Contact Line Mobility Research Articles

De-wetting of evaporating drops on regular patterns of triangular posts.

Directional wicking and spreading of liquids can be achieved by regular micro-patterns of specifically designed topographic features that break the reflection symmetry of the underlying pattern. The present study aims to understand the formation and stability of wetting films during the evaporation of volatile liquid drops on surfaces with a micro-pattern of triangular posts arranged in a rectangular lattice. Depending on the density and aspect ratio of the posts, we observe either spherical-cap shaped drops with a mobile three-phase contact line or the formation of circular or angular drops with a pinned three-phase contact line. Drops of the latter class eventually evolve into a liquid film extending to the initial footprint of the drop and a shrinking cap-shaped drop sitting on the film. The drop evolution is controlled by the density and aspect ratio of the posts, while no influence of the orientation of the triangular posts on the contact line mobility becomes evident. Our experiments corroborate previous results of systematic numerical energy minimization, predicting that conditions for a spontaneous retraction of a wicking liquid film depend weakly on the orientation of the film edge relative to the micro-pattern.

Coalescence-induced droplet spreading: Experiments aboard the International Space Station

We report experiments of centimeter-sized sessile drop coalescence aboard the International Space Station, where microgravity conditions enable inertial-capillary spreading motions to be explored for a range of hydrophobic wetting conditions. Observations of the time traces of the coalescence event and projected areas compare favorably to numerical simulations, which employ the Davis–Hocking contact line (CL) condition with contact line mobility M parameter independently measured using the resonant-frequency scan technique of Xia and Steen [“Moving contact-line mobility measured,” J. Fluid Mech. 841, 767–783 (2018)]. This observation suggests that M is a material parameter, and that the Davis–Hocking model is an appropriate CL model for inertial-capillary spreading.

Is contact-line mobility a material parameter?

Dynamic wetting phenomena are typically described by a constitutive law relating the dynamic contact angle θ to contact-line velocity UCL. The so-called Davis–Hocking model is noteworthy for its simplicity and relates θ to UCL through a contact-line mobility parameter M, which has historically been used as a fitting parameter for the particular solid–liquid–gas system. The recent experimental discovery of Xia & Steen (2018) has led to the first direct measurement of M for inertial-capillary motions. This opens up exciting possibilities for anticipating rapid wetting and dewetting behaviors, as M is believed to be a material parameter that can be measured in one context and successfully applied in another. Here, we investigate the extent to which M is a material parameter through a combined experimental and numerical study of binary sessile drop coalescence. Experiments are performed using water droplets on multiple surfaces with varying wetting properties (static contact angle and hysteresis) and compared with numerical simulations that employ the Davis–Hocking condition with the mobility M a fixed parameter, as measured by the cyclically dynamic contact angle goniometer, i.e. no fitting parameter. Side-view coalescence dynamics and time traces of the projected swept areas are used as metrics to compare experiments with numerical simulation. Our results show that the Davis–Hocking model with measured mobility parameter captures the essential coalescence dynamics and outperforms the widely used Kistler dynamic contact angle model in many cases. These observations provide insights in that the mobility is indeed a material parameter.

Detailed modelling of contact line motion in oscillatory wetting

The experimental results of Xia and Steen for the contact line dynamics of a drop placed on a vertically oscillating surface are analyzed by numerical phase field simulations. The concept of contact line mobility or friction is discussed, and an angle-dependent model is formulated. The results of numerical simulations based on this model are compared to the detailed experimental results of Xia and Steen with good general agreement. The total energy input in terms of work done by the oscillating support, and the dissipation at the contact line, are calculated from the simulated results. It is found that the contact line dissipation is almost entirely responsible for the dissipation that sets the amplitude of the response. It is argued that angle-dependent line friction may be a fruitful interpretation of the relations between contact line speed and dynamic contact angle that are often used in practical computational fluid dynamics.

From coffee stains to uniform deposits: Significance of the contact-line mobility

HypothesisContact-line motion upon drying of a sessile droplet strongly affects the solute transport and solvent evaporation profile. Hence, it should have a strong impact on the deposit formation and might be responsible for volcano-like, dome-like and flat deposit morphologies. ExperimentsA method based on a thin-film interference was used to track the drop height profile and contact line motion during the drying. A diverse set of drying scenarios was obtained by using inks with different solvent compositions and by adjusting the substrate wetting properties. The experimental data was compared to the predictions of a phenomenological model. FindingsWe highlight the essential role of contact-line mobility on the deposit morphology of solution-based inks. A pinned contact line produces exclusively ring-like deposits under normal conditions. On the contrary, drops with a mobile contact line can produce ring-, flat- or dome-like morphology. The developed phenomenological model shows that the deposit morphology depends on solvent evaporation profile, evolution of the drop radius relative to its contact angle, and the ratio between initial and maximal (gelling) solute concentration. These parameters can be adjusted by the ink solvent composition and substrate wetting behaviour, which provides a way for deposition of uniform and flat deposits via inkjet printing.

Numerical study of natural oscillations of supported drops with free and pinned contact lines

The oscillation of droplets supported by solid surfaces is important for a wide variety of applications such as dropwise condensation. In the present study, the axisymmetric natural oscillations of a liquid drop supported by a flat surface are investigated by direct numerical simulation. The liquid–gas interface is captured using a geometric volume-of-fluid method. A parametric study is carried out by varying the equilibrium contact angle and the gravitational Bond number (Bo). Both positive and negative gravities are considered, and thus the results cover both pendant and sessile drops. To incorporate the effect of contact line mobility, the two asymptotic limits, namely, the pinned contact line (PCL) and free contact line (FCL) conditions, are considered and their effects on the drop oscillation features are characterized. The predicted oscillation frequencies for PCL and FCL serve as the upper and lower bounds for general situations. The drop oscillation is initiated by increasing the gravity magnitude for a short time. The first mode due to the drop centroid translation dominates the excited oscillation. The oscillation frequency scales with the capillary frequency, and the normalized frequency monotonically decreases with the equilibrium contact angle. For zero gravity, the computed frequencies for all contact angles agree remarkably well with the inviscid theory for both the PCL and FCL conditions. The kinetic energy correction factor is introduced to account for the additional contribution of the oscillation-induced internal flow to the overall kinetic energy of the drop. Both the frequency and the kinetic energy correction factor increase with Bo, decrease with the contact angle, and increase when the contact line condition changed from FCL to PCL. The variation of oscillation frequency due to the change of Bo is particularly significant when the contact angle is large, suggesting that the gravity effect must be incorporated to accurately predict the oscillation frequency for drops supported by hydrophobic or superhydrophobic surfaces.

Natural oscillations of a sessile drop on flat surfaces with mobile contact lines

Oscillation of sessile drops is important to many applications. In the present study, the natural oscillation of a sessile drop on flat surfaces with free contact lines (FCL) is investigated through numerical and theoretical analysis. The FCL condition represents a limit of contact line mobility, i.e. the contact angle remains constant when the contact line moves. In the numerical simulation, the interfaces are captured by the volume-of-fluid method and the contact angle at the boundary is specified using the height-function method. The oscillation frequencies for sessile drops with FCL are mainly controlled by the contact angle and the Bond number and a parametric study is carried out to characterize their effects on the frequencies for the first and high-order modes. Particular attention is paid to the frequency of the first mode, since it is usually the dominant mode. An inviscid theoretical model for the first mode is developed. The model yields an explicit expression for the first-mode frequency as a function of the contact angle and the Bond number, with all parameters involved fully determined by the equilibrium drop theory and the simulation. The predicted frequencies for a wide range of contact angles agree very well with the simulation results for small Bond numbers. The frequencies for both the first and high-order modes decrease with the contact angle and increase with the Bond number. For the high-order modes, the frequencies for different modes generally scale with the Rayleigh frequencies. The scaling relation performs better for small Bond numbers and large contact angles. A simple model is proposed to predict the frequencies of high-order modes for large contact angles and a good agreement with the simulation results is observed.

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.

Paradox of Contact Angle Selection on Stretched Soft Solids.

The interfacial mechanics of soft elastic networks plays a central role in biological and technological contexts. Yet, effects of solid capillarity have remained controversial, primarily due to the strain-dependent surface energy. Here we derive the equations that govern the selection of contact angles of liquid drops on elastic surfaces from variational principles. It is found that the substrate's elasticity imposes a nontrivial condition that relates pinning, hysteresis, and contact line mobility to the so-called Shuttleworth effect. We experimentally validate our theory for droplets on a silicone gel, revealing an enhanced contact line mobility when stretching the substrate.

Depinning of bubble contact line on a biphilic surface in subatmospheric boiling: Revisiting the theories of bubble departure

Boiling suffers from inefficient intermittent cycles of bubble generation under subatmospheric conditions. Such deterioration in heat transfer rates can be alleviated but not completely eliminated by use of mixed-wettability (biphilic) surfaces. Here we study bubble dynamics on a single hydrophobic spot in low-pressure pool boiling. The results reveal an interesting transition in bubble departure behavior from the surface-driven mode to the drag-driven mode, which correlates closely with the dynamic state of the three-phase contact line on the surface. Based on the force-balance argument, a simple model is derived to map the contact-line mobility during bubble growth. It is found that below a certain threshold pressure, the bubble base expansion is increasingly likely to overcome the strong pinning of the contact line at the interface between the hydrophobic and hydrophilic regions. That could lead to total removal of vapor residues from the surface and cause deactivation of the nucleation site, which portends the eventual takeover of intermittent boiling on the biphilic surface.

Moving contact-line mobility measured

Contact-line mobility characterizes how fast a liquid can wet or unwet a solid support by relating the contact angle $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}$ to the contact-line speed $U_{CL}$. The contact angle changes dynamically with contact-line speeds during rapid movement of liquid across a solid. Speeds beyond the region of stick–slip are the focus of this experimental paper. For these speeds, liquid inertia and surface tension compete while damping is weak. The mobility parameter $M$ is defined empirically as the proportionality, when it exists, between $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}$ and $U_{CL}$, $M\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}=U_{CL}$. We discover that $M$ exists and measure it. The experimental approach is to drive the contact line of a sessile drop by a plane-normal oscillation of the drop’s support. Contact angles, displacements and speeds of the contact line are measured. To unmask the mobility away from stick–slip, the diagram of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}$ against $U_{CL}$, the traditional diagram, is remapped to a new diagram by rescaling with displacement. This new diagram reveals a regime where $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}$ is proportional to $U_{CL}$ and the slope yields the mobility $M$. The experimental approach reported introduces the cyclically dynamic contact angle goniometer. The concept and method of the goniometer are illustrated with data mappings for water on a low-hysteresis non-wetting substrate.

Enhancement of contact line mobility by means of infrared laser illumination. II. Numerical simulations

A droplet that moves on a solid substrate with a velocity higher than a certain critical velocity disintegrates, i.e., leaves behind residual droplets. Infrared laser illumination can be used to increase the droplet mobility and suppress the shedding of droplets. By means of two-dimensional numerical simulations, we studied the effect of a non-uniform temperature distribution on the dynamics of straight receding contact lines. A streamfunction-vorticity model is used to describe the liquid flow in the vicinity of the receding contact line. The model takes into account the thermocapillary shear stress and the temperature-dependent liquid viscosity and density. A second, coupled model describes the laser-induced displacement of the contact line. Our results show that the reduction of the liquid viscosity with increasing temperature is the dominant mechanism for the increase of the critical velocity. Thermocapillary shear stresses are important primarily for low substrate speeds.

Enhancement of contact line mobility by means of infrared laser illumination. I. Experiments

The shape of a droplet moving on a solid substrate is largely determined by the mobility of its contact line. Above a certain critical velocity, the droplet typically disintegrates and leaves residual liquid behind. We studied experimentally how localized infrared laser illumination can stabilize it against the progressive loss of liquid. We systematically varied the substrate velocity, the laser power, and the laser spot position and found that the critical velocity of the droplet can be substantially increased. These results are potentially important for increasing throughput in immersion lithography systems.

Response of driven sessile drops with contact-line dissipation.

A partially-wetting sessile drop is driven by a sinusoidal pressure field that produces capillary waves on the liquid/gas interface. Response diagrams and phase shifts for the droplet, whose contact-line moves with contact-angle that is a smooth function of the contact line speed, are reported. Contact-line dissipation originating from the contact-line speed condition leads to damping for drops with finite contact-line mobility, even for inviscid fluids. The critical mobility and associated driving frequency to generate the largest contact-line dissipation is computed. Viscous dissipation is approximated using the irrotational flow and the critical Ohnesorge number bounding regions beyond which a given mode becomes over-damped is computed. Regions of modal coexistence where two modes can be simultaneously excited by a single forcing frequency are identified. Predictions compare favorably to related experiments on vibrated drops.

Dynamics of sessile drops. Part 1. Inviscid theory

AbstractA sessile droplet partially wets a planar solid support. We study the linear stability of this spherical-cap base state to disturbances whose three-phase contact line is (i) pinned, (ii) moves with fixed contact angle and (iii) moves with a contact angle that is a smooth function of the contact-line speed. The governing hydrodynamic equations for inviscid motions are reduced to a functional eigenvalue problem on linear operators, which are parameterized by the base-state volume through the static contact angle and contact-line mobility via a spreading parameter. A solution is facilitated using inverse operators for disturbances (i) and (ii) to report frequencies and modal shapes identified by a polar $k$ and azimuthal $l$ wavenumber. For the dynamic contact-line condition (iii), we show that the disturbance energy balance takes the form of a damped-harmonic oscillator with ‘Davis dissipation’ that encompasses the dynamic effects associated with (iii). The effect of the contact-line motion on the dissipation mechanism is illustrated. We report an instability of the super-hemispherical base states with mobile contact lines (ii) that correlates with horizontal motion of the centre-of-mass, called the ‘walking’ instability. Davis dissipation from the dynamic contact-line condition (iii) can suppress the instability. The remainder of the spectrum exhibits oscillatory behaviour. For the hemispherical base state with mobile contact line (ii), the spectrum is degenerate with respect to the azimuthal wavenumber. We show that varying either the base-state volume or contact-line mobility lifts this degeneracy. For most values of these symmetry-breaking parameters, a certain spectral ordering of frequencies is maintained. However, because certain modes are more strongly influenced by the support than others, there are instances of additional modal degeneracies. We explain the physical reason for these and show how to locate them.

On the onset of motion of sliding drops

In this article we numerically investigate the onset of motion of liquid drops in contact with a plane and homogeneous substrate with contact angle hysteresis. The drops are driven by a body force F = ρgV, where ρ is the density of the liquid, g is the acceleration of gravity, and V is the volume of the drop. We compare two protocols to vary the bond number Bo = λ(v)/λ(c) by changes of either the drop size λ(v) = V(1/3) or the capillary length λ(c) = (γ/ρg)(1/2) where γ is the interfacial tension, revealing that the transition between pinned and steady moving states can be either continuous or discontinuous. In a certain range both pinned and moving states can be found for a given bond number Bo, depending on the history of the control parameters g and V. Our calculations are extended to arbitrary combinations of static advancing and receding contact angles and provide a comprehensive picture of the depinning transition induced by a quasi-static variation of the control parameters. Finally, we demonstrate that the particular form of the contact line mobility in our model has an impact on the interfacial shape of steady moving drops.

Modeling dissolution and spreading of Bi–Sn alloy drops on a Bi substrate

A dissolutive wetting model is employed to study the dynamics of axisymmetric Bi–Sn alloy drops spreading on Bi. The liquid drop surface is assumed to be a spherical cap, while the isothermal model for solute transport with simplified hydrodynamics computes the evolution of the solid–liquid interface. Simulations are performed on a millimeter scale to model experiments. The evolution of the drop radius, the flow and solute concentration profiles near the triple junction, the contact line mobility relationship, the apparent contact angles and the shape of the dissolution boundary during spreading are investigated as functions of initial Sn concentration, model parameters and initial conditions. Good agreement is obtained when compared with the experiments for the cases where diffusion is the dominant transport mode. The shape of the solid–liquid interface is predicted better when the initial experimentally determined dissolution boundary is implemented as the starting point of the subsequent simulation.

Contact line mobility in liquid droplet spreading on rough surface

We quantitatively estimate the effect of the substrate roughness on the liquid droplet spreading. Since the droplet size is in the order of millimeters, the surface energy becomes the dominant factor. A nonequilibrium thermodynamics framework [Y.X. Gao, H. Fan, Z. Xiao, Acta Mater. 48 (2000) 863–874] seems feasible for describing the millimeter size droplet spreading on a solid substrate. Within the framework, there are two system constants, namely the mobilities of liquid/air surface and the triple joint contact line that need to be determined from experimental testing. In the present paper, we demonstrate the experimental process of determining the mobility of the contact line via a droplet spreading on a steel substrate. Particularly, we obtained the contact line mobility on a steel surface with various roughness values. It is shown that the mobility value is lower for a rougher surface.

Dissolutive wetting in the Bi–Sn system

The isothermal dissolutive wetting behavior of various Bi–Sn alloys on pure Bi substrates was studied at 250 °C using the sessile drop technique. Wetted radius and apparent dynamic contact angle were recorded as a function of time. Sequential quenching and metallographic cross-sectioning were employed to investigate the temporal evolution of the S/L interface and the complete configuration of the contact line region. The macroscopic total contact angle was found to be close to 90° during spreading for the majority of Bi–Sn alloys investigated. The contact line mobility relationship, based on apparent dynamic contact angle, showed certain similarities to a universal correlation for partially wetting inert systems. Additional discussions of grain boundary effects and mass transport are presented.

Wetting in the Au–Sn System

The isothermal spreading of liquid Sn on Au substrates was studied over the temperature range of 250–430 °C using the sessile drop technique in a gaseous flux atmosphere. Digital video imaging was used to record the spreading kinetics, and rapid quenching and metallographic cross-sectioning was used to characterize dissolution and compound formation. Observations showed that three temperature/composition regimes gave rise to their own unique spreading behaviors. At low temperatures/Au concentrations, limited spreading was accompanied by nearly complete isothermal solidification. At intermediate temperatures/Au concentrations, rapid spreading and concurrent appearance of a transient intermetallic compound were observed. At higher temperatures/Au concentrations, extensive spreading was observed in the form of a thick film (∼20 μm) supplied by a central dissolution well whose L/V interface undergoes a global curvature reversal. We discuss the contact line mobility and other mechanisms that affect wetting and spreading behaviors: fluid flow, mass transport and metallurgical reaction.