Numerical investigation of vibration-induced droplet shedding on microstructured superhydrophobic surfaces

The vibration-induced droplet shedding mechanism on microstructured superhydrophobic surfaces was simulated using the lattice Boltzmann method. The numerical simulations of natural droplet oscillations for various surface structures show that the natural frequency of the droplet is strongly dependent on surface morphology. The results show good agreement with basic theoretical values. Furthermore, simulations of the motion of the droplet subjected to vertical surface vibration demonstrate that droplets in the Cassie wetting state are easily removed from the surface, whereas for Wenzel state droplets, pinch-off occurs and only partial removal is possible. Microstructure spacing was found to be a key factor in the shedding process. On a surface with small microstructure spacing, the increased surface adhesion leads to a decrease of droplet departure velocity. In contrast, for large roughness spacing, the droplet is impaled on the microstructures, which causes the departure velocity to decrease. Reperforming the simulations under different vibration intensities reveals that as the vibration amplitude is increased, the optimum frequency for droplet removal decreases. The findings of this study shed light on the underlying mechanisms involved in forced vibrations of droplets and can be helpful in engineering applications in which droplet shedding processes are critical.

Figure 1

Dimensionless maximum droplet height $h_m{R}^{-1}$, at different prescribed equilibrium contact angles ${\theta }_Y$. The solid line represents the theoretical droplet height given by [49], and the LBM predicted droplet height in the present work is shown by triangular symbols.

Figure 2

Snapshots of the interface in the Rayleigh-Taylor instability problem.

Figure 3

Temporal evolution of the interface in the Rayleigh-Taylor instability problem, $\mathrm{Re}=3000$, $\mathrm{Ca}=0.26$, $\mathrm{Pe}=1000$, ${\rho }_h/{\rho }_l=3$, and ${\eta }_h/{\eta }_l=1$. The light and heavy fluid front positions are shown by the dashed line and the solid line, respectively. The front position is equal to $y\left(x=0\right)/l_0$ for the light fluid and $y\left(x=l_0/2\right)/l_0$ for the heavy fluid.

Figure 4

Schematic of the problem setup, displaying the boundary conditions and important geometric features.

Figure 5

Dimensionless droplet displacement $Y^{*}$ in free vibration for different droplet radii on a microstructured surface immediately after applying the impulse. The droplet is in the Cassie wetting state, $W^{*}=30$, and $P^{*}=0.117$.

Figure 6

Droplet displacement vs dimensionless time on microstructured and flat surfaces for $R=600\mu \mathrm{m}$ ($\mathrm{Oh}=4.8\times {10}^{-3}$). The contact angle is $\theta =147.8^{\circ }$ for the microstructured and $\theta ={145}^{\circ }$ for the flat case. The dashed lines are exponential fits which show the vibration decay rate. The inverse of peak-to-peak time (period) is equal to the resonance frequency of the droplet.

Figure 7

Square of the resonance frequency vs the inverse of the droplet mass for different surface morphologies. The dashed line represents the theory [Eq. (23))] obtained by Noblin et al. [59] for the $j=2$ vibration mode.

Figure 8

Temporal evolution of the dimensionless droplet displacement and velocity. The dimensionless droplet displacement $Y^{*}$ is shown by the blue line with square symbols, and the dimensionless droplet velocity $U^{*}$ and wall velocity $U_w^{*}$ are represented by the red line with triangular symbols and the dash-dotted line, respectively. The dashed line segments show that the droplet is not in contact with the surface.

Figure 9

Comparison of droplet response to vertical vibration for different wetting states. The prescribed contact angle is ${\theta }_{\mathrm{eq}}={100}^{\circ }$ and $\mathrm{Oh}=4.8\times {10}^{-3}\left(R=600\mu \mathrm{m}\right)$, $A^{*}=0.2$, and $\mathrm{St}=0.225$ in all cases. (a) The droplet is in the Cassie state and easily detaches from the surface. (b) The Wenzel state droplet is compressed and stretched, but detachment does not occur. The triple line is stationary due to the pinning effect. (c) On the flat surface the droplet shows similar behavior to (b), but the triple line is free to move.

Figure 10

Maximum droplet stretch at different dimensionless vibration amplitudes. For $A^{*}=1$, the negative curvature of the droplet causes pinch-off to occur.

Figure 11

Snapshots of the pinch-off phenomenon for the Wenzel state droplet at $A^{*}=1$ and $\mathrm{St}=0.225$. At $t^{*}=2.2$, necking starts to happen, which eventually leads to pinch-off. As seen in $t^{*}=4.7$, the contact diameter remains unchanged due to pinning effects.

Figure 12

Effects of vibration amplitude and frequency on dimensionless droplet departure velocity. $\mathrm{Oh}=4.8\times {10}^{-3}$, $W^{*}=30$, and $P^{*}=0.117$. $\mathrm{S}{\mathrm{t}}_r$ is the Strouhal number corresponding to the droplet resonance frequency, $f_r$, obtained in Sec. 4a.

Figure 13

Impact of microstructure pitch on the droplet departure velocity. $\mathrm{Oh}=4.8\times {10}^{-3}$, $W^{*}=30$, $A^{*}=0.5$, and $\mathrm{St}=0.225$. The dimensionless departure velocity is shown by red triangular symbols and the phase difference between the droplet movement and the surface movement is represented by blue circles.

Figure 14

The dimensionless droplet departure velocity vs the Ohnesorge number. $W^{*}=30$, $P^{*}=0.117$, $A^{*}=0.5$, and $\mathrm{St}=\mathrm{S}{\mathrm{t}}_r$.