Vibration spectroscopy of a sessile drop and its contact line

Resonance Modes of Water Drops Pinned to a Vibrating Rectangular Post

We studied the effects of vertical vibrations on a water drop that was pinned to the sharp edges of a rectangular post. By varying the frequency and amplitude of the vertical displacement, distinct resonance peaks were observed using a simple optical technique. The vibrational spectra of the first two modes exhibited two closely spaced peaks, which corresponded to standing waves that exist along the major and minor contour lengths of the drops. The values of the resonance frequencies can be explained rather well by a simple model, which was originally proposed for axially symmetric drops.

Air/Water Interface Rheology Probed by Thermal Capillary Waves

Atomic force microscopy (AFM) was used to study the interfacial rheology of air/water interfaces by investigating the thermal capillary fluctuations of surfactant-loaded interfaces. These interfaces are formed by depositing an air bubble on a solid substrate immersed in a surfactant (Triton X-100) solution. An AFM cantilever, in contact with the north pole of the bubble, probes its thermal fluctuations (amplitude of the vibration versus the frequency). The measured power spectral density of the nanoscale thermal fluctuations presents several resonance peaks corresponding to the different vibration modes of the bubble. The measured damping versus the surfactant concentration of each mode presents a maximum and then decreases to a saturation value. The measurements are in good agreement with the model developed by Levich for the damping of capillary waves in the presence of surfactants. Our results show that the AFM cantilever in contact with a bubble is a powerful tool to probe the rheological properties of air/water interfaces.

Surface Tension of the Oxide Skin of Gallium-Based Liquid Metals

Gallium-based alloys have garnered considerable attention in the scientific community, particularly as they are in an atypical liquid state at and near room temperature. Though physical parameters, such as thermal conductivity, electrical conductivity, viscosity, yield stress, and surface tension, of these alloys are broadly known, the surface tension (surface free energy) of the oxide skin remains intangible due to the high yield stress of the oxide skin. In this article, we propose to employ gradually attenuated vibrations to obtain equilibrium shapes, which are analyzed along the lines of the puddle height method. The surface tension of the oxide skin was determined on quartz glass and liquid metal-phobic diamond coating to be around 350-365 mN/m, thus independent of the substrate surface or employed liquid metal (i.e., eutectic Ga-In (EGaIn) and galinstan). The similarity of the surface tension for different alloys was ascribed to the composition of the oxide skin, which predominantly comprises gallium oxides due to thermodynamic constraints. We envision that this method can also be applied to other liquid metal alloys and liquid metal marble systems facilitating modeling, simulation, and optimization processes.

Enhancing Li Ion Battery Performance by Mechanical Resonance

The quest for safe and high-performance Li ion batteries (LIBs) motivates intense efforts seeking a high-energy but reliable anode, cathode, and nonflammable electrolyte. For any of these, exploring new electrochemistry methods that enhance safety and performance by employing well-designed electrodes and electrolytes are required. Electrolyte wetting, governed by thermodynamics, is another critical issue in increasing Li ion transport through the separator. Herein, we report an approach to enhancing LIB performance by applying mechanical resonant vibration to increase electrolyte wettability on the separator. Wetting is activated at a resonant frequency with a capillary wave along the surface of the electrolyte, allowing the electrolyte to infiltrate into the porous separator by inertia force. This mechanical resonance, rather than electrochemistry, leads to the high specific capacity, rate capability, and cycling stability of LIBs. The concept of the mechanical approach is a promising yet simple strategy for the development of safer LIBs using liquid electrolytes.

Audible Sound from Vibrating Sessile Droplets for Monitoring Chemicals and Reactions in Liquid

To reduce environmental impact and sensor footprint, researchers need cost-effective and small-size surface tension and viscosity measurement devices. New measurement principles are needed for such sensors. We demonstrate that a sessile droplet's mechanical vibration can be transformed to audible sound, by recording the ultrasonic Doppler frequency shift in the form of an acoustic signal. The recorded sound wave reveals a droplet's surface tension and its viscosity, through its frequency spectrum and attenuation rate of the signal, respectively. Based on such sensors, two chemical measurements inside sessile droplets are shown: (I) titration of a Ni²⁺ and Co²⁺ mixture with a surface-active indicator (using surface tension) and (II) measurement of the molecular weight of a polymer in solution (using viscosity). Unlike the commercial technique, our ultrasound-based sensor is cost-effective in terms of equipment price and sample volume.

Thermodynamic and Kinetic Pathways to Agitated and Spontaneous Emulsification

Emulsification of an oil (dodecane and diesel fuel) in salinized water was studied under turbulent and agitation-free conditions in the presence of a mixture of an ionic and a nonionic surfactant. The properties of the air-water and the oil-water interfaces were investigated using the methods of du-Nouy ring, drop resonance vibrometry, and Langmuir film balance that allowed pinpointing the relevance of certain interfacial properties in emulsification. Estimation of the droplet size and its distribution from the nanometer-to-micrometer range was carried out with optical microscopy, acoustic attenuation spectroscopy, and continuous hydrodynamic flow fractionation. These measurements provided the platform for the comparison of the emulsion droplet size with those predicted from the fluctuation of the dynamic stress in the turbulent water via a capillary hydrodynamic model. While such a comparison was reasonably meaningful for micron size emulsion droplets, production of nanometer size droplets was beyond such a rudimentary expectation. We thus carried out systematic investigations into other factors that contribute to emulsification under both agitated and agitation-free conditions. An important finding of these studies is that the infusion of air bubbles that profoundly enhance the hydrodynamic fluctuation produces mainly submicroscopic emulsion droplets, while a fluctuation inhibiting water-soluble polymer has the opposite effect. Furthermore, while a hydrophilic polymer dissolved in water enhances the ripening of the droplets with time, hydrophobic polymer in oil thwarts aging, plausibly by osmotic backpressure and interfacial stiffening, which, upon compression, acts against surface tension, thereby decreasing the chemical potential of the trapped oil molecules inside the droplet. These effects are similarly observed in spontaneous emulsifications, that is, when a layer of oil containing the additives is deposited upon the surface of the aqueous phase in the absence of any external work input.

An active self-cleaning surface system for photovoltaic modules using anisotropic ratchet conveyors and mechanical vibration

The purpose of this work is to develop an active self-cleaning system that removes contaminants from a solar module surface by means of an automatic, water-saving, and labor-free process. The output efficiency of a solar module can be degraded over time by dust accumulation on top of the cover glass, which is often referred to as "soiling". This paper focuses on creating an active self-cleaning surface system using a combination of microsized features and mechanical vibration. The features, which are termed anisotropic ratchet conveyors (ARCs), consist of hydrophilic curved rungs on a hydrophobic background. Two different ARC systems have been designed and fabricated with self-assembled monolayer (SAM) silane and fluoropolymer thin film (Cytop). Fabrication processes were established to fabricate these two systems, including patterning Cytop without degrading the original Cytop hydrophobicity. Water droplet transport characteristics, including anisotropic driving force, droplet resonance mode, cleaning mechanisms, and system power consumption, were studied with the help of a high-speed camera and custom-made test benches. The droplet can be transported on the ARC surface at a speed of 27 mm/s and can clean a variety of dust particles, either water-soluble or insoluble. Optical transmission was measured to show that Cytop can improve transmittance by 2.5~3.5% across the entire visible wavelength range. Real-time demonstrations of droplet transport and surface cleaning were performed, in which the solar modules achieved a 23 percentage-point gain after cleaning.

Vibration Induced Transport of Enclosed Droplets

The droplet response to vibrations has been well characterized on open substrates, but microfluidic applications for droplets on open systems are limited by rapid evaporation rates and prone to environmental contamination. However, the response of enclosed droplets to vibration is less understood. Here, we investigate the effects of a dual-plate enclosure on droplet transport for the anisotropic ratchet conveyor system. This system uses an asymmetric pattern of hydrophilic rungs to transport droplets with an applied vibration. Through this work, we discovered that the addition of a substrate on top of the droplet, held in place with a 3D printed fixture, extends the functional frequency range for droplet transport and normalizes the device performance for droplets of different volumes. Furthermore, we found that the edge movements are anti-phasic between top and bottom substrates, providing a velocity profile that is correlated to vibration frequency, unlike the resonance-dependent profiles observed on open systems. These results expand the capabilities of this system, providing avenues for new applications and innovation, but also new insights for droplet mechanics in response to applied vibration.

Evaporation-Oscillation Driven Assembly: Microtailoring the Spatial Ordering of Particles in Sessile Droplets

This work explores the physical mechanism that can be used to control the final residual pattern of nanoparticles obtained from an evaporating-oscillating sessile droplet. To that end, the substrate is vibrated in the vertical direction with a constant amplitude, while the frequency of excitation is varied. It is found that evaporation progressively shifts the mode number of the oscillating droplet to lower values, while the oscillations enhance the rate of solvent loss, causing a reduction in the droplet lifetime. The coupling between evaporation and oscillation drives the internal flow through two distinct regimes. Initially, oscillation leads to inner flow recirculation, which delays the evaporation driven edge deposition of particles. Subsequently at lower modes, caused by solvent depletion, the effect of oscillation is weakened, which allows evaporation-driven flow to gain prominence and thus transport the dispersed particles to the contact line. We demonstrate here how this delay in particle migration can be controlled to engineer morphological changes in not just the resulting macroscopic aspect of the deposit but also its microstructure. We especially focus on the relatively unexplored microstructural pattern of deposits from evaporating-oscillating droplets. Using scanning electron micrograph and Voronoi tessellation of the final deposit, we show unique spatial variation in particle ordering at macro-micro length scales. Thus, droplet oscillation tunes the spatial extent of the particle ordering crucial in applications like photonic crystals and photonic glass.

Transport velocity of droplets on ratchet conveyors

Anisotropic ratchet conveyors (ARC) are a type of digital microfluidic system. Unlike electrowetting based systems, ARCs transport droplets through a passive, micro-patterned surface and applied orthogonal vibrations. The mechanics of droplet transport on ARC devices has yet to be as well characterized and understood as on electrowetting systems. In this work, we investigate how the design of the ARC substrate affects the droplet response to vibrations and perform the first characterization of transport velocity on ARC devices. We discovered that the design of the ARC device has a significant effect on both the transport efficiency and velocity of actuated droplets, and that the amplitude of the applied vibration can modulate the velocity of transported droplets. Finally, we show that the movement of droplet edges is not continuous but rather the sum of quantized steps between features of the ARC device. These results provide new insights into the behavior of droplets vibrated on asymmetric surface patterns and will serve as the foundation for the design and development of future lab-on-a-chip systems.

Converting Vertical Vibration of Anisotropic Ratchet Conveyors into Horizontal Droplet Motion

An anisotropic ratchet conveyor is an asymmetric, periodic, micropatterned surface that propels droplets when vibrated with a sinusoidal signal at certain frequencies and amplitudes. For each input frequency, there is a threshold amplitude beyond which the droplet starts to move. In this paper, we study the parameters that initiate droplet motion and the relationship between the input frequency and threshold amplitude among droplets with different volume, density, viscosity, and surface tension. Through this investigation we demonstrate how nondimensionalization reveals consistent behavior for droplets of different volumes. Finally, we propose a compact model that captures the essential features of the system to describe how a pure vertical vibration results in horizontal droplet motion. This model provides an intuitive understanding of the underlying physics and explains how the surface asymmetry is the key for lateral droplet motion.

Drops on a Superhydrophobic Hole Hanging On under Evaporation

Drops with larger volumes placed over a superhydrophobic (SH) surface with a hole do not fall through unless they are evaporated to a size that is small enough. This feature offers the ability to preconcentrate samples for biochemical analysis. In this work, the influence of pinning on the behavior of drops placed on a 0.1 mm thick SH substrate with a 2 mm diameter hole as they evaporated was investigated. With 16 μL of water dispensed, the sessile drop component volume was initially higher than that of the overhanging drop component and maintained this until the later stages where almost identical shapes were attained and full evaporation was achieved without falling off the hole. With 15 μL of water dispensed, the volume of the sessile drop was initially higher than that of the overhanging drop component but the liquid body was able to squeeze through the hole after 180 s due to the contact line not having sufficient pinning strength when it encountered the edge of the hole. This resulted in the liquid body either falling through the hole or remaining pinned with an oval-like shape. When it did not fall-off, the liquid body had volume and contact angle characteristics for the sessile drop and overhanging drop components that were reversed. In the later stages, however, nearly identical shapes were again attained and full evaporation was achieved without falling off the hole. The effects of pinning, despite the substrate being SH, offer another path toward achieving practical outcomes with liquid bodies without the need for chemical surface functionalization. Similarities and differences could be seen in the behavior of a sessile drop on a SH plate that was inclined at 30° to the horizontal and evaporated.

Acoustic levitation of liquid drops: Dynamics, manipulation and phase transitions

The technique of acoustic levitation normally produces a standing wave and the potential well of the sound field can be used to trap small objects. Since no solid surface is involved it has been widely applied for the study of fluid physics, nucleation, bio/chemical processes, and various forms of soft matter. In this article, we survey the works on drop dynamics in acoustic levitation, focus on how the dynamic behavior is related to the rheological properties and discuss the possibility to develop a novel rheometer based on this technique. We review the methods and applications of acoustic levitation for the manipulation of both liquid and solid samples and emphasize the important progress made in the study of phase transitions and bio-chemical analysis. We also highlight the possible open areas for future research.

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.

Generation of Motion of Drops with Interfacial Contact

A liquid drop moves on a solid surface if it is subjected to a gradient of wettability or temperature. However, the pinning defects on the surface manifested in terms of a wetting hysteresis, or first-order nonlinear friction, limit the motion in the sense that a critical size has to be exceeded for a drop to move. The effect of hysteresis can, however, be mitigated by an external vibration that can be either structured or stochastic, thereby creating a directed motion of the drop. Many of the well-known features of rectification, amplification, and switching that are generic to electronics can be engineered with such types of movements. A specific case of interest is the random coalescence of drops on a surface that gives rise to self-generated noise. This noise overcomes the pinning potential, thereby generating a random motion of the coalesced drops. Randomly moving coalesced drops themselves exhibit a directed diffusive flux when a boundary is present to eliminate them by absorption. With the presence of a bias, the coalesced drops execute a diffusive drift motion that can have useful applications in various water and thermal management technologies.

Mechanical vibrations of pendant liquid droplets

A simple optical deflection technique was used to monitor the vibrations of microlitre pendant droplets of deuterium oxide, formamide, and 1,1,2,2-tetrabromoethane. Droplets of different volumes of each liquid were suspended from the end of a microlitre pipette and vibrated using a small puff of nitrogen gas. A laser was passed through the droplets and the scattered light was collected using a photodiode. Vibration of the droplets resulted in the motion of the scattered beam and time-dependent intensity variations were recorded using the photodiode. These time-dependent variations were Fourier transformed and the frequencies and widths of the mechanical droplet resonances were extracted. A simple model of vibrations in pendant/sessile drops was used to relate these parameters to the surface tension, density and viscosity of the liquid droplets. The surface tension values obtained from this method were found to be in good agreement with results obtained using the standard pendant drop technique. Damping of capillary waves on pendant drops was shown to be similar to that observed for deep liquid baths and the kinematic viscosities obtained were in agreement with literature values for all three liquids studied.

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%.

Effect of roughness geometry on wetting and dewetting of rough PDMS surfaces

Rough PDMS surfaces comprising 3 μm hemispherical bumps and cavities with pitches ranging from 4.5 to 96 μm have been fabricated by photolithographic and molding techniques. Their wetting and dewetting behavior with water was studied as model for print surfaces used in additive manufacturing and printed electronics. A smooth PDMS surface was studied as control. For a given pitch, both bumpy and cavity surfaces exhibit similar static contact angles, which increase as the roughness ratio increases. Notably, the observed water contact angles are shown to be consistently larger than the calculated Wenzel angles, attributable to the pinning of the water droplets into the metastable wetting states. Optical microscopy reveals that the contact lines on both the bumpy and cavity surfaces are distorted by the microtextures, pinning at the lead edges of the bumps and cavities. Vibration of the sessile droplets on the smooth, bumpy, and cavity PDMS surfaces results in the same contact angle, from 110°-124° to ∼91°. The results suggest that all three surfaces have the same stable wetting states after vibration and that water droplets pin in the smooth area of the rough PDMS surfaces. This conclusion is supported by visual inspection of the contact lines before and after vibration. The importance of pinning location rather than surface energy on the contact angle is discussed. The dewetting of the water droplet was studied by examining the receding motion of the contact line by evaporating the sessile droplets of a very dilute rhodamine dye solution on these surfaces. The results reveal that the contact line is dragged by the bumps as it recedes, whereas dragging is not visible on the smooth and the cavity surfaces. The drag created by the bumps toward the wetting and dewetting process is also visible in the velocity-dependent advancing and receding contact angle experiments.

Equilibrium contact angle or the most-stable contact angle?

It is well-established that the equilibrium contact angle in a thermodynamic framework is an "unattainable" contact angle. Instead, the most-stable contact angle obtained from mechanical stimuli of the system is indeed experimentally accessible. Monitoring the susceptibility of a sessile drop to a mechanical stimulus enables to identify the most stable drop configuration within the practical range of contact angle hysteresis. Two different stimuli may be used with sessile drops: mechanical vibration and tilting. The most stable drop against vibration should reveal the changeless contact angle but against the gravity force, it should reveal the highest resistance to slide down. After the corresponding mechanical stimulus, once the excited drop configuration is examined, the focus will be on the contact angle of the initial drop configuration. This methodology needs to map significantly the static drop configurations with different stable contact angles. The most-stable contact angle, together with the advancing and receding contact angles, completes the description of physically realizable configurations of a solid-liquid system. Since the most-stable contact angle is energetically significant, it may be used in the Wenzel, Cassie or Cassie-Baxter equations accordingly or for the surface energy evaluation.

Understanding (sessile/constrained) bubble and drop oscillations

The diffuse literature on drop oscillation is reviewed, with an emphasis on capillary wave oscillations of constrained drops. Based on the review, a unifying conceptual framework is presented for drop and bubble oscillations, which considers free and constrained drops/bubbles, oscillation of the surface or the bulk (i.e. center of mass) of the drop/bubble, as well as different types of restoring forces (surface tension, gravity, electromagnetic, etc). Experimental results (both from literature and from a new set of experiments studying sessile drops in cross flowing air) are used to test mathematical models from literature, using a novel whole profile analysis technique for the new experiments. The cause of oscillation (cross flowing air, vibrated surface, etc.) is seen not to affect oscillation frequency. In terms of models, simplified models are seen to poorly predict oscillation frequencies. The most advanced literature models are found to be relatively accurate at predicting frequency. However it is seen that no existing models are reliably accurate across a wide range of contact angles, indicating the need for advanced models/empirical relations especially for drops undergoing the lowest frequency mode of oscillation (the order 1 degree 1 non-axisymmetric 'bending' mode that corresponds to a lateral 'rocking' motion of the drop).

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.

Direct measurement of the surface tension of a soft elastic hydrogel: exploration of elastocapillary instability in adhesion

An adhesively stressed thin film of a soft hydrogel confined between two rigid flat substrates autoroughens with its dominant wavelength (λ) exhibiting pronounced dependence on the film thickness (H). A linear stability analysis confirmed that this long wavelength instability (λ ~ 7H) is due to an elastocapillary effect, the implementation of which required direct measurements of the surface tension and the elasticity of the gel. The surface tension of the gel was estimated from the fundamental spherical harmonic of a hemispherical cap of the gel that was excited by an external noise. The shear modulus (μ) of the gel was determined from its resonant shear mode in a confined geometry. During the course of this study, it was found that a high density steel ball submerges itself inside the gel by balancing its excess weight with the accumulated strain induced elastic force that allows another estimation of its elastic modulus. The large ratio (1.8 mm) of the surface tension to its elasticity ascertains the role of elastocapillarity in the adhesion-induced pattern formation with such gels. Experimental results are in accord with a linear stability analysis that predicts that the rescaled wavelength λ(μH/γ)(0.27) is linear with H, which also modifies the conventional stress to pull a flat rigid object out of a very soft film by a multiplicative factor: (γ/μH)(1/4). The analysis also suggests some new results related to the role of the finite dilation of a material in interfacial pattern formation that may have nontrivial consequences in the adhesive delamination of very thin and/or soft elastic films via self-generated cracks.

Vibrational modes of elongated sessile liquid droplets

Vibrations of small (microliter) sessile liquid droplets were studied using a simple optical deflection technique. The droplets were made to elongate in one direction by taking advantage of the anisotropic wetting of the liquids on structured diffraction grating surfaces. They were vibrated by applying a puff of nitrogen gas. Motion of the droplets was monitored by scattering laser light from their surfaces. The scattered light was collected using a photodiode, and the resulting time-dependent intensity signals were Fourier-transformed to obtain the vibrational response of the drops. The vibrational spectra of elongated sessile drops were observed to contain two closely spaced peaks. A simple model that considers the frequency of capillary wave fluctuations on the surfaces of the drops was used to show that the vibrational frequencies of these peaks correspond to standing wave states that exist along the major and minor profile lengths of the droplets.

Microscale capillary wave turbulence excited by high frequency vibration

Low frequency (O(10 Hz-10 kHz)) vibration excitation of capillary waves has been extensively studied for nearly two centuries. Such waves appear at the excitation frequency or at rational multiples of the excitation frequency through nonlinear coupling as a result of the finite displacement of the wave, most often at one-half the excitation frequency in so-called Faraday waves and twice this frequency in superharmonic waves. Less understood, however, are the dynamics of capillary waves driven by high-frequency vibration (>O(100 kHz)) and small interface length scales, an arrangement ideal for a broad variety of applications, from nebulizers for pulmonary drug delivery to complex nanoparticle synthesis. In the few studies conducted to date, a marked departure from the predictions of classical Faraday wave theory has been shown, with the appearance of broadband capillary wave generation from 100 Hz to the excitation frequency and beyond, without a clear explanation. We show that weak wave turbulence is the dominant mechanism in the behavior of the system, as evident from wave height frequency spectra that closely follow the Rayleigh-Jeans spectral response η ≈ ω(-17/12) as a consequence of a period-halving, weakly turbulent cascade that appears within a 1 mm water drop whether driven by thickness-mode or surface acoustic Rayleigh wave excitation. However, such a cascade is one-way, from low to high frequencies. The mechanism of exciting the cascade with high-frequency acoustic waves is an acoustic streaming-driven turbulent jet in the fluid bulk, driving the fundamental capillary wave resonance through the well-known coupling between bulk flow and surface waves. Unlike capillary waves, turbulent acoustic streaming can exhibit subharmonic cascades from high to low frequencies; here it appears from the excitation frequency all the way to the fundamental modes of the capillary wave at some four orders of magnitude in frequency less than the excitation frequency, enabling the capillary weakly turbulent wave cascade to form from the fundamental capillary wave upward.

Activated drops: Self-excited oscillation, critical speeding and noisy transport

A small drop (∼10μl) of water exhibits critical speeding dynamics on an inclined super-hydrophobic pillared surface, in that it moves very slowly at first, but speeds up rapidly after a critical velocity is reached. During the mobile phase, some of the natural vibration modes of the drops are self-excited on a pillared surface, but not on a smooth hydrophobic surface. Additional experiments were carried out with glycerin and the solutions of water and glycerin that allowed their density and surface tension to be held more or less constant, while their viscosity could be varied. The terminal velocities of these drops following the critical speeding did not exhibit the expected decrease with increasing viscosity, but showed a highly non-linear behavior, exhibiting a maximum at an intermediate viscosity. Any of these drops moves steadily on a sub-critically inclined pillared substrate when it is subjected to a mechanical noise, the dynamics of which is remarkably similar to that obtained from another designed experiment in which the drops were made to cross a physical barrier assisted by an external noise. The dynamics of the low viscosity (1mPa s to 5.3mPa s) drops are amenable to a Kramers-like transition rate in the low friction limit, although the overall dynamics is found to be sub-Arrhenius. This work highlights the importance of the fluctuation of a drop that is either self-excited or that induced by an external noise in its motion on a surface.