Is contact-line mobility a material parameter?

Dynamic wetting phenomena are typically described by a constitutive law relating the dynamic contact angle θ to contact-line velocity UCL. The so-called Davis-Hocking model is noteworthy for its simplicity and relates θ to UCL through a contact-line mobility parameter M, which has historically been used as a fitting parameter for the particular solid-liquid-gas system. The recent experimental discovery of Xia & Steen (2018) has led to the first direct measurement of M for inertial-capillary motions. This opens up exciting possibilities for anticipating rapid wetting and dewetting behaviors, as M is believed to be a material parameter that can be measured in one context and successfully applied in another. Here, we investigate the extent to which M is a material parameter through a combined experimental and numerical study of binary sessile drop coalescence. Experiments are performed using water droplets on multiple surfaces with varying wetting properties (static contact angle and hysteresis) and compared with numerical simulations that employ the Davis-Hocking condition with the mobility M a fixed parameter, as measured by the cyclically dynamic contact angle goniometer, i.e. no fitting parameter. Side-view coalescence dynamics and time traces of the projected swept areas are used as metrics to compare experiments with numerical simulation. Our results show that the Davis-Hocking model with measured mobility parameter captures the essential coalescence dynamics and outperforms the widely used Kistler dynamic contact angle model in many cases. These observations provide insights in that the mobility is indeed a material parameter.

Electro-Osmotic Gripper Characterization for Layered Assembly.

Layered assembly is a voxel-based additive manufacturing process that relies on parallel grasping of voxels to produce multi-material parts. Although there exists substantial diversity in mechanisms of gripping, there still exists a lack of consistency, accuracy, and efficacy in positioning very large numbers of milli-, micro-, and nano-scale objects. We demonstrate the use of parallel electro-osmotic grippers to selectively transport multiple millimeter-sized voxels simultaneously. In contrast to previous research focused on using arrays of droplets to grab a single substrate, each element in the array is individually controlled via capillary effects, which are, in turn, controlled by an electric field to create predetermined patterns of droplets to pick and place selected objects. The demonstrated fluidic pick-and-place method has two key advantages: It is suitable for transport of fragile and complex objects due to the lack of mechanical contact, and it easily parallelizes to arbitrary array sizes for massively parallel pick-and-place. This work demonstrates a 25-element parallel assembly of 1.5-mm spheres with 95-98% grasping reliability, in a variety of geometric patterns. Experimental performance was validated against both analytical and computational models. The results suggest that electro-osmotic droplet arrays may enable the additive manufacturing of multi-material objects containing millions of components in the same print bed.

The draining of capillary liquids from containers with interior corners aboard the ISS.

In this work, we analyze liquid drains from containers in effective zero-g conditions aboard the International Space Station (ISS). The efficient draining of capillary fluids from conduits, containers, and media is critical in particular to high-value liquid samples such as minuscule biofluidics processing on earth and enormous cryogenic fuels management aboard spacecraft. The amount and rate of liquid drained can be of key concern. In the absence of strong gravitational effects, system geometry, and liquid wetting dominate capillary fluidic behavior. During the years 2010-2015, NASA conducted a series of handheld experiments aboard the ISS to observe "large" length scale capillary fluidic phenomena in a variety of irregular containers with interior corners. In this work, we focus on particular single exit port draining flows from such containers and digitize hours of archived NASA video records to quantify transient interface profiles and volumetric flow rates. These data are immediately useful for theoretical and numerical model benchmarks. We demonstrate this by making comparisons to lubrication models for slender flows in simplified geometries which show variable agreement with the data, in part validating certain geometry-dependent dynamical interface curvature boundary conditions while invalidating others. We further compare the data for the draining of complex vane networks and identify the limits of the current theory. All analyzed data is made available to the public as MATLAB files, as detailed within.

Switchable Wettability for Condensation Heat Transfer.

Condensation proceeds as dropwise or filmwise depending on the wettability of the condensing surface. These two modes of condensation have disparate heat transfer coefficients, with dropwise often exceeding filmwise. This work reports a surface with switchable superhydrophilic to hydrophobic wetting behavior that can exhibit both modes of condensation. Relative to the highly wetting state, which yields filmwise condensation, the nonwetting state exhibits dropwise condensation and twice the heat transfer coefficient. Relevance to thermal management is additionally discussed.

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.

Dissipation of oscillatory contact lines using resonant mode scanning.

Moving contact-lines (CLs) dissipate. Sessile droplets, mechanically driven into resonance by plane-normal forcing of the contacting substrate, can exhibit oscillatory CL motions with CL losses dominating bulk dissipation. Conventional practice measures CL dissipation based on the rate of mechanical work of the unbalanced Young's force at the CL. Typical approaches require measurements local to the CL and assumptions about the "equilibrium" contact angle (CA). This paper demonstrates how to use scanning of forcing frequency to characterize CL dissipation without any dependence on measurements from the vicinity of the CL. The results are of immediate relevance to an International Space Station (ISS) experiment and of longer-term relevance to Earth-based wettability applications. Experiments reported here use various concentrations of a water-glycerol mixture on a low-hysteresis non-wetting substrate.

Simulating Heat Transfer During Transient Dropwise Condensation on a Low-Thermal-Conductivity Substrate.

The instantaneous heat-transfer performance of a surface is dictated by the number and sizes of drops on the surface. While performance averaged over longer times is of interest from a technology standpoint, accurate simulation of the transient state is important in condenser design because the maximum heat rejection of the surface occurs in this range. Steady-state dropwise condensation can be thought of as a collection of transient dropwise condensation cycles occurring in parallel. Traditional simulation of dropwise condensation has focused on making comparisons with experimental drop-size distributions at later times, after the process has reached a statistical stationary phase where the heat transfer is lower. Understanding how to model and simulate transient dropwise condensation where a maximum in heat transfer occurs will help us design higher heat-rejecting surfaces. Additionally, a constant temperature difference between the steam and the surface below the drop is assumed. While often valid, there are some cases where this is not valid, and measuring the drop growth rate is required. We report a way to simulate transient dropwise condensation using a measured population averaged drop growth rate. The simulation reasonably predicts the time evolution of the number density of drops, fractional coverage, normalized condensate volume, and median drop radius for pendant mode dropwise condensation experiments on a cooled, horizontal, dodecyltrichlorosilane-coated glass surface. It was also found that assuming a constant temperature difference grossly underpredicts the heat transfer. Modification of the single-drop heat-transfer model to include substrate conduction and a thermal boundary layer shows that in the limit of low thermal conductivity the drop growth rate is constant for large drops. Additionally, a comparison between experiments and simulation shows that condensation might be initialized by nucleation onto fixed sites and then transitions to random nucleation as more sites become activated and more favorable. Understanding how a substrate's thermal properties affect the progression of dropwise condensation is important in determining the removal performance of the surface. With the commercialization of 3D printing, it is possible to fabricate low-cost, lightweight, plastic substrates with physical texturing for condensation applications where mass and cost savings are critical.

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.

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.

Mass production of shaped particles through vortex ring freezing.

A vortex ring is a torus-shaped fluidic vortex. During its formation, the fluid experiences a rich variety of intriguing geometrical intermediates from spherical to toroidal. Here we show that these constantly changing intermediates can be 'frozen' at controlled time points into particles with various unusual and unprecedented shapes. These novel vortex ring-derived particles, are mass-produced by employing a simple and inexpensive electrospraying technique, with their sizes well controlled from hundreds of microns to millimetres. Guided further by theoretical analyses and a laminar multiphase fluid flow simulation, we show that this freezing approach is applicable to a broad range of materials from organic polysaccharides to inorganic nanoparticles. We demonstrate the unique advantages of these vortex ring-derived particles in several applications including cell encapsulation, three-dimensional cell culture, and cell-free protein production. Moreover, compartmentalization and ordered-structures composed of these novel particles are all achieved, creating opportunities to engineer more sophisticated hierarchical materials.

Adaptive adhesion by a beetle: manipulation of liquid bridges and their breaking limits.

A drop brought into contact with a nearby substrate can wet and spread against the substrate, forming a liquid bridge that exerts a capillary force. This force due to surface tension can be used to "grab" the substrate, pulling it toward the drop. "Wet" adhesion results from the parallel action of an array of small liquid bridges. The Florida palm beetle, Hemisphaerota cyanea, uses wet adhesion to defend itself against attacking predators by adhering to the palm leaf using an array of about 120,000 µm-sized liquid bridges. The beetle's survival depends on the strength of adhesion which, in turn, depends on how liquid bridges break. Individual bridges break when they go unstable, according to their response curves. However, the ultimate strength of an individual bridge depends on the class of disturbances to which it is subjected, and it has been speculated that the beetle may have some control over this class. The authors experimentally study families of liquid bridge equilibria for their breaking limits when subjected to constant-length (L) and constant-force (F) disturbances. While to control constant-L disturbances is straightforward, to apply and control constant-F disturbances on a liquid bridge requires more ingenuity. The authors introduce an apparatus with a lever-arm and a ball-bearing slide. The authors then compare our experimentally measured bridge response curves to the force trace from experiments on the beetle (prior literature) to infer the mode of beetle detachment. Under normal loads, the beetle detaches as a constant-L instability for smaller loads and as a constant-F instability for larger loads. The beetle's ability to adjust the type and magnitude of loading in real time is not only crucial to its survival but has implications for the design of various engineering devices.

Condensation on surface energy gradient shifts drop size distribution toward small drops.

During dropwise condensation from vapor onto a cooled surface, distributions of drops evolve by nucleation, growth, and coalescence. Drop surface coverage dictates the heat transfer characteristics and depends on both drop size and number of drops present on the surface at any given time. Thus, manipulating drop distributions is crucial to maximizing heat transfer. On earth, manipulation is achieved with gravity. However, in applications with small length scales or in low gravity environments, other methods of removal, such as a surface energy gradient, are required. This study examines how chemical modification of a cooled surface affects drop growth and coalescence, which in turn influences how a population of drops evolves. Steam is condensed onto a horizontally oriented surface that has been treated by silanization to deliver either a spatially uniform contact angle (hydrophilic, hydrophobic) or a continuous radial gradient of contact angles (hydrophobic to hydrophilic). The time evolution of number density and associated drop size distributions are measured. For a uniform surface, the shape of the drop size distribution is unique and can be used to identify the progress of condensation. In contrast, the drop size distribution for a gradient surface, relative to a uniform surface, shifts toward a population of small drops. The frequent sweeping of drops truncates maturation of the first generation of large drops and locks the distribution shape at the initial distribution. The absence of a shape change indicates that dropwise condensation has reached a steady state. Previous reports of heat transfer enhancement on chemical gradient surfaces can be explained by this shift toward smaller drops, from which the high heat transfer coefficients in dropwise condensation are attributed to. Terrestrial applications using gravity as the primary removal mechanism also stand to benefit from inclusion of gradient surfaces because the critical threshold size required for drop movement is reduced.

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.

Capillarity-based switchable adhesion.

Drawing inspiration from the adhesion abilities of a leaf beetle found in nature, we have engineered a switchable adhesion device. The device combines two concepts: The surface tension force from a large number of small liquid bridges can be significant (capillarity-based adhesion) and these contacts can be quickly made or broken with electronic control (switchable). The device grabs or releases a substrate in a fraction of a second via a low-voltage pulse that drives electroosmotic flow. Energy consumption is minimal because both the grabbed and released states are stable equilibria that persist with no energy added to the system. Notably, the device maintains the integrity of an array of hundreds to thousands of distinct interfaces during active reconfiguration from droplets to bridges and back, despite the natural tendency of the liquid toward coalescence. We demonstrate the scaling of adhesion strength with the inverse of liquid contact size. This suggests that strengths approaching those of permanent bonding adhesives are possible as feature size is scaled down. In addition, controllability is fast and efficient because the attachment time and required voltage also scale down favorably. The device features compact size, no solid moving parts, and is made of common materials.

Coarsening of capillary drops coupled by conduit networks.

A system of n spherical-cap drops, coupled by a network of conduits, coarsens due to surface tension forces. The total interfacial energy drives the fluid through the conduits such that, with time, the volume becomes increasingly localized into fewer large drops. The coarsening rate is predicted heuristically for drops coupled by orthogonal networks, a porous medium, and fractal networks of various dimensions. The predicted coarsening law as it depends upon the type and dimension of network, total number of drops, and initial drop volume is compared against numerical simulations of large n . Additionally, distributions of large drop volumes are obtained using a Lifshitz-Slyozov-Wagner (LSW) model. The predicted distributions are independent of network topology; in contrast, simulation results depend weakly on the network dimension. The heuristic coarsening rate laws are recovered using the LSW model for all but a square network topology.

Determination of the zeta potential of porous substrates by droplet deflection: II. Generation of electrokinetic flow in a nonpolar liquid.

Here we study the nature and extent of free electrical charges in nonpolar liquids, using a recently introduced technique of observing droplet deflection generated by electrokinetic flow in a porous substrate. In the presence of dispersed water, surfactant molecules agglomerate and inverted micelles are generated which may act as charge carriers. In the present work, the conductivities of solutions of a nonpolar liquid with several concentrations of a dissolved surfactant are measured by electrical transients. The induced current densities are proportional to the applied voltage, indicating that the solutions represent an ohmic system. The conductivity does not scale simply with the surfactant concentration, though. It is inferred that different micellization mechanisms exist depending on the surfactant concentration, and a model is sketched. Further experiments reveal that flows of such solutions can be generated within saturated porous substrates when they are subjected to moderate electric fields. An investigation of the phenomena leads to the conclusion that these flows exist due to the presence of an electrical double layer; that is, they are of electrokinetic (electroosmotic) origin. Hence, the measured electrokinetic flow rates can be related to the zeta potential of the porous substrate saturated with the solution. Plotting the zeta potential against the logarithm of the ionic strength reveals a linear relationship.

Determination of the zeta potential of porous substrates by droplet deflection. I. The influence of ionic strength and pH value of an aqueous electrolyte in contact with a borosilicate surface.

This paper presents a new method to determine the zeta potential of porous substrates in contact with a liquid. Electroosmosis, arising near the solid/liquid boundaries within a fully saturated porous substrate, pumps against the capillary pressure arising from the surface tension of a droplet placed in series with the pump. The method is based on measuring the liquid/gas interface deflection due to the imposed electric potential difference. The distinguishing features of our technique are accuracy, speed, and reliability, accomplished with a straightforward and cost-effective setup. In this particular setup, a bistable configuration of two opposing droplets is used. The energy barrier between the stable states defines the range of capillary resistance and can be tuned by the total droplet volume. The electroosmotic pump is placed between the droplets. The large surface area-to-volume ratio of the porous substrate enables the pumping strength to exceed the capillary resistance even for droplets small enough that their shapes are negligibly influenced by gravity. Using a relatively simple model for the flow within the porous substrate, the zeta potential resulting from the substrate-liquid combination is determined. Extensive measurements of a borosilicate substrate in contact with different aqueous electrolytes are made. The results of the measurements clarify the influence of the ionic strength and pH value on the zeta potential and yield an empirical relationship important to engineering approaches.

The electroosmotic droplet switch: countering capillarity with electrokinetics.

Electroosmosis, originating in the double-layer of a small liquid-filled pore (size R) and driven by a voltage V, is shown to be effective in pumping against the capillary pressure of a larger liquid droplet (size B) provided the dimensionless parameter sigmaR(2)/epsilon|zeta|VB is small enough. Here sigma is surface tension of the droplet liquid/gas interface, epsilon is the liquid dielectric constant, and zeta is the zeta potential of the solid/liquid pair. As droplet size diminishes, the voltage required to pump electroosmotically scales as V approximately R(2)/B. Accordingly, the voltage needed to pump against smaller higher-pressure droplets can actually decrease provided the pump poresize scales down with droplet size appropriately. The technological implication of this favorable scaling is that electromechanical transducers made of moving droplets, so-called "droplet transducers," become feasible. To illustrate, we demonstrate a switch whose bistable energy landscape derives from the surface energy of a droplet-droplet system and whose triggering derives from the electroosmosis effect. The switch is an electromechanical transducer characterized by individual addressability, fast switching time with low voltage, and no moving solid parts. We report experimental results for millimeter-scale droplets to verify key predictions of a mathematical model of the switch. With millimeter-size water droplets and micrometer-size pores, 5 V can yield switching times of 1 s. Switching time scales as B(3)/VR(2). Two possible "grab-and-release" applications of arrays of switches are described. One mimics the controlled adhesion of an insect, the palm beetle; the other uses wettability to move a particle along a trajectory.