Dynamics of Droplet Motion under Electrowetting Actuation

This work focuses on the numerical study of electrohydrodynamic multiphase fluid flow problems. Liquid bodies experience deformations caused by strong external electric fields. These deformations can be used in industrial applications to accurately control the motion of single droplets. In the limiting case, strong electric fields can force atomisation of fine microdroplets from much larger liquid bulks. Due to differences in the electrical properties of the different fluid phases, the flow affects in turn the electric field distribution. The analysis of such coupled flows requires numerical modelling of the interdependent hydrodynamic and electrodynamic problems. Since most liquids exhibit some conductivity due to intrinsic ionic species and dissolved impurities, the electrodynamic problem must be modelled by an electroquasistatic model taking into account capacitive, resistive and convective electrical currents. This electroquasistatic problem is coupled to an incompressible fluid flow problem described by the Navier-Stokes equations. Both problems are solved on the same computational grid, using the finite volume method. The fluid-fluid interface is modelled using the volume of fluid method. The resulting diffuse interface captures the motion of the phase boundaries while efficiently handling topology changes. The motion of contact lines is furthermore modelled using a dynamic contact angle model including hysteresis effects. Pinned contact lines and stick-slip contact line motion in transient problems can therefore be represented. The developed solver is readily applicable to a large range of electrohydrodynamic flow problems. This work investigates electrohydrodynamic flows occurring in three technical applications. First, the dynamics of sessile droplets subjected to an AC electric field on the surface of an insulator are considered. The dynamics of the oscillating droplets are compared with experimental data. Partial discharge inception fields are then estimated for similar configurations. Secondly, the detachment dynamics of two liquids in an on-demand droplet generator, where droplet detachment is enforced by electric fields, are considered and compared to experimental data. The different conductivities of the liquids are shown to lead to substantially different detachment dynamics. Relevant parameters in the detachment dynamics are extracted from the simulations. Finally, transient electrosprays in the cone-jet mode are characterised for a number of liquids with different electromechanical properties. The charge-radius correlations of the first ejections are found to obey the scaling laws reported in the literature. Moreover, additional scaling laws are found for the subsequent ejections.

Self-propelled motion of sessile droplets on gradient surfaces is key to the advancement of microfluidic, nanofluidic, and surface fluidic technologies. Precise control over droplet dynamics, which often involves liquid–vapor phase transitions, is crucial for a variety of applications, including thermal management, self-cleaning surfaces, biochemical assays, and microreactors. Understanding how specific phase changes like condensation and evaporation affect droplet motion is essential for enhancing droplet manipulation and improving transport efficiency. We use the thermal Navier–Stokes–Korteweg equations to investigate the effects of condensation and evaporation on the motion and internal dynamics of droplets migrating across a surface with a linear surface energy profile. The study focuses on the early dynamics of self-propelled motion of a phase changing droplet at sub-micron scale before viscous forces are comparable with the gradient forces. Our results demonstrate that phase change significantly affects the self-propelled motion of droplets by reshaping interfacial mass flux distributions and internal flow dynamics. Condensation increases droplet volume and promotes extensive spreading toward regions of higher wettability, while evaporation reduces both volume and spreading. These changes in droplet shape and size directly affect the driving forces of motion, augmenting self-propulsion through condensation and suppressing it during evaporation. Additionally, each phase change type generates distinct internal flow patterns within the droplet, with condensation and evaporation exhibiting unique circulatory movements driven by localized phase changes near the contact lines.

Advancing and receding angles are physical quantities frequently measured to characterize the wetting properties of a rough surface. Thermodynamically, the advancing and receding angles are often interpreted as the maximum and minimum contact angles that can be formed by a droplet without losing its stability. Despite intensive research on wetting of rough surfaces, the gravitational effect on these angles has been overlooked because most studies have considered droplets smaller than the capillary length. In this study, however, by combining theoretical and numerical modeling, we show that the shape of a droplet smaller than the capillary length can be substantially modified by gravity under advancing and receding conditions. First, based on the Laplace pressure equation, we predict the shape of a two-dimensional Cassie-Baxter droplet on a textured surface with gravity at each pinning point. Then, the stability of the droplet is tested by examining the interference between the liquid surface and neighboring pillars and analyzing the free energy change upon depinning. Interestingly, it turns out that the apparent contact angles under advancing and receding conditions are not affected by gravity, while the overall shape of a droplet and the position of the pinning point are affected by gravity. In addition, the advancing and receding of the droplet with continuously increasing or decreasing volume are analyzed, and it is shown that the gravitational effect plays a key role in the movement of the droplet tip. Also, the gravitational effect on the degree of the stability of the droplet upon the external effect such as vibration is discussed. Finally, the theoretical predictions were validated against line tension-based front tracking modeling (LTM) that seamlessly captures the attachment and detachment between the liquid surface and the solid substrate. Our findings provide a deeper understanding on the advancing and receding phenomena of a droplet and essential insight into designing devices that utilize the wettability of rough surfaces.

The behavior of water droplets placed on the surface of a hydrophobic polymer plate under a DC field was investigated experimentally. In order to confirm the effect of hydrophobicity on the movement of the water droplet, three kinds of hydrophobic insulating materials-PTFE, silicone rubber (SR) and ethylene propylene rubber (EPDM)-were used. Water droplets were placed on the insulating plate in contact with the aluminum foil electrode or placed between the thin foil electrodes. A single droplet placed in contact with the electrode on the SR or EPDM plate was elongated and a water filament extended along the surface by the electrostatic force, while that on the PTFE plate left the electrode. The water droplet placed on the border of electrode at the PTFE plate was deformed and formed a semi-cone at a triple junction. The apparent contact angle varied with time for a constant applied voltage and some amounts of charge were emitted by micro discharge or by ejection of small droplets. The behavior of the droplets under DC field strongly depends on not only the hydrophobicity but also the surface property of an insulating plate.

Numerical simulation of bubble formation on orifice plates with a moving contact line

Numerical simulation of liquid water emerging and transport in the flow channel of PEMFC using the volume of fluid method

Prediction and analysis of the shapes of liquid-vapor interface of droplets under the influence of external forces is critical for various applications. In this regard, a geometric model that can capture the macroscopic shape of the liquid-vapor interface in tandem with the subtleties near the contact line, particularly in the regime where the droplet shape deviates significantly from the idealized spherical cap geometry, is desirable. Such deviations may occur when external forces such as gravity or centrifugal dominate over the surface tension force. Here we use vector parametrized cubic spline representation for axisymmetric fluid-fluid interfaces along with a novel thermodynamic free energy minimization based heuristic to determine the shape of liquid-vapor interface of droplets. We show that the current scheme can easily predict the shapes of sessile and pendant droplets under the action of centrifugal force over a broad range of surface contact angle values and droplet sizes encountered in practical applications. Finally, we show that the cubic spline based modeling approach makes it convenient to perform the inverse analysis as well, i.e., predict interfacial properties from the shape of a droplet under the action of various types of external forces including gravity and centrifugal. We believe that this versatile modeling approach can be extended to model droplet shapes under various other external forces including electric and acoustic. In addition, the simple shape analysis approach is also promising for the development of inexpensive interfacial analysis tools such as surface tensiometers.

Lattice Boltzmann Modeling (LBM) simulations were performed on the dynamic behavior of liquid droplets on chemically striped patterned surfaces, ultimately with the aim to develop a predictive tool enabling reliable design of future experiments. The simulations accurately mimic experimental results, which have shown that water droplets on such surfaces adopt an elongated shape due to anisotropic preferential spreading. Details of the contact line motion such as advancing of the contact line in the direction perpendicular to the stripes exhibit pronounced similarities in experiments and simulations. The opposite of spreading, i.e., evaporation of water droplets, leads to a characteristic receding motion first in the direction parallel to the stripes, while the contact line remains pinned perpendicular to the stripes. Only when the aspect ratio is close to unity, the contact line also starts to recede in the perpendicular direction. Very similar behavior was observed in the LBM simulations. Finally, droplet movement can be induced by a gradient in surface wettability. LBM simulations show good semiquantitative agreement with experimental results of decanol droplets on a well-defined striped gradient, which move from high- to low-contact angle surfaces. Similarities and differences for all systems are described and discussed in terms of the predictive capabilities of LBM simulations to model direction wetting.

The dynamic spreading of nanofluids on solid surfaces – Role of the nanofilm structural disjoining pressure

Analyses of the dynamic behavior of droplets are significant for a broad range of industrial applications. However, previous works have primarily studied the dynamic behavior on solid flat plates. Therefore, they could consider both the effect of geometric structure and energy simultaneously. In order to analyze the coupled-dynamic behavior of droplets, SMA membranes with microsphere arrays were used to apply the effects of both geometry and energy to droplet surfaces. The shape of the droplet changed harmonically with time during actuation. In the simulation, 50 and 200 Hz excitations related to the first and second droplet frequencies, respectively were used in order to investigate the coupled-behavior of the droplet. COMSOL Multiphasics™ was used to numerically analyze the dynamic behavior of droplets. The simulation results showed that the dynamic behavior of the water droplet was strongly affected by the natural frequency, and the interface and behavior were determined by both the mode shape and vibrating energy. As the actuation frequency increased, the energy equilibrium changed at the interface and occasionally broke. And then, the contact line on the actuated surface was changed to create new stable state.

<p style='text-indent:20px;'>We investigate numerically the advancing motion of 3D droplets spreading on physically flat chemically heterogeneous surfaces with periodic structures. We use the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions to model the motion of droplets. Based on a convex splitting scheme, we have done numerical simulations and compared the results between different surface patterns quantitatively. We study the effect of pattern property on the advancing motion of three phase contact lines, the critical volume at the contact line jump and the effective advancing angles. By increasing the volume of droplet slowly on heterogeneous surfaces with different pattern property, we find that the advancing contact line approaches an equiangular octagon for the patterned surface with periodic squares separated by channels and approaches a regular hexagon for the patterned surface with periodic circles in regular hexagonal arrays. The shape of three-phase contact line is much more determined by the macro structure of the pattern than the micro structure of the pattern in each period.

Anisotropy in the wetting of rough surfaces

Using a continuum model capable of describing the one-component liquid-gas hydrodynamics down to the contact line scale, we carry out numerical simulation and physical analysis for the droplet motion driven by thermal singularity. For liquid droplets in one-component fluids on heated or cooled substrates, the liquid-gas interface is nearly isothermal. Consequently, a thermal singularity occurs at the contact line and the Marangoni effect due to temperature gradient is suppressed. Through evaporation or condensation in the vicinity of the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature. This effect on the contact angle can be used to move the droplets on substrates with thermal gradients. Our numerical results for this kind of droplet motion are explained by a simple fluid dynamical model at the droplet length scale. Since the mechanism for droplet motion is based on the change of contact angle, a separation of length scales is exhibited through a comparison between the droplet motion induced by a wettability gradient and that by a thermal gradient. It is shown that the flow field at the droplet length scale is independent of the statics or dynamics at the contact line scale.

We develop a mathematical model for heat transfer and fluid flow near a contact line on a heated surface in the presence of thermocapillary flow and evaporation. The coupled heat transfer and flow problem is reduced to an equation for local thickness, which is then solved numerically. The steady-state results indicate that thermocapillary stresses act to reduce the rate of liquid flow towards the contact line and increase interfacial curvature there. We also discuss solutions than involve moving contact lines, applicable to studies of start-up and shut-down operations of heat pipes. The velocity of the contact line and the apparent contact angle are found as functions of the Marangoni number. Thermocapillary effect is shown to reduce contact line speed and increase the apparent contact angle. Finally, the local solution is incorporated into global solutions for curvature variations of an evaporating three-dimensional meniscus in a corner. This configuration is typically encountered in proposed designs of micro heat pipes. Interface curvature is found as a function of the axial coordinate for the case of linear axial temperature variation in the corner.

The interaction of pure capillary waves with boundaries that constrain the contact line are of interest for problems involving liquids contained by minimal solid contact for applications in low gravity and at small scales in normal gravity. Time-harmonic capillary waves on a liquid cylinder axially incident on and scattered by an infinitesimal concentric barrier are considered theoretically in the inviscid limit. The barrier is taken to be infinitesimally small in the sense that its immersed depth is of the order of the amplitude of contact-line motion. Edge conditions on the barrier that are investigated include a pinned contact line and a moving contact line by an effective-slip model, assuming that contact-line velocity is proportional to the deviation of the contact angle from equilibrium multiplied by a slip coefficient. The incident waves are taken to be those with wavelengths short enough to be stable on the liquid cylinder. Scattering and dissipation by the contact line are determined as a function of wavenumber and slip coefficient. Zero transmission is approached in the long-wave limit. The short-wave limit agrees with established results for the scattering of planar gravity–capillary waves on a deep liquid by a surface-piercing vertical barrier in the limit of zero barrier depth and zero gravity. We find that contact-line dissipation at the barrier is a maximum for incident waves whose phase speed is of the order of the slip coefficient, which is interpreted as an effect of impedance matching. Transmission past an infinitesimal barrier is found to be low over all parameter space, illustrating the importance of contact-line constraints.