Generation and structural characterization of Debye random media

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semianalytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks) and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here.

Figure 1

Representative digitized images of Debye random media obtained from the construction algorithm at different volume fractions. Each sample consists of $N^2={501}^2$ pixels and the length scale $a=5$ pixels. The volume fractions of the solid (yellow) phase are (a) ${\varphi }_2=0.1$. (b) ${\varphi }_2=0.2$. (c) ${\varphi }_2=0.3$. (d) ${\varphi }_2=0.4$. (e) ${\varphi }_2=0.5$.

Figure 2

Comparison between the simulated $S_2\left(r\right)$'s of the constructed Debye random media (shown as dashed curves) and the targeted ones (shown as solid black curves) for different volume fractions of phase 2: Curves from top to bottom span from ${\varphi }_2=0.5$ to ${\varphi }_2=0.1$ in increments of 0.1. The simulated two-point functions are in excellent agreement with the corresponding target functions.

Figure 3

Surface-void correlation function $F_{\mathrm{sv}}\left(r\right)$ for Debye random media at different volume fractions. (a) ${\varphi }_2=0.1$. (b) ${\varphi }_2=0.2$. (c) ${\varphi }_2=0.3$. (d) ${\varphi }_2=0.4$. (e) ${\varphi }_2=0.5$.

Figure 4

Surface-surface correlation function $F_{\mathrm{ss}}\left(r\right)$ for Debye random media at different volume fractions. (a) ${\varphi }_2=0.1$. (b) ${\varphi }_2=0.2$. (c) ${\varphi }_2=0.3$. (d) ${\varphi }_2=0.4$. (e) ${\varphi }_2=0.5$.

Figure 5

(a) The pore-size probability density function $P\left(\delta \right)$ for Debye random media for selected volume fractions ${\varphi }_2=0.1,0.2,0.3,0.4$, and 0.5. (b) The mean pore size $\langle \delta \rangle$ as a function of volume fraction.

Figure 6

(a) The lineal-path function $L\left(z\right)$ for Debye random media for different volume fractions of phase 2, ${\varphi }_2$. (b) The fitted $L_w$ as a function of ${\varphi }_2$.

Figure 7

The matrix chord-length probability density function $p\left(z\right)$ for Debye random media.

Figure 8

Representative images of the two models of particle dispersions (each with ${\varphi }_2=0.3$) discussed in Sec. 6. (a) Overlapping disks. (b) Equilibrium hard disks.

Figure 9

Comparison of (a) autocovariance function ${\chi }_{{}_V}\left(r\right)$, (b) surface-void correlation function $F_{\mathrm{sv}}\left(r\right)$, and (c) surface-surface correlation function $F_{\mathrm{ss}}\left(r\right)$ for Debye random media, overlapping disks, and equilibrium hard disks at ${\varphi }_2=0.3$. The radial distance $r$ is scaled by the specific surface $s$.

Figure 10

The pore-size probability density function $P\left(\delta \right)$ for three different systems, Debye random media, overlapping disks, and equilibrium hard disks at ${\varphi }_2=0.3$. The pore-size distance $\delta$ is scaled by the specific surface $s$.

Figure 11

The lineal-path function $L\left(z\right)$ for three different systems, Debye random media, overlapping disks, and equilibrium hard disks at ${\varphi }_2=0.3$. Note that curves for $L\left(z\right)$ for overlapping disks and equilibrium hard disks are the same after we scale $z$ by $s$.

Figure 12

The matrix chord-length density function $p\left(z\right)$ for three different systems, Debye random media, overlapping disks and equilibrium hard disks at ${\varphi }_2=0.3$. Note that by scaling $z$ by $s$, the curves for $p\left(z\right)$ for overlapping disks and equilibrium hard disks are the same.