Oscillations of Drops with Mobile Contact Lines on the International Space Station: Elucidation of Terrestrial Inertial Droplet Spreading

We analyze shape oscillations of sessile water drops with fully mobile contact lines (CL) aboard the International Space Station. The unique microgravity environment enables the study of centimeter-sized droplets with associated inertial-capillary motions. Plane-normal substrate vibrations induce resonance behaviors quantified by frequency scans from which the natural frequencies and mode shapes are identified for nine different hydrophobic surfaces. Experimental observations agree well with, and validate, a recent spectral prediction of mobile CL sessile drop oscillations. The experimental findings help elucidate terrestrial droplet inertial spreading, a poorly understood phenomenon pervasive in many processes.

Figure 1

(a) Definition sketch of driven sessile drop with base radius $a$ and contact-angle $\alpha$ whose contact line obeys (b) a dynamic contact-line law which relates $\alpha$ to the contact-line speed $u_{\mathrm{CL}}$ through the contact-line mobility parameter $\mathrm{\Lambda }$ and advancing ${\alpha }_a$ and receding ${\alpha }_r$ contact angles. (c) Under resonant conditions, the drop will oscillate with characteristic frequency $\mathrm{\Omega }$ and shape defined by the mode number pair $[k,l]$ classified using spherical harmonic terminology into zonal [6,0], tesseral [6,2], and sectoral [6,6] shapes. (d) Theoretical prediction of the scaled oscillation frequency $\widehat{\mathrm{\Omega }}$ for the first three axisymmetric modes against mobility $\mathrm{\Lambda }$ for $\alpha =105°$ illustrates the limiting cases of the fully mobile $\mathrm{\Lambda }=0$ and pinned $\mathrm{\Lambda }=\infty$ contact line, with the shadowed region denoting the experimental parameter range.

Figure 2

Droplet driven at 1.6 Hz oscillating in the [2,0] mode on substrate $F_7$ exhibits a freely moving contact line.

Figure 3

Experimental schematic.

Figure 4

Resonance identification from overlaid images for sequential driving frequencies. Top row: the [4, 0] mode persists over the entire frequency range exhibiting maximal extension at resonance $f_d=4.0\mathrm{Hz}$ for substrate $F_4$. Bottom row: the [5, 1] mode exists only in a small frequency window at $f_d=6.2\mathrm{Hz}$ that lies between axisymmetric shapes for substrate $F_5$.

Figure 5

Experimentally-observed modes $[k,l]$ overlaid throughout an oscillation cycle.

Figure 6

Frequency (Hz) against static contact angle $\alpha (°)$ contrasting experiment (symbols) with theoretical predictions with free contact lines $\mathrm{\Lambda }=0$ (solid lines) for modes $[k,l]$ on substrates $F_1-F_8$. Inset: a plot of frequency $\mathrm{\Omega }$ for substrate $P_1$ for pinned drops. Experimental error $\pm 0.2\mathrm{Hz}$ is defined by the symbol size.