Controlling the shape and position of moving and pinned droplets on a solid surface is an important feature often found in microfluidic applications. However, automating them, e.g., for high-throughput applications, does rarely involve model-based optimal control strategies. In this work, we demonstrate the optimal control of both the shape and position of a droplet sliding on an inclined surface. This basic test case is a fundamental building block in plenty of microfluidic designs. The static contact angle between the solid surface, the surrounding gas, and the liquid droplet serves as the control variable. By using several control patches, e.g., like done in electrowetting, the contact angles are allowed to vary in space and time. In computer experiments, we are able to calculate mathematically optimal contact angle distributions using gradient-based optimization. The dynamics of the droplet are described by the Cahn-Hilliard-Navier-Stokes equations. We anticipate our demonstration to be the starting point for more sophisticated optimal design and control concepts.
Manipulating surface topography is one of the most promising strategies for increasing the efficiency of numerous industrial processes involving droplet contact with superheated surfaces. In such scenarios, the droplets may immediately boil upon contact, splash and boil, or could levitate on their own vapor in the Leidenfrost state. In this work, we report the outcomes of water droplets coming in gentle contact with designed nano/micro-textured surfaces at a wide range of temperatures as observed using high-speed optical and X-ray imaging. We report a paradoxical increase in the Leidenfrost te…
Passive droplet transport mechanisms, in which continuous external energy input is not required for motion, have received significant attention in recent years. Experimental studies of such mechanisms often ignore, or use careful treatments to minimize, contact angle hysteresis, which can impede droplet motion, or even arrest it completely. Here, we consider the effect of contact angle hysteresis on bendotaxis, a mechanism in which droplets spontaneously deform an elastic channel via capillary pressure and thereby move. Here, we seek to understand when contact angle hysteresis prevents bendota…
Evaporation of water droplets on a superhydrophobic substrate, on which the contact line is pinned, is investigated. While previous studies mainly focused on droplets with contact angles smaller than 90^\circ, here we analyze almost the full range of possible contact angles (10^\circ -150^\circ). The greater contact angles and pinned contact lines can be achieved by the use of superhydrophobic Carbon Nanofiber substrates. The time-evolutions of the contact angle and the droplet mass are examined. The experimental data is in good quantitative agreement with the model presented by Popov (Physica…
We theoretically investigate the apparent contact angle of droplets on liquid infused surfaces as a function of the relative size of the wetting ridge and the deposited droplet. We provide an intuitive geometrical interpretation whereby the variation in the apparent contact angle is due to the rotation of the Neumann triangle. We also derive linear and quadratic corrections to the apparent contact angle as power series expansion in terms of pressure differences between the lubricant, droplet and gas phases. These expressions are much simpler and more compact compared to those previously derive…
Moving contact lines of more than two phases dictate a large number of interfacial phenomena. Despite its significance to fundamental and applied processes, the contact lines at a junction of four-phases (two immiscible liquids, solid and gas) have been addressed only in a few investigations. Here, we report an intriguing phenomenon that follows after the four phases of oil, water, solid and gas make contact through the coalescence of two different three-phase contact lines. We combine experimental study and theoretical analysis to reveal and rationalize the dynamics exhibited upon the coalesc…
When a droplet gently lands on an atomically smooth substrate, it will most likely contact the underlying surface in about 0.1 s. However, theoretical estimation from fluid mechanics predicts a contact time of 10 to 100 s. What causes this large discrepancy, and how does nature speed up contact by 2 orders of magnitude? To probe this fundamental question, we prepare atomically smooth substrates by either coating a liquid film on glass or using a freshly cleaved mica surface, and visualize the droplet contact dynamics with 30 nm resolution. Interestingly, we discover two distinct speed-up appro…
Line tension from dual-geometry sessile droplet measurements: Combining contact-angle size-dependence data for axisymmetric and cylindrical droplets to determine the line tension
We consider the effect of droplet geometry on the early-stages of coffee ring formation during the evaporation of a thin droplet with an arbitrary simple, smooth, pinned contact line. We perform a systematic matched asymptotic analysis of the small-capillary number, large-solutal Peclet number limit for two evaporative models: a kinetic model, in which the evaporative flux is constant across the droplet, and a diffusive model, in which the evaporative flux is singular at the contact line. For both evaporative models, solute is transported to the contact line by a capillary flow while, local to…
In recent years the Smoothed Particle Hydrodynamics (SPH) approach gained popularity in modeling multiphase and free-surface flows. In many situations, due to certain reasons, interface and free-surface fragmentation occurs. As a result single SPH particle solids/droplets of one phase can appear and travel through other phases. In this paper we investigate this issue focusing on a movement of such single SPH particles. The main questions we try to answer here are: is movement of such particles physically correct? What is its physical size? How numerical parameters affect on it? With this in mi…
Gravitational settling of a droplet in air onto a soft substrate is a ubiquitous event relevant to many natural processes and engineering applications. We study this phenomenon by developing a three-phase lubrication model of droplet settling onto a solid substrate coated by a thin soft layer represented by a viscous film, an elastic compressible layer and an elastic sheet supported by a viscous film. By combining scaling analysis, analytical methods, and numerical simulations we elucidate how the resulting droplet dynamics is affected by the nature of the soft layer. We show that these soft l…
This paper reports the dynamic wetting behavior and heat transfer characteristics for impinging droplets on heated bi-phobic surfaces (superhydrophobic matrix with hydrophobic spots). A non-patterned superhydrophobic and a sticky hydrophobic surface acted as control wettability surfaces. As expected, differences in wetting and heat transfer dynamics were noticeable for all surfaces, with the most pronounced variation during the receding phase. During spreading, inertia from the impact dominated the droplet dynamics and heat transfer was dominated by convection at the contact line and internal …
Electrowetting-on-dielectric (EWOD) is a powerful tool in many droplet-manipulation applications with a notorious weakness caused by contact-angle saturation (CAS), a phenomenon limiting the equilibrium contact angle of an EWOD-actuated droplet at high applied voltage. In this paper, we study the spreading behaviours of droplets on EWOD substrates with the range of applied voltage exceeding the saturation limit. We experimentally find that at the initial stage of spreading, the driving force at the contact line still follows the Young-Lippmann law even if the applied voltage is higher than the…
The dynamic behavior of a partially wetting polymer droplet driven over a nanostructured interface is studied using molecular dynamics simulations. We consider the bead-spring model to represent a polymeric liquid that partially wets a rough surface composed of a periodic array of spherical particles. It is shown that at sufficiently small values of the external force, the droplet remains pinned at the particles' surface, while above the threshold, its motion consists of alternating periods of pinning and rapid displacements between neighboring particles. The latter process involves large peri…
An arbitrary Lagrangian--Eulerian (ALE) finite element scheme for computations of soluble surfactant droplet impingement on a horizontal surface is presented. The numerical scheme solves the time-dependent Navier--Stokes equations for the fluid flow, scalar convection-diffusion equation for the surfactant transport in the bulk phase, and simultaneously, surface evolution equations for the surfactants on the free surface and on the liquid-solid interface. The effects of surfactants on the flow dynamics are included into the model through the surfactant-dependent surface tension and dynamic cont…
Droplets on fibers have been extensively studied in the recent years. Although the equilibrium shapes of simple droplets on fibers are well established, the situation becomes more complex for compound fluidic systems. Through experimental and numerical investigations, we show herein that compound droplets can be formed on fibers and that they adopt specific geometries. We focus on the various contact lines formed at the meeting of the different phases and we study their equilibrium state. It appears that, depending on the surface tensions, the triple contact lines can remain separate or merge …
Depending on the contact line motion, colloid-rich drolets evaporation can leave a ring-like or a spot-like residue. Herein, we determine this outcome by controlling the contact line motion using coplanar direct current electrowetting-on-dielectrics (DC-EWOD). Combined with theoretical calculations of the droplet shape and its evaporation rate, the time-dependent actuation voltage is first derived from experiments and simulations. Thanks to the additional control over the contact angle, the contact line can be maintained in pinned state even on surfaces that exhibit little contact angle hyster…
The motion of a contact line is examined, and comparisons drawn, for a variety of models proposed in the literature. Pressure and stress behaviours at the contact line are examined in the prototype system of quasistatic spreading of a thin two-dimensional droplet on a planar substrate. The models analysed include three disjoining pressure models based on van der Waals interactions, a model introduced for polar fluids, and a liquid-gas diffuse-interface model; Navier-slip and two non-linear slip models are investigated, with three microscopic contact angle boundary conditions imposed (two of th…
Building on the recent theoretical work of Wray, Duffy and Wilson [J. Fluid Mech. 884, A45 (2020)] concerning the competitive diffusion-limited evaporation of multiple thin sessile droplets in proximity to each other, we obtain theoretical predictions for the spatially non-uniform densities of the contact-line deposits (often referred to as "coffee stains" or "ring stains") left on the substrate after such droplets containing suspended solid particles have completely evaporated. Neighbouring droplets interact via their vapour fields, which results in a spatially non-uniform "shielding" effect.…
If a droplet is placed on a substrate with a conical shape it spontaneously starts to spread in the direction of a growing fibre radius. We describe this capillary spreading dynamics by developing a lubrication approximation on a cone and by the perturbation method of matched asymptotic expansions. Our results show that the droplet appears to adopt a quasi-static shape and the predictions of the droplet shape and spreading velocity from the two mathematical models are in excellent agreement for a wide range of slip lengths, cone angles and equilibrium contact angles. At the contact line region…
Experimental and modelling study is presented for the effect of a wide range of liquid viscosities on the droplet impact on a smooth solid surface at atmospheric pressure. A non-monotonic variation of threshold between droplet deposition and splash was observed experimentally. Specifically, the critical Weber number separating deposition from splashing decreases and then increases with increasing the Ohnesorge number. The observations of splash in low viscosity region and receding breakup in high viscosity region were analyzed qualitatively from the perspectives of Kelvin-Helmholtz instability…
We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. Our deriv…
The analogy between evaporating surface droplets in air to dissolving long-chain alcohol droplets in water is worked out. We show that next to the three known modi for surface droplet evaporation or dissolution (constant contact angle mode, constant contact radius mode, and stick-slide mode), a fourth mode exists for small droplets on supposedly smooth substrates, namely the stick-jump mode: intermittent contact line pinning causes the droplet to switch between sticking and jumping during the dissolution. We present experimental data and compare them to theory to predict the dissolution time i…
In this paper, the pinning and depinning mechanism of the contact line during droplet evaporation on chemically stripe-patterned surfaces is numerically investigated using a thermal multiphase lattice Boltzmann (LB) model with liquid-vapor phase change. A local force balance in the context of diffuse interfaces is introduced to explain the equilibrium states of droplets on chemically patterned surfaces. It is shown that, when the contact line is pinned on a hydrophobic-hydrophilic boundary, different contact angles can be interpreted as the variation of the length of the contact line occupied …
Droplet deposition onto a hydrophobic surface is studied experimentally and numerically. A wide range of droplet sizes can result from the same syringe, depending strongly on the needle retraction speed. Three regimes are identified according to the motion of the contact line. In Region I, at slow retraction speeds, the contact line expands and large droplets can be achieved. In Region II, at moderate needle speeds, a quasi-cylindrical liquid bridge forms resulting in drops approximately the size of the needle. Finally, at high speeds (Region III), the contact line retracts and droplets much s…
A droplet that impacts onto a solid substrate deforms in a complex dynamics. To extract the principal mechanisms that dominate this dynamics we deploy numerical simulations based on the phase field method. Direct comparison with experiments suggests that a dissipation local to the contact line limits the droplet spreading dynamics and its scaled maximum spreading radius $β_\mathrm{max}$. By assuming linear response through a drag force at the contact line, our simulations rationalize experimental observations for droplet impact on both smooth and rough substrates, measured through a single con…
Liquid droplets usually wet smooth and homogeneous substrates isotropically. Recent research works have revealed that droplets sit, slide and spread anisotropically on uniaxially stretched soft substrates, showing an enhanced wettability and contact line mobility along the stretching direction. This phenomenon arises from the anisotropic deformation of the substrate below the contact line. Here, we investigate how the stretching of soft substrates affects droplets drying. We observe that water droplet evaporates with an elongated non-circular contact line on the stretched substrates and switch…
Freezing or solidification of impacting droplets is omnipresent in nature and technology, be it a rain droplet falling on a supercooled surface, be it in inkjet printing where often molten wax is used, be it in added manufacturing or in metal production processes or in extreme ultraviolet lithography (EUV) for the chip production where molten tin is used to generate the EUV radiation. For many of these industrial applications, a detailed understanding of the solidification process is essential. Here, by adopting a totally new optical technique in the context of freezing, namely TIR (Total-Inte…
We predict and analyze the drying time of respiratory droplets from a COVID-19 infected subject, which is a crucial time to infect another subject. The drying of the droplet is predicted by diffusion-limited evaporation model for a sessile droplet placed on a partially-wetted surface with a pinned contact line. The variation of droplet volume, contact angle, ambient temperature, and humidity are considered. We analyze the chances of the survival of the viruses present in the droplet, based on the lifetime of the droplets in several conditions, and find that the chances of survival of the virus…
Coalescence-induced alignment of anisotropic particles in drying sessile droplets
Forschungszentrum Jülich（于利希研究中心）；Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IET-2)（亥姆霍兹埃尔朗根-纽伦堡可再生能源研究所（IET-2））；Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
Evaporation of surfactant laden sessile droplets is omnipresent in nature and industrial applications such as inkjet printing. Soluble surfactants start to form micelles in an aqueous solution for surfactant concentrations exceeding the critical micelle concentration (CMC). Here, the evaporation of aqueous sodium dodecyl sulfate (SDS) sessile droplets on hydrophobic surfaces were experimentally investigated for SDS concentrations ranging from 0.025 to 1 CMC. In contrast to the constant contact angle of an evaporating sessile water droplet, we observed that, at the same surface the contact angl…
A finite element model was developed to compute the fluid flow inside a sessile evaporating droplet on hydrophilic substrate in ambient conditions. The evaporation is assumed as quasi-steady and the flow is considered as axisymmetric with a pinned contact line. The Navier-Stokes equations in cylindrical coordinates were solved inside the droplet. Galerkin weight residual approach and velocity pressure formulation was used to discretise the governing equations. Six node triangular mesh and quadratic shape functions were used to obtain higher accuracy solutions. Radial velocity profiles in axial…
New exact solutions for the evaporation flux density of a small droplet on a flat horizontal substrate with a contact angle in the range of 135-180 degrees
When a droplet is brought in contact with an undercooled surface, it wets the substrate and solidifies at the same time. The interplay between the phase transition effects and the contact-line motion, leading to its arrest, remains poorly understood. Here we reveal the early solidification patterns and dynamics of spreading hexadecane droplets. Total internal reflection (TIR) imaging is employed to temporally and spatially resolve the early solidification behaviour. With this, we determine the conditions leading to the contact-line arrest. We quantify the overall nucleation behaviour, \textit{…
This paper presents an experimental study on thermal transport to single water droplets evaporating on heated bi-phobic surfaces consisting of a superhydrophobic matrix with a circular hydrophobic pattern with strong contact line pinning. A single water droplet of 8 ul volume is placed on a preheated surface and allowed to evaporate in an open laboratory environment. We investigate the influence of substrate orientation (horizontal and vertical) on evaporation dynamics. Using optical and infrared imaging, we report droplet fluid dynamics and heat transfer characteristics of the evaporating dro…
Young's law fails on soft solid and liquid substrates where there are substantial deformations near the contact line. On liquid substrates, this is captured by Neumann's classic analysis, which provides a geometrical construction for minimising the interfacial free energy. On soft solids, the total free energy includes an additional contribution from elasticity. A linear-elastic model incorporating an out-of-plane restoring force due to solid surface tension was recently shown to accurately predict the equilibrium shape of a thin elastic film due to a large sessile droplet. Here, we extend thi…
Understanding flow patterns and coupled transport phenomena during evaporation of droplets loaded with colloidal particles is central to design technical applications such as organizing proteins/DNA on a solid surface. We review recent reports on evaporating sessile droplets of colloidal suspensions on a solid surface. Starting from the classical mechanism of formation of a ring-like deposit, we discuss the influence of several problem parameters. Notably, thermal or solutal Marangoni effect, particle size, particle concentration, particle shape, substrate wettability, pH of the suspension etc…
Spreading of respiratory diseases, such as COVID-19, from contaminated surfaces is dependent on the drying time of the deposited droplets containing the virus. The evaporation rate depends on environmental conditions, such as ambient temperature and relative humidity and physical properties (e.g., droplet volume, contact angle and composition). The respiratory droplets contain salt (NaCl), protein (mucin), and surfactant (dipalmitoylphosphatidylcholine) in addition to water, which are expected to influence the evaporation in a big way. A diffusion-based theoretical model for estimating the dry…
Fast contact-line motion of a droplet spreading on a solid substrate under the electrowetting effect generates strong capillary waves on the droplet's surface. The capillary waves may be strong enough to induce ejection of a satellite droplet from the primary one. In this study, we show that the size of the satellite droplet and the ejection time are not only dependent on the contact-line velocity, which directly relates to the applied voltage enabling the electrowetting effect, but also affected by the ejection dynamics. We derive a theoretical model of the criteria for droplet ejection and e…
This paper investigates the properties of a three dimensional shear flow overpassing a hemispherical droplet resting on a plane wall. The exact solution is computed as a function of the viscosity ratio between the droplet and the surrounding fluid and generalizes the solution for the hemispherical no-slip bump given in an earlier paper by Price (1985). Several expressions including the torque and the force acting on the drop will be considered as well as the importance of the deformations on the surface for small Capillary numbers.
The evaporation of sessile droplets on a flat surface involves a complex interplay between phase change, diffusion, advection and surface forces. In an attempt to significantly reduce the complexity of the problem and to make it manageable, we propose a simple model hinged on a surface free energy-based relaxation dynamics of the droplet shape, a diffusive evaporation model and a contact line pinning mechanism governed by a yield stress. Our model reproduces the known dynamics of droplet shape relaxation and of droplet evaporation, both in the absence and in the presence of contact line pinnin…
We construct a novel model for the steady-state contact angles of liquid droplets at the wetted substrate. The non-removable, thin liquid film covering the substrate is governed by the intermolecular forces between molecules of liquid and solid, which we describe using the standard disjoining pressure approximation. Balancing the disjoining pressure against the surface tension we find the smooth shape of the surface of the liquid. We show that we can extract an effective contact angle from the region where the film and the droplet meet. Crucially, we find that for large droplets the contact an…
Water droplet freezing is a common phenomenon in our daily life. In both natural scenarios and industrial production, different surface inclinations bring distinctive deformation and freezing dynamics to frozen droplets. We explore the freezing of pendent and sessile droplets at different Bond number regimes. The effect of gravity on the droplet freezing process is analyzed by considering droplet morphology, freezing front dynamics, and freezing time. It is found that gravity can significantly influence droplet freezing processes via shaping the initial droplet, resulting in the flattening or …
We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading -- including Tanner's Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, whic…
The objective of this work is to investigate the coupling of fluid dynamics, heat transfer and mass transfer during the impact and evaporation of droplets on a heated solid substrate. A laser-based thermoreflectance method is used to measure the temperature at the solid-liquid interface, with a time and space resolution of 100 μs and 20 μm, respectively. Isopropanol droplets with micro- and nanoliter volumes are considered. A finite-element model is used to simulate the transient fluid dynamics and heat transfer during the droplet deposition process, considering the dynamics of wetting as well…
Motivated by recent experiments, we consider theoretically the compression of droplets pinned at the bottom on a surface of finite area. We show that if the droplet is sufficiently compressed at the top by a surface, it will always develop a shape instability at a critical compression. When the top surface is flat, the shape instability occurs precisely when the apparent contact angle of the droplet at the pinned surface is pi, regardless of the contact angle of the upper surface, reminiscent of past work on liquid bridges and sessile droplets as first observed by Plateau. After the critical c…
The motion of the three-phase contact line between two immiscible fluids and a solid surface arises in a variety of wetting phenomena and technological applications. One challenge in continuum theory is the effective representation of molecular phenomena close to the contact line. Here, we characterize the molecular processes of the moving contact line to assess the accuracy of two different continuum two-phase models. Specifically, molecular dynamics (MD) simulations of a two-dimensional droplet between two moving plates are used to create reference data for different capillary numbers and co…
The pinning of the contact line adversely influences the electrowetting performance of sessile liquid droplets. In this paper, we report the electrowetting hysteresis characteristics of 100 mM aq. KCl sessile liquid droplet placed on a hydrophobic PDMS surface. The effect of nanoparticles on the contact angle hysteresis under the imposed electric potential is further investigated. This study reveals that the contact angle hysteresis decreases beyond a certain threshold value of nanoparticles concentration. Therefore, nanoparticle suspension in the liquid droplet can be used to enhance or suppr…
