Nanometer-Resolved Collective Micromeniscus Oscillations through Optical Diffraction

We study the dynamics of periodic arrays of micrometer-sized liquid-gas menisci formed at superhydrophobic surfaces immersed into water. By measuring the intensity of optical diffraction peaks in real time, we are able to resolve nanometer-scale oscillations of the menisci with submicrosecond time resolution. Upon driving the system with an ultrasound field at variable frequency, we observe a pronounced resonance at a few hundred kilohertz, depending on the exact geometry. By modeling the system using the unsteady Stokes equation, we find that this low resonance frequency is caused by a collective mode of the acoustically coupled oscillating menisci.

Figure 1

(a) Schematic figure of a single micromeniscus. (b) Pattern of an array of micromenisci. Arrows denote the primitive translations and nearest neighbor distance $a$.

Figure 2

(a) Schematic figure of the experiment. (b) Inverted photograph of the diffraction pattern ($\vartheta \approx 60°$). Numbers indicate Miller indices of diffraction orders. (c) ac component of the light intensity measured in the $(1,0)$ diffraction order at the beginning of an ultrasound wave train.

Figure 3

(a) Time-resolved intensity of the $(1,0)$ diffraction order, corresponding to small ultrasound pressure ($\Delta P\approx 400\mathrm{Pa}$) (solid line) and large ultrasound pressure ($\Delta P\approx 800\mathrm{Pa}$) (dashed line). (b) Calculated intensity of the first diffraction order as a function of meniscus deflection for a rectangular surface profile with groove width $w=2R$ ($\vartheta =66°$). The data are displayed simultaneously in terms of the displacement $\zeta$ and the nondimensionalized curvature $\kappa R$. Both are related geometrically by $\zeta =2/\kappa (1-\sqrt{1-R^2{\kappa }^2/4})$. The equilibrium curvature ${\kappa }_{0}R=0.04$ corresponds to ${\zeta }_0=30\mathrm{nm}$.

Figure 4

Frequency response of an array of micromenisci with a hexagonal pattern, $a=15\mu \mathrm{m}$, and $R=3\mu \mathrm{m}$. Crosses show experimental data. The dashed-dotted line shows the theory for a single meniscus. Theoretical data for the array are displayed in terms of the mean (solid line) and the root mean square deflection (dashed line).

Figure 5

Amplitude and phase of an array of micromenisci with a hexagonal pattern, $a=15\mu \mathrm{m}$, and $R=3\mu \mathrm{m}$ (a) at the fundamental collective mode (159 kHz) and (b) at the second collective mode (290 kHz). Note that the color scale for the phase is narrower in (a) as compared to (b).