1. Introduction
The interplay between geometric microstructure characteristics and physical properties of materials motivates a major stream of current research activities. A better understanding of this relationship can help to increase the overall performance of functional materials as diverse as such used in solid oxide fuel cells, batteries and solar cells. This paper provides tools for mathematical modeling and statistical analysis of two crucial quantities: geodesic tortuosity and constrictivity.
In materials science, diverse definitions of tortuosity are used for the characterization of microstructures (Clennell, 1997), where tortuosity is sometimes defined as an effective transport property. The notion of geodesic tortuosity measures the lengths of shortest transportation paths with respect to the materials thickness and is particularly wellsuited for algorithmic estimations from 3D image data (Peyrega and Jeulin, 2013). Although this characteristic merely depends on the geometry of the underlying microstructure, it essentially influences effective transport properties of the material. Indeed, if the microstructure is set up such that the transportation paths between two interfaces are long and highly winded, then this serves as a first indication for poor quality of effective transport properties.
Despite the importance of path lengths, conductivity of materials is also influenced by more finegrained characteristics of the material along the trajectories. For instance, transport processes are substantially obstructed by the presence of frequent narrow bottlenecks of transportation paths. To capture this effect, constrictivity quantifies the appearance of bottlenecks resulting from narrow passages. In the setting of tubes with periodically appearing bottlenecks, constrictivity is intimately related to the concept of effective diffusivity (Petersen, 1958). More recently, the definition of constrictivity was extended to complex microstructures based on the continuous pore size distribution (Holzer et al., 2013). The latter characteristic is directly related to the notion of granulometry in mathematical morphology (Matheron, 1975).
Virtual materials testing – that is, the combination of stochastic microstructure modeling, image analysis and numerical simulation – made it possible to empirically derive a quantitative relationship between volume fraction , mean geodesic tortuosity , constrictivity and the ratio of effective conductivity to intrinsic conductivity (Stenzel et al., 2016). Validation with experimental data from literature showed that the relationship can be used to predict effective conductivity in real microstructures by geometric microstructure characteristics.
In order to understand geometric properties of the microstructure in materials via statistical analysis and simulation, stochastic geometry has emerged as a powerful, scalable and versatile framework, see e.g. (Chiu et al., 2013; Kendall and Molchanov, 2010; Ohser and Schladitz, 2009). Despite their numerous applications in the materials science literature, the notions of tortuosity and constrictivity have not yet been analyzed from a mathematical point of view. In fact, even a rigorous definition in the framework of stationary random closed sets is not available. This raises the question whether the empirical estimates in the literature for both, mean geodesic tortuosity (Gommes et al., 2009; Peyrega and Jeulin, 2013; Soille, 2003; Stenzel et al., 2016) and constrictivity (Holzer et al., 2013; Stenzel et al., 2016)
have any statistical underpinning, or whether the findings are purely heuristical.
In the present paper, we address this issue by providing mathematically precise definitions of these quantities as well as consistent estimators. The most naive version of the estimators is difficult to implement in practice, as it requires to take into account paths reaching arbitrarily far outside of the considered sampling window. Therefore, we establish sufficient conditions under which the estimators remain consistent in the setting where only a small amount of plussampling of the sampling window is required. Additionally, we illustrate how the conditions can be verified in the case of mollified Poisson relative neighborhood graphs, which have recently been used to model microstructures in solid oxide fuel cells (Neumann et al., 2016). Realizations of this model are simulated and the estimators of geodesic tortuosity and constrictivity are computed in order to illustrate the theoretical results.
The paper is organized as follows. In Section 2, the notions of geodesic tortuosity and constrictivity as well as their estimators are defined in the framework of random closed sets. Moreover, in this section, the main results are presented. The corresponding proofs are given in Sections 3 and 4. In Section 5, the influence of edge effects on the estimation of geodesic tortuosity and constrictivity is analyzed for a certain class of random closed sets using percolation theory. Finally, numerical simulations illustrate the theoretical results in Section 6.
2. Definitions and main results
The proofs of the results stated in this section are postponed to Section 4.
2.1. Preliminaries
We consider the dimensional Euclidean space, . For each the interior, the closure and the boundary of are denoted by and respectively. Let and denote the families of closed and compact sets in , respectively. Furthermore, let be the dimensional Lebesgue measure and be the dimensional Hausdorff measure for . By
we denote the Borel algebra with respect to the Fell topology (Fell, 1962) on . The open and closed balls with radius centered at are denoted by and , respectively. By and
we denote the hyperplanes orthogonal to the
th standard unit vector
at distance and to the origin. The set of all paths going from to via the interior of is denoted byNote that the Lipschitz condition ensures that paths are rectifiable, see (Federer, 1969, p. 251). Finally,
denotes the set of all points connected to a closed set through . In other words, describes the union of the connected components of intersecting . Throughout the paper, we consider a complete probability space , which is not further specified.
2.2. Definitions
2.2.1. Mean geodesic tortuosity
The notion of mean geodesic tortuosity for stationary random closed sets quantifies the windedness of directional paths through the considered sets. Among the various concepts of tortuosity in materials science, the concept of mean geodesic tortuosity constitutes an important microstructure characteristic for the prediction of effective conductivity in multiphase materials (Stenzel et al., 2016).
In the following, we assume that the direction of transport is given by . Tortuosity in other directions reduces to this setting after a suitable rotation of the underlying random closed set. Intuitively speaking, mean geodesic tortuosity of a stationary random closed set in is defined as the expectation of the length of the shortest path in from the origin to the hyperplane under the condition that at least one such path exists. For normalization, mean geodesic tortuosity is divided by . An illustration of a shortest path is given in Figure 1, left. After rescaling suitably by , i.e., considering , we may assume that in the following.
In order to define the mean geodesic tortuosity of a random closed set in rigorously, we need to be able to measure lengths of connection paths in and determine whether different parts of are contained in a common connected component. Recall that describes the union of the connected components of intersecting a set . Furthermore,
denotes the length of the shortest path contained in the interior of from to a target set .
Although we consider the paths in the interior of the random closed set for technical reasons, this definition is sensible for applications since transport through infinitely thin paths is not possible. In applications, it is sometimes assumed that paths are restricted in the halfspace of nonnegative last coordinate and many of our results carry over to this setting.
A random closed set is called stationary if it is stationary with respect to shifts by vectors of the linear subspace .
Definition 1.
Let be an stationary random closed set. The mean geodesic tortuosity of is then defined by
(1) 
Since, a priori, there are uncountably many paths, it is not clear that
is a random variable and
is a measurable set. This is discussed in Proposition 9 and in the remark thereafter. Note that in the definition of , it is sufficient to assume that is stationary. In case that is stationary, but not stationary, is a local characteristic of . This can be relevant for the investigation of microstructures exhibiting a structural gradient. If is additionally stationary, is invariant under translations of the origin and thus is a global characteristic of .2.2.2. Constrictivity
The notion of constrictivity of a stationary random closed set measures the strength bottleneck effects. This characteristic was introduced in materials science for tubes with periodically appearing bottlenecks in (Petersen, 1958), where constrictivity has been defined in dimension as the ratio of the minimum and maximum area, through which transport goes. In (Petersen, 1958) the minimum as well as the maximum area are circular areas with radii and . Thus constrictivity is defined as . Note that the concept of constrictivity can be transfered from simple geometries to complex microstructures (Holzer et al., 2013), i.e. and are defined for complex microstructures based on the concept of the continuous pore size distribution (Münch and Holzer, 2008), which is directly related to the granulometry function of mathematical morphology (Matheron, 1975). As in Section 2.2.1 we assume that the transport direction is . Constrictivity with respect to other transport directions can be reduced to this setting by a suitable rotation of the random closed set. When generalizing the concept of constrictivity to an arbitrary dimension, the definition changes to The exponent appears in the definition of constrictivity as transport in towards a predefined direction goes through dimensional crosssections.
In the following denotes the erosion of a set by for each . The Minkowski addition of two sets is denoted by . To quantify bottleneck effects in a closed set , we consider the set consisting of all such that can be reached by a path in the interior of starting at . Then, is the subset of that can be filled by spheres starting from , i.e. the centers of spheres are in , and rolling freely in . For an illustration, see Figure 1, right.
Next, for an stationary random closed set by
we denote the largest radius such that in expectation at least half of can be filled by an intrusion of balls with radius , i.e., by an intrusion from to . Note that the intrusion in transport direction determining is strongly influenced by bottlenecks in .
In order to obtain a quantity invariant under rescaling of , the radius must be related to the overall thickness of . More precisely, writing for the opening of let
denote the largest radius such that in expectation at least half of can be covered by balls of radius entirely contained within .
Definition 2.
Let be an stationary random closed set. The constrictivity of is then defined by
Since the constrictivity of is identical to the constrictivity of the scaled set , we only consider the case from now on. Conceptually is a measure for the strength of bottleneck effects. Typically, there are many narrow constrictions in if is close to 0, whereas if , then there are no constrictions at all. If is almost surely connected, then and . Otherwise, it can happen , as can be seen in the case where is almost surely a union of disjoint sets of diameter smaller than 1.
2.3. Construction of consistent estimators
In the following, let be an stationary and ergodic random closed set. That is, is stationary and ergodic with respect to the group of translations , where is a invariant mapping such that . Now, we construct estimators for mean geodesic tortuosity and constrictivity of a random closed set observed in a bounded sampling window for some integer .
2.3.1. Mean geodesic tortuosity
To estimate the mean geodesic tortuosity , we consider paths starting in the window Then, we define the estimator for mean geodesic tortuosity as
We prove strong consistency of as for
stationary and ergodic random closed sets under some moment condition of the shortest pathlengths.
Theorem 3.
Let . Then, defines a strongly consistent estimator of . That is, converges almost surely to as .
Using the estimator requires information about the length of all shortest paths from to through . In practice, is observed in a bounded sampling window, which does not necessarily contain all shortest paths that are required to compute . Thus we consider a further estimator for . For estimating mean geodesic tortuosity based on a bounded sampling window, we observe paths from to going through the dilated window
for some . Concerning all paths going through a dilated sampling window reduces the edge effects. Indeed, we also take paths from to into account leaving , which tend to be longer than paths completely contained in .
Hence, we consider the estimator
(2) 
In contrast to estimation of by means of takes those shortest paths into account, which are completely contained in the extended sampling window . Under some further assumptions regarding the shortest paths in we obtain a result on consistency of the estimator . Therefore, we define as the event that all shortest paths going from to are completely contained in , where denotes the set of rational numbers.
Corollary 4.
Let . Then, the following statements are true:

If there exists an almost surely finite random variable such that the event occurs for all , then the estimator is strongly consistent as .

If , then the estimator is weakly consistent as .
2.3.2. Constrictivity
In order to estimate the constrictivity , we introduce estimators for and . In particular, for , we define the estimator
Theorem 5.
If there exists at most one with
(3) 
then the estimator is strongly consistent as .
To estimate , we put and define the estimator
Theorem 6.
If there exists at most one with
(4) 
then the estimator is strongly consistent as .
In practice, it might be difficult to verify Conditions (3) and (4) for a given random closed set. Thus, we present a further more accessible sufficient condition.
Corollary 7.
For the usage of the complete information about is required for each . In applications, is observed in a bounded sampling window. Then has to be determined based on the observation of . This is taken into account by the estimator
(6) 
where is used instead of to estimate , since can be determined based on a observation of in . Under further assumptions on the connected components of we obtain a consistency result of the estimator . Therefore, we define for each the event that each can either be connected to by a path within or is contained in a connected component, which does not intersect
Corollary 8.
Let Condition (4) be fulfilled.

If there exists an almost surely finite random variable such that the event occurs for all , then the estimator is strongly consistent as .

If , then the estimator is weakly consistent as .
3. Measurability
3.1. Geodesic tortuosity
In the following we show the welldefinedness of geodesic tortuosity as formalized in the following result.
Proposition 9.
The functions and are measurable. In particular, the conditional expectation given in (1) is well defined.
Remark.
In Section 3.2, it is shown that if is a random closed set, then so is . Note that this result can be analogously obtained for .
That is, we show that is a random variable with values in and that for each random closed set . For this purpose, continuous paths from to through are approximated by line segments, which is a common approach in the literature Davis and Sethuraman (2017). In the following, we denote the line segment between and by
for all in .
Since our proof of measurability of geodesic tortuosity relies on an approximation by line segments, we first show that the event that a line segment is contained in a random closed set is measurable.
Lemma 10.
Let Define the mapping by
Then, is measurable, where
Proof.
Since is a piecewise constant function, it suffices to show that is measurable with respect to . Considering the rational approximation
we have
which completes the proof. ∎
Proof of Proposition 9.
Since it suffices to prove measurability of . By the previous remark on rectifiability, the length of a path can be approximated by the sum of the length of line segments connecting points on the curve . Moreover, since contains only paths contained in the interior of , the line segments can be assumed to be contained in . That is,
Thus, defining to be the family of all piecewise affine linear functions that have coefficients in and satisfy , , we can approximate paths by line segments to obtain that
Hence, can be represented via nested infima and suprema of countably many functions that are measurable by Lemma 10. Thus is measurable as an infimum of countably many measurable functions. Since is measurable, is measurable. Moreover this is which leads to the claim. ∎
3.2. Constrictivity
In the following it is shown that the constrictivity is well defined for random closed sets . The welldefinedness of can be deduced directly from basic properties of random closed sets.
Lemma 11.
Let be arbitrary. Then, and are random closed sets.
Proof.
The assertions follow from (Molchanov, 2005, Chapter 1, Thm. 2.25) since and are closed for all . ∎
Showing that is well defined involves further arguments and is summarized in the following result.
Proposition 12.
The function is measurable.
The idea of the proof is to represent as sublevel set of a measurable lower semicontinuous function and apply (Molchanov, 2005, Chapter 5, Proposition 3.6). For this purpose, we define the function by
To simplify notation, we write and as in Proposition 9 it can be shown that is measurable in for each Then, can be expressed in terms of , in particular
(7) 
The following two lemmas are used to show that is measurable.
Lemma 13.
The function is lower semicontinuous. That is,
Lemma 14.
The function is measurable.
Before proving the auxiliary results, we explain how to complete the proof of Proposition 12.
Proof of Proposition 12.
It remains to establish lower semicontinuity and measurability of .
Proof of Lemma 13.
It suffices to consider the case, where . Let now with Furthermore, let be a sequence in with Since there exists an such that there is an with for each and . Then it holds that for each . Thus is continuous at . ∎
Proof of Lemma 14.
The proof is strongly leaned on the method used for the proof of (Gowrisankaran, 1972, Theorems 2 and 3). For each , define the strict sublevel set
Then, if . Furthermore, for every we have
where the third equality follows from the openness of . Finally, for every the mapping is continuous at each with Let . Note that
which leads to for each . ∎
4. Proofs
Proof of Theorem 3.
First, we rewrite as
Now, the individual ergodic theorem (Chiu et al., 2013, Theorem 6.2) implies that the last expression converges almost surely to
This shows the strong consistency of ∎
Next, we prove Corollary 4.
Proof of Corollary 4.
Since, equals given the event we obtain that
Moreover, by Theorem 3 the second summand tends to 0 almost surely as . Since in cases (i),(ii) the first summand tends to 0 almost surely respectively in probability, we conclude the proof. ∎
Proof of Theorem 5.
We define a family of stochastic processes with index set by
for all . It is sufficient to show that for each and for each . First, let . Due to ergodicity as , the random variables converge almost surely to which by assumption (3) is strictly positive. The case follows verbatim. ∎
Proof of Theorem 6.
Note that is stationary and ergodic as for each Due to the ergodicity, the proof is analogous to the proof of Theorem 5. ∎
Proof of Corollary 7.
5. Analysis of edge effects
The question arises to which extend plus sampling is required to estimate and sufficiently precisely by the estimators given in (2) and (6). In this section it is shown that the amount of required plus sampling is asymptotically negligible in comparison to the size of the sampling window for a certain type of a twophase randomset model that has been considered previously in an application to materials science (Neumann et al., 2016). Loosely speaking, this model results from a Voronoi mollification of a union of parametric proximity graphs, to be more precise, of betaskeletons (Kirkpatrick and Radke, 1985). Here we consider the special case, where the parameters of the betaskeletons are chosen such that each of them coincides with the relative neighborhood graph.
First, in Section 5.1, we provide a precise definition of this model. Then, in Section 5.2, we show that starting from a sampling window of side length the paths that are relevant for estimation of tortuosity are contained in an environment with high probability, where is an arbitrary positive number. The results obtained in Section 5.2 are valid for an arbitrary dimension Finally, in Section 5.3, we show that the issue of edge effects for constrictivity estimation can be translated into questions of an appropriately constructed continuum percolation model. We analyze this model first rigorously for very large and very small erosion radii, and then via simulation for intermediate values in Section 6 to conclude that also for constrictivity estimation only an asymptotically negligible amount of plus sampling is required. Moreover, to ensure that constrictivity can be estimated strongly consistently in the multiphase RNG model, a verification of Conditions (3) and (5) is necessary. As the corresponding proof is based on a rather technical construction, it is postponed to Appendix A. Note that the results regarding the estimation of constrictivity are only valid for dimension .
5.1. Definition of a multiphase RNG model
In this section, we provide a mathematically precise definition of the multiphase model under consideration as well as consistency results regarding the estimation of geodesic tortuosity and constrictivity.
In materials science, microstructures, in which transport processes take place, exhibit a high degree of connectivity. Such highly connected structures can be represented by models based on connected random geometric graphs, such as the relative neighborhood graph (RNG) (Jaromczyk and Toussaint, 1992; Neumann et al., 2016). Note that results regarding the lengths of shortest paths in the RNG itself have been discussed in (Aldous and Shun, 2010).
Loosely speaking, the phases are based on skeletons given by independent Poisson RNG, i.e., by RNG with vertices given by Poisson point processes. Then, we use a Voronoi mollification to associate the skeleton with a fulldimensional random closed set. In the following, we provide a more detailed description of both construction steps. First, we motivate the use of the RNG.
The relative neighborhood graph is a graph on the locally finite vertex set where two nodes are connected by an edge if and only if there does not exist such that . In order to model a phase material, , we first build RNG based on independent homogeneous Poisson point processes with some intensities
In a second step, we use a Voronoi mollification to construct for each of the graphs a fulldimensional random closed set representing the corresponding phase. More precisely, if is a collection of locally finite subsets of , we define the Voronoi mollification of with respect to by
where we use the notation for each . That is, is the set of all points that are closer to the graph than to any other of the graphs , see Figure 2. Furthermore, put for each
Theorem 15.
Let be arbitrary and
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