Substrate constraint modifies the Rayleigh spectrum of vibrating sessile drops

Anomalous Oscillation Modes of Superfluid Pendant Droplets

Droplets should exhibit various dynamical phenomena when adhered to a surface; not all of them are realized in classical fluids. Visualization of superfluid ^{4}He pendant droplets revealed that the droplets were horizontally translated on a flat surface, bouncing off at the corner, known as the Noether mode that reflects the translation symmetry. The droplets exhibited another mode in vertical oscillations with high amplitude that included oscillation of the droplet edge. The oscillation period remained constant even as the droplets grew, exhibiting an anomalously weak size dependence. The high mobility of the droplet edges owing to the superfluidity was a crucial factor for the appearance of these anomalous modes.

Advances in Microscale Droplet Generation and Manipulation

Microscale droplet generation and manipulation have widespread applications in numerous fields, from biochemical assays to printing and additive manufacturing. There are several techniques for droplet handling. Most techniques, however, can generate and work with only a limited range of droplet sizes. Furthermore, there are constraints regarding the workable variety of fluid properties (e.g., viscosity, surface tension, mass loading, etc.). Recent works have focused on developing techniques to overcome these limitations. This feature article discusses advances in this area that cover a wide range of droplet sizes from subpicoliter to microliter.

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.

Oscillation Dynamics of Drops on Immiscible Thin Liquid Films

While drop oscillation dynamics has been widely studied for many decades, the influence of a moving contact line on the oscillation modes of drops remains underexplored. Herein, we report the oscillation dynamics of drops on thin liquid films with different viscosities where lower viscosities provide a slipping surface and higher viscosities immobilize the contact line. A gently deposited drop onto an oil film undergoes shape oscillations due to capillarity, where the frequency, amplitude, and apparent contact angle are tracked via a high-speed camera. This study demonstrates that restraining the mobility of the drop contact line by increasing the viscosity of a thin oil film underneath the drop increases the extent of the drop oscillation time as well as affecting the natural frequency of the drop oscillation. The drop oscillation time was defined by the time at which the changes in the drop height dropped to values less than 1% of the equilibrium height. The experimental results for the first longitudinal mode oscillation frequencies as a function of the equilibrium contact angles for the pinning and slipping contact lines were in good agreement with previously reported numerical simulations and model predictions.

Sounds Stimulation on In Vitro HL1 Cells: A Pilot Study and a Theoretical Physical Model

Mechanical vibrations seem to affect the behaviour of different cell types and the functions of different organs. Pressure waves, including acoustic waves (sounds), could affect cytoskeletal molecules via coherent changes in their spatial organization and mechano-transduction signalling. We analyzed the sounds spectra and their fractal features. Cardiac muscle HL1 cells were exposed to different sounds, were stained for cytoskeletal markers (phalloidin, beta-actin, alpha-tubulin, alpha-actinin-1), and studied with multifractal analysis (using FracLac for ImageJ). A single cell was live-imaged and its dynamic contractility changes in response to each different sound were analysed (using Musclemotion for ImageJ). Different sound stimuli seem to influence the contractility and the spatial organization of HL1 cells, resulting in a different localization and fluorescence emission of cytoskeletal proteins. Since the cellular behaviour seems to correlate with the fractal structure of the sound used, we speculate that it can influence the cells by virtue of the different sound waves’ geometric properties that we have photographed and filmed. A theoretical physical model is proposed to explain our results, based on the coherent molecular dynamics. We stress the role of the systemic view in the understanding of the biological activity.

Nonspherical modes nondegeneracy of a tethered bubble

When excited at sufficiently high acoustic pressures, a wall-attached bubble may exhibit asymmetric nonspherical modes. These vibration modes can be decomposed over the set of spherical harmonics Y_{nm}(θ,ϕ) for a degree n and order m. We experimentally capture the time-resolved dynamics of asymmetric bubble oscillations in a top-view configuration. A spatiotemporal modal analysis is performed and allowed recovering the set of zonal (m=0), tesseral (0<m<n), and sectoral (m=n) spherical harmonics that develop at the bubble interface. The analysis of the surface instability thresholds reveals that the frequencies of asymmetric modes differ from the standard Lamb spectrum. In addition, the nondegeneracy of asymmetric modes for a given degree n is evidenced by noncompletely overlapping resonance bands. Finally, the coexistence between zonal and sectoral modes is analyzed through their modal interaction, amplitude interplay and relation of phase, as well as their geometric compatibility.

Excitation of Faraday-like body waves in vibrated living earthworms

Biological cells and many living organisms are mostly made of liquids and therefore, by analogy with liquid drops, they should exhibit a range of fundamental nonlinear phenomena such as the onset of standing surface waves. Here, we test four common species of earthworm to demonstrate that vertical vibration of living worms lying horizontally on a flat solid surface results in the onset of subharmonic Faraday-like body waves, which is possible because earthworms have a hydrostatic skeleton with a flexible skin and a liquid-filled body cavity. Our findings are supported by theoretical analysis based on a model of parametrically excited vibrations in liquid-filled elastic cylinders using material parameters of the worm's body reported in the literature. The ability to excite nonlinear subharmonic body waves in a living organism could be used to probe, and potentially to control, important biophysical processes such as the propagation of nerve impulses, thereby opening up avenues for addressing biological questions of fundamental impact.

Steiner triangular drop dynamics

Steiner's circumellipse is the unique geometric regularization of any triangle to a circumscribed ellipse with the same centroid, a regularization that motivates our introduction of the Steiner triangle as a minimal model for liquid droplet dynamics. The Steiner drop is a deforming triangle with one side making sliding contact against a planar basal support. The center of mass of the triangle is governed by Newton's law. The resulting dynamical system lives in a four dimensional phase space and exhibits a rich one-parameter family of dynamics. Two invariant manifolds are identified with "bouncing" and "rocking" periodic motions; these intersect at the stable equilibrium and are surrounded by nested quasiperiodic motions. We study the inherently interesting dynamics and also find that this model, however minimal, can capture space-time symmetries of more realistic continuum drop models.

Harmonic and subharmonic waves on the surface of a vibrated liquid drop

Liquid drops and vibrations are ubiquitous in both everyday life and technology, and their combination can often result in fascinating physical phenomena opening up intriguing opportunities for practical applications in biology, medicine, chemistry, and photonics. Here we study, theoretically and experimentally, the response of pancake-shaped liquid drops supported by a solid plate that vertically vibrates at a single, low acoustic range frequency. When the vibration amplitudes are small, the primary response of the drop is harmonic at the frequency of the vibration. However, as the amplitude increases, the half-frequency subharmonic Faraday waves are excited parametrically on the drop surface. We develop a simple hydrodynamic model of a one-dimensional liquid drop to analytically determine the amplitudes of the harmonic and the first superharmonic components of the linear response of the drop. In the nonlinear regime, our numerical analysis reveals an intriguing cascade of instabilities leading to the onset of subharmonic Faraday waves, their modulation instability, and chaotic regimes with broadband power spectra. We show that the nonlinear response is highly sensitive to the ratio of the drop size and Faraday wavelength. The primary bifurcation of the harmonic waves is shown to be dominated by a period-doubling bifurcation, when the drop height is comparable with the width of the viscous boundary layer. Experimental results conducted using low-viscosity ethanol and high-viscocity canola oil drops vibrated at 70Hz are in qualitative agreement with the predictions of our modeling.

Observation of the Resonance Frequencies of a Stable Torus of Fluid

We report the first quantitative measurements of the resonance frequencies of a torus of fluid confined in a horizontal Hele-Shaw cell. By using the unwetting property of a metal liquid, we are able to generate a stable torus of fluid with an arbitrary aspect ratio. When subjected to vibrations, the torus displays azimuthal patterns at its outer periphery. These lobes oscillate radially, and their number n depends on the forcing frequency. We report the instability "tongues" of the patterns up to n=25. These resonance frequencies are well explained by adapting to a fluid torus the usual drop model of Rayleigh. This approach could be applied to the modeling of large-scale structures arisen transiently in vortex rings in various domains.

Droplet motions fill a periodic table

Drawing parallels to the symmetry breaking of atomic orbitals used to explain the periodic table of chemical elements; here we introduce a periodic table of droplet motions, also based on symmetry breaking but guided by a recent droplet spectral theory. By this theory, higher droplet mode shapes are discovered and a wettability spectrometer is invented. Motions of a partially wetting liquid on a support have natural mode shapes, motions ordered by kinetic energy into the periodic table, each table characteristic of the spherical-cap drop volume and material parameters. For water on a support having a contact angle of about 60°, the first 35 predicted elements of the periodic table are discovered. Periodic tables are related one to another through symmetry breaking into a two-parameter family tree.

Axisymmetric and Nonaxisymmetric Oscillations of Sessile Compound Droplets in an Open Digital Microfluidic Platform

Manipulating droplets of biological fluids in an electrowetting on dielectric (EWOD)-based digital microfluidic platform is a significant challenge because of biofouling and surface contamination. This problem is often addressed by operating in an oil environment. We study an alternate configuration of sessile compound droplets having an aqueous core surrounded by a smaller oil shell. In contrast to the conventional EWOD platform, an open digital microfluidic platform enabled by the core-shell configuration will allow electrical, mechanical, or optical probes to get unrestricted access to the droplet, thus enabling highly flexible and dynamically reconfigurable lab-on-chip systems. Understanding droplet oscillations is essential as they are known to enhance mixing. To our knowledge, this is the first study of axisymmetric and nonaxisymmetric oscillations of compound droplets actuated using EWOD platforms. Mode shapes for both axisymmetric and nonaxisymmetric oscillations were studied and explained. Enhancement in the axisymmetric oscillation of the core by decreasing the shell volume was obtained experimentally and modeled theoretically. Smaller shell volumes reduce the damping losses, allowing the appearance of nonaxisymmetric modes over a larger range of operating parameters. The oscillation frequency regime for obtaining prominent nonaxisymmetric oscillations for different shell volumes was identified. Compound droplets provide a mechanism to reduce biofouling, sample contamination, and evaporation. We demonstrate axisymmetric and nonaxisymmetric oscillations of compound droplets with the biological core of red blood cells, providing crucial first steps for promoting applications such as rapid efficient assays, mixing of biological fluids, and fluidic photonics on hysteresis-free surfaces.

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.

Response of driven sessile drops with contact-line dissipation

A partially-wetting sessile drop is driven by a sinusoidal pressure field that produces capillary waves on the liquid/gas interface. Response diagrams and phase shifts for the droplet, whose contact-line moves with contact-angle that is a smooth function of the contact line speed, are reported. Contact-line dissipation originating from the contact-line speed condition leads to damping for drops with finite contact-line mobility, even for inviscid fluids. The critical mobility and associated driving frequency to generate the largest contact-line dissipation is computed. Viscous dissipation is approximated using the irrotational flow and the critical Ohnesorge number bounding regions beyond which a given mode becomes over-damped is computed. Regions of modal coexistence where two modes can be simultaneously excited by a single forcing frequency are identified. Predictions compare favorably to related experiments on vibrated drops.

How Natural Evaporation Temporally Self-Tunes an Oscillating Sessile Droplet To Resonate at Different Modes

We report the dynamics and underlying physics of evaporation driven transitions and autotuning of oscillation modes in sessile droplets subject to substrate perturbations. We have shown that evaporation controls temporal transition of the oscillation mode with a spatially downward shift of nodes (surface locations with zero displacement) toward the three-phase contact line. We have explained the physical mechanism using two parameters: the first quantifies evaporation driven tuning for resonance detection, and the second parameter characterizes mode lifetime which is found to be governed by evaporation dynamics. It is desirable to achieve autotuning of the oscillation modes in sessile droplets that essentially self-evolves in a spatiotemporal manner with continued evaporation. The insights suggest control of mode resonances is possible, which in turn will allow precision manipulations at droplet scale crucial for many applications such as surface patterning and others.

Controlling particle deposit morphologies in drying nano-particle laden sessile droplets using substrate oscillations

Sessile water droplets containing nano-silica particles are allowed to evaporate in the presence of driven substrate oscillations at chosen frequencies.

Nonlinear oscillations of a sessile drop on a hydrophobic surface induced by ac electrowetting

We examine the nature of ac electrowetting (EW)-driven axisymmetric oscillations of a sessile water drop on a dielectric substrate. In ac EW, small-amplitude oscillations of a drop differ from the Rayleigh linear modes of freely oscillating drops. In this paper, we demonstrate that changes in the time-averaged contact angle of the sessile drop attributed to the presence of an electric field and a solid substrate mainly caused this discrepancy. We combine the domain perturbation method with the Lindsted-Poincaré method to derive an asymptotic formula for resonant frequency. Theoretical analysis shows that the resonant frequency is a function of the time-averaged contact angle. Each mode of the resonance frequency is a linear function of ɛ(1), which is the magnitude of the cosine of the time-averaged contact angle. The most dominant mode in this study, that is, the fundamental mode n=2, decreases linearly with ɛ(1). The results of the theoretical model are compared with those of both the experiments and numerical simulations. The average resonant frequency deviation between the perturbation solutions and numerical simulations is 4.3%, whereas that between the perturbation solutions and the experiments is 1.8%.

Electrowetting-driven oscillating drops sandwiched between two substrates

Drops sandwiched between two substrates are often found in lab-on-chip devices based on digital microfluidics. We excite azimuthal oscillations of such drops by periodically modulating the contact line via ac electrowetting. By tuning the frequency of the applied voltage, several shape modes can be selected one by one. The frequency of the oscillations is half the frequency of the contact angle modulation by electrowetting, indicating a parametric excitation. The drop response to sinusoidal driving deviates substantially from sinusoidal behavior in a "stop and go" fashion. Although our simple theoretical model describes the observed behavior qualitatively, the resonances appear at lower frequencies than expected. Moreover, the oscillations produce a nonperiodic fluid transport within the drop with a typical velocity of 1 mm/s. In digital microfluidic devices, where the typical drop size is less than 1 mm, this flow can result in very fast mixing on the spot.