler-Poincaré feature, is considered to assess connectivity. Sinuosity is the last notion involved in tortuosity, evaluating how sinuous a path is, which could be seen as the integral of the local curvature along the path.

2.1 Morphological insights in material science

Geometric tortuosity characterization consists in assessing the impact of a microstructure’s morphology over percolating paths. Originally, geometric tortuosity is defined for a path, i.e., a pore, connecting two specific locations, entry and exit, then focusing on sinuosity only [17]. The definition used in [18], based on rotation-invariant operators such as distance transforms, is defined as the ratio of the geodesic distance, i.e., constrained path, by the Euclidean distance between the entry and the exit. In other words, the geometric tortuosity of a unique pore is the normalization of the geodesic distance by the Euclidean distance,(1)

Consequently, the issue to address is how to extend this local definition in order to characterize a whole microstructure X with its sinuosity and connectivity. Decker et al. [18] proposes to analyze the probability distribution function of the geodesic distances over the network between parallel faces. The issue of reducing this distribution to a scalar value is then addressed.

The first idea is to assess the minimal geodesic distance of a constrained path starting from an entry, a plane for instance, to the exit, the opposite plane, normalized by the Euclidean distance between the two planes L (see Fig. 2a). In this case, a propagation direction is clearly defined,(2)with D_G the geodesic distance map at the exit plane.

Nevertheless, considering only the minimal path can be too restrictive; computing the average is more representative of the microstructure, giving rise to this second definition (see Fig. 2b),(3)

These previous definitions take implicitly pore width into consideration, i.e., two very similar pores, one being a slightly bigger version of the other, have distinct tortuosity as shown in Figure 3. The use of homothopic skeleton Sk [19], defining geometric tortuosities according to Stenzel et al. [16] in contrast to geodesic tortuosities computed over the “transporting phase”, leads to invariance with respect to the scale or to the size of a moving particle percolating through the porosity (see Fig. 3). Moreover, the skeleton provides a less scattered distribution of distances, as shown in [14].

Fig. 3 Geodesic paths according to the pore width. Illustrations of the dependency of the geodesic path (blue dashed lines) to the pore width, contrary to the skeleton (orange lines) being the same for the two pores despite their distinct width.

However, no map of tortuosity can be extracted explicitly from these definitions. Indeed, if we consider the geodesic map of the whole structure divided by L, only the values of the exit plane correspond to proper tortuosity values. Peyrega et al. [20] propose a distinct definition, considering forward propagation from a plane A and backward propagation from an opposite one B (see Fig. 2c), in order to provide tortuosity of each point p of the microstructure,(4)with the geodesic distance map in the direction A toward B, A being the entry or the source plane.

This definition increases the amount of information while providing a tortuosity map, analyzed by means of histograms for each of the three propagation directions. This tortuosity map has the advantage to be constrained by the orientation only, i.e., the axis, as forward and backward propagations are performed.

However, these definitions of tortuosity based on a propagation direction between opposite planes and an approximation of the Euclidean distance (L), are sensitive to relative deviation. In other words, a tortuosity higher than 1 will be attributed to a straight leaning pore, whose tortuosity value will depend on the angle of inclination with respect to the propagation direction. In order to tackle this issue, a return to point definition seems necessary, which is done in the following definition.

In fracture analysis, Berrocal et al. [21] propose to weight the tortuosity of a fracture, i.e., a path, by the geodesic distance in order to gather several tortuosities in a final mean value promoting sinuous paths. The consideration of the original definition (Eq. (8)) leads to invariance with respect to relative deviation.(5)

Distinct interpretations of tortuosity have been proposed in [22], considering a propagation direction between opposite planes but the real Euclidean distance D_E. Tortuosity is first seen as the limit toward infinity of the original ratio (Eq. (8)),(6)

Practically, τ_G is assessed by the initial slope of a parabolic fit of D_G vs. D_E, both computing from a source plane. A second definition considers morphological reconstruction, direct and geodesic, to define tortuosity as a ratio of volumes; volume of the direct reconstruction V_D by volume of the geodesic reconstruction V_G,(7)with n the step in the reconstruction processes.

The main advantage of this second definition where Euclidean and geodesic quantities are reversed to provide a value upper than 1 [23], is its straightforward extension to gray-scale images, avoiding a delicate step of segmentation. These definitions give access to tortuosity maps too, as geodesic and Euclidean values are computed for each point, but constrained by the propagation direction.

Consider the primary application target of this article: the 3D images of irregular samples of real materials, such as the one presented in Figure 1b. Directional definitions are not adapted to this kind of images, samples being too small to extract a representative cube from them. To this end, a pore-to-pore tortuosity map is proposed by Moreaud et al. [24], by identifying the different entries of the porous microstructure, to assess the material’s accessibility while extracting tortuosity matrices quantifying each entry in relation to the others.

2.2 Limitations and challenges

Despite these various definitions, none can be applied on both microstructures of Figure 1: digitalized cubes of materials and irregular small samples of materials. In the context of characterizing the microstructure of Figure 1b, only the definition of [24] can be used for practical purposes, but cannot be compared to the other definitions which are commonly used in physicochemical analyses. Recently, the M-tortuosity overcomes this issue (see Fig. 2d) [14], providing the ability of characterizing irregular samples of real materials to reach internal nanometric porosity analysis, as shown by Batista et al. [25]. In contrast to the state-of-the-art definitions presented above, the M-tortuosity is the only definition applicable on both microstructures of Figure 1. In what follows, we first recall the original definition of M-tortuosity, before proposing an extended definition of this notion.

3 M-tortuosity; original definitions

Initially, the M-tortuosity has been defined to overcome the complexity of materials such as γ-alumina sample in Figure 1b [14, 25], by probing the microstructure in random directions and at various scales. Its principle is the following; several locations are probed, their local insights are gathered into a final representative viewpoint of the whole microstructure. In the literature, the M-tortuosity and the H-tortuosity are defined, providing scalar value and curve as final representative viewpoint of the microstructure, respectively. Both are based on the same mathematical formulation [14, 15].

3.1 Stochasticity for complexity overcoming

If we consider the material of Figure 1b, entry and exit planes are not available. To overcome this pitfall, a random drawing of locations, i.e. points inside the microstructure X, is considered [26]. The N probed locations are defined randomly through a stratified sampling, ending to the points set definition . Consequently, the tortuosity is computed between points instead of planes. Considering two connected points in X p_n, the source point, and p_M, the tortuosity is,(8)with D_E(p_m, p_n) the value of the Euclidean distance map at point p_m from p_n, and D_G(p_m, p_n; X) the value of the geodesic distance map in X at point p_m from p_n. For brevity, and .

3.2 Second set and tortuosity dimension

The local insights are based on S₂ a collection of sets of points, each assigned to a point p_n ∈ S₁, and . For the M-tortuosity, giving rise to a scalar local insight L_n (the M-coefficient in [14]). For the H-tortuosity, with representing the surrounding neighborhoods of p_n as a function of the Euclidean distance d, M being the number of accessible points located at a distance of d from p_n, leading to a curve as local insight L_n (the H-coefficients in [15]).

In other words, for each p_n, the M-tortuosity considers the geodesic paths to all the other points of S₁, whereas the H-tortuosity simulates a wave propagating from p_n in the microstructure. These local insights L_n are then gathered into the final representative tortuosities T (M-scalar or H-scalars), which keeps the dimension of L_n.

3.3 Weighted averages

The local insights L_n and the final representative tortuosities T are defined by weighted average, more specifically the harmonic mean, defined by the reciprocal of the arithmetic mean of the reciprocals of the values, which is an ideal candidate for disconnections handling as well as isolation. Disconnection arises when no path in X exists between two given points, and the geodesic distance of two non-connected points of X is infinite. With harmonic mean these two points do not interfere in the computation, leading to zero contribution. Isolation is the extension of disconnection and arises when there exists no point in a subset connected to a given point p_n.

The weights considered in L_n and T both promote certain geodesic paths of the whole microstructure. Consequently, the point tortuosities values τ_n,m in L_n are weighted by the corresponding geodesic distance , and the L_n values in T are weighted by the Euclidean distance between p_n and p_c, the center of inertia of X. The latter weighting is based on the following idea, the eccentricity of p_n is connected to the representativity of its local insight L_n.

3.4 Alternative definitions

To improve the computational efficiency, a graph-based definition is given to the M-tortuosity by Hammoumi et al. [27]. Deterministic definitions are proposed in [25] by imposing S₁ and S₂ to meet the specific application goal of this article. S₁ is composed of the centers of inertia p_n of particles inside the porous medium and , leading to analyze the relative locations of particles with respect to the microstructure constraint. Finally, the A-protocol defined in [28] and embedded the M-tortuosity, gives rise to accessibility consideration by simulating a probe of given size travelling through the network [14, 15].

4 M-tortuosity; extended definition

M-tortuosity and H-tortuosity are gathered into a single manifold tortuosity called M-tortuosity. This extended definition is enriched of new parameters, correcting some issues in the original definition while increasing the versatility. The first parameters named α allow to reach proper promotion of long geodesic paths among other things and provide the user the opportunity to adapt the definition to its own application by controlling the promotion of certain paths. The second parameters, named ρ and embedding percolation, lead to proper disconnection and isolation insensitivity, which is a major distinction with the state-of-the-art definitions. Consequently, we consider the arithmetic mean as an alternative to the harmonic one. Both are defined.

4.1 Local insights

M-tortuosity versatility relies on S₁ and S₂ definition. S₂ has a key role in the very meaning of the resulting descriptor, its structure inducing the type of the operator, as shown above. Formally speaking, the collection S₂ is a set of rank s defining the dimension of the local insights being equal to s − 2. This rank to three, resulting in a local insight of dimension one or zero, fitting with the literature definitions.

Consequently, keeping the previous notations, and S_d = {p_m} but d is no longer necessarily a distance, for generalization purposes. For n ∈ ⟦0, N − 1⟧, local insights set L_n attached to each p_n ∈ S₁ are defined as a function of d using the harmonic or the arithmetic mean of the geometric tortuosities between p_n and each p_m. Then, for all d ∈ ⟦0, D − 1⟧, using the harmonic mean,(9)with,

(10)and using the arithmetic mean,(11)with,(12)The power factor parameter α₁ ∈ ℤ is one of the additional parameters, modifying the original weighting; the larger |α₁|, the more the longest (α₁ > 0) or shortest (α₁ < 0) geodesic paths are promoted with the arithmetic definition (to be reversed with harmonic definition). The Boolean matrix ρ₁ allows to reach the disconnections insensitivity. is equal to 1 if and only if there exists a path connected p_n ∈ S₁ to p_m ∈ S₂.

4.2 Representative tortuosity

The representative viewpoint T of the whole microstructure is defined for all d ∈ ⟦0, D − 1⟧, considering p_c and the second power factor α₂ ∈ ℤ, by(13)with,

(14)if the harmonic mean is considered, and if the arithmetic one is used, by,(15)with,(16)

The second power factor α₂ extends the original definition, the latter corresponding to α₂ = 1; the larger |α₂|, the more the furthest (α₂ > 0) or nearest (α₂ < 0) points from p_c are promoted with the arithmetic definition (to be reversed with harmonic definition). The Boolean vector ρ₂ allows to reach the isolations insensitivity. is equal to 1 if and only if there exists a path connected ρ_n ∈ S₁ to any ρ_m ∈ S₂.

This new formulation gathers the previous definitions. For a scalar analysis, is a points set, then d = {0} and . For curve analysis, is a collection of sets of points, then d > 0 and . In the case of [15], d is a distance, D is the maximal distance from each source point for microstructure probing, and M is the number of points in X accessible from a given source point p_n of S₁ and located at distance d from it.

4.3 Versatility for broad applications

In the original definition, the isolation insensitivity was not properly reached. The weight at the numerator associated to an isolated point p_n ∈ S₁(L_n = 0) is still taken into account. This statement suggests that the M-tortuosity is sensitive to isolation. fixes this issue by being equal to 1 if and only if p_n is not isolated. Our contribution improves the versatility of the M-tortuosity thanks to two additional parameters pairs, denoted as and , while preserving the benefits of the original definitions. The first element of each pair highlights the consideration of percolation using Boolean values and the second element being power factors serving for path length and excentricity promotions, respectively. This novel definition is properly insensitive to disconnections and to isolations. To highlight the increase of verstatility, two aspects are discussed.

First, let consider Figures 1a and 1b. In the case of scalar tortuosimetric characterization, by imposing α₂ > 0, long geodesic paths are promoted a second time as points closer to the image boundaries have a higher probability to get longer geodesic paths in their local insight than points close to p_c. In contrast, in case of characterization using tortuosity curves, α₂ < 0 promotes points closer to p_c being less impacted by the boundary effect. As a conclusion, the α₂ value allows to fit with the application.

Second, the harmonic mean provides lower values than the arithmetic one. The choice between the two means can be motivated by the application. Indeed, the harmonic mean, usually used when rates and ratios are involved, brings us closer to the acoustic tortuosity definition [29, 30]. Indeed, distance values can be interpreted as time values instead of spatial values.

5 Results and discussion

This section deals with binary microstructures analysis by means of tortuosimetric numerical operators, focusing on scalar characterization only. Moreover, all computations are performed over the whole microstructure, without skeleton preprocessing. First, the M-tortuosity is analyzed and the harmonic and arithmetic definitions are compared while the impact of the parameters (α₁, α₂) is assessed. Second, the comparison to the classical descriptors is performed on synthetic microstructures. The synthetic microstructures are generated from random model as described below.

5.1 Random models of porous microstructures

Boolean models generates homogeneous microstructures, i.e., structure possessing a unique scale of grains’ density, made of isotropic or anisotropic grains A’, located at Poisson points [31]. They are defined by a single volume fraction of grains V_v and by the grains’ morphology. In order to analyze heterogeneity, multi-scale microstructures are generated by using Cox models [32, 33], providing materials with aggregates or inclusions, i.e., areas of higher or lower density of grains. In the following, two-scale models are defined by three volume fractions: V_a the volume fraction of aggregates, V_i and V_o, the volume fraction of grains inside and outside the aggregates, respectively. Aggregates are defined as spheres of radius R_a. A unique grain’s morphology is considered for the homogeneous and heterogeneous cases, the platelet shape [34], being defined by three parameters, length l, width w and height h, all equal to 10 (units are in pixels). For each model, 40 realizations, i.e., volumes, of size 400³ are generated. Figures 4a and 4b show the shapes of the platelets and the aggregates, respectively.

Fig. 4 Synthetic microstructures illustrations. (a) The grains shape (cubic platelet) and (b) the aggregate shape. Volume representation and 2D slice of a realization of Boolean models presenting the two isotropic cases, homogeneous: (1) Vv = 0.4 and (3) Vv = 0.6, and heterogeneous: (5) Vi = 0.65 and Vo = 0.15, and the anisotropic case: (1’) Vv = 0.4. Volumes and slices generated and rendered using [35].

Tortuosity as a function of the volume faction of homogeneous microstructures is first considered; 3 models ((1)–(3)) are generated, represented by the three first columns in Table 1 (Fig. 4, two first microstructures). Second, aggregation impact on tortuosity is assessed by two additional heterogeneous microstructures, named (4) and (5), represented by the two last columns in Table 1 (Fig. 4, third microstruture). The homogeneous models are then transformed into anisotropic models, named (1’), (2’) and (3’), by simulating a compression of the materials in the x direction by the suppression of one plane over two (Fig. 4, last microstruture).

Figure 4 illustrates the models by displaying the shapes of grains and aggregates, together with some realizations of the models, volumes and slices, highlighting the considered situations: increasing of volume fraction of grains V_v, increasing of the heterogeneity and impact of the anisotropy. The complementary set of the grains set represents the porosity (black areas in Fig. 4). Let V_p be the porous volume fraction, being define in the homogeneous case by V_p = 1 − V_v.

Confidence intervals with 95% confidence level are equal to with σ the standard deviation over the n_r realizations. Finally, in this case, the tortuosimetric analysis provides averaged assessments. Let τ(M), be a given descriptor value for a given model (M), τ(M) is the averaged value over all realizations of the set of tortuosity values of each realization.

5.2 M-tortuosity

Considering the M-tortuosity, the focus is on the scalar version of the representative tortuosity, named , by defining and S₁ being drawn randomly by a stratified stochastic process as in [14]. The impacts of (α₁, α₂), as well as the choice of the mean, are analyzed; α₁ = α₂ = {−10, −5, −2, −1, 0, 1, 2, 5, 10}. The models (3) is considered in Figure 5, for these purposes.

Fig. 5 M-tortuosity as a function of α1 and α2. Average M-tortuosity values with their corresponding confidence intervals lσ, computed on 40 realizations of model (3). Screening of parameters α1 = {−10, −5, −2, −1, 0, 1, 2, 5, 10} and α2 = {−10, 0, 10}. (a) Harmonic means representative tortuosities TH. (b) Arithmetic means representative tortuosities TA.

As expected, the harmonic version of the M-tortuosity (Fig. 5a) provides slightly smaller values than the arithmetic one (Fig. 5b), but this difference is negligible when compared to the differences with the state-of-the-art definitions. Moreover, the impacts of α₁ and α₂ are reversed between the harmonic and the arithmetic definitions. When α₁ tends towards infinity the long geodesic paths are promoted in the arithmetic M-tortuosity (short geodesic paths with the harmonic M-tortuosity), which is reversed if α₁ tends towards minus infinity. When α₂ tends towards infinity the source points the furthest of p_c are promoted (the nearest of p_c with the harmonic M-tortuosity), which is reversed if α₂ tends towards minus infinity. Globally, the short geodesic paths promotion makes increase the tortuosity, but the confidence interval l_σ too. This comment is in good agreement with the results of Chaniot et al. [15], who highlight that larger tortuosities are obtained for short paths.

For the rest of the analysis, we will considered the arithmetic M-tortuosity only.

5.3 Tortuosimetric analysis

Some definitions are adapted to perform the comparison analysis. All definitions considered are presented in Table 2. Equations (2) and (3) are unchanged. Considering the definition of Peyrega et al. [20] (Eq. (4)), the mean over the tortuosity map provides the final scalar value used hereafter. The weighted tortuosity (Eq. (5)), of which the weighting is considered in the M-tortuosity definition, is adapted to plane to plane propagation. The computational processes of Gommes et al. [22] are used for ending on scalar values. The definition of Moreaud et al. [24] is not considered as it is not adapted to this comparison analysis. The state-of-the-art tortuosities is computed in the x, y, z directions and their arithmetic mean is used for comparison with the M-tortuosity (Fig. 6).

Fig. 6 Tortuosities with respect to porous volume fraction, morphological heterogeneity and structural anisotropy. Average tortuosities with their corresponding confidence intervals lσ, computed on 40 realizations of each model. (a) Porous volume fraction Vp decreasing with models (1), (2) and (3). (b) Morphological heterogeneity increasing with models (4) and (5). (c)–(d) Structural anisotropy with in (c) decreasing Vp with models (1’), (2’) and (3’), and in (d) average arithmetic M-tortuosity values of model (3’) with a screening of parameters α1 = {−10, −5, −2, −1, 0, 1, 2, 5, 10} and α2 = {−10, 0, 10}.

Tortuosities are separated into two groups: the definitions based on propagation direction definition, named plane-based tortuosities, and the arithmetic M-tortuosity based on stochastic points process (see Table 2). The two original definitions of tortuosity making use of minimal geodesic path τ_g and averaging geodesic paths τ_m are used as arbitrary reference. Four scenarios are considered: in the case of isotropic microstructures, the impacts of decreased V_p and of increased heterogeneity, and in the case of anisotropic microstructures, the impacts decreased V_p and of structural anisotropy at constant V_p. Tortuosity behaviors are evaluated and the various approaches are compared in Figure 6. Tortuosity values are given in Tables 3 and 4. For the sake of brevity, we only provide the necessary values to support our statements. Confidence intervals are sometimes too small to be clearly visible, attesting the representativity of the results.

5.3.1 Isotropic microstructures

Figures 6a and 6b present the behavior of tortuosity with the decreasing of the porous volume fraction V_p and with the increasing of heterogeneity, respectively.

Considering Figure 6a, despite the fact that the M-tortuosity values are globally smaller, close to the values of τ_g, it behaves similarly to the state-of-the-art definitions. The values of almost all the plane-based definitions, excepted τ_g, are very similar when compared to τ_g and the M-tortuosity (see Table 3). The smaller V_p, the bigger the tortuosity. As mentioned above, attests of bigger values of l_σ than , because of short paths promotion. These values of (α₁, α₂) correspond to the minimal and maximal values of the M-tortuosity, providing an indication of its range of values. Consequently, the lowest M-tortuosity values are obtained for long geodesic paths of source points near p_c, probably shorter than the longest ones, and the biggest M-tortuosity are obtained for short geodesic paths of source points near p_c, probably the shortest ones.

Considering Figure 6b, for the plane-based tortuosities, the more heterogeneous the microstructure, the bigger the tortuosity, excepted for the specific case of τ_g which decreases. The behavior of τ_g in Figure 6b is expected. Considering that the volume outside and inside the aggregates is equal, the more heterogeneous the microstructure, the more porous the outside of the aggregates increasing the probability to find a straight path connecting the entry plane to the exit plane. The M-tortuosity behaves differently. The long geodesic paths promotion () leads to a behavior similar to plane-based definitions but with lower values. The short geodesic paths () attest of a large diversity of values, seen through l_σ. Indeed, the M-tortuosity values of models (1), (4) and (5) are indistinguishable considering the confidence intervals. This apparent similarity between models (1) and (4) persists until α₁ = −1. For α₁ ≥ 0 (short paths not promoted) and whatever α₂, the M-tortuosity increases with the heterogeneity.

5.3.2 Anisotropic microstructures

As a recall, a compression is simulated in the x direction. Figures 6c and 6d present the anisotropic case with, in Figure 6c the decreasing of V_p and in Figure 6d the analysis of the impacts of (α₁, α₂) (to be compared to Fig. 5b).

Considering Figure 6c, the behaviors are still similar, i.e., increasing of the tortuosity with the decreasing of V_p. In comparison to the isotropic situation, the plane-based tortuosities behave similarly while being bigger (Fig. 6a). The M-tortuosity attests of a bigger sensitivity by showing a larger range of values; bigger maximal values () and smaller minimal values ().

The (α₁, α₂) pair corresponding to the maximal tortuosities is different from the isotropic case. Indeed, the maximal M-tortuosities are obtained for the promotion of short geodesic paths far from p_c, probably longer the shortest ones, while the minimal ones correspond to the promotion of long geodesic paths near p_c as in the isotropic situation (Fig. 6a). Figure 6d provides explanations. Indeed the evolution of the arithmetic M-tortuosity for the model (3’) as a function of α₁ ∈ {−10, −5, −2, −1, 0, 1, 2, 5, 10} and according to α₂ ∈ {−10, 0, 10}, behaves similarly to the isotropic situation for long geodesic paths promotion (close to α₁ = 10); the classification reversing between the different curves corresponding to the various α₂ values. However, an additional reversing is noticed for α₁ between −2 and −1, when short geodesic paths start to be promoted. Similarly to the plane-based tortuosities which identify anisotropy by the differences between the x, y, z directions (x and y are given in Table 3), this second classification reversing could be a solution for anisotropy detection.

5.3.3 Discriminative power

Discriminative power is the ability of quantitatively distinguishing between two situations, i.e., two distinct microstructures. The tortuosity contrast focuses on small parameters differences by considering the contrast between neighboring models (here multiplied by 1000 to ease the reading). Consequently, the discriminative power is here seen through the absolute value of the tortuosity contrast; the sign provides indications about the tortuosity evolution, i.e., its behavior. The results are presented in Tables 5 and 6; the same parameters selection as above is considered.

Considering the discriminative power and all scenarios, the tortuosity contrasts highlight some differences between the plane-based tortuosities and the M-tortuosity. Despite the M-tortuosity is inherently complementary to the usual descriptors, some values of (α₁, α₂) allow the M-tortuosity to be in the same range of values as the state-of-the-art definitions. For the isotropic and the anisotropic homogeneous scenarios, the discriminative power is inversely proportional to V_p. For the isotropic heterogeneous scenario, the discriminative power increases. For the last scenario, comparing isotropic and anisotropic situations at constant V_p, the discriminative power increases as V_p decreases. Moreover, as a global statement, this monotonic evolution of the tortuosity discriminative power seems to tend toward infinity when models tend toward the morphological limit; the percolation threshold ρ of the V_p for the volume fraction based scenarios, isotropic and anisotropic, complete aggregation for heterogeneity based scenario and complete compression for isotropy vs. anisotropy scenario. In other words, the discriminative power increases as a function of microstructure parameters, whatever the tortuosity definition or the type of microstructure, among the tortuosity and the models considered.

The above comments about the tortuosities behaviors are underscored by the focus on the contrast sign. Comparing isotropic and anisotropic microstructures at constant V_p (three last lines of Tables 5 and 6), the plan-based tortuosities decrease in the y direction, similarly to the M-tortuosity when long geodesic paths are promoted, and increase in the x direction, the one of the compression, similarly to the M-tortuosity when short geodesic paths are promoted. Globally, the mean values over the x, y, z directions increase with the anisotropy which have been highlighted in Figures 6a and 6c. The M-tortuosity (Table 6) increases similarly for α₁ < 0 but starts decreasing at α₁ = 0, whatever α₂, meaning that long geodesic paths are less tortuous in average for anisotropic microstructures than for isotropic ones.

5.4 Overall view

Considering all scenarios, the plane-based tortuosities are equivalent, excepted τ_g which is always smaller and the only tortuosity to decrease in the heterogenity scenario. On closer inspection, τ_B is almost equal to the classical τ_m, τ_P is a little bit bigger than τ_B while τ_G and are generally the biggest; is bigger than τ_G except for the heterogeneity scenario. In the anisotropic scenario, the differences between the tortuosity values according to the direction is a way to detect anisotropy in a microstructure. In this case, considering tortuosity behaviors as functions of V_p, there is a contradiction for the plane-based definitions; in the x direction (compression direction) the tortuosities are bigger than the isotropic case, in the y direction the tortuosities are smaller. Globally, according to the mean tortuosity, the anisotropy induces an increasing of the tortuosity.

Let consider the M-tortuosity with the arithmetic definition. Globally, the short geodesic paths makes increase the tortuosity (α₁ tends toward minus infinity), while increasing the uncertainty about the mean value. For isotropic scenarios (Figs. 6a and 6b), it provides smaller tortuosity values than the other plane-based definitions, excepted for the model (3). For anisotropic scenario, the situation is different. The power factor α₁ provides a certain control of the sensitivity to geodesic paths length. This is a complementary insight over the whole microstructure. Moreover, beside this first benefit, the M-tortuosity probes the microstructures in random directions, not only the x, y, z ones, and succeeds in detecting the anisotropy (Fig. 6d). Consequently, the M-tortuosity reveals to be a good candidate to characterize thoroughly complex microstructures where propagation direction is delicate to impose. Moreover, the choices of (α₁, α₂) could be motivated by the application, if the local diffusion is to be analyzed as in [25].

Finally, if the purpose is to analyze microstructures using a unique scalar value, it is of interest to identify the values of (α₁, α₂) to simulate the state-of-the-art definitions in applications they do not fit. As the discriminative power of the M-tortuosity is often larger than the usual definitions, no perfect matching is reachable. In the isotropic scenarios, α₁ = −10 leads to closer tortuosity values to the plane-based ones. Still in the isotropic scenarios, for the specific case of τ_g, the closest curves are obtained for α₁ around −2 for the volume fraction scenario. For the heterogeneity one, the behaviors are too different; α₁ ≥ 0 ensures the M-tortuosity to increase. The value of α₂ impacts less the final result, which is expected with isotropic microstructures. Nonetheless, it seems that when α₂ tends toward minus infinity the tortuosity increases but large values of l_σ add uncertainty in the average value. Indeed, α₂ has an influence over the confidence interval; in the volume fraction scenario as well as in the heterogeneity one, α₂ = 0 provides the lowest l_σ values. In the anisotropic situation, the range of M-tortuosity values encompasses the state-of-the-art tortuosity values. The closest values are obtained for α₁ = α₂ = −10. For the specific case of τ_g, the closest curves are obtained for α₁ = −1, whatever α₂.

6 Conclusion

Based on a stochastic process, the M-tortuosity fits with the characterization of complex microstructures where propagation directions are delicate to impose, such as irregular piece of materials or atypical contexts. The extension presented in this work provides versatility through additional parameters making explicit the consideration of percolation while giving the opportunity to the user to adapt the characterization to the application. A brief review of morphological visions of tortuosity is presented and the M-tortuosity is compared to these state-of-the-art descriptors.

The new parameter α₁ added to the original M-tortuosity definition provides a certain sensitivity to geodesic paths lengths, allowing to promote long or short geodesic paths. α₂ promotes or inhibits eccentricity in microstructure probing. The other parameters, named ρ, embed percolation to get a proper disconnection and isolation insensitivity, one of the distinctions with the state-of-the-art. Moreover, the M-tortuosity probes the microstructures in random directions, not only the x, y, z ones, particularly adapted to applicative situations where propagation direction is undefined or if only local diffusion is to be quantified. The M-tortuosity is compared to some state-of-the-art definitions in three specific situations thanks to Boolean models. The tortuosity behavior with respect to porous volume fraction, morphological heterogeneity and structural anisotropy is evaluated.

As a result, the M-tortuosity behavior is equivalent to the state-of-the-art definitions while being inherently complementary thanks to the new parameters. Globally, the short geodesic paths promotion (α₁ < 0) leads to an increase in tortuosity, contrary to the long geodesic paths promotion. In the isotropic scenario, the M-tortuosity behaves similarly to the plan-based tortuosities but its values are smaller. However, while the anisotropy does not affect the behavioral aspect, the M-tortuosity sensivity leads to a larger range of values, encompassing the state-of-the-art tortuosities. The optimal values of (α₁, α₂) to get closer to the plan-based tortuosities are discussed. The M-tortuosity is a potential candidate to replace these definitions in situations where they are not adapted to and used in diffusion and conductivity analyses. Moreover, one of the advantages of the M-tortuosity relies in the microstructure characterization as a function of the parameters (α₁, α₂), allowing, among other things, to detect anisotropy without imposed propagation direction.

These statements are supported by the discriminative power analysis, based on contrast in tortuosity values of pairs of neighboring models. This highlights once again the similarity of the M-tortuosity to the classical definitions considering isotropic microstructures while providing additional details about tortuosity behavior at different scales, especially in the heterogeneity case. Last but not least, anisotropic microstructures point out the differences to the classical definitions. This specific situation of structural anisotropy highlights the contradiction in tortuosity behaviors considering plane-based tortuosities; tortuosity increases or decreases with anisotropy according to the propagation direction. The mean value of the x, y, z directions is considered for comparison with the M-tortuosity. Considering the M-tortuosity and the standard tortuosties, the detection of the anisotropy is connected to how it impacts the tortuosity according to the length of the geodesic path. The anisotropy leads to increase the tortuosity of short geodesic paths, similarly to its impact in the x direction (compression direction), and to decrease the tortuosity of long geodesic paths, similarly to its impact in the y or z directions.

The versatility of the M-tortuosity relies on the various operators it could provide, as demonstrated in [15, 25] which is now enriched by parameters allowing to adapt the computations to the applications. In the future, two main points will be investigated. The first one is the heterogeneity case in the results and discussion section, which points out certain limitations of considering only scalar values to represent the tortuosity of complex microstructures. In this case, a dimension adapted to the required description using 3D maps, curves or histograms could turn the M-tortuosity a manyfold tortuosity. The second point concerns its use on real microstructure samples as the one of Figure 1.

The M-tortuosity and the state-of-the-art tortuosities discussed in this article are freely available in [35].

References

All Tables

All Figures

Fig. 1 Microstructures illustrations. (a) Boolean schemes of spheres generated in a cube where straightforward definition of propagation direction for tortuosity calculations is possible. (b) real γ-alumina sample, imaged by electron tomography (resolution around 1 nm·vx−1, vx standing for voxel), where a proper propagation direction is delicate to define. Volume (a) was generated using [35] and volumes (a) and (b) were rendered using [35].

In the text

Fig. 2 Strategies to extend geometric tortuosity from paths to microstructures. Illustrations of various ways to characterize whole microstructures based on geometric tortuosity definition (Eq. (8)). (a) path of minimal length connecting entry (green) and exit (red) planes (Eq. (2)), (b) average of paths connecting entry and exit planes (Eq. (3)), (c) point tortuosities, i.e., tortuosity map, assessed by forward and backward propagations between two opposite planes (Eq. (4)), (d) M-tortuosity averaging local tortuosities between random locations.

In the text

	Fig. 3 Geodesic paths according to the pore width. Illustrations of the dependency of the geodesic path (blue dashed lines) to the pore width, contrary to the skeleton (orange lines) being the same for the two pores despite their distinct width.
In the text

Fig. 4 Synthetic microstructures illustrations. (a) The grains shape (cubic platelet) and (b) the aggregate shape. Volume representation and 2D slice of a realization of Boolean models presenting the two isotropic cases, homogeneous: (1) Vv = 0.4 and (3) Vv = 0.6, and heterogeneous: (5) Vi = 0.65 and Vo = 0.15, and the anisotropic case: (1’) Vv = 0.4. Volumes and slices generated and rendered using [35].

In the text

	Fig. 5 M-tortuosity as a function of α1 and α2. Average M-tortuosity values with their corresponding confidence intervals lσ, computed on 40 realizations of model (3). Screening of parameters α1 = {−10, −5, −2, −1, 0, 1, 2, 5, 10} and α2 = {−10, 0, 10}. (a) Harmonic means representative tortuosities TH. (b) Arithmetic means representative tortuosities TA.
In the text

Fig. 6 Tortuosities with respect to porous volume fraction, morphological heterogeneity and structural anisotropy. Average tortuosities with their corresponding confidence intervals lσ, computed on 40 realizations of each model. (a) Porous volume fraction Vp decreasing with models (1), (2) and (3). (b) Morphological heterogeneity increasing with models (4) and (5). (c)–(d) Structural anisotropy with in (c) decreasing Vp with models (1’), (2’) and (3’), and in (d) average arithmetic M-tortuosity values of model (3’) with a screening of parameters α1 = {−10, −5, −2, −1, 0, 1, 2, 5, 10} and α2 = {−10, 0, 10}.

In the text