Percolation through voids around toroidal inclusions

In the case of media comprised of impermeable particles, fluid flows through voids around impenetrable grains. For sufficiently low concentrations of the latter, spaces around grains join to allow transport on macroscopic scales, whereas greater impenetrable inclusion densities disrupt void networks and block macroscopic fluid flow. A critical grain concentration ${\rho }_c$ marks the percolation transition or phase boundary separating these two regimes. With a dynamical infiltration technique in which virtual tracer particles explore void spaces, we calculate critical grain concentrations for randomly placed interpenetrating impermeable toroidal inclusions; the latter consist of surfaces of revolution with circular and square cross sections. In this manner, we study continuum percolation transitions involving nonconvex grains. As the radius of revolution increases relative to the length scale of the torus cross section, the tori develop a central hole, a topological transition accompanied by a cusp in the critical porosity fraction for percolation. With a further increase in the radius of revolution, as constituent grains become more ringlike in appearance, we find that the critical porosity fraction converges to that of high-aspect-ratio cylindrical counterparts only for randomly oriented grains.

Figure 1

Cutaway view of tori of circular cross section with (a) ${\tilde{r}}_1=0.47$, (b) ${\tilde{r}}_1=0.5$, and (c) ${\tilde{r}}_1=0.53$.

Figure 2

Randomly oriented tori for aspect ratios (a) ${\tilde{r}}_1=0.30$, (b) ${\tilde{r}}_1=0.50$, (c) ${\tilde{r}}_1=0.70$, and (d) ${\tilde{r}}_1=0.95$.

Figure 3

Randomly oriented tori of square cross section for (a) $r_1^{*}=0.70$, (b) $r_1^{*}=0.90$, (c) cylinders for $r_{\mathrm{A}}=8$, and (d) square prisms for $r_{\mathrm{A}}=8$.

Figure 4

Voxels visited by tracer particles shown in red for ${10}^7$ interactions of virtual tracers with tori of circular cross section for ${\tilde{r}}_1=0.70$. Small red cubes are voxels visited by tracer particles; the simulation volume is indicated by the large green cube.

Figure 5

Sample data collapse for randomly oriented tori of circular cross section for ${\tilde{r}}_1=0.75$. The red line is an analytical curve, while symbols represent Monte Carlo data corresponding to subinterval times $t_i=iT/16$ ($T$ being the total dwell time) as indicated in the legend. The inset shows a crossing of effective diffusion exponents with the vertical blue line indicating ${\rho }_c$ obtained from a quantitative data collapse.

Figure 6

Critical porosity fractions for aligned (closed squares) and randomly oriented (open diamonds) tori of circular cross section in the main graph and tori of square cross section in the graph inset.

Figure 7

Critical porosity fractions for aligned (closed squares) and rotated (open diamonds) square prisms.

Figure 8

Critical porosity fraction versus aspect ratio for (a) aligned tori of circular cross section and cylinders, (b) randomly oriented tori of circular cross section and cylinders, (c) aligned tori of square cross section and square prisms, and (d) randomly oriented tori of square cross section and square prisms.

Figure 9

Critical concentration ratios for calculations with ${10}^7$ and ${10}^6$ interactions of virtual tracer particles with grains (a), (c), and (e) aligned and (b), (d), and (f) randomly oriented. Results are shown for (a) and (b) tori of circular cross section, (c) and (d) tori of square cross section, and (e) and (f) square prisms.

Figure 10

Exponent ratios $k/x$ with black closed circles and green closed squares corresponding to calculations for ${10}^7$ and ${10}^6$ interactions of virtual tracer particles with (a), (c), and (e) aligned grains and (b), (d), and (e) randomly oriented inclusions. Results are shown for tori of (a) and (b) circular cross section, (c) and (d) square cross section, and (e) and (f) square prisms. Dashed horizontal lines indicate the 0.667 ratio based on universal scaling arguments.