Substrate constraint modifies the Rayleigh spectrum of vibrating sessile drops

In this work, we study the resonance behavior of mechanically oscillated, sessile water drops. By mechanically oscillating sessile drops vertically and within prescribed ranges of frequencies and amplitudes, a rich collection of resonance modes are observed and their dynamics subsequently investigated. We first present our method of identifying each mode uniquely, through association with spherical harmonics and according to their geometric patterns. Next, we compare our measured resonance frequencies of drops to theoretical predictions using both the classical theory of Lord Rayleigh and Lamb for free, oscillating drops, and a prediction by Bostwick and Steen that explicitly considers the effect of the solid substrate on drop dynamics. Finally, we report observations and analysis of drop mode mixing, or the simultaneous coexistence of multiple mode shapes within the resonating sessile drop driven by one sinusoidal signal of a single frequency. The dynamic response of a deformable liquid drop constrained by the substrate it is in contact with is of interest in a number of applications, such as drop atomization and ink jet printing, switchable electronically controlled capillary adhesion, optical microlens devices, as well as digital microfluidic applications where control of droplet motion is induced by means of a harmonically driven substrate.

Figure 1

Eigenmodes of hemispherical drops predicted by the theory of Rayleigh and Lamb are spherical harmonics. Zonal modes are in row 1 ($l$ $=$ 0), sectoral modes are along the diagonal ($k$ $=$ $l$), and tesseral modes are where $k$ ≠ $l$. Note that the shape at (0,0) is the static drop shape.

Figure 2

Schematic of imaging platform. 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 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 zonal, sectoral, and tesseral modes, where the numbers in the brackets are degree $k$ and order $l$ of the associated Legendre function specified in Eq. (2), according to which the 3D surfaces are rendered in rows 1 and 2. Images in rows 3 and 4 are top-view snapshots from experiments. The two snapshots for each mode differ temporally by one half-period of oscillation.

Figure 4

Demonstration of distinction between harmonic and subharmonic modes: Zonal modes exhibit $f_d\approx f_o$ and therefore are harmonic modes. In contrast, the sectoral and tesseral modes exhibit $f_o\approx 0.5f_d$ and are subharmonic modes.

Figure 5

Summary of mode shapes observed in experiments. For reference, the picture for ($n$,$l$) $=$ (1,0) is the top view of a static drop. Modes are arranged by physical appearance and nature frequency responses (i.e., subharmonic and harmonic): As partitioned by the double vertical line, zonal modes ($l$ $=$ 0) are axisymmetric and harmonic modes; others are subharmonic modes lacking axisymmetry. Sectoral modes only differ from tesseral modes by having $n$ $=$ 1.

Figure 6

Comparison of normalized resonance frequencies and frequency scaling prediction by Rayleigh and Lamb [1] [(a)–(c)] and Bostwick and Steen [(d)–(f)]: (a), (d) zonal modes; (b), (e) sectoral modes; and (c), (f) tesseral modes with $k$ $=$ $l$+2.

Figure 7

Image sequence of a relatively pure (5,3) mode. The image of $t$ $=$ 1.75 ms corresponds to that with ($n$,$l$) $=$ (2,3) in Fig. 5.

Figure 8

Image sequence of a ($k$,$l$) $=$ (5,3) mode mixing with a ($k$,$l$) $=$ (8,0) mode. The (8,0) appears twice in one period of the (5,3) and oscillates with the same frequency as the driving signal. The (8,0) is boxed to highlight that mode. The shape is confirmed by comparing the boxed images to that of ($n$,$l$) $=$ (5,0) in Fig. 5.

Figure 9

Examples of measured accelerations. Driving and sampling frequencies, $f_d$ and $f_s$, are (a) $f_d$ $=$ 573 Hz, $f_s$ $=$ 25 kHz; (b) $f_d$ $=$ 658 Hz, $f_s$ $=$ 10 kHz.

Figure 10

Comparison of $z$[$i$] defined in Eq. (A1) and any inner-product-based norm: (a) both detect pattern translation, but (b) only $z$[$i$] detect brightness variation.

Figure 11

An example of estimating $N$ and $N_c$ with Eq. (A2): (a) two notches in the plot gives $N$ $=$ 2; (b) the position of the second notch yields $N_c$ $=$ 175. Together with the 5000-Hz frame rate of the corresponding image sequence, the resonance frequency is 5000×2/175 $=$ 57.14 Hz.

Figure 12

Demonstration of harmonic and subharmonic modes: $f_o/f_d=1$ (harmonic) for all zonal modes, and 0.5 (subharmonic) otherwise.

Figure 13

Direct comparison of experimental data with the Rayleigh and Lamb theory according to Eq. (5). The comparison yields a uniform underestimation by the theory.

Figure 14

Additional results of comparing experimental data to theories: (a)–(c) comparison with the Rayleigh and Lamb theory using the normalized scheme; (d)–(f) direct comparison to the Bostwick and Steen theory.

Figure 15

Image sequence of a ($k$,$l$) $=$ (7,7) mode mixing with a ($k$,$l$) $=$ (10,0). The latter appears twice in one period of the former and oscillates with the same frequency as the driving signal, and hence is boxed to highlight. The shape is confirmed by comparing the boxed images to that of ($n$,$l$) $=$ (6,0) in Fig. 5, as the former possess the same number (=3) of circular regions where the mesh pattern is magnified (peaks) as the pattern is miniaturized (troughs) in the latter.

Figure 16

Image sequence of a ($k$,$l$) $=$ (8,6) mode mixing with a ($k$,$l$) $=$ (12,0). The latter is boxed for the same reason as the (10,0) in Fig. 15. The shape is confirmed by comparing the boxed images to that of ($n$,$l$) $=$ (7,0) in Fig. 5.