Dynamics of liquid films exposed to high-frequency surface vibration

We derive a generalized equation that governs the spreading of liquid films under high-frequency (MHz-order) substrate vibration in the form of propagating surface waves and show that this single relationship is universally sufficient to collectively describe the rich and diverse dynamic phenomena recently observed for the transport of oil films under such substrate excitation, in particular, Rayleigh surface acoustic waves. In contrast to low-frequency (Hz- to kHz-order) vibration-induced wetting phenomena, film spreading at such high frequencies arises from convective drift generated by the viscous periodic flow localized in a region characterized by the viscous penetration depth ${\beta }^{-1}\equiv {(2\mu /\rho \omega )}^{1/2}$ adjacent to the substrate that is invoked directly by its vibration; $\mu$ and $\rho$ are the viscosity and the density of the liquid, respectively, and $\omega$ is the excitation frequency. This convective drift is responsible for driving the spreading of thin films of thickness $h\ll k_l^{-1}$, which spread self-similarly as $t^{1/4}$ along the direction of the drift corresponding to the propagation direction of the surface wave, $k_l$ being the wave number of the compressional acoustic wave that forms in the liquid due to leakage of the surface wave energy from the substrate into the liquid and $t$ the time. Films of greater thicknesses $h\sim k_l^{-1}\gg {\beta }^{-1}$, in contrast, are observed to spread with constant velocity but in a direction that opposes the drift and surface wave propagation due to the attenuation of the acoustic wave in the liquid. The universal equation derived allows for the collective prediction of the spreading of these thin and thick films in opposing directions.

Figure 1

Schematic illustration of the spreading observed for liquids on high-surface-energy substrates vibrating at high frequencies under Rayleigh surface acoustic wave (SAW) excitation generated by interdigitated transducer (IDT) electrodes deposited on a piezoelectric substrate. (a) Thin submicron films spread along the propagation direction of the SAW, whereas films of intermediate thicknesses spread in the other direction, opposing that of the SAW propagation. (b) Sessile drops (or thick submillimeter films), on the other hand, advance along the SAW propagation direction.

Figure 2

Schematic of the two-dimensional liquid film under excitation by surface waves propagating along the solid substrate.

Figure 3

Schematic illustrations of (a) a thin film ($h^{*}\lesssim {\beta }^{{*}^{-1}}\ll k_l^{{*}^{-1}}$) and (b) a thick film ($h^{*}\lesssim k_l^{{*}^{-1}}$), indicating where the SW penetrates the liquid and is attenuated. $d{x}_0/dt$ denotes the Eulerian film spreading velocity at the film's leading edge. Note that the attenuation of the SAW is negligible for the case of the thin film.

Figure 4

Dimensionless acoustic radiation pressure and its gradient as a function of the scaled film thickness.

Figure 5

Variation in the linear film thickness growth rate $\sigma$ with the disturbance wave number $n$ for positive values in the acoustic radiation pressure slope (${\partial }_h{p}_r{|}_{h=h_0}>0$), which renders the film thickness unstable ($\sigma >0$) under sufficiently small disturbance wave numbers ($n/\sqrt{{\partial }_h{p}_r{|}_{h=h_0}}<1$); we exclude complex contributions to $\sigma$.