Contact Line Speed Research Articles

Direct simulation of multiphase flows with modeling of dynamic interface contact angle

We describe a modeling technique for dynamic contact angle between a phase interface and a solid wall using a generalized Navier boundary condition in the context of a front-tracking-based multiphase method. The contact line motion is determined by the generalized Navier slip boundary condition in order to eliminate the infinite shear stress at the contact line. Applying this slip boundary condition only to the interface movement with various slip ratios shows good agreement with experimental results compared to allowing full fluid slip along the solid surface. The interface slip model performs well on grid convergence tests using both the slip ratio and slip length models. A detailed energy analysis was performed to identify changes in kinetic, surface, and potential energies as well as viscous and contact line dissipation with time. A friction coefficient for contact line dissipation was obtained based on the other computed energy terms. Each energy term and the friction coefficient were compared for different grid resolutions. The effect of varying the slip ratio as well as the contact angle distribution versus contact line speed was analyzed. The behavior of drop impact on a solid wall with different advancing and receding angles was investigated. Finally, the proposed dynamic contact model was extended to three dimensions for large-scale parallel calculations. The impact of a droplet on a solid cylinder was simulated to demonstrate the capabilities of the proposing formulation on general solid structures. Widely different contact angles were tested and showed distinctive characteristic behavior clearly.

Scaling and mechanism of droplet array formation on a laser-ablated superhydrophobic grid

Superhydrophobic grid patterns were ablated on a hydrophobic polydimethylsiloxane (PDMS) substrate using a nanosecond pulsed laser. The substrate with a wettability contrast (superhydrophobic-hydrophobic) was vertically immersed in a pool of water and withdrawn at different speeds to form droplet arrays spontaneously. The size of droplets was tuned by controlling the withdrawal speed (Sw), which resulted in different contact line speeds on non-irradiated cell (NIC) and laser-irradiated grid (LIG) areas with different surface properties of no-slip and partial-slip conditions, respectively. The ratio of the contact line speeds on NIC and LIG areas (SNIC/SLIG) decreased exponentially with the withdrawal speed, leading to formation of larger droplets on NIC. A geometric scaling model predicted the size of droplets formed on NIC arrays over a range of contact line speeds on NIC and LIG, the ratio of the contact line speeds, the withdrawal speed, and the size of NIC.

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.

A semi-analytical method to estimate the effective slip length of spreading spherical-cap shaped droplets using Cox theory

The Cox–Voinov law on dynamic spreading relates the difference between the cubic values of the apparent contact angle (θ) and the equilibrium contact angle to the instantaneous contact line speed (U). Comparing spreading results with this hydrodynamic wetting theory requires accurate data of θ and U during the entire process. We consider the case when gravitational forces are negligible, so that the shape of the spreading drop can be closely approximated by a spherical cap. Using geometrical dependencies, we transform the general Cox law in a semi-analytical relation for the temporal evolution of the spreading radius. Evaluating this relation numerically shows that the spreading curve becomes independent from the gas viscosity when the latter is less than about 1% of the drop viscosity. Since inertia may invalidate the made assumptions in the initial stage of spreading, a quantitative criterion for the time when the spherical-cap assumption is reasonable is derived utilizing phase-field simulations on the spreading of partially wetting droplets. The developed theory allows us to compare experimental/computational spreading curves for spherical-cap shaped droplets with Cox theory without the need for instantaneous data of θ and U. Furthermore, the fitting of Cox theory enables us to estimate the effective slip length. This is potentially useful for establishing relationships between slip length and parameters in numerical methods for moving contact lines.

Nonthermal ice nucleation observed at distorted contact lines of supercooled water drops.

Ice nucleation is the crucial step for ice formation in atmospheric clouds and therefore underlies climatologically relevant precipitation and radiative properties. Progress has been made in understanding the roles of temperature, supersaturation, and material properties, but an explanation for the efficient ice nucleation occurring when a particle contacts a supercooled water drop has been elusive for over half a century. Here, we explore ice nucleation initiated at constant temperature and observe that mechanical agitation induces freezing of supercooled water drops at distorted contact lines. Results show that symmetric motion of supercooled water on a vertically oscillating substrate does not freeze, no matter how we agitate it. However, when the moving contact line is distorted with the help of trace amounts of oil or inhomogeneous pinning on the substrate, freezing can occur at temperatures much higher than in a static droplet, equivalent to ∼10^{10} increase in nucleation rate. Several possible mechanisms are proposed to explain the observations. One plausible explanation among them, decreased pressure due to interface curvature, is explored theoretically and compared with the observational results quasiquantitatively. Indeed,theobserved freezing-temperature increase scales with contact line speed in a manner consistent with the pressure hypothesis. Whatever the mechanism, the experiments demonstrate a strong preference for ice nucleation at three-phase contact lines compared to the two-phase interface, and they also show that movement and distortion of the contact line are necessary contributions to stimulating the nucleation process.

Thermocapillary migration characteristics of self-rewetting drop

The thermocapillary migration characteristics of a self-wetting drop on the non-uniformly heated, horizontal, solid substrate are investigagted by numerical simulation. Based on the lubrication theory, an evolution equation for the height of the two-dimensional drop is established. The substrate underlying the drop is subjected to a temperature gradient which induces surface tension gradient-driven drop deformation and migration. The self-rewetting fluid has non-monotonic dependence of the surface tension on temperature with a well-defined minimum, and the position of the minimum corresponding to the temperature on the substrate is called the critical point. The effect of the relationship between the critical point and the drop position on drop dynamics is analyzed. With the temperature sensitivity coefficient of three interfaces under the same condition, the substrate is illustrated with constant wettability. The direction of drop migration will alter as the initial drop location moves to the left relative to the critical point position, resulting from the variation of the interplay among thermocapillary, gravity, and capillarity forces within the drop. But the drop always migrates toward the high interfacial tension region due to the thermocapillary force. In the presence of substrate wettability variations, the drop migrates toward the low temperature region no matter where the drop is placed relative to the critical point. This is due to the fact that the deterioration of substrate wettability on the right side of the drop prevents the drop from migrating toward the hot region. Under the critical point being on the left or within the drop, as the initial drop location moves to the left relative to the critical point position, the enhancement of the thermocapillary force toward the left leads to increased moving speed of the left contact line and increased spreading area. When the critical point is positioned on the outer right side of the drop, the speed of the left contact line sharply decreases at t=6103, caused by the suddenly deteriorating substrate wettability. Hence, it is effective to manipulate the self-wetting drop movement by regulating the relationship between the critical point and the initial drop location. To inhibit the migration of the drop toward the cold region, the drop should be placed on the right side of the critical point.

Spreading of a distilled water droplet over polished and laser-treated aluminum surfaces

Spreading of a distilled water droplet over polished and laser-treated aluminum surfaces is studied experimentally. We present an attempt to apply the molecular-kinetic theory to the experimental dependences of the dynamic contact angle on the contact line speed. Advancing and receding contact angles at different liquid flow rates are obtained experimentally, and their values are used to calculate the contact angle hysteresis on each substrate. The bottom-up approach is used to form a droplet on the surface when liquid is supplied through an opening in the substrate. This makes it possible to control wetting and dewetting by increasing and decreasing the droplet volume. The main geometrical parameters are obtained using the shadow method by Axisymmetric Drop Shape Analysis-Profile. Experimental data on the dynamic contact angle on a laser-treated surface are found to be a better fit to the theoretical equation in comparison with the results on a smooth polished surface. We have formulated a hypothesis that the theoretical and experimental values of the dynamic contact angle are in good agreement due to the physics of the process (smaller hysteresis on the laser-treated surface).

Evaporation of liquid droplets on solid substrates. I. Flat substrate with pinned or moving contact line

Evaporation of drops on flat substrates is studied for a pinned contact line and a moving contact line with a fixed contact angle. For the former, the local contact angle goes to zero prior to total evaporation. For the latter, the droplet keeps its initial shape and the contact line speed is constant.

Wetting dynamics of a collapsing fluid hole

The collapse dynamics of an axisymmetric fluid cavity that wets the bottom of a rotating bucket bound by vertical sidewalls are studied. Lubrication theory is applied to the governing field equations for the thin film to yield an evolution equation that captures the effect of capillary, gravitational and centrifugal forces on this converging flow. The focus is on the quasi-static spreading regime, whereby contact-line motion is governed by a constitutive law relating the contact-angle to the contact-line speed. The collapse time, as it depends upon the initial hole size, is reported showing that gravity accelerates the collapse process. Surface tension forces dominate the collapse dynamics for small holes leading to a universal power law whose exponent compares favorably to experiments in the literature. Volume dependence is predicted and compared with experiment. Centrifugal forces slow the collapse process and lead to complex dynamics characterized by stalled spreading behavior that separates the large and small hole asymptotic regimes.

Forced spreading over superhydrophobic and copper surfaces

Dynamic spreading over superhydrophobic and copper surfaces was studied experimentally under the condition of contact line movement with speed greater than 1 mm/sec. Three modes of spreading of distilled water drop over copper surfaces with sufficient typical roughness (0.591, 5.190 and 6.210 μM) were detected. The first one is drop formation when the contact line speed and dynamic contact angle increase sharply. The second mode is spreading of a drop, which is characterized by a monotonic decrease in the contact line speed and dynamic contact angle. The third one is a formation of an equilibrium contact angle at a constant wetted area (the contact line speed tends to zero, and spreading of a drop occurs as long as the driving force is greater than zero). Some features in spreading were detected on superhydrophobic surface with parameter roughness of 0.751μm compared to other substrates. During drop formation after sharp increase in the contact line speed and dynamic contact angle, there is a mode which is accompanied by a decrease in the contact line speed and monotonic increase in the advancing dynamic contact angle.

Mechanism of Contact Line Movement of a Droplet Spreading Over a Solid Surface

The contact line movement, advancing and receding, of the distilled water droplet was studied on the polished and laser patterned aluminum surfaces. The obtained dependences of the dynamic contact angles on the contact line speed were fitted by the combined theory consisting of molecular-kinetic and hydrodynamic theories. Results obtained on laser patterned surface are found to fit better to the theory in comparison with polished surface. The assumption was made that this result is connected with the low contact angle hysteresis on the patterned surface compared to the polished one. It means that combined theory does not consider the contact angle hysteresis.

On the motion of a sessile drop on an incline: Effect of non-monotonic thermocapillary stresses

We studied the short-time contact-line dynamics of a self-rewetting sessile droplet sliding “freely” on a silicone oil layer, on an inclined, uniformly heated substrate under non-isothermal conditions (liquid–solid). The effect of thermocapillarity and the contribution of surface tension gradients (Marangoni effect) to the droplet motion was investigated. The temperature of the substrate in conjunction with the non-monotonic surface tension/temperature dependence of the deformed self-rewetting droplet was found to significantly affect the early-stage inertial-capillary spreading regime. Infrared (IR) thermography images were also acquired to investigate the generation of thermal patterns at the liquid surface due to the strong surface-tension gradients. Our results demonstrate that the presence of strong surface tension driven flows at the liquid interface combined with droplet deformation (contact-angle hysteresis) gives rise to complex droplet dynamics. The interplay between thermocapillary stresses and body forces results in enhanced spreading rates, temporal non-monotonic dependence of the contact-line speed, as well as the droplet motion overcoming gravity in some instances.

Forced versus Spontaneous Spreading of Liquids

Two sets of experiments are performed, one for the free spreading of a liquid drop on a glass substrate and the other for the forced motion of a glass plate through a gas-liquid interface. The measured macroscopic advancing contact angle, θA, versus the contact line speed, U, differ markedly between the two configurations. The hydrodynamic theory (HDT) and the molecular kinetic theory (MKT) are shown to apply separately to the two systems. This distinction has not been previously noted. Rules of thumb are given that for an experimentalist involve a priori knowledge of the expected behavior.

Numerical analysis of contact line dynamics passing over a single wettable defect on a wall

In this study, the dynamics of a contact line passing a single defect, which was represented by a locally wettable part (whose static contact angle is less than the other part, namely, chemically heterogeneous and physically flat part), was analyzed using numerical simulations employing the front-tracking method and the generalized Navier boundary condition. We observed that the contact line was distorted with a logarithmic shape far from the defect; however, the distortion was dependent on the wall velocity. The apparent (averaged) dynamic contact angle of the wall with a defect was evaluated using a macroscopic energy balance. The apparent dynamic contact angles estimated from the energy balance agree well with the arithmetic averaged angles obtained from the present simulations. The macroscopic energy balance is useful to consider the effect of heterogeneity or roughness of the wall on the relation between the dynamic contact angle and contact line speed.

Thermal singularity and contact line motion in pool boiling: Effects of substrate wettability.

The dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)] is employed to model the growth of a single vapor bubble in a superheated liquid on a flat homogeneous substrate. The bubble spreading dynamics in the pool boiling regime has been numerically investigated for one-component van der Waals fluids close to the critical point, with a focus on the effect of the substrate wettability on bubble growth and contact line motion. The substrate wettability is found to control the apparent contact angle and the rate of bubble growth (the rate of total evaporation), through which the contact line speed is determined. An approximate expression is derived for the contact line speed, showing good agreement with the simulation results. This demonstrates that the contact line speed is primarily governed by (1) the circular shape of interface (for slow bubble growth), (2) the constant apparent contact angle, and (3) the constant bubble growth rate. It follows that the contact line speed has a sensitive dependence on the substrate wettability via the apparent contact angle which also determines the bubble growth rate. Compared to hydrophilic surfaces, hydrophobic surfaces give rise to a thinner shape of bubble and a higher rate of total evaporation, which combine to result in a much faster contact line speed. This can be linked to the earlier formation of a vapor film and hence the onset of boiling crisis.

Contact angle hysteresis of a drop spreading over metal surfaces

The paper presents experimental data on the contact angle hysteresis of the distilled water drop spreading over the surfaces of non-ferrous metals. The measurements of the advancing and receding contact angles were carried out by method of sitting drop on the horizontal surface during increasing and decreasing drop volume with a syringe pump. It was found that the contact line speed has a great influence on the hysteresis of the polished non-elastic substrates. The mechanism of spreading was described using the balance of the forces from the physical point of view.

Dynamic Wetting and Dewetting: Comparison of Experiment with Theories

The dynamics wetting/dewetting of a metal surface by distilled water drop was studied experimentally. The advancing and receding dynamic contact angles were obtained as a function of a contact line speed. The hydrodynamic and molecular-kinetic models have been applied to the experimental data to interpret the obtained results. The independent variables of the molecular-kinetic and hydrodynamic models, and the determination coefficient were determined by fitting procedure. The receding contact angles are found to be fitted better to the wetting models in comparison with the advancing dynamic contact angles.

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.

Evaporation, viscous flow, and electrostatic interaction of charged interfaces in the apparent contact line region

We consider evaporation of an aqueous solution near an apparent contact line separating a macroscopically dry area of a heated solid substrate and a constant-curvature meniscus far away from the substrate. Viscous flow, described by a lubrication-type model, is coupled to the interaction of electrical double layers formed near the solid-liquid and liquid-vapor interfaces. The electrostatic interaction is described using the nonlinear Poisson-Boltzmann equation and is shown to affect both normal and shear stress balances at the deformable interface. For steady configurations, we find that the apparent contact line region becomes wider and the total evaporation rate there increases as the substrate potential is increased. Motion of the apparent contact line in response to changes in the substrate temperature is also investigated. The contact line speed is found to increase when the electrostatic effects are incorporated into the model.

Dynamic contact angle and three-phase contact line of water drop on copper surface

Nowadays there is a lack of experimental data describing the physical process of drop spreading on a solid metal surface for developing wetting and spreading theory. The experimental data obtained by using the high speed video-recording will allow to identify unknown previously spreading modes as well as the change of the dynamic contact angle and the three-phase contact line. The purpose of the work is to determine the effect of the drop growth rate and the copper substrate surface roughness on the dynamic contact angle and the three-phase contact line speed at distilled water drop spreading. Shadow and Schlieren methods are used to obtain experimental data. Three drop spreading modes on the rough surfaces were identified. Time dependences of the dynamic contact angle and contact line speed were obtained. Experimental results can be used for assessing the validity of the developed mathematical models of wetting and spreading processes in the field of micro- and nanoelectronics, ink jet printing, thin-film coatings, spray cooling, and optoelectronics.