Droplet motion driven by humidity gradients during evaporation and condensation.

The motion of droplets on solid surfaces in response to an external gradient is a fundamental problem with a broad range of applications, including water harvesting, heat exchange, mixing and printing. Here we study the motion of droplets driven by a humidity gradient, i.e. a variation in concentration of their own vapour in the surrounding gas phase. Using lattice-Boltzmann simulations of a diffuse-interface hydrodynamic model to account for the liquid and gas phases, we demonstrate that the droplet migrates towards the region of higher vapour concentration. This effect holds in situations where the ambient gradient drives either the evaporation or the condensation of the droplet, or both simultaneously. We identify two main mechanisms responsible for the observed motion: a difference in surface wettability, which we measure in terms of the Young stress, and a variation in surface tension, which drives a Marangoni flow. Our results are relevant in advancing our knowledge of the interplay between gas and liquid phases out of thermodynamic equilibrium, as well as for applications involving the control of droplet motion.

Droplet Self-Propulsion on Slippery Liquid-Infused Surfaces with Dual-Lubricant Wedge-Shaped Wettability Patterns.

Young's equation is fundamental to the concept of the wettability of a solid surface. It defines the contact angle for a droplet on a solid surface through a local equilibrium at the three-phase contact line. Recently, the concept of a liquid Young's law contact angle has been developed to describe the wettability of slippery liquid-infused porous surfaces (SLIPS) by droplets of an immiscible liquid. In this work, we present a new method to fabricate biphilic SLIP surfaces and show how the wettability of the composite SLIPS can be exploited with a macroscopic wedge-shaped pattern of two distinct lubricant liquids. In particular, we report the development of composite liquid surfaces on silicon substrates based on lithographically patterning a Teflon AF1600 coating and a superhydrophobic coating (Glaco Mirror Coat Zero), where the latter selectively dewets from the former. This creates a patterned base surface with preferential wetting to matched liquids: the fluoropolymer PTFE with a perfluorinated oil Krytox and the hydrophobic silica-based GLACO with olive oil (or other mineral oils or silicone oil). This allows us to successively imbibe our patterned solid substrates with two distinct oils and produce a composite liquid lubricant surface with the oils segregated as thin films into separate domains defined by the patterning. We illustrate that macroscopic wedge-shaped patterned SLIP surfaces enable low-friction droplet self-propulsion. Finally, we formulate an analytical model that captures the dependence of the droplet motion as a function of the wettability of the two liquid lubricant domains and the opening angle of the wedge. This allows us to derive scaling relationships between various physical and geometrical parameters. This work introduces a new approach to creating patterned liquid lubricant surfaces, demonstrates long-distance droplet self-propulsion on such surfaces, and sheds light on the interactions between liquid droplets and liquid surfaces.

Cassie's Law Reformulated: Composite Surfaces from Superspreading to Superhydrophobic.

In 1948, Cassie provided an equation describing the wetting of a smooth, heterogeneous surface. He proposed that the cosine of the contact angle, θc, for a droplet on a composite surface could be predicted from a weighted average using the fractional surface areas, fi, of the cosines of contact angles of a droplet on the individual component surfaces, i.e., cos θc = f1 cos θ1 + f2 cos θ2. This was a generalization of an earlier equation for porous materials, which has recently proven fundamental to underpinning the theoretical description of wetting of superhydrophobic and superoleophobic surfaces. However, there has been little attention paid to what happens when a liquid exhibits complete wetting on one of the surface components. Here, we show that Cassie's equation can be reformulated using spreading coefficients. This reformulated equation is capable of describing composite surfaces where the individual surface components have negative (droplet state/partial wetting) or positive (film-forming/complete wetting) spreading coefficients. The original Cassie equation is then a special case when the combination of interfacial tensions results in a droplet state on the composite surface for which a contact angle can be defined. In the case of a composite surface created from a partial wetting (droplet state) surface and a complete wetting (film-forming) surface, there is a threshold surface area fraction at which a liquid on the composite surface transitions from a droplet to a film state. The applicability of this equation is demonstrated from literature data including data on mixed self-assembled monolayers on copper, silver, and gold surfaces that was regarded as definitive in establishing the validity of the Cassie equation. Finally, we discuss the implications of these ideas for super-liquid repellent surfaces.

Suppression of crystallization in saline drop evaporation on pinning-free surfaces.

For sessile droplets of pure liquid on a surface, evaporation depends on surface wettability, the surrounding environment, contact angle hysteresis, and surface roughness. For non-pure liquids, the evaporation characteristics are further complicated by the constituents and impurities within the droplet. For saline solutions, this complication takes the form of a modified partial vapor pressure/water activity caused by the increasing salt concentration as the aqueous solvent evaporates. It is generally thought that droplets on surfaces will crystallize when the saturation concentration is reached, i.e., 26.3% for NaCl in water. This crystallization is initiated by contact with the surface and is thus due to surface roughness and heterogeneities. Recently, smooth, low contact angle hysteresis surfaces have been created by molecular grafting of polymer chains. In this work, we hypothesize that by using these very smooth surfaces to evaporate saline droplets, we can suppress the crystallization caused by the surface interactions and thus achieve constant volume droplets above the saturation concentration. In our experiments, we used several different surfaces to examine the possibility of crystallization suppression. We show that on polymer grafted surfaces, i.e., Slippery Omniphobic Covalently Attached Liquid-like (SOCAL) and polyethyleneglycol(PEGylated) surfaces, we can achieve stable droplets as low as 55% relative humidity at 25 °C with high reproducibility using NaCl in water solutions. We also show that it is possible to achieve stable droplets above the saturation concentration on other surfaces, including superhydrophobic surfaces. We present an analytical model, based on water activity, which accurately describes the final stable volume as a function of the initial salt concentration. These findings are important for heat and mass transfer in relatively low humidity environments.

The Liquid Young's Law on SLIPS: Liquid-Liquid Interfacial Tensions and Zisman Plots.

Slippery liquid-infused porous surfaces (SLIPS) are an innovation that reduces droplet-solid contact line pinning and interfacial friction. Recently, it has been shown that a liquid analogue of Young's law can be deduced for the apparent contact angle of a sessile droplet on SLIPS despite there never being contact by the droplet with the underlying solid. Since contact angles on solids are used to characterize solid-liquid interfacial interactions and the wetting of a solid by a liquid, it is our hypothesis that liquid-liquid interactions and the wetting of a liquid surface by a liquid can be characterized by apparent contact angles on SLIPS. Here, we first present a theory for deducing liquid-liquid interfacial tensions from apparent contact angles. This theory is valid irrespective of whether or not a film of the infusing liquid cloaks the droplet-vapor interface. We show experimentally that liquid-liquid interfacial tensions deduced from apparent contact angles of droplets on SLIPS are in excellent agreement with values from the traditional pendant drop technique. We then consider whether the Zisman method for characterizing the wettability of a solid surface can be applied to liquid surfaces created using SLIPS. We report apparent contact angles for a homologous series of alkanes on Krytox-infused SLIPS and for water-IPA mixtures on both the Krytox-infused SLIPS and a silicone oil-infused SLIPS. The alkanes on the Krytox-infused SLIPS follow a linear relationship in the liquid form of the Zisman plot provided that the effective droplet-vapor interfacial tension is used. All three systems follow a linear relationship on a modified Zisman plot. We interpret these results using the concept of the critical surface tension (CST) for the wettability of a solid surface introduced by Zisman. In our liquid surface case, the obtained critical surface tensions were found to be lower than the infusing liquid-vapor surface tensions.

Self-Assembled, Hierarchical Structured Surfaces for Applications in (Super)hydrophobic Antiviral Coatings.

A versatile method for the creation of multitier hierarchical structured surfaces is reported, which optimizes both antiviral and hydrophobic (easy-clean) properties. The methodology exploits the availability of surface-active chemical groups while also manipulating both the surface micro- and nanostructure to control the way the surface coating interacts with virus particles within a liquid droplet. This methodology has significant advantages over single-tier structured surfaces, including the ability to overcome the droplet-pinning effect and in delivering surfaces with high static contact angles (>130°) and good antiviral efficacy (log kill >2). In addition, the methodology highlights a valuable approach for the creation of mechanically robust, nanostructured surfaces which can be prepared by spray application using nonspecialized equipment.

Friction Coefficients for Droplets on Solids: The Liquid-Solid Amontons' Laws.

The empirical laws of dry friction between two solid bodies date back to the work of Amontons in 1699 and are pre-dated by the work of Leonardo da Vinci. Fundamental to those laws are the concepts of static and kinetic coefficients of friction relating the pinning and sliding friction forces along a surface to the normal load force. For liquids on solid surfaces, contact lines also experience pinning and the language of friction is used when droplets are in motion. However, it is only recently that the concept of coefficients of friction has been defined in this context and that droplet friction has been discussed as having a static and a kinetic regime. Here, we use surface free energy considerations to show that the frictional force per unit length of a contact line is directly proportional to the normal component of the surface tension force. We define coefficients of friction for both contact lines and droplets and provide a droplet analogy of Amontons' first and second laws but with the normal load force of a solid replaced by the normal surface tension force of a liquid. In the static regime, the coefficient of static friction, defined by the maximum pinning force of a droplet, is proportional to the contact angle hysteresis, whereas in the kinetic regime, the coefficient of kinetic friction is proportional to the difference in dynamic advancing and receding contact angles. We show the consistency between the droplet form of Amontons' first and second laws and an equation derived by Furmidge. We use these liquid-solid Amontons' laws to describe literature data and report friction coefficients for various liquid-solid systems. The conceptual framework reported here should provide insight into the design of superhydrophobic, slippery liquid-infused porous surfaces (SLIPS) and other surfaces designed to control droplet motion.

Lattice Boltzmann Simulations of Multiphase Dielectric Fluids.

The dynamic effect of an electric field on dielectric liquids is called liquid dielectrophoresis. It is widely used in several industrial and scientific applications, including inkjet printing, microfabrication, and optical devices. Numerical simulations of liquid-dielectrophoresis are necessary to understand the fundamental physics of the phenomenon, but also to explore situations that might be difficult or expensive to implement experimentally. However, such modeling is challenging, as one needs to solve the electrostatic and fluid dynamics equations simultaneously. Here, we formulate a new lattice-Boltzmann method capable of modeling the dynamics of immiscible dielectric fluids coupled with electric fields within a single framework, thus eliminating the need of using separate algorithms to solve the electrostatic and fluid dynamics equations. We validate the numerical method by comparing it with analytical solutions and previously reported experimental results. Beyond the benchmarking of the method, we study the spreading of a droplet using a dielectrowetting setup and quantify the mechanism driving the variation of the apparent contact angle of the droplet with the applied voltage. Our method provides a useful tool to study liquid-dielectrophoresis and can be used to model dielectric fluids in general, such as liquid-liquid and liquid-gas systems.

Controlling the breakup of toroidal liquid films on solid surfaces.

The breakup of a slender filament of liquid driven by surface tension is a classical fluid dynamics stability problem that is important in many situations where fine droplets are required. When the filament is resting on a flat solid surface which imposes wetting conditions the subtle interplay with the fluid dynamics makes the instability pathways and mode selection difficult to predict. Here, we show how controlling the static and dynamic wetting of a surface can lead to repeatable switching between a toroidal film of an electrically insulating liquid and patterns of droplets of well-defined dimensions confined to a ring geometry. Mode selection between instability pathways to these different final states is achieved by dielectrophoresis forces selectively polarising the dipoles at the solid-liquid interface and so changing both the mobility of the contact line and the partial wetting of the topologically distinct liquid domains. Our results provide insights into the wetting and stability of shaped liquid filaments in simple and complex geometries relevant to applications ranging from printing to digital microfluidic devices.

Contact-Angle Hysteresis and Contact-Line Friction on Slippery Liquid-like Surfaces.

Contact-line pinning and dynamic friction are fundamental forces that oppose the motion of droplets on solid surfaces. Everyday experience suggests that if a solid surface offers low contact-line pinning, it will also impart a relatively low dynamic friction to a moving droplet. Examples of such surfaces are superhydrophobic, slippery porous liquid-infused, and lubricant-impregnated surfaces. Here, however, we show that slippery omniphobic covalently attached liquid-like (SOCAL) surfaces have a remarkable combination of contact-angle hysteresis and contact-line friction properties, which lead to very low droplet pinning but high dynamic friction against the motion of droplets. We present experiments of the response of water droplets to changes in volume at controlled temperature and humidity conditions, which we separately compare to the predictions of a hydrodynamic model and a contact-line model based on molecular kinetic theory. Our results show that SOCAL surfaces offer very low contact-angle hysteresis, between 1 and 3°, but an unexpectedly high dynamic friction controlled by the contact line, where the typical relaxation time scale is on the order of seconds, 4 orders of magnitude larger than the prediction of the classical hydrodynamic model. Our results highlight the remarkable wettability of SOCAL surfaces and their potential application as low-pinning, slow droplet shedding surfaces.

Self-propelled droplet transport on shaped-liquid surfaces.

The transport of small amounts of liquids on solid surfaces is fundamental for microfluidics applications. Technologies allowing control of droplets of liquid on flat surfaces generally involve the generation of a wettability contrast. This approach is however limited by the resistance to motion caused by the direct contact between the droplet and the solid. We show here that this resistance can be drastically reduced by preventing direct contact with the help of dual-length scale micro-structures and the concept of "liquid-surfaces". These new surfaces allow the gentle transport of droplets along defined paths and with fine control of their speed. Moreover, their high adhesion permits the capture of impacting droplets, opening new possibilities in applications such as fog harvesting and heat transfer.

Lattice-Boltzmann simulations of the dynamics of liquid barrels.

We study the relaxation towards equilibrium of a liquid barrel-a partially wetting droplet in a wedge geometry-using a diffuse-interface approach. We formulate a hydrodynamic model of the motion of the barrel in the framework of the Navier-Stokes and Cahn-Hilliard equations of motion. We present a lattice-Boltzmann method to integrate the diffuse-interface equations, where we introduce an algorithm to model the dynamic wetting of the liquid on smooth solid boundaries. We present simulation results of the over-damped dynamics of the liquid barrel. We find that the relaxation of the droplets is driven by capillary forces and damped by friction forces. We show that the friction is determined by the contribution of the bulk flow, the corner flow near the contact lines and the motion of the contact lines by comparing simulation results for the relaxation time of the barrel. Our results are in broad agreement with previous analytical predictions based on a sharp interface model.

Slippery when wet: mobility regimes of confined drops in electrowetting.

The motion of confined droplets in immiscible liquid-liquid systems strongly depends on the intrinsic relative wettability of the liquids on the confining solid material and on the typical speed, which can induce the formation of a lubricating layer of the continuous phase. In electrowetting, which routinely makes use of aqueous drops in ambient non-polar fluids that wet the wall material, electric stresses enter the force balance in addition to capillary and viscous forces and confinement effects. Here, we study the mobility of droplets upon electrowetting actuation in a wedge-shaped channel, and the subsequent relaxation when the electrowetting actuation is removed. We find that the droplets display two different mobility regimes: a fast regime, corresponding to gliding on a thin film of the ambient fluid, and a slow regime, where the film is replaced by direct contact between the droplet and the channel walls. Using a combination of experiments and numerical simulations, we show that the cross-over between these regimes arises from the interplay between the small-scale dynamics of the thin film of ambient fluid and the large-scale motion of the droplet. Our results shed light on the complex dynamics of droplets in non-uniform channels driven by electric actuation, and can thus help the rational design of devices based on electrowetting-driven droplet transport.

Droplet Retention and Shedding on Slippery Substrates.

A significant limitation for droplet mobility on solid surfaces is to overcome the inherent pinning of the droplet's contact line that occurs because of chemical/physical heterogeneities. A recent innovation is to use surface texture or porosity to create a stabilized lubricant surface. Droplets on such slippery liquid-infused porous surfaces (SLIPS)/lubricant-impregnated surfaces (LIS) are highly mobile because of the lubricant layer. Low pinning of the contact line reduces the energy required to move a droplet; however, it makes it difficult to accurately position the droplet or to stop its motion altogether. In this paper, a simple structure (step), as small as a few microns in height, is used to introduce controlled droplet pinning on a slippery substrate. The key effect is identified as the capillary force, arising from the interaction between the lubricant menisci created by the step and droplet. The effect of changing step height, lubricant thickness, and initial position on step-droplet interactions has been investigated, showing that droplets can both be repelled from and attracted to the step. To measure the adhesion strength, we report droplet detachment angle measurements under gravity and scaling of force with the lubricant thickness/step height ratio. Under certain conditions, the interaction strength is sufficient to ensure droplet-step attachment even when the surface is rotated to an upside-down orientation. These findings can motivate the design of SLIPS structures, capable of shedding or retaining droplets preferentially, for example, according to the size or wettability, relevant to applications from microfluidics to fog harvesting.

Lattice-Boltzmann Simulations of Electrowetting Phenomena.

When a voltage difference is applied between a conducting liquid and a conducting (solid) electrode, the liquid is observed to spread on the solid. This phenomenon, generally referred to as electrowetting, underpins a number of interfacial phenomena of interest in applications that range from droplet microfluidics to optics. Here, we present a lattice-Boltzmann method that can simulate the coupled hydrodynamics and electrostatics equations of motion of a two-phase fluid as a means to model the electrowetting phenomena. Our method has the advantage of modeling the electrostatic fields within the lattice-Boltzmann algorithm itself, eliminating the need for a hybrid method. We validate our method by reproducing the static equilibrium configuration of a droplet subject to an applied voltage and show that the apparent contact angle of the drop depends on the voltage following the Young-Lippmann equation up to contact angles of ≈50°. At higher voltages, we observe a saturation of the contact angle caused by the competition between electric and capillary stresses, similar to previous experimental observations. We also study the stability of a dielectric film trapped between a conducting fluid and a solid electrode and find a good agreement with analytical predictions based on lubrication theory. Finally, we investigate the film dynamics at long times and report observations of film breakup and entrapment similar to previously reported experimental results.

Apparent Contact Angles on Lubricant-Impregnated Surfaces/SLIPS: From Superhydrophobicity to Electrowetting.

A fundamental limitation of liquids on many surfaces is their contact line pinning. This limitation can be overcome by infusing a nonvolatile and immiscible liquid or lubricant into the texture or roughness created in or applied onto the solid substrate so that the liquid of interest no longer directly contacts the underlying surface. Such slippery liquid-infused porous surfaces (SLIPS), also known as lubricant-impregnated surfaces, completely remove contact line pinning and contact angle hysteresis. However, although a sessile droplet may rest on such a surface, its contact angle can be only an apparent contact angle because its contact is now with a second liquid and not a solid. Close to the solid, the droplet has a wetting ridge with a force balance of the liquid-liquid and liquid-vapor interfacial tensions described by Neumann's triangle rather than Young's law. Here, we show how, provided the lubricant coating is thin and the wetting ridge is small, a surface free energy approach can be used to obtain an apparent contact angle equation analogous to Young's law using interfacial tensions for the lubricant-vapor and liquid-lubricant and an effective interfacial tension for the combined liquid-lubricant-vapor interfaces. This effective interfacial tension is the sum of the liquid-lubricant and the lubricant-vapor interfacial tensions or the liquid-vapor interfacial tension for a positive and negative spreading power of the lubricant on the liquid, respectively. Using this approach, we then show how Cassie-Baxter, Wenzel, hemiwicking, and other equations for rough, textured or complex geometry surfaces and for electrowetting and dielectrowetting can be used with the Young's law contact angle replaced by the apparent contact angle from the equivalent smooth lubricant-impregnated surface. The resulting equations are consistent with the literature data. These results enable equilibrium contact angle theory for sessile droplets on surfaces to be used widely for surfaces that retain a thin and conformal SLIPS coating.

Pinning-Free Evaporation of Sessile Droplets of Water from Solid Surfaces.

Contact-line pinning is a fundamental limitation to the motion of contact lines of liquids on solid surfaces. When a sessile droplet evaporates, contact-line pinning typically results in either a stick-slip evaporation mode, where the contact line pins and depins from the surface in an uncontrolled manner, or a constant contact-area mode with a pinned contact line. Pinning prevents the observation of the quasi-equilibrium constant contact-angle mode of evaporation, which has never been observed for sessile droplets of water directly resting on a smooth, nontextured, solid surface. Here, we report the evaporation of a sessile droplet from a flat glass substrate treated with a smooth, slippery, omni-phobic covalently attached liquid-like coating. Our characterization of the surfaces shows high contact line mobility with an extremely low contact-angle hysteresis of ∼1° and reveals a step change in the value of the contact angle from 101° to 105° between a relative humidity (RH) of 30 and 40%, in a manner reminiscent of the transition observed in a type V adsorption isotherm. We observe the evaporation of small sessile droplets in a chamber held at a constant temperature, T = (25.0 ± 0.1) °C and at constant RH across the range RH = 10-70%. In all cases, a constant contact-angle mode of evaporation is observed for most of the evaporation time. Furthermore, we analyze the evaporation sequences using the Picknett and Bexon ideal constant contact-angle mode for diffusion-limited evaporation. The resulting estimate for the diffusion coefficient, DE, of water vapor in air of DE = (2.44 ± 0.48) × 10-5 m2 s-1 is accurate to within 2% of the value reported in the literature, thus validating the constant contact-angle mode of the diffusion-limited evaporation model.

Snap evaporation of droplets on smooth topographies.

Droplet evaporation on solid surfaces is important in many applications including printing, micro-patterning and cooling. While seemingly simple, the configuration of evaporating droplets on solids is difficult to predict and control. This is because evaporation typically proceeds as a "stick-slip" sequence-a combination of pinning and de-pinning events dominated by static friction or "pinning", caused by microscopic surface roughness. Here we show how smooth, pinning-free, solid surfaces of non-planar topography promote a different process called snap evaporation. During snap evaporation a droplet follows a reproducible sequence of configurations, consisting of a quasi-static phase-change controlled by mass diffusion interrupted by out-of-equilibrium snaps. Snaps are triggered by bifurcations of the equilibrium droplet shape mediated by the underlying non-planar solid. Because the evolution of droplets during snap evaporation is controlled by a smooth topography, and not by surface roughness, our ideas can inspire programmable surfaces that manage liquids in heat- and mass-transfer applications.

Using evaporation to control capillary instabilities in micro-systems.

The instabilities of fluid interfaces represent both a limitation and an opportunity for the fabrication of small-scale devices. Just as non-uniform capillary pressures can destroy micro-electrical mechanical systems (MEMS), so they can guide the assembly of novel solid and fluid structures. In many such applications the interface appears during an evaporation process and is therefore only present temporarily. It is commonly assumed that this evaporation simply guides the interface through a sequence of equilibrium configurations, and that the rate of evaporation only sets the timescale of this sequence. Here, we use Lattice-Boltzmann simulations and a theoretical analysis to show that, in fact, the rate of evaporation can be a factor in determining the onset and form of dynamical capillary instabilities. Our results shed light on the role of evaporation in previous experiments, and open the possibility of exploiting diffusive mass transfer to directly control capillary flows in MEMS applications.

Energy Invariance in Capillary Systems.

We demonstrate the continuous translational invariance of the energy of a capillary surface in contact with reconfigurable solid boundaries. We present a theoretical approach to find the energy-invariant equilibria of spherical capillary surfaces in contact with solid boundaries of arbitrary shape and examine the implications of dynamic frictional forces upon a reconfiguration of the boundaries. Experimentally, we realize our ideas by manipulating the position of a droplet in a wedge geometry using lubricant-impregnated solid surfaces, which eliminate the contact-angle hysteresis and provide a test bed for quantifying dissipative losses out of equilibrium. Our experiments show that dissipative energy losses for an otherwise energy-invariant reconfiguration are relatively small, provided that the actuation time scale is longer than the typical relaxation time scale of the capillary surface. We discuss the wider applicability of our ideas as a pathway for liquid manipulation at no potential energy cost in low-pinning, low-friction situations.