# Making a splash with water repellency

Original entry: Sujit S. Datta, APPHY 225, Fall 2009.

## Reference

C. Duez, C. Ybert, C. Clanet, and L. Bocquet, Nature Physics 3, 180 (2007).

## Keywords

wetting, contact angle, splashing, superhydrophobic

## Key Points

What determines how strongly a solid body rushing into a liquid will splash? (For example, consider a sphere being dropped into water). This splash occurs when a pocket of air gets trapped in the fluid after the sphere rushes in, and intuitively, one imagines (correctly) that inertia plays a key role. The purpose of this work was to demonstrate that the surface wettability of the 'intruding' sphere also plays a fundamental role.

This is illustrated by a simple experiment: Duez and co-workers took two identical glass beads, coated one with a very thin layer of silane chains (to make it hydrophobic), and oxidized the other one in a strong etchant (to make it hydrophilic). When both were dropped in the same manner, the hydrophobic one made a larger splash, both visually and in the amplitude of the resulting sound. Furthermore, splashing only occurred for sufficiently large impact velocities - this "threshold" velocity in turn was observed to depend on the surface wettability of the sphere (it was larger for spheres that were more easily wet by the liquid), as well as a characteristic capillary velocity given by $\gamma/\mu$, where $\mu$ is the viscosity of the liquid.

But why does surface wettability matter? After all, these experiments are done in a regime where inertia dominates over any capillary effects (that is, the Weber number $\rho U^2 a/\gamma$ is large, where $\rho$ is fluid density, $U$ and $a$ are the sphere velocity and size, and $\gamma$ is the fluid surface tension). The answer, of course, lies in exactly how an air pocket is formed after the sphere falls into the water. A clear schematic is shown in figure 3c of the paper: when a sphere falls in, a thin liquid film is formed around it, "climbing" up its body as it continues to fall into the water. The extent to which this film "hugs" the sphere should in some way depend on how well the liquid wets the surface of the sphere: if the film "hugs" the sphere tightly, an air pocket is less likely to form. This explains why the threshold velocity increases with increasing wettability: it becomes harder for an air pocket to form.

What sets this threshold velocity, then? As tends to be the case in the physics of fluids, previous work by de Gennes leads the way. By considering the dynamics of the "triple line" at the intersection of the air, the surface of the sphere, and the surface of the water -- in particular, by balancing the force of traction pulling the air towards the liquid with the force due to the moving triple line -- de Gennes predicted a critical Capillary number ($Ca=\mu v/\gamma$, where $v$ is the speed of the triple line, typically on the order of $U$) for the hydrophobic case, with Ca proportional to the cube of $\pi-\theta$, where $\theta$ is the sphere-fluid contact angle. This suggests that for decreasing wettability ($\theta$ increasing), the critical capillary number (and hence the critical value of $U$) decreases - as expected. Indeed, this cubic dependence is observed in the experiments. Note that the idea here is similar to "forced" wetting that we considered in class - for example, the idea that for a solid immersed in a liquid, if the solid is taken out at sufficiently high velocity then it drags along with it a film of finite thickness. The only modification is that here we are considering a dewetting process: so the roles of the air and the fluid are reversed.

This isn't the entire story, however: one needs to modify this argument to incorporate viscous dissipation at the triple line (this is simply proportional to $\mu$ and $v$). When this is included into the picture, one obtains a threshold sphere velocity (in the hydrophobic case) of $U\sim(\pi-\theta)^{3}\cdot\gamma/\mu$. The data for the hydrophobic case supports this.

While a similar analytic solution does not exist for the hydrophilic case, similar arguments suggest a similar relation, with one modification: because in the case of a wettable sphere $\theta$ is small, it is likely that the critical speed is only weakly dependent on the contact angle -- as Duez and co-workers observe.