Acoustogravitational balance in climbing films

We study the contributions of acoustical and gravitational stresses to the dynamic and static wetting states of liquid atop a vertical substrate, which is in contact with a liquid reservoir, as a model for the acoustic-enhanced coating process. The propagation of MHz-frequency surface acoustic waves (SAWs) on the substrate and into a liquid reservoir gives rise to the continuous climb of a thin film of oil or to the finite climb of a water meniscus. Theory indicates that the local thickness of the climbing oil film balances the gravitational stress by supporting resonance-enhanced radiation pressure. Indeed, the measured rate of oil climb over the substrate appears to be independent of gravity and insensitive to the height of the liquid film above the reservoir. It is solely determined by a balance between acoustical and viscous stresses in the liquid. However, the partially wetting water meniscus is curved and does not support acoustical resonance and corresponding high values of radiation pressure. The meniscus climbs over the solid substrate to a finite height, which is determined from an interplay between acoustical, capillary, and gravitational stresses.

Figure 1

(a) A schematic sketch of the experimental system and (b) a magnification therein of the SAW device (side projection), where a SAW propagates into a liquid reservoir along a vertical solid substrate, supporting a climbing liquid film/meniscus against the action of gravity.

Figure 2

Images corresponding to the sketch in Fig. 1 and showing (a)–(c) the steady menisci formed by 8 mM surfactant solutions of water at different voltage levels (given in terms of $U)$ and (d)–(f) a sequence of temporal frames of a silicon oil film $(\nu =50$ cSt) climbing atop a vertical substrate, where the red dotted lines highlight the three-phase contact line of the climbing liquid and the scale bar is 5 mm in length. Note that the liquids climb solely along the part of the substrate under the IDT, which supports the downward-propagating SAW, and do not climb elsewhere on the substrate, where the SAW is absent (the IDT is shown as a silver square at the top of each image).

Figure 3

SAW velocity amplitude $\left(U\right)$ variations of the steady climbing height, $\mathrm{\Delta }L$, by the menisci of water and surfactant solutions atop the vertical substrate (relative to the menisci height at rest) for different surfactant concentrations $C$, where experimental data are given as symbols (characteristic error of measurement: 0.037 mm) and theory is given as continuous lines in corresponding colors. Moreover, in experiment $\mathrm{\Delta }L=0$ for $U\le 54$ mm/s in all cases due to contact angle hysteresis. We account for contact angle hysteresis in theory phenomenologically by shifting the theoretical prediction to the right (on the $x$-axis) by the threshold value of $U=54$ mm/s, which is required to overcome contact angle hysteresis in experiment. Inset: A 2D sketch of the liquid meniscus climbing atop the vertical solid substrate.

Figure 4

(a) Vertical velocity of the climbing oil films, $V$, vs the normal SAW velocity amplitude, $U$, for different oil viscosities, where the dots are experimental data (characteristic error of measurement: 0.0017 mm/s), which are connected by dashed lines to guide the eye, and the corresponding continuous curves are obtained according to the theoretical formula. (b) Normalized interface height, $L/(k{U}^2/4g)$, vs the normalized thickness of the climbing film, $h_0/{\lambda }_l$.