Delayed coalescence of droplets with miscible liquids: Lubrication and phase field theories

Drops on Polymer Brushes: Advances in Thin-Film Modeling of Adaptive Substrates

We briefly review recent advances in the hydrodynamic modeling of the dynamics of droplets on adaptive substrates, in particular, solids that are covered by polymer brushes. Thereby, the focus is on long-wave and full-curvature variants of mesoscopic hydrodynamic models in gradient dynamics form. After introducing the approach for films/drops of nonvolatile simple liquids on a rigid smooth solid substrate, it is first expanded to an arbitrary number of coupled degrees of freedom before considering the specific case of drops of volatile liquids on brush-covered solids. After presenting the model, its usage is illustrated by briefly considering the natural and forced spreading of drops of nonvolatile liquids on a horizontal brush-covered substrate, stick-slip motion of advancing contact lines as well as drops sliding down a brush-covered incline. Finally, volatile liquids are also considered.

Binding potential and wetting behavior of binary liquid mixtures on surfaces

We present a theory for the interfacial wetting phase behavior of binary liquid mixtures on rigid solid substrates, applicable to both miscible and immiscible mixtures. In particular, we calculate the binding potential as a function of the adsorptions, i.e., the excess amounts of each of the two liquids at the substrate. The binding potential fully describes the corresponding interfacial thermodynamics. Our approach is based on classical density functional theory. Binary liquid mixtures can exhibit complex bulk phase behavior, including both liquid-liquid and vapor-liquid phase separation, depending on the nature of the interactions among all the particles of the two different liquids, the temperature, and the chemical potentials. Here we show that the interplay between the bulk phase behavior of the mixture and the properties of the interactions with the substrate gives rise to a wide variety of interfacial phase behaviors, including mixing and demixing situations. We find situations where the final state is a coexistence of up to three different phases. We determine how the liquid density profiles close to the substrate change as the interaction parameters are varied and how these determine the form of the binding potential, which in certain cases can be a multivalued function of the adsorptions. We also present profiles for sessile droplets of both miscible and immiscible binary liquids.

Numerical Study of the Coalescence and Mixing of Drops of Different Polymeric Materials

In this study, we employ direct numerical simulation (DNS) to investigate the solutal hydrodynamics dictating the three-dimensional coalescence of microscopic, identical-sized sessile drops of different but miscible shear-thinning polymeric liquids (namely, PVAc or polyvinyl acetate and PMMA or polymethylmethacrylate), with the drops being in partially wetted configuration. Despite the ubiquitousness of the interaction of different dissimilar droplets in a variety of engineering problems ranging from additive manufacturing to understanding the behavior of photonic crystals, coalescence of drops composed of different polymeric and non-Newtonian materials has not been significantly explored. Interaction of such dissimilar droplets often involves simultaneous drop spreading, coalescence, and mixing. The mixing dynamics of the dissimilar drops are governed by interphase diffusion, the residual kinetic energy of the drops stemming from the fact that coalescence starts before the spreading of the drops have been completed, and the solutal Marangoni convection. We provide the three-dimensional velocity fields and velocity vectors inside the completely miscible, dissimilar coalescing droplets. Our simulations explicate the relative influence of these different effects in determining the flow field at different locations and at different time instances and the consequent mixing behavior inside the interacting drops. We also show the non-monotonic (in terms of the direction of migration) propagation of the mixing front of the miscible coalescing drops over time. We also establish that the overall mixing (on either side of the mixing front) speeds up as the Marangoni effects dictate the mixing. We anticipate that our study will provide an important foundation for studying miscible multi-material liquid systems, which will be crucial for applications such as inkjet or aerosol jet printing, lab-on-a-chip, polymer processing, etc.

Gas-Phase Induced Marangoni Flow Causes Unstable Drop Merging

The merging of drops plays a key role in many processes from simple rain to complex coating applications. In technical applications, often liquids with different surface tensions merge on a substrate like inkjet printing. For a suitable set of surface tensions, one drop can form a stable wetting film that is covering the other drop. Here, we explore the dynamics of driven wetting films and show a route toward their instability when these wetting films are driven by an external source of energy, which is Marangoni stress in our case. The wetting becomes unstable via a fingering instability and can be observed in various liquid combinations. The vapor of the liquid with the lower surface tension induces a Marangoni driven flow inside the other drop that pulls the wetting film. The concentration of the driving vapor can be controlled through the spreading velocity of the corresponding drop. We use this dependence to map out the characteristics of the instability. For very high or very low spreading velocities, no instability is observed. This is summarized in a stability diagram, which has three different regimes. A detailed analysis reveals a strong coupling of the characteristics of the fingering instability to the spreading velocity. The use of the spreading velocity as a control parameter is justified by a simplified 1D model that motivates how the spreading velocity controls the concentration profile of the second liquid vapor before and at contact. The strength of the Marangoni flow that drives the instability depends on the exposure time of the sitting drop to the vapor concentration profile around the spreading drop.

Coexistence and Sudden Entrapment between Two Dissimilar, Miscible Oil Lenses

The property of substrates is one of the important factors determining the interaction between two lenses (droplets). There likely exist different interactions between two dissimilar oil lenses (droplets) floating on the surface of a liquid phase from the interaction between two dissimilar oil droplets on a rigid substrate, for example, coalescence or coexistence. The interaction between two dissimilar oil lenses (droplets) is dependent on the intrinsic properties of both oil lenses (droplets) and external environmental factors. In this work, we investigate the contact interaction between two dissimilar, miscible oil lenses (toluene and silicone oil) on the surface of deionized water (DI water). The morphological evolution of two dissimilar, miscible oil lenses during the interaction under different experimental conditions is recorded and analyzed. The effects of the volume ratio of two dissimilar, miscible oil lenses, temperature of DI water, and viscosity of silicone oil on characteristic parameters are systematically studied. A sudden "entrapment" of a toluene lens into a silicone oil lens occurs after a period of the "mass exchange" (coexistence) between these two oil lenses. Several characteristic parameters, including the duration of the "mass exchange" and critical sizes of the toluene lens at the onset of the entrapment and after the entrapment, are found to be dependent on experimental conditions.

Insights into Drying of Noncircular Sessile Nanofluid Droplets toward Multiscale Surface Patterning Using a Wall-Less Confinement Architecture

Surface patterning with functional colloids is an important research area because of its widespread applicability in domains such as nanoelectronics, pharmaceutics, semiconductors, and photovoltaics among others. In this endeavor, we propose a low-cost patterning technique that aspires to eliminate the more expensive methodologies that are presently in practice. Using a simple document stamp on which patterns of any geometry can be embossed, we are able to print 2D millimeter-scale "wall-less confinement" using an ink-based hydrophobic fence on any plasma-treated superhydrophilic surface. The confinement is subsequently filled with nanocolloidal liquid(s). Using confinement geometry, we are able to control the 3D shape of the droplet to exhibit multiple interfacial curvatures. The droplet in the "wall-less confinements" evaporates naturally, exhibiting unique geometry (curvature)-induced flow structures that induce the nanoparticles to self-assemble into functional patterns. We have also shown that by modifying the geometry of the pattern, evaporation, flow, and particle deposition dynamics get altered, leading to precipitate topologies from macro- to microscales. We present two such geometrical designs that demonstrate the capability of modifying both macroscopic and microscopic features of the final precipitate. We have also provided a description of the physical mechanisms of the drying process by resolving the unique flow pattern using a combination of imaging and microparticle image velocimetry. These provide insights into the coupled dynamics of evaporation and flow responsible for the evolution of particle deposition pattern. Precipitate characterization using scanning electron microscopy and dark-field microscopy highlights the transformation in the deposit morphology.

Recent Advances in Controlling the Depositing Morphologies of Inkjet Droplets

Inkjet printing has been widely used in functional material patterning for fabrication of optical/electrical devices. The depositing morphologies of inkjet droplets are critical to the resolution and performance of resulted functional patterns. This review summarizes various strategies to control the depositing morphologies of inkjet droplets, including suppressing and utilizing coffee-ring effect, employing liquid substrates, developing patterned substrates and controlling droplets coalescence. Moreover, the remaining challenges in controlling inkjet droplets are presented, and the broad research and application prospects of controlling nanomaterial patterning by inkjet printing are proposed.

Can vibrations control drop motion?

We discuss a mechanism for controlled motion of drops with applications for microfluidics and microgravity. The mechanism is the following: a solid plate supporting a liquid droplet is simultaneously subject to lateral and vertical harmonic oscillations. In this way the symmetry of the back-and-forth droplet movement along the substrate under inertial effects is broken and thus will induce a net driven motion of the drop. We study the dependency of the traveled distance on the oscillation parameters (forcing amplitude, frequency, and phase shift between the two perpendicular oscillations) via phase field simulations. The internal flow structure inside the droplet is also investigated. We make predictions on resonance frequencies for drops on a substrate with a varying wettability.

Partial coalescence of sessile drops with different miscible liquids

Using computer simulations in three spatial dimensions, we examine the interaction between two deformable drops consisting of two perfectly miscible liquids sitting on a solid substrate under a given contact angle. Driven by capillarity and assisted by Marangoni effects at the droplet interfaces, several distinct coalescence regimes are achieved after the droplets' collision.

Noncoalescence of sessile drops from different but miscible liquids: hydrodynamic analysis of the twin drop contour as a self-stabilizing traveling wave

Capillarity always favors drop fusion. Nevertheless, sessile drops from different but completely miscible liquids often do not fuse instantaneously upon contact. Rather, intermediate noncoalescence is observed. Two separate drop bodies, connected by a thin liquid neck, move over the substrate. Supported by new experimental data, a thin film hydrodynamic analysis of this state is presented. Presumably advective and diffusive volume fluxes in the neck region establish a localized and temporarily stable surface tension gradient. This induces a local surface (Marangoni) flow that stabilizes a traveling wave, i.e., the observed moving twin drop configuration. The theoretical predictions are in excellent agreement with the experimental findings.

Coalescence preference depends on size inequality

During bubble or droplet coalescence, there is a puzzling tendency for the coalesced bubble or droplet to be preferentially placed closer to the larger of its two parents. We confirm that this preference is a function of parent size ratio by directly visualizing coalescing air bubbles on an oil-water interface and coalescing water droplets immersed in oil. We find that the final position of the coalesced sphere is controlled by surface energy release and is related to the parent size ratio by a power-law relationship. 

On the coalescence of sessile drops with miscible liquids

Sessile drops sitting on highly wettable solid substrates fuse in qualitatively different ways after contact, depending on the surface tension gradients between the mixing droplets. In early time evolution the drop coalescence can be fast or delayed (intermittent). In long time evolution a secondary drop formation can occur. We study numerically droplet dynamics during coalescence in two and three spatial dimensions, within a phase field approach. We discuss criteria to distinguish different coalescence regimes. A comparison with recent experiments will be done.