Deep spontaneous penetration of a water droplet into hot granular materials

The Levitation of a droplet on a hot surface due to the rapid vaporization of the liquid at the proximity of the heat source is known as the Leidenfrost effect. This effect has been demonstrated when the surface is a solid or a liquid. Here, we explore the interaction between a liquid droplet and a hot granular material. The granular material is, by essence, a solid but adopts some behaviors similar to fluids. Surprisingly, we found that the droplet deeply penetrated the hot granular material. During the penetration process, the vapor production is so intense that the grains underneath the droplet are ejected. Consequently, the droplet moved downward under the action of gravity. A mechanism based on the Leidenfrost effect is proposed considering that the granular material can be modeled as a rough surface that can be eroded when the vapor speed is sufficient.

Figure 1

(a) 3D setup: A cylindrical container was filled with SiC grains (SiC-III). The temperature was controlled with a thermostatic plate. The penetration of the water droplet was recorded with a high speed camera under an angle of ${45}^{\circ }$. (b) A drop released onto a hydrophilic granular bed at $T_s={165}^{\circ }\mathrm{C}$ (regime I). The sand boiling effect was observed as a grain fountain nearly instantaneously occurred after contact between the droplet and grains. The droplet boiled near the interface of the bed. When the grain fountain settled (nearly 700 ms), small holes remained at the surface. They were found along the initial perimeter of the droplet and were regularly spaced. (as shown in the fourth image in panel (b) and Movie S2 [18]). (c) The grain bed ($d_g=$ 55 $\mu \mathrm{m}$) was heated up to a temperature of $T_s={300}^{\circ }\mathrm{C}$. Water was completely vaporized after 12 s. After the complete vaporization of the droplet, a large amount of grains was expelled across a long distance from the impact. A crater was then obtained after the full vaporization of the droplet (please refer to Movie S3 [18]). In both cases, panels (b) and (c), a spherical assembly of grains was found below the surface (see Fig. S5). (d) As a comparison, the same grains were coated with hydrophobic molecules ($T_s={300}^{\circ }\mathrm{C}$). Because of this treatment, the water droplet no longer penetrated the surface of the granular material (as shown in Movie S5 [18]).

Figure 2

(a) Side view of the 2D setup: A Hele-shaw cell was constructed and filled with SiC grains. One of the vertical walls was heated and maintained at a constant temperature, while the opposite wall consisting of sapphire enabled us to record the motion of the droplet below the surface with a high speed camera. (b) Front view of the 2D setup. (c) Time lapse of a typical penetration process (also shown in Movie S6 [18]). The temperature of the granular bed was $T_b$ = 300${}^{\circ }\mathrm{C}$. The pictures are separated by 2 s. The maximum digging depth was $H_{\text{max}}=39.0$ mm. The droplet moved at a constant speed until $t=10$ s. Above the droplet, the granular bed was fluidized along a vertical cylinder whose diameter corresponded to the diameter of the droplet until 10 mm below the surface. Then the fluidized granular part formed a funnel at the surface, i.e., the crater. The crater diameter increased over time. At $t>10$ s, the droplet no longer moved downward. The grains remained fluidized until the complete vaporization of water.

Figure 3

(a) Depth reached by the droplet as a function of time at the different temperatures between 200 and ${450}^{\circ }\mathrm{C}$. The legend is corresponding to the superheat $\mathrm{\Delta }T=(T_b-T^{*})$, where $T^{*}$=100${}^{\circ }\mathrm{C}$ is the boiling point of water. The grains are SiC-I grains (please refer to Supplemental Material Sec. S4 for information on the other types of grains [18]). Three steps are identified: the time to cross the interface ${\tau }_i$, the deep digging process during the time ${\tau }_d$ and the standing state during ${\tau }_s$. As shown in purple in panel (a), the different step durations are indicated in the case of $T_b={400}^{\circ }\mathrm{C}$. (b) Distribution of the duration of the different steps (${\tau }_i$, ${\tau }_d$, and ${\tau }_s$ are marked with the blue circles, blue triangles, and brown squares, respectively) during the digging process as a function of the superheat $\mathrm{\Delta }T$. The total duration of the process remains constant. The time to cross the interface increases with the superheat. In contrast, since the standing time (or equivalently ${\tau }_i+{\tau }_d$ linearly increases) linearly decreases with $\mathrm{\Delta }T$, the digging duration reaches its maximum value st $\mathrm{\Delta }T={250}^{\circ }$ C. (c) Based on the $H\left(t\right)$, the maximum depth reached by the droplet is determined. The blue squares, orange triangles and purple triangles indicate $H_{\text{max}}$ for the SiC-I, SiC-II, and SiC-III grains, respectively as a function of the temperature. (d) The speed $v_d$ during the digging process is shown as a function the granular bed superheat. The symbols are the same as those in panel (c). The speed continuously decreases with the temperature. The curves correspond to the model described in Sec. 4 considering $r_{\text{rug}}$=69.5 $\pm 0.5$, 71.5 $\pm 0.5$, and 75 $\pm 0.5\mu \mathrm{m}$ for the SiC-I, SiC-II, and SiC-III, respectively. The predicted lifetime of the droplet is presented in the inset as a function of the superheat.

Figure 4

(a) The digging droplet is modeled as a droplet surrounded by an air-grain mixed layer ejected under the action of the produced vapor. (b) Zoom of the side view of the film. The cyan spheres indicate the spheres (radius $r_{\text{rug}}$) adopted to represent the roughness. Heat is transferred via conduction [1]. The purple dashed line indicates the height of the vapor film calculated on an idealistic smooth surface [2]. This line intersects the rough spheres [3]. The intersection with the spheres induces a supplemental amount of vapor that allows us to calculate the speed of the vapor $v_v\left[4\right]$. The vapor then entrains grains [5]. (c) Top view of intersection between the liquid and rough spheres. The contact is indicated as purple discs.