Porous Superhydrophobic Membranes: Hydrodynamic Anomaly in Oscillating Flows

We have fabricated and characterized a novel superhydrophobic system, a meshlike porous superhydrophobic membrane with solid area fraction ${\Phi }_s$, which can maintain intimate contact with outside air and water reservoirs simultaneously. Oscillatory hydrodynamic measurements on porous superhydrophobic membranes as a function of ${\Phi }_s$ reveal surprising effects. The hydrodynamic mass oscillating in phase with the membranes stays constant for $0.9\lesssim {\Phi }_s\le 1$, but drops precipitously for ${\Phi }_s<0.9$. The viscous friction shows a similar drop after a slow initial decrease proportional to ${\Phi }_s$. We attribute these effects to the percolation of a stable Knudsen layer of air at the interface.

Figure 1

(a) Top view of a porous membrane chip ($a\times a=600\times 600\mu {\mathrm{m}}^2$). (b),(c) Optical micrographs of ${\Phi }_s=0.78$ and 0.34 membranes, respectively. (d) A drop of water placed on a larger membrane ($a\times a=2\times 2{\mathrm{mm}}^2$ and ${\Phi }_s=0.48$) showing the superhydrophobicity of the surface. (e) Vacuum mechanical resonances of a $600\times 600\mu {\mathrm{m}}^2$ porous membrane with ${\Phi }_s=0.34$. Nearly degenerate modes ($m$, $n$) and ($n$, $m$) are observed when $m\ne n$. Single standard deviations in the data are smaller than the symbols.

Figure 2

(a) Schematic of the measurement cell in cross-sectional and isometric views. The cell is filled with water and is placed on top of the membrane chip. A heterodyne Michelson interferometer probes the motion from below. (b) Thermal noise spectra of the fundamental mode of three different membranes with water atop. From top to bottom, ${\Phi }_s=0.48$, 0.6, and 1. The frequency axis is normalized with the respective resonance frequencies in water. The inset shows the shape of the fundamental mode for the ${\Phi }_s=0.6$ membrane. (c) Fundamental-mode resonance frequencies in vacuum and with water atop, and linewidths ${\gamma }_w/2\pi$ with water atop. Error bars represent the associated single standard deviations and are only shown when larger than the symbols.

Figure 3

(a) Measured $M_w$ as a function of ${\Phi }_s$. The dashed line segment is the hydrodynamic mass of the entire water layer. (b) Average friction force and (c) the normalized friction coefficient. The plate prediction is calculated from Eq. (1) using experimental velocities and frequencies where needed. The normalized friction coefficient for the plate model is $\approx {\Phi }_s$. Error bars represent the associated single standard deviations and are only shown when larger than the symbols.