Rolling and slipping of droplets on superhydrophobic surfaces

The leaves of many plants are superhydrophobic, a property that may have evolved to clean the leaves by encouraging water droplets to bead up and roll off. Superhydrophobic surfaces can also exhibit reduced friction, and liquids flowing over such surfaces have been found to slip in apparent violations of the classical no-slip boundary condition. Here we introduce slip into a model for rolling droplets on superhydrophobic surfaces and investigate under what conditions slip might be important for the steady-state motion. In particular, we examine three limiting cases in which dissipation in the rolling droplet is dominated by viscous dissipation, surface friction, or contact-line dissipation. We find that in molecular-dynamics simulations of droplets on ideal superhydrophobic surfaces with large effective slip lengths, contact-line dissipation dominates droplet motion. However, on real leaves, droplet motion is likely to be dominated by viscous shear, and slip, for the most part, can be neglected.

Figure 1

A droplet in the Cassie state on a superhydrophobic surface. Although the droplet radius is below the capillary length $R\ll {\kappa }^{-1}$, the droplet is depressed by a distance $\delta$ by gravity (as gravitational potential energy is converted to surface energy) to form a contact zone with the surface of radius $\ell$.

Figure 2

A droplet rolling down a superhydrophobic surface tilted at an angle $\alpha$ to the horizontal with respect to the direction of the body force $m\vec{g}$. The center-of-mass velocity of the droplet, $U$, is assumed to be equal to the sum of rolling velocity $U_r$ and any slip velocity $U_s$.

Figure 3

The height $h$, width $w$, and period $L$ of the surface features for the ridged (a) and pillared (b) surfaces are presented above, with all values given in terms of the lattice spacing $\sigma$. The particle diameter is shown as $1\sigma$, matching the lattice spacing. Part (c) shows a snapshot of a simulated droplet ($R=10\sigma$) rolling in the Cassie state on a ridged superhydrophobic surface. Although the droplet radius is below the capillary length $R\ll {\kappa }^{-1}$, the droplet is compressed a distance $\delta$ by gravity (gravitational potential energy is converted to surface energy) to form a contact zone with the surface of radius $\ell$, spanning multiple surface features. The image is rotated so that the applied body force is directed vertically down.

Figure 4

The time-averaged velocity field (relative to the center-of-mass velocity) and average velocity profile for a droplet ($R=16\sigma$) in steady-state motion on a ridged surface with $g=5.7\times {10}^{-4}\sigma {\tau }^{-2}$ and ${\epsilon }_{\text{SL}}=0.07\epsilon$. A straight line has been fitted to the top half of the profile to calculate the rolling velocity $U_r$ (projected toward the surface for reference). A nonzero slip velocity $U_s$ is evident at the pattern surface (indicated by the solid line at $z=2\sigma$).

Figure 5

The evolution of $U$, $U_r$, and $U_s$ over the course of a single simulation run for a $16\sigma$ droplet on a ridged surface is shown. Each point was calculated using averages of 100 consecutive frames of data, with frames generated every 200 time steps. While $U$ remains reasonably consistent, varying by up to 5% relative to the mean, $U_r$ (and by extension $U_s$) varies by as much as 30%.

Figure 6

The velocity profiles of droplets with initial radii of $11\sigma$ (blue circles), $16\sigma$ (green triangles), and $20\sigma$ (orange squares) in steady-state motion on a ridged surface with $g=5.7\times {10}^{-4}\sigma {\tau }^{-2}$ and ${\epsilon }_{\text{SL}}=0.07\epsilon$. Straight lines have been fitted to the velocities as in Fig. 4, illustrating deviations from a purely rolling motion. The resulting shear becomes more apparent as the droplet size increases.

Figure 7

Plot of time-averaged droplet center-of-mass velocities $U$ vs the various droplet radii on a ridged surface with $g=5.7\times {10}^{-4}\sigma {\tau }^{-2}$ and ${\epsilon }_{\text{SL}}=0.07\epsilon$. Droplets larger than the radius indicated by the right-hand vertical line are larger than the capillary length ${\kappa }^{-1}$ and so they can be expected to deform and depart from spherical shape assumed by the model. Error bars are determined by the standard error associated with the $U$ values taken from 4000 simulation frames, and $U_s$ and $U_r$ as constructed from 40 100-frame bins.

Figure 8

A log-log plot of the steady-state center-of-mass velocities for droplets with radii fixed at $16\sigma$ and ${\epsilon }_{\text{SL}}=0.07\epsilon$, as $g$ is increased from $5.7\times {10}^{-5}$ to $5.7\times {10}^{-3}\sigma {\tau }^{-2}$. The simulations were performed on the pillared surface. The apparent linear region between $\sim 2.3\times {10}^{-4}\sigma {\tau }^{-2}$ and $1.14\times {10}^{-3}\sigma {\tau }^{-2}$ corresponds to a power law $U\propto g^{\beta }$ with an exponent $\beta \approx 0.59$. Error bars are calculated as in Fig. 7.

Figure 9

The steady-state center-of-mass droplet velocities scaled by radius for radii between $10\sigma$ and $18\sigma$ on a ridged surface with ${\epsilon }_{\text{SL}}=0.07\epsilon$, as $g$ is increased from $1.14\times {10}^{-4}$ to $5.7\times {10}^{-4}\sigma {\tau }^{-2}$. The log-log plot confirms the $R/{\kappa }^{-1}$ scaling of the center-of-mass velocity for droplets smaller than the capillary length. The standard errors, calculated as in Fig. 7, are too small to be visible.

Figure 10

$U$ and $U_s$ are found to decrease as the strength of the liquid interaction with the ridged surface, characterized by ${\epsilon }_{\text{SL}}$, grows larger. Here, $g$ is $5.7\times {10}^{-4}\sigma {\tau }^{-2}$ and $R$ is $16\sigma$. Error bars are calculated as in Fig. 7.

Figure 11

The estimated effective slip length $b\sim \frac{R{U}_s}{U_r}$ increasing with increasing droplet radius calculated for the $U_s$ and $U_r$ data shown in Fig. 7 (on a ridged surface, with $g=5.7\times {10}^{-4}\sigma {\tau }^{-2}$ and ${\epsilon }_{\text{SL}}=0.07\epsilon$). We interpret the increase in effective slip length with $\epsilon$ as a decreasing contribution of the contact-line friction as the droplet size increases.