On Variability and Interdependence of Local Porosity and Local Tortuosity in Porous Materials: a Case Study for Sack Paper

Abstract

The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.

Appendix

In addition to Section 3.1, we provide the numerical values of the fitted parameters regarding the univariate distributions of porosity, τ⁽⁰⁾, τ^(1.5), and τ⁽³⁾ in Tables 4, 5, 6, and 7, respectively. Moreover, the corresponding p-values of the Kolmogorov-Smirnov test indicating the goodness of fit are shown. Since our sample consists of 204 observations, the p-values are simulated based on the procedure proposed in Marsaglia et al. (2003), rather than using the ones of the asymptotic case. For this purpose, we make use of the implementation in the statistical software package R (R Core Team 2015).

Table 4 Estimated values $\widehat {a}$ and $\widehat {b}$ for the parameters a and b of the beta-distribution modeling the univariate distribution of porosity. Moreover, the p-values corresponding to the Kolmogorov-Smirnov test for different cutout sizes are shown

Table 5 Estimated values $\widehat {\alpha }, \widehat {d},$ and $\widehat {k}$ for the parameters α,d and k of the shifted generalized gamma distribution modeling the univariate distribution of τ⁽⁰⁾. Moreover, the p-values corresponding to the Kolmogorov-Smirnov test for different cutout sizes are shown

Table 6 Estimated values $\widehat {\alpha }, \widehat {d},$ and $\widehat {k}$ for the parameters α,d and k of the shifted generalized gamma distribution modeling the univariate distribution of τ^(1.5). Moreover, the p-values corresponding to the Kolmogorov-Smirnov test for different cutout sizes are shown

Table 7 Estimated values $\widehat {\alpha }, \widehat {d},$ and $\widehat {k}$ for the parameters α,d and k of the shifted generalized gamma distribution modeling the univariate distribution of τ^(3.0). Moreover, the p-values corresponding to the Kolmogorov-Smirnov test for different cutout sizes are shown