Fixed Contact Line Research Articles

Physics of a strongly oscillating axisymmetric air-water interface with a fixed boundary condition

A free-slip air-water interface with a fixed contact line boundary condition, when oscillating at certain frequencies with a large amplitude, can efficiently induce a fast-speed, far-propagating streaming jet. The extraordinary characters of the streaming jet can be employed to address engineering challenges across multiple disciplines.

Dripping–jetting transition of liquid stream from plate-type micro-orifice affected by wetting and dewetting

The liquid stream from an orifice typically undergoes a transition process from dripping to jetting under increased pressure. The dripping–jetting (D–J) transition performance from a thin-walled tube or an orifice with a wide diameter has been well studied, where the triple contact line of the stream is pinned on the edge of the orifice and the flow behaviors are analyzed accordingly with a precondition of the fixed contact line. The D–J transition of the liquid stream from plate-type micro-orifices was investigated in our study of droplet generation for extreme ultraviolet source and wafer cleaning. The triple contact line of the stream keeps moving as pressure changes as a result of the liquid wetting and dewetting around the micro-orifice. The transition from dripping to jetting involves three processes: wetting to generate pendant droplet and dripping, jetting from pendant droplet, and pendant droplet dewetting to form jetting directly. The aforementioned performance is different from boundary-pinned conditions. To reveal the transition characteristic, critical pressure causing the D–J transition under different wetting conditions was determined. The process of wetting to generate pendant droplet and dripping, and the effects of the contact angle and the pressure difference were formulated. Analysis of the mechanism causing jetting from the pendant droplet was implemented by combining a model formulating the jet ejecting outside the droplet and numerical simulation of the stream inside the pendant droplet. Finally, the normalized model was built to illustrate the critical pressure of D–J transition concerning the triple contact line diameter.

Strength of the monolithic coverings with non-fixable formwork of steel fibro-tuffcrete elements

The article presents the strength study results of the tuffcrete flexible monolithic structures with fixed formwork, combined reinforced rods and fibers. The nature and degree of the fibers percentage influence in the tuffcrete element of the fixed formwork, the dimensions of the fixed formwork elements’ fiber-reinforced section and the strength of monolithic tuffcrete on the normal strength to the longitudinal axis of the monolithic combined-reinforced bending floor structures, as well as on the adhesion strength of the fixed form contact line with concrete monolithic, are analyzed. The experimental dependences that make it possible to more economically design the structures under consideration, taking into account the strength and quantity parameters, as well as the geometric dimensions of their individual components, are proposed.

О влиянии внутреннего давления на колебания цилиндрического газового пузырька

Natural and forced oscillations of a gas bubble are studied. The bubble has the shape of a round cylinder in the state of equilibrium. It is bounded in the axial direction by two parallel solid surfaces and is surrounded by an incompressible liquid of a finite volume with a free outer surface. The entire system is under an alternating pressure field. The velocity of the contact line of three media (gas-liquid-solid substrate) is proportional to the deviation of the contact angle from the equilibrium value. The frequency of eigenmodes of a gas bubble can increase with an increase in the Hocking parameter, in contrast to the frequencies of an incompressible liquid drop, which only decrease. It is shown that radial oscillations of a cylindrical bubble are possible only in a finite volume of liquid. The effect of crossing the modes of natural oscillations is considered for the dissipative case. The amplitude-frequency characteristics are constructed for different values of the internal gas pressure. Resonance phenomena are found. It is shown that the external influence excites, first of all, volumetric oscillations of the bubble. Variations in shape are caused by the movement of the contact line. Expressions are found for the vibration amplitude in the case of a fixed contact line and a fixed contact angle.

Double-Emulsion Drop Evaporation and Formation of a Daughter Droplet.

In this study, we present experimental and theoretical analyses of evaporating a double-emulsion drop resting on a substrate. Multistage evaporation of the outer and inner droplet is witnessed. The complete evaporation of the outer drop and the initialization of the inner drop evaporation demonstrate an interesting transition dynamics. After the apparent completion of evaporation of the inner phase of a double-emulsion drop, surprisingly, formation of a daughter droplet is observed. We further investigated to hypothesize this phenomenon and achieved the formation of the daughter droplet for a single-phase drop as well. While engineering the "daughter drop formation" phenomena, we also proposed a way to obtain prolonged fixed contact line evaporation for a single-phase drop.

GNBC-based front-tracking method for the three-dimensional simulation of droplet motion on a solid surface

Previous front-tracking (FT) method-based models to simulate droplet motion on a solid surface with a moving contact line (MCL) are limited to two-dimensional models in which the Navier boundary condition (NBC) is employed for the MCL. In this paper, we develop a three-dimensional FT method that integrates the generalized Navier boundary condition (GNBC) to model the MCL. This GNBC-based FT method addresses several key issues, such as the integration of GNBC for the dynamic description of the MCL and its coupling with the surrounding flow, the accurate updating of the density and viscosity of the two-phase fluid near the contact line, and the restructuring of the Lagrangian mesh for tracking the drop surface, especially near the contact line. The stability and accuracy of the present numerical method are validated by several tests: (1) numerical performance tests, (2) simulation of the transient and steady-state shapes of droplets under flow with a fixed contact line, and (3) simulation of a droplet spreading under gravity and moving under a shear flow with MCLs. Excellent agreement is achieved between the results obtained by our model and the data obtained by other theoretical and numerical approaches.

Sodium chloride crystallization from thin liquid sheets, thick layers, and sessile drops in microgravity

Crystallization from aqueous sodium chloride solutions as thin liquid sheets, 0.2–0.7mm thick, with two free surfaces supported by a wire frame, thick liquid layers, 4–6mm thick, with two free surfaces supported by metal frame, and hemispherical sessile drops, 20–32mm diameter, supported by a flat polycarbonate surface or an initially flat gelatin film, were carried out under microgravity on the International Space Station (ISS). Different crystal morphologies resulted based on the fluid geometry: tabular hoppers, hopper cubes, circular [111]-oriented crystals, and dendrites. The addition of polyethylene glycol (PEG-3350) inhibited the hopper growth resulting in flat-faced surfaces. In sessile drops, 1–4mm tabular hopper crystals formed on the free surface and moved to the fixed contact line at the support (polycarbonate or gelatin) self-assembling into a shell. Ring formation created by sessile drop evaporation to dryness was observed but with crystals 100 times larger than particles in terrestrially formed coffee rings. No hopper pyramids formed. By choosing solution geometries offered by microgravity, we found it was possible to selectively grow crystals of preferred morphologies.

A fully coupled numerical simulation of sessile droplet evaporation using Arbitrary Lagrangian–Eulerian formulation

A transient sessile droplet evaporation numerical model based on Arbitrary-Lagrangian–Eulerian (ALE) formulation is developed, taking into account of the coupled transport processes in solid, liquid and gas phases as well as the evolution of free surface. The ALE formulation can trace the sharp two-phase interface and therefore can calculate evaporative flux and surface tension force accurately. The numerical simulations of axisymmetric sessile droplets evaporation at fixed contact line mode are conducted to verify the numerical model by comparing with the previous quasi-steady-state numerical simulation results and experimental data. It is shown that the present numerical model can predict the evaporation rate at quasi-steady-state, the droplet surface temperature distribution and the droplet volume variation with time correctly. The importance of evaporative cooling is emphasized and the influence of Marangoni flow on evaporation rate and surface temperature distribution are analyzed under different contact angles. The numerical model developed in this study can easily be extended to simulate more complicated droplet evaporation situations, such as contact line dynamics during droplet evaporation, droplet impingement cooling and 3-D droplet movement driven by a thermocapillary force on a substrate with surface temperature gradients.

Bubble dynamics atop an oscillating substrate: Interplay of compressibility and contact angle hysteresis

We consider a sessile hemispherical bubble sitting on the transversally oscillating bottom of a deep liquid layer and focus on the interplay of the compressibility of the bubble and the contact angle hysteresis. In the presence of contact angle hysteresis, the compressible bubble exhibits two kinds of terminal oscillations: either with the stick-slip motion of the contact line or with the completely immobile contact line. For the stick-slip oscillations, we detect a double resonance, when the external frequency is close to eigenfrequencies of both the breathing mode and shape oscillations. For the regimes evolving to terminal oscillations with the fixed contact line, we find an unusual transient resembling modulated oscillations.

Modeling Approach To Assess Clustering Impact on Release Rates of Pesticides from Microencapsulated Products

The pesticide release rate from polymer-encapsulated microcapsules is controlled by diffusion across the polymer membrane, membrane thickness, and pesticide loading. However, conditions for microcapsule clustering following conventional application practices and the impact of clustering on the overall release rate are often ignored. Microcapsules are delivered to target surfaces using water droplets as a carrier, and capillary-driven velocities arise within the deposited sessile drop as the water evaporates. This work describes experimental observations and a quasi-static mathematical approach used to elucidate microcapsule clusters that remain following the drop-drying process, along with the anticipated patterns for pesticide release as a function of cluster geometries. The dynamic behavior of a drying sessile drop was modeled as a trajectory of constrained quasi-static equilibrium shapes having a fixed contact line. Observed monolayer clustering of microcapsules was a function of the initial contact angle and size of the sessile drop, evaporation rate, and the microcapsule number density within the drop. The net pesticide release rate from microcapsule clusters was calculated from superposition of smaller decomposed cluster geometries for which release rates were determined by numerical solution to the mass balance governing equation, thereby coupling the self-assembly of microcapsule and microcapsule clustering with environmental release rate predictions.

Linear oscillations of a compressible hemispherical bubble on a solid substrate

The linear natural and forced oscillations of a compressible hemispherical bubble on a solid substrate are under theoretical consideration. The contact line dynamics is taken into account by application of the Hocking condition, which eventually leads to nontrivial interaction of the shape and volume oscillations. Resonant phenomena, mostly pronounced for the bubble with the fixed contact line or with the fixed contact angle, are found out. A double resonance, where independent of the Hocking parameter, an unbounded growth of the amplitude occurs, is detected. The limiting case of weakly compressible bubble is studied. The general criterion identifying whether the compressibility of a bubble can be neglected is obtained.

Frequencies of gravity–capillary waves on highly curved interfaces with edge constraints

A recently developed technique to calculate the natural frequencies of gravity–capillary waves in a confined liquid mass with a possibly highly curved free surface is extended to the case where the contact line is pinned. The general technique is worked out in detail for the cases of rectangular and cylindrical containers of circular section, the cases for which experimental data are available. The results of the present method are in excellent agreement with all earlier experimental and theoretical data for the flat static interface case [Benjamin and Scott, 1979. Gravity–capillary waves with edge constraints. J. Fluid Mech. 92, 241–267; Graham-Eagle, 1983. A new method for calculating eigenvalues with applications to gravity–capillary waves with edge constraints. Math. Proc. Camb. Phil. Soc. 94, 553–564; Henderson and Miles, 1994. Surface-wave damping in a circular cylinder with a fixed contact line. J. Fluid Mech. 275, 285–299]. However, the present method is applicable even when the contact angle is not π / 2 and the static interface is curved. As a consequence we are able to work out the effects of a curved meniscus on the results of Cocciaro et al. [1993. Experimental investigation of capillary effects on surface gravity waves: non-wetting boundary conditions. J. Fluid Mech. 246, 43–66] where the measured contact angle was 62 ∘ . We find that the meniscus does indeed account, as suggested by Cocciaro et al., for the earlier discrepancy between theory and experiment of about 20 mHz and there is now excellent agreement between the two.

Fixed contact line helical interfaces in zero gravity

Fluid interfaces supported in microgravity by a helical structure are shown to have a more robust stability than more common structures such as liquid bridges. In particular, helical interfaces can take the form of infinite right circular cylinders over a broad range of configurations. In the case of a single fixed contact line support, the infinite cylinder is stable for all cases in which the pitch to diameter ratio is less than π∕3 (more tightly coiled interfaces). When there are two or more equally spaced fixed contact line supports, the infinite cylinder is stable for all configurations. Furthermore, in the two support case (the double helix), stability persists for all volumes from the cylinder to zero volume, when the pitch to diameter ratio is greater than 2.082 (more loosely coiled interfaces). The equivalent to the axisymmetric Young-Laplace equation is derived for helical interfaces. Interfacial stability is determined from equilibrium branch structure following the application of Maddocks’ method by Lowry and Steen [Proc. R. Soc. London, Ser. A 449, 411 (1995)]. Perturbations to finite wavelength disturbances are considered for the case of a single helical support. Overall stability envelopes are presented for single and multiple support cases. Limited experimental results verify the infinite length stability limit for the single helical support case.

Axisymmetric Surface Oscillations in a Cylindrical Container with Compensated Gravity

In this study the first mode of axisymmetric surface oscillations of a free liquid interface in a partly filled right circular cylinder under compensated gravity is investigated numerically. The situation is similar to a spacecraft that enters a ballistic flight at the end of thrust. A reorientation of the liquid of the free liquid surface toward the new equilibrium position takes place. The characteristics of this flow, such as the frequency and the damping, are governed by the behavior of the contact line. The investigations focus on the dependence on the contact line boundary condition, including contact angle variation, and on viscous effects. The effect on the natural frequency and the damping is considered. An analytical approximation for the frequency behavior for a frictionless liquid for a free and a fixed contact line condition is given. Further analytical approximations for the damping behavior in dependence on the contact angle and the Ohnesorge number are determined. The numerical results show very good agreement with experimental data.

A robust algorithm for the simultaneous parameter estimation of interfacial tension and contact angle from sessile drop profiles

The pendant and sessile drop profile analysis using the finite element method (PSDA-FEM) is an algorithm which allows simultaneous determination of the interfacial tension ( γ) and contact angle ( θ c ) from sessile drop profiles. The PSDA-FEM algorithm solves the nonlinear second-order spherical coordinate form of the Young–Laplace equation. Thus, the boundary conditions at the drop apex and contact position of the drop with the substrate are required to solve for the drop profile coordinates. The boundary condition at the position where the drop contacts the substrate may be specified as a fixed contact line or fixed contact angle. This paper will focus on the fixed contact angle boundary condition for sessile drops on a substrate and how this boundary condition is used in the PSDA-FEM curve-fitting algorithm. The PSDA-FEM algorithm has been tested using simulated drop shapes with and without the addition of random error to the drop profile coordinates. The random error is varied to simulate the effect of camera resolution on the estimates of γ and θ c values obtained from the curve-fitting algorithm. The error in the experimental values for γ from sessile drops of water on acrylic and Mazola corn oil on acrylic falls within the predicted range of errors obtained for γ values from simulated sessile drop profiles with randomized errors that are comparable in magnitude to the resolution of the experimental setup.

Effects of static contact angles on inviscid gravity-capillary waves

The effects of a static contact angle θ0 on the natural frequencies of inviscid gravity-capillary waves depend on the contact-line conditions. The cases of free-end and pinned-end edge conditions are analyzed in the limit of large Bond number B. The ratio (ω2−ω02)∕ω02 (where ω0 and ω are the natural frequencies without and with meniscus, respectively) is proportional to B−1 or B−1 in the case of fixed contact angle or fixed contact line, respectively, and then the effect of the meniscus is greater for pinned-end edges than for free-end ones. When the contact angle is fixed (and for circular cylinders of aspect ratio not small) the natural frequency ω increases with increasing contact angle. When the contact line is fixed the natural frequency ω is smaller than ω0 in both cases of hydrophilic (θ0<π∕2) and hydrophobic surfaces (θ0>π∕2). It is shown that the reduction in frequency due to the meniscus can exceed the reduction associated with viscous effects. The results of this work compare better with experimental measurements, than those reported in the literature previously.

Non-axisymmetric oscillations of a hemispherical drop

Natural oscillations of a hemispherical drop on a solid substrate are considered. The Hocking condition is used to take into account the contact angle dynamics. The natural oscillations attenuate due to dissipation on the contact line and the degeneracy in the azimuthal number is eliminated. Forced oscillations of the drop are studied for tangential vibrations of the substrate. For the case of a fixed contact line, the amplitude of the forced oscillations grows without bound near the fundamental requency. In other cases, the amplitude is finite.

Shaping of sessile liquid metal drops using high-frequency magnetic fields

An analytical model for the electromagnetic shaping of sessile liquid metal drops in high-frequency magnetic fields is presented. We deal with both an infinitely long drop and a circular drop. In each case, the arrangement of drop and inductor is symmetric. Applying the skin depth approximation reduces the Lorentz forces induced in the liquid metal drop to a magnetic pressure on the drop surface. We neglect the coupling between drop contour and magnetic field distribution. In this case, the magnetic field can be calculated analytically applying the mirror-current method. Finally, we achieve an analytical solution of the static drop contours with the help of Green's functions. The theory is applied to three problems: (i) squeezing a drop while conserving its volume, (ii) drops with a fixed contact line, (iii) pumping up of drops. The results demonstrate the suitability of high-frequency magnetic fields for the shaping of liquid metals.

Contact angle dependence of solid probe–liquid drop forces in AFM measurements

In order to obtain the force between an atomic force microscopy probe particle and a liquid drop, we require the small change δ R in the mean radius of curvature of the drop, which is proportional to the small change in liquid pressure inside the drop. These quantities depend on the total force F, the contact angle θ made by the unperturbed drop at the substrate, and the nature of the boundary conditions imposed at the contact line. Chan et al. [J. Colloid Interface Sci. 236 (2001) 141] developed a semi-analytical method for calculating the force in the fixed contact line case, while more recently Attard and Miklavcic [Langmuir 17 (2001) 8217] considered the fixed contact angle condition. Due to errors in perturbation theory calculations, the results presented by these authors for δ R are incorrect, and their results [J. Colloid Interface Sci. 236 (2001) 141; Langmuir 17 (2001) 8217; J. Colloid Interface Sci. 247 (2002) 255] for the force are thus also in error. In contrast with the results of the above authors, we find that δ R is singular at θ=0, and perturbation theory breaks down at small θ. Consequently we find that, for angles of practical interest, δ R and thus the change in fluid pressure inside the drop can be orders of magnitude larger than previous calculations suggest. Modifying the method from [J. Colloid Interface Sci. 236 (2001) 141] accordingly, we present indicative numerical force F( X) curves (where X is the central probe-stage separation) for both boundary conditions. The general perturbation expansions are invalid near θ= π/2. We therefore extend the results into this experimentally significant region with a separate analysis at θ= π/2. Additionally, a nonperturbative matched expansion is developed to extend the results to small θ and to verify all analytical results.

Sufficient conditions for the stability and instability of a fluid boundary subjected to local stress

Sufficient conditions for either stability or instability of the interface of a fluid drop subject to localized surface stresses are presented. The stated conditions pertain to the case of an axisymmetric sessile drop having a fixed contact line and subject to axisymmetric forces acting on the surface of the drop. These conditions, appearing for the first time in the literature, are in the form of pointwise inequalities. They have quite general applicability, as they do not rely on explicit knowledge of the specific nature of an externally applied surface force. Supplementary integral inequalities are also provided and discussed.