Wetting of a drop on a sphere

Particulate-Droplet Coalescence and Self-Transport on Superhydrophobic Surfaces

Particulate transport from surfaces governs a variety of phenomena including fungal spore dispersal, bioaerosol transmission, and self-cleaning. Here, we report a previously unidentified mechanism governing passive particulate removal from superhydrophobic surfaces, where a particle coalescing with a water droplet (∼10 to ∼100 μm) spontaneously launches. Compared to previously discovered coalescence-induced binary droplet jumping, the reported mechanism represents a more general capillary-inertial dominated transport mode coupled with particle/droplet properties and is typically mediated by rotation in addition to translation. Through wetting and momentum analyses, we show that transport physics depends on particle/droplet density, size, and wettability. The observed mechanism presents a simple and passive pathway to achieve self-cleaning on both artificial as well as biological materials as confirmed here with experiments conducted on butterfly wings, cicada wings, and clover leaves. Our findings provide insights into particle-droplet interaction and spontaneous particulate transport, which may facilitate the development of functional surfaces for medical, optical, thermal, and energy applications.

Droplet Spreading and Adhesion on Spherical Surfaces

Adhesion of a liquid droplet to a solid surface is a result of solid surface interactions with surrounding fluids, affected by its wettability and morphology. Unfortunately, the direct measurements of adhesion forces are rarely reported in the scientific literature, especially for solids with curvatures. In this study, by using a high-sensitivity microelectronic mechanical balance which vertically deposits and then pulls liquid droplets, the spreading and adhesion forces for water and ethylene glycol droplets on spherical surfaces of polyethylene terephthalate (PET) with radii of curvature from 2 to 8 mm were recorded. Results show that the surface curvature does not affect the advancing and most-stable contact angles but affects the extent of spreading and maximum adhesion forces. The solid surface curvature affects both surface tension and Laplace pressure forces at the spreading point, whereas it mainly affects the Laplace pressure force at the maximum adhesion point.

Droplet Characteristics at the Maximum Adhesion on Curved Surfaces

For applications involving droplet detachment from solid surfaces, it is vital to study the droplet characteristics (e.g., contact angle and base width) when the droplet is experiencing the maximum force that detaches the droplet (maximum adhesion state). Historically, such investigations were mainly conducted on flat two-dimensional surfaces and the characteristics on curved surfaces with the third dimension remain unknown. Thus, the generalized description of such characteristics has not been established yet. Here, by vertically pulling liquid droplets using a microbalance, we study the droplet characteristics at the maximum adhesion on curved homogeneous surfaces. Variables in this study include liquid surface tension, initial droplet base area, and the asymmetry in solid surface curvature. Results show that the contact angle is identical everywhere along the droplet perimeter on curved surfaces irrespective of the asymmetry in surface curvature. In addition, we found that the droplet base is nonaxisymmetric (not circular) at the maximum adhesion, opposing previous understanding that was formulated for flat surfaces. As a result, we propose a more generalized and quantitative description of the droplet characteristics at the maximum adhesion, derived from the component of the surface tension force acting along the droplet perimeter.

Contact angle as a powerful tool in anisotropic colloid synthesis

Nucleation and growth is a technique widely used to prepare colloids, in which droplets are adsorbed onto substrate particles. Changing the contact angle of the substrates can greatly alter the morphology of the product particles. Here, we investigate the nucleation and growth of 3-methacryloxypropyltrimethoxysilane (MPTMS) both onto Stöber spheres and onto (cross-linked) MPTMS* spheres. The former results in 'snowman' particles with a cap-shaped MPTMS* compartment, and we show that their morphology is highly controllable via the MPTMS content in the reaction mixture. The contact angle of the MPTMS* compartment decreases with droplet diameter, suggesting that this wetting process is affected not only by surface tension but also by line tension. In contrast to Stöber spheres, MPTMS* substrate particles yield highly reproducible and tuneable 'engulfed-sphere' colloids with an internal reference axis (but a homogeneous mass distribution). These engulfed-sphere particles can be fully index-matched for confocal microscopy on account of their homogeneous refractive index. Suitable index-matching mixtures of polar and of low-polar media are presented, where cyclohexyl iodide (CHI) is introduced as a new medium for colloids of high refractive index. Finally, the index-matched engulfed-sphere colloids are self-assembled into (close-packed and long-range) plastic phases, and the particles' rotational diffusion inside the crystal phases is tracked via confocal microscopy.

Dynamical insights into the mechanism of a droplet detachment from a fiber

Quantifying the detachment behavior of a droplet from a fiber is important in many applications such as fog harvesting, oil-water separation, or water management in fuel cells. When the droplets are forcibly removed from hydrophilic fibers, the ease of detachment strongly depends on droplet volume and the rate of the process controlled by the applied force. Experiments, conducted on a ferrofluid under magnetic force, as well as continuum level calculations from fluid mechanics have so far been unable to resolve the time-dependent dynamics of droplet detachment and, most importantly, to assess the role of the applied force as the key determinant of the volume of the droplet residue remaining on the fiber after detachment. In the present work, we study the mechanism of water droplet detachment and retention of residual water on smooth hydrophilic fibers using nonequilibrium molecular dynamics simulations. We investigate how the applied force affects the breakup of a droplet and how the minimal detaching force per unit mass decreases with droplet size. We extract scaling relations that allow extrapolation of our findings to larger length scales that are not directly accessible by molecular models. We find that the volume of the residue on a fiber varies nonmonotonically with the detaching force, reaching the maximal size at an intermediate force and associated detachment time. The strength of this force decreases with the size of the drop, while the maximal residue increases with the droplet volume, V, sub-linearly, in proportion to the V2/3.

Maximum Spreading and Rebound of a Droplet Impacting onto a Spherical Surface at Low Weber Numbers

The spreading and rebound patterns of low-viscous droplets upon impacting spherical solid surfaces are investigated numerically. The studied cases consider a droplet impinging onto hydrophobic and superhydrophobic surfaces with various parameters varied throughout the study, and their effects on the postimpingement behavior are discussed. These parameters include impact Weber number (through varying the surface tension and impingement velocity), the size ratio of the droplet to the solid surface, and the surface contact angle. According to the findings, the maximum spreading diameter increases with the impact velocity, with an increase of the sphere diameter, with a lower surface wettability, and with a lower surface tension. Typical outcomes of the impact include (1) complete rebound, (2) splash, and (3) a final deposition stage after a series of spreading and recoiling phases. Finally, a novel, practical model is proposed, which can reasonably predict the maximum deformation of low Reynolds number impact of droplets onto hydrophobic or superhydrophobic spherical solid surfaces.

An alternative method to implement contact angle boundary condition and its application in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces

We propose an alternative method to implement the contact angle boundary condition on a solid wall and apply it in hybrid lattice-Boltzmann finite-difference simulations of two-phase flows with immersed surfaces in which the flow equations are solved by the lattice-Boltzmann method and the interface equations are solved by the finite-difference method. Using the hyperbolic tangent profile of the order parameter across an interface in phase-field theory, we were able to obtain its unknown value at a ghost point from the information at only one point in the fluid domain. This is in contrast with existing approaches relying on interpolations involving several points. The special feature allows it to be more easily implemented on immersed surfaces cutting through the grid lines. It was properly incorporated into the framework of the hybrid lattice-Boltzmann finite-difference simulation, and applied successfully for several problems with different levels of complexity. First, the equilibrium shapes of a droplet on a sphere with different contact angles and radii were studied under cylindrical geometry and a good agreement with theoretical predictions was found. Preliminary studies on a three-dimensional droplet spreading on a sphere were also performed and the results agreed well with the corresponding axisymmetric results. Second, the spreading of a two-dimensional drop on an embedded inclined wall with a given contact angle was investigated and the results matched those on a flat wall at the domain boundary under the same condition. Third, capillary filling in a cylindrical tube was studied and the speed of the interface in the tube was found to follow Washburn's law. Fourth, a droplet impacting on a sphere was investigated and several different outcomes were captured depending on the Reynolds number, the viscosity ratio, and the wettability and radius of the sphere. Finally, the proposed method was shown to be capable of studying even more complicated problems involving the interaction between a droplet and multiple objects of different sizes and contact angles.

Topography- and topology-driven spreading of non-Newtonian power-law liquids on a flat and a spherical substrate

The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids on a flat and a spherical rough and textured substrate is theoretically studied in the capillary-controlled spreading regime. A droplet whose scale is much larger than that of the roughness of substrate is considered. The equilibrium contact angle on a rough substrate is modeled by the Wenzel and the Cassie-Baxter model. Only the viscous energy dissipation within the droplet volume is considered, and that within the texture of substrate by imbibition is neglected. Then, the energy balance approach is adopted to derive the evolution equation of the contact angle. When the equilibrium contact angle vanishes, the relaxation of dynamic contact angle θ of a droplet obeys a power-law decay θ∼t^{-α} except for the Newtonian and the non-Newtonian shear-thinning liquid of the Wenzel model on a spherical substrate. The spreading exponent α of the non-Newtonian shear-thickening liquid of the Wenzel model on a spherical substrate is larger than others. The relaxation of the Newtonian liquid of the Wenzel model on a spherical substrate is even faster showing the exponential relaxation. The relaxation of the non-Newtonian shear-thinning liquid of Wenzel model on a spherical substrate is fastest and finishes within a finite time. Thus, the topography (roughness) and the topology (flat to spherical) of substrate accelerate the spreading of droplet.

Six Stages of Microdroplet Detachment from Microscale Fibers

The detachment of droplets from cylindrical fibers is of fundamental importance for both scientific research and engineering applications. Due to the challenges to determine dynamic contact angles on the fiber surface, the process of the droplet detachment from a fiber is not well understood. In this paper, a multibody dissipative particle dynamics (MDPD) method, a particle-based mesh-free method that can automatically capture the dynamic contact angles through direct modeling of liquid-solid particle interactions, was applied to study the detachment process of a liquid microdroplet from a cylindrical solid fiber pulled by an atomic force microscopy (AFM) tip under a constant velocity. After the validation of the numerical results through comparison with experiments in a benchmark case, the same numerical tool was applied to analyze the droplet detachment mechanisms. Based on the slope of the time history curve for the displacement of the droplet mass center, the detachment process can be divided into six stages. The change of slope in each stage can be explained from the change of surface energy. The results can greatly advance the fundamental understanding of the detachment process of microdroplets from cylindrical fibers.

Spreading law of non-Newtonian power-law liquids on a spherical substrate by an energy-balance approach

The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids, both shear-thickening and shear-thinning liquids, that completely wet a spherical substrate is theoretically investigated in the capillary-controlled spreading regime. The crater-shaped droplet model with the wedge-shaped meniscus near the three-phase contact line is used to calculate the viscous dissipation near the contact line. Then the energy balance approach is adopted to derive the equation that governs the evolution of the contact line. The time evolution of the dynamic contact angle θ of a droplet obeys a power law θ∼t^{-α} with the spreading exponent α, which is different from Tanner's law for Newtonian liquids and those for non-Newtonian liquids on a flat substrate. Furthermore, the line-tension dominated spreading, which could be realized on a spherical substrate for late-stage of spreading when the contact angle becomes low and the curvature of the contact line becomes large, is also investigated.

Spreading law on a completely wettable spherical substrate: The energy balance approach

The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied. The nonequilibrium thermodynamic formulation is used to derive the thermodynamic driving force of spreading including the line-tension effect. Then the energy balance approach is adopted to derive the evolution equation of the spreading droplet. The time evolution of the contact angle θ of a droplet obeys a power law θ∼t^{-α} with the exponent α, which is different from that derived from Tanner's law on a flat substrate. Furthermore, the line tension must be positive to promote complete wetting on a spherical substrate, while it must be negative on a flat substrate.

Self-removal of condensed water on the legs of water striders

The ability to control drops and their movements on phobic surfaces is important in printing or patterning, microfluidic devices, and water-repellent materials. These materials are always micro-/nanotextured, and a natural limitation of repellency occurs when drops are small enough (as in a dew) to get trapped in the texture. This leads to sticky Wenzel states and destroys the superhydrophobicity of the material. Here, we show that droplets of volume ranging from femtoliter (fL) to microliter (μL) can be self-removed from the legs of water striders. These legs consist of arrays of inclined tapered setae decorated by quasi-helical nanogrooves. The different characteristics of this unique texture are successively exploited as water condenses, starting from self-penetration and sweeping effect along individual cones, to elastic expulsion between flexible setae, followed by removal at the anisotropic leg surface. We envision that this antifogging effect at a very small scale could inspire the design of novel applicable robust water-repellent materials for many practical applications.

Bubbles and drops on curved surfaces

Surface curvature affects the shape, stability, and apparent contact angle of sessile and pendant drops. Here, we develop an approximate analytical solution for non-axisymmetric perturbations to small spherical drops on a flat substrate, assuming a fixed contact angle and fixed drop volume. The analytical model is validated using numerical solutions of the Laplace equation from the Surface Evolver software. We investigate the effects of surface curvature on drop shape, pressure, and surface energy, ascertaining the energy-gradient force that drives lateral drop migration. By balancing this force with the viscous resistance/drag force, in the lubrication approximation, we predict velocities of the order of 0.1 mm s(-1) for 1 mm diameter drops of water with a 30° contact angle on a substrate with a curvature gradient of 0.01 mm(-2), achieved, for example, on a harmonic surface with a wavelength of 4 cm and an amplitude of 4 mm.

Buoyant droplets on functional fibers

In the absence of gravity, the wetting of droplets on fibers is characterized by the competition between an axisymmetric barrel morphology engulfing the fiber and a symmetry-broken clamshell morphology with the droplet sitting on the side of the fiber. In the generic case of nonzero buoyancy the cylindrical symmetry of the barrel morphology is broken, yet barrels and clamshells can still be distinguished based on their different interfacial topologies being multiply and simply connected, respectively. Next to contact angle and droplet size the capillary length appears as a third parameter controlling the droplet morphology. For droplets of variable size, contact angle and buoyancy are independently varied in experiments by use of electrowetting and density mismatch. This approach--together with the complementary numerical calculations--provides new insights into the gradual shifts of the stability limits in the presence of an external volume force. Overall, the parameter space for stable clamshells is found to expand with increasing gravitational forces, gradually shrinking the regimes of stable barrels and bistability. In addition, a new stability limit is introduced for the clamshell morphology related to a partial detachment of the wetting liquid from the fiber, appearing toward higher droplet volumes.

Colloidal dynamics near a particle-covered surface

How the diffusive dynamics of colloidal spheres changes in the vicinity of a particle-coated surface is of importance for industrial challenges such as fouling and sedimentation as well as for fundamental studies into confinement effects. We addressed this question by studying colloidal dynamics in a partially coated surface layer, using video microscopy. Particle mean squared displacement (MSD) functions were measured as a function of a (local) effective volume fraction (EVF), which was varied by making use of gravity settling. Comparison of MSDs at the bare and coated surfaces for EVF of 0.2-0.4 revealed that at the latter surface the motion amplitudes are strongly reduced, accompanied by a sharp transition from diffusive to nearly caged motion. This clearly indicates that the surface-attached particles cannot be taken into account via volume fraction and that their immobility has a distinct effect. For EVF > 0.45, the caging becomes dominated by the suspended particles, making the dynamics at the bare and coated surfaces similar.