Role of contact-angle hysteresis for fluid transport in wet granular matter

The stability of sand castles is determined by the structure of wet granulates. Experimental data on the size distribution of fluid pockets are ambiguous with regard to their origin. We discovered that contact-angle hysteresis plays a fundamental role in the equilibrium distribution of bridge volumes, and not geometrical disorder as commonly conjectured. This has substantial consequences on the mechanical properties of wet granular beds, including a history-dependent rheology and lowered strength. Our findings are obtained using a model in which the Laplace pressures, bridge volumes, and contact angles are dynamical variables associated with the contact points. While accounting for contact line pinning, we track the temporal evolution of each bridge. We observe a crossover to a power-law decay of the variance of capillary pressures at late times and a saturation of the variance of bridge volumes to a finite value connected to contact line pinning. Large-scale simulations of liquid transport in the bridge network reveal that the equilibration dynamics at early times is well described by a mean-field model. The spread of final bridge volumes can be directly related to the magnitude of contact-angle hysteresis.

Figure 1

Effect of contact-angle hysteresis on the equilibration dynamics of two capillary bridges labeled $i=1,2$ between beads with identical radii in mechanical contact.

Figure 2

Schematic of the capillary pressure $P$ against the volume $V$ of a bridge. The lower and upper branches (dashed lines) correspond to a receding (lower) and advancing (upper) contact line. The left and right branches (solid lines) describe bridges with a pinned contact line with a small (left) and large (right) opening angle.

Figure 3

Relaxation of individual capillary bridge volumes as a function of time in experiments (symbols) taken from Ref. [10] and in our simulations for $\mathrm{\Delta }\theta ={25}^{\circ }$ in the full network model (“NW,” black dashed lines) and the mean-field model (“MF,” red solid lines). The inset shows the equilibration dynamics in the network model with $\mathrm{\Delta }\theta =0^{\circ }$. The receding contact angle is ${\theta }_{\mathrm{r}}=7^{\circ }$ in both cases.

Figure 4

Probability distribution function (PDF) of final bridge volumes $V$ for vanishing contact-angle hysteresis according to setup A. Contact angle $\theta =7^{\circ }$. Inset: variance of volumes in the final state for different polydispersities $r_{\text{pol}}$.

Figure 5

Distributions for data in the network model (NW) shown in Fig. 3: (a) PDF of bridge volume $V$ and (b) average contact angle $\langle \theta \rangle$ of bridges with volume $V$. (c) PDF to find bridges in bins of contact angles $\theta$ close to the advancing, receding, or intermediate range at different times $t$.

Figure 6

(a) Squared variance of capillary pressure, ${\sigma }_p^2$, and (b) of bridge volumes, ${\sigma }_v^2$, as a function of time $t$ for $\mathrm{\Delta }\theta =0^{\circ }$ (red solid lines) and $\mathrm{\Delta }\theta ={25}^{\circ }$ in the network model and in the mean-field approximation (dashed lines) for setup B. Insets show the same data at early times in linear scale.

Figure 7

Temporal evolution of the squared variance ${\sigma }_v^2$ of bridge volumes in the full network model for different magnitudes of contact-angle hysteresis, $\mathrm{\Delta }\theta =0^{\circ },4^{\circ },8^{\circ },{15}^{\circ },{25}^{\circ }$. Inset: double logarithmic plot of ${\sigma }_v^2(t=\infty )$ against $\mathrm{\Delta }\theta$ in comparison to a power law $\sim \mathrm{\Delta }{\theta }^3$ (dashed line).