Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

The quantitative characterization of the microstructure of random heterogeneous media in $d$-dimensional Euclidean space ${\mathbb{R}}^d$ via a variety of $n$-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface $F_{\text{ss}}$ and surface-void $F_{\text{sv}}$ correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-$r$ behaviors of these functions, which are linked to the mean curvature of the system. Then we demonstrate that one can reduce the computational complexity of the problem, without sacrificing accuracy, by extracting the necessary interfacial information from a cut of the $d$-dimensional statistically homogeneous and isotropic system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated “stealthy” patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures.

Figure 1

(a) A schematic two-phase medium showing the surface-surface correlation function $F_{\text{ss}}\left(r\right)$, surface-void correlation function $F_{\text{sv}}\left(r\right),$ and void-void correlation function $F_{\text{vv}}\left(r\right)$ [or $S_2\left(r\right)]$, where the blue phase is the “solid” phase. (b) Corresponding schematic showing the scattering of radiation by a two-phase medium, where the blue phase indicates the bulk and the red region indicates the interface.

Figure 2

(a) Schematic that illustrates the small-$r$ asymptotic behavior of the surface-surface correlation function in three dimensions in which the vicinity of the $p_0$ is approximated by a plane, where the area of interface contained in the shell is shaded. (b) Schematic that illustrates the small-$r$ asymptotic behavior of the surface-void correlation function in two dimensions, where $r_c$ is the local radius of curvature of the interface.

Figure 3

An illustration of the previous algorithm that computes $F_{\text{ss}}\left(r\right)$, where $\epsilon$ is the dilation thickness. Then $F_{\text{ss}}\left(r\right)$ is computed by $S_2(r;\epsilon )/{\epsilon }^2$ as $\epsilon \to 0$.

Figure 4

A schematic plot that elucidates our algorithm that computes surface correlation functions. Here the sampling straight line intersects with the interface at the points $P_1$, $P_2$, $P_3$, and $P_4$.

Figure 5

Comparison of theoretical and simulation results of surface-surface correlation function $F_{\text{ss}}\left(r\right)$ and surface-void correlation function $F_{\text{sv}}\left(r\right)$ for overlapping spheres in three dimensions, where $D$ is the diameter of the sphere. The simulations are carried out using 250 000 overlapping spheres in a cubic box under periodic boundary conditions using 1 000 000 line samples. The particle-phase volume fraction $\varphi$ is 0.649.

Figure 6

Simulation results of $F_{\text{ss}}\left(r\right)$ of a digitized Gaussian random field with level cut $F_0=0$ computed under different resolutions. The resolutions shown here are $1000\times 1000$, $2000\times 2000$, $4000\times 4000$ and 10 000$\times 10$ 000. The continuum result is computed directly using the analytic expression of the scalar field. One can see that as the resolution increases, the numerical result is closer to the continuum result.

Figure 7

An illustration of the sampling scenario. The unit circle is the interface and the solid straight line samples along the direction that is perpendicular to itself by varying $x$.

Figure 8

A comparison of simulation results of $F_{\text{ss}}\left(r\right)$ of a digitized Gaussian random field with level cut $F_0=0$ computed with and without applying a threshold $\delta$. The resolution is $1000\times 1000$, and the threshold $\delta$ is 100. The continuum result is computed directly using the analytic expression of the scalar field. By applying a threshold, the fluctuations are largely suppressed and the result is much closer to the continuum result, even comparable to the ones with much higher resolutions.

Figure 9

(a) Digitized random overlapping disks with particle-phase volume fraction 0.677. (b) Corresponding scalar field of (a) by using a Gaussian kernel (39) with $b=0.042D$.

Figure 10

Simulation results of $F_{\text{ss}}\left(r\right)$ of a digitized overlapping disk configuration by applying Gaussian kernels for different values of $b$.

Figure 11

(a) A comparison of the local surface-area variances ${\sigma }_{{}_S}^2\left(R\right)$ with the volume-fraction variances ${\sigma }_{{}_V}^2\left(R\right)$ for three-dimensional overlapping spheres of radius $a$ as functions of window radius $R$ at particle-phase volume fraction $\varphi =0.649$. Note the surface-area fluctuation is much larger at small $R$, suggesting that it is a more sensitive microstructure descriptor. (b) A comparison of rescaled local surface-area variances with the volume-fraction variances from (a).

Figure 12

A comparison of local surface-area variances ${\sigma }_{{}_S}^2\left(R\right)$ with volume-fraction variances ${\sigma }_{{}_V}^2\left(R\right)$ for three-dimensional hard spheres of radius $a$ as functions of window radius $R$ in equilibrium at different packing fractions $\varphi$; (a) $\varphi =0.1$. (b) $\varphi =0.3$. (c) $\varphi =0.5$.

Figure 13

(a) A comparison of volume-fraction variances of hard spheres at equilibrium at different packing fractions, it is noteworthy that there is a crossover at small $R$. (b) A comparison of surface-area variances of hard spheres at equilibrium at different packing fractions, which clearly reflect the increase of short-range order as packing fraction increases.

Figure 14

A comparison of surface-area variances ${\sigma }_{{}_S}^2\left(R\right)$ of three-dimensional hard spheres in equilibrium and overlapping spheres of radius $a$ as functions of window radius $R$ at different volume fractions. (a) $\varphi =0.1$. (b) $\varphi =0.5$.

Figure 15

Surface-surface correlation functions $F_{\text{ss}}$ of stealthy patterns decorated with spheres with different diameters: (a) $D=0.05c$. (b) $D=0.07c$. (c) $D=0.08c$. (d) $D=0.1c$.

Figure 16

The interfaces of two binary mixtures undergoing a phase separation at critical quench (volume fraction ratio of two phases is 1:1) and off critical quench (volume fraction ratio of two phases is 2:8), respectively. The system size is $1000\times 1000$.

Figure 17

(a) The scaled surface-surface correlation function $F_{\text{ss}}\left(r\right)/k_1{\left(t\right)}^2$ versus $r{k}_1\left(t\right)$ at different time stages associated with the spinodal decomposition pattern shown in the left panel of Fig. 16. (b) The scaled surface-surface correlation function $F_{\text{ss}}\left(r\right)/k_1{\left(t\right)}^2$ versus $r{k}_1\left(t\right)$ at different time stages associated with the spinodal decomposition pattern shown in the right panel of Fig. 16. One can see that they both collapse onto a single curve respectively after the rescaling. The shapes of two curves are significantly different.

Figure 18

(a) The scaled surface-void correlation function $F_{\text{sv}}\left(r\right)/k_1\left(t\right)$ versus $r{k}_1\left(t\right)$ at different time stages associated with the spinodal decomposition pattern shown in the left panel of Fig. 16. We see that $F_{\text{sv}}\left(r\right)$ is a constant (flat function), which is consistent with the exact expression (35). (b) The scaled surface-void correlation function $F_{\text{sv}}\left(r\right)/k_1\left(t\right)$ versus $r{k}_1\left(t\right)$ at different time stages associated with the spinodal decomposition pattern shown in the right panel of Fig. 16. One can see that they both collapse onto a single curve respectively after the rescaling. The shapes of two curves are significantly different.

Figure 19

(a) The autocovariance function ${\chi }_V\left(r\right)$ versus $r{k}_1$ at $t=100000$ associated with the spinodal decomposition pattern shown in the left panel of Fig. 16, where $r$ is scaled by the characteristic wave number $k_1$. (b) The autocovariance function ${\chi }_V\left(r\right)$ versus $r{k}_1$ at $t=100000$ associated with the spinodal decomposition pattern shown in the right panel of Fig. 16. There is no significant difference between these two curves in (a) and (b).

Figure 20

Surface-surface correlation function $F_{\text{ss}}\left(r\right)$ of patterns generated from the Swift-Hohenberg equation with two different values of the parameter $k_0$.

Figure 21

Local surface-area variances ${\sigma }_{{}_S}^2\left(R\right)$ as functions of window radius $R$ computed from Eq. (10) for patterns generated from the Swift-Hohenberg equation with two different values of the parameter $k_0$, with comparison of different scalings.

Figure 22

An illustration of evaluating the two-body correction to the small $r$ behavior of $F_{\text{sv}}\left(r\right)$, where the upper left sphere is the “invading” sphere and the sphere in dotted line is the “test” sphere.

Figure 23

(a) Simulation results of $F_{\text{ss}}\left(r\right)$ for systems of three-dimensional overlapping spheres computed with different thresholds 10, 100, 1000 and without threshold. We use the same system in Sec. 4, except that only 10 000 sampling lines are used. Clearly, applying thresholds does not change the overall shape of the curve, but significantly reduce the fluctuations. (b) Red circles: Average of $F_{\text{ss}}\left(r\right)$ in the interval $[D,11D]$ for different thresholds. The blue line is $s^2$, which is the theoretical value of $F_{\text{ss}}\left(r\right)$ when $r>D$. Green squares: the largest deviation of $F_{\text{ss}}\left(r\right)$ from $s^2$ in the interval $[D,11D]$ for different thresholds. The average of $F_{\text{ss}}\left(r\right)$ is always very close to the expected value (within $1\%$ for all cases). However, by applying a more stringent threshold, there is a trend to reduce the abnormal peaks significantly.