Footprint geometry and sessile drop resonance

In this work, we examine experimentally the resonance of a sessile drop with a square footprint (square drop) on a flat plate. Two families of modal behaviors are reported. One family is identified with the modes of sessile drops with circular footprints (circular drop), denoted as “spherical modes.” The other family is associated with Faraday waves on a square liquid bath (square Faraday waves), denoted as “grid modes.” The two families are distinguished based on their dispersion behaviors. By comparing the occurrence of the modes, we recognize spherical modes as the characteristic of sessile drops, and grid modes as the constrained response. Within a broader context, we further discuss the resonance modes of circular sessile drops and free spherical drops, and we recognize various modal behaviors as surface waves under different extents of constraint. From these, we conclude that sessile drops resonate according to how wave-number selection by footprint geometry and capillarity compete. For square drops, a dominant effect of footprint constraint leads to grid modes; otherwise, the drops exhibit spherical modes, the characteristic of sessile drops on flat plates.

Figure 1

Modes on the free surfaces of liquid baths and sessile drops for circular and square boundary symmetries. The motivating question of this work is as follows: What are the modes of a vibrated sessile drop with a square footprint?

Figure 2

Schematic of our experimental platform [21, 22]. Shown in the red (rightmost) box are the key components: mesh pattern and LED light source under the drop. Light rays from LEDs are refracted by the drop's deforming surface, reflected into the high-speed camera by mirrors A, B, and C, and they convey a deformed mesh pattern to the computer, thereby visualizing the deformation of the drop's surface. A signal generator (not shown) oscillates the surface sinusoidally in the direction perpendicular to the plane of the surface.

Figure 3

Examples of a zonal, a sectoral, and two tesseral modes identified by their number of layer(s) $n$ and number of sector(s) $l$ [22]. For each mode, the side-view profile on the upper right and the top-view surface on the lower right are predicted by Bostwick-Steen theory [12]. The image on the lower left is an experimental top-view snapshot of the mode [22].

Figure 4

Top-view snapshots of circular footprint (top row) and square footprint (bottom row) mode shapes, with two columns per mode comparing full-period and half-period shapes. The represented modes are (1, 2), (1, 2)${}_s$, (1, 3), (1, 3)${}_s$, (1, 4), and (1, 4)${}_s$. Corresponding forcing frequencies and accelerations are shown in the table below. As a guide to the eye, yellow bars are superimposed to highlight the symmetry of the mode shapes. The footprints of circular drops are 5 mm in diameter. Each edge of a square drop's footprint is 4.43 mm. A video of these modes is supplied in the Supplemental Material [45].

Figure 5

Measured frequency bands of circular and square footprint modes as volume varies: (1, 2), (1, 2)${}_s$, (1, 3), (1, 3)${}_s$, (1, 4), and (1, 4)${}_s$ modes.

Figure 6

A partial catalog of observed grid modes paired with simulated patterns. For each pair, to the left is shown the mode shape on a pinned, flat square surface according to Eq. (1). The brightness indicates the height. To the right is shown the corresponding snapshot on a square drop, from experiment. The drops are $10-16\mu \mathrm{L}$. Among the observations, (1, 3), (1, 5), and (1, 7) are harmonic modes that oscillate at their forcing frequencies. All others are half-frequency subharmonics. Marked with an X as missing is the mode $(p,q)=(2,3)$ because it has never been observed in our experiments. A video of selected grid modes is provided in our Supplemental Material [45].

Figure 7

Dispersion relation of grid modes for square drops of 10–20 $\mu \mathrm{L}$. (a) Observed ($◯$) and predicted ($\mathrm{\Delta }$) frequencies. (b) The fitting parameter $\overline{h}$ for different drop volumes.

Figure 8

Exponent $\beta$ for sessile drop modes. Grid-Sqr: grid modes on square drops. Sectoral-Cir: sectoral modes on circular drops. Zonal-Cir: zonal modes on circular drops.

Figure 9

Continuum of rest shapes with corresponding resonance modes labeled below, from spherical harmonic waves on free drops (Rayleigh capillary waves [38]) to rectangular waves on flat surfaces in rectangular containers (Faraday waves [13]). The dashed line segments denote the transition between spherical and grid modes at contact angles of approximately ${75}^{\circ }-{85}^{\circ }$.

Figure 10

Variations of contact angle with the volume of a sessile drop. For square drops, the contact angle is calculated at the midpoint of a square drop's edge.