Axisymmetric Droplet Research Articles

Slip of Spreading Viscoplastic Droplets.

The spreading of axisymmetric viscoplastic droplets extruded slowly on glass surfaces is studied experimentally using shadowgraphy and swept-field confocal microscopy. The microscopy furnishes vertical profiles of the radial velocity using particle image velocimetry (PIV) with neutrally buoyant tracers seeded in the fluid. Experiments were conducted for two complex fluids: aqueous solutions of Carbopol and xanthan gum. On untreated glass surfaces, PIV demonstrates that both fluids experience a significant amount of effective slip. The experiments were repeated on glass that had been treated to feature positive surface charges, thereby promoting adhesion between the negatively charged polymeric constituents of the fluids and the glass surface. The Carbopol and xanthan gum droplets spread more slowly on the treated surface and to a smaller radial distance. PIV demonstrated that this reduced spreading was associated with a substantial reduction in slip. For Carbopol, the effective slip could be eliminated entirely to within the precision of the PIV measurements; the reduction in slip was less effective for xanthan gum, with a weak slip velocity remaining noticeable.

Determination of surface tension and contact angle by the axisymmetric bubble and droplet shape analysis

The algorithms of solution to the Young–Laplace equation, describing the shape of an axisymmetric droplet on a flat horizontal surface, with various ways of setting the initial data and geometric parameters of a droplet, were derived and tested. Analysis of the Young–Laplace equation showed that a family of curves that form the droplet surface is the single-parametric one with the accuracy of up to the scale factor, whose role is played by the capillary length, and the contact angle determines the curve turn at a contact point, but it does not affect the shape of the curve. The main natural parameter defining the family of the forming curve is the curvature at the droplet top. The droplet shape is uniquely determined by three independent geometric parameters. This fact allows us to calculate the physical properties, such as the capillary length and contact angle, measuring three independent values: height, droplet diameter, and diameter of the droplet base or the area of the axial cross section of the droplet or its volume.

Interfacial instability of thin ferrofluid films under a magnetic field

AbstractWe study magnetically induced interfacial instability of a thin ferrofluid film subjected to an applied uniform magnetic field and covered by a non-magnetizable passive gas. Governing equations are derived using the long-wave approximation of the coupled static Maxwell and Stokes equations. The contact angle is imposed via a disjoining/conjoining pressure model. Numerical simulations show the patterning resulting from unstable perturbations and dewetting of the ferrofluid film. We find that the subtle competition between the applied field and the van der Waals induced dewetting determines the appearance of satellite droplets. The results suggest a new route for generating self-assembled ferrofluid droplets from a thin film using an external magnetic field. An axisymmetric droplet on a surface is also studied, and we demonstrate the deformation of the droplet into a spiked cone, in agreement with experimental findings.

A parametric study of the hydrophobicity of rough surfaces based on finite element computations

Several biological and artificial hydrophobic surfaces exhibit self-cleaning mechanisms, which ensure smooth sliding/rolling of liquid droplets, allowing them to sweep pollutant particles away from the surface. While enhancing of the self-cleaning property is a major topic in numerous industrial applications, the underlying physics is not yet fully understood. The first step towards analyzing this mechanism is studying the surface hydrophobicity, which is characterized by the contact angle and surface topography. In this article, we investigate the wetting of hydrophobic surfaces at three different length-scales, over a range of surface and droplet parameters. A mathematical model describing multi-scale surface topographies is presented, using exponential functions. The contact between liquid droplets and these surfaces is numerically studied using a finite element (FE) droplet model. The introduced models are verified numerically and experimentally. Numerical examples are shown for axisymmetric droplets of different size on surfaces with various contact angles and various levels of surface roughness, representing different length-scales. The corresponding wetted area is computed in order to evaluate the hydrophobicity of the surface. Significant differences are observed in contact angles and wetted areas captured at different length-scales. This highlights the importance of the multi-scale structure of hydrophobic surfaces on wetting. Furthermore, the presented work provides guidelines for the design of artificial hydrophobic surfaces.

Dynamic wetting behavior of a triple-phase contact line in several experimental systems

Dynamic wetting behavior was investigated experimentally. First, the movement of a liquid column in a capillary tube was observed. Using the measured dynamic contact angles dependent on the contact line velocity, the solution of the modified Lucas–Washburn equation predicts the time variance of the liquid height in the capillary efficiently. Second, the dynamic contact angles were measured for a two-dimensional meniscus and an axisymmetric droplet to detect the influence of contact line curvature. The measured angles coincide well with each other for a two-dimensional meniscus and droplets of various radii. The dynamic wetting behavior is independent of the triple-phase contact line curvature. The dynamic contact angles measured in this study distribute widely from 10° to 170°. The results cannot be arranged by a single function of contact line velocity proposed by Hoffmann. It is suggested that the roughness or impurities on the solid surface could play an important role on the dynamic wetting behavior.

Automated Drop-on-Fiber contact angle measurement using a microrobotic platform

Abstract Measuring contact angles on single fibers enables the separation of structural factors from surface chemistry factors. Current Drop-on-Fiber (DOF) contact angle measurement methods for natural fibers generally, and for pulp/paper fibers specifically, present a number of challenges. These are manipulation and mounting of the microscale fibers, straightening natural fibers to produce repeatable axisymmetric droplet conformation, measuring the droplet volume excluding the fiber volume and also human errors due to manually performed tasks. This paper presents a novel method to measure contact angles in DOF systems and overcome the above mentioned challenges using microrobots. The proposed microrobotic platform is capable of handling natural and synthetic fibrous materials in microscale, and dispensing probe liquid droplets down to 12 nl. It measures contact angle values using computer vision and a droplet-profiledetection algorithm. It reports the contact angle values as a function of volume of the droplet. The paper validates the capabilities of the proposed platform by applying three commonly used probe liquids: deionized water, ethylene glycol and diiodomethane for measuring contact angles on glass and pulp fibers. Finally, the results are compared with a picoliter contact angle measurement approach.

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.

Evaporation-induced saline Rayleigh convection inside a colloidal droplet

Inside evaporating two-component sessile droplets, a family of the Rayleigh convection exists, driven by salinity gradient formed by evaporation of solvent and solute. In this work, the characteristic of the flow inside an axisymmetric droplet is investigated. A stretched coordinate system is employed to account for the effect of boundary movement. A scaling analysis shows that the flow velocity is proportional to the (salinity) Rayleigh number (Ras) at the small-Rayleigh-number limit. A numerical analysis for a hemispherical droplet exhibits the flow velocity is proportional to the non-dimensional number \documentclass[12pt]{minimal}\begin{document}$Ra_s^{1/2}$\end{document}Ras1/2, at high Rayleigh numbers. A self-similar condition is established for the concentration field irrespective of the Rayleigh numbers after a moderate time, and the flow field is invariant with time at this stage. The scaling relation for the high Rayleigh numbers is verified experimentally by using aqueous NaCl droplets.

Validation and modification of asymptotic analysis of slow and rapid droplet spreading by numerical simulation

AbstractUsing a slip-length-based level-set approach with adaptive mesh refinement, we have simulated axisymmetric droplet spreading for a dimensionless slip length down to $O(1{0}^{\ensuremath{-} 4} )$. The main purpose is to validate, and where necessary improve, the asymptotic analysis of Cox (J. Fluid Mech., vol. 357, 1998, pp. 249–278) for rapid droplet spreading/dewetting, in terms of the detailed interface shape in various regions close to the moving contact line and the relation between the apparent angle and the capillary number based on the instantaneous contact-line speed, $\mathit{Ca}$. Before presenting results for inertial spreading, simulation results are compared in detail with the theory of Hocking & Rivers (J. Fluid Mech., vol. 121, 1982, pp. 425–442) for slow spreading, showing that these agree very well (and in detail) for such small slip-length values, although limitations in the theoretically predicted interface shape are identified; a simple extension of the theory to viscous exterior fluids is also proposed and shown to yield similar excellent agreement. For rapid droplet spreading, it is found that, in principle, the theory of Cox can predict accurately the interface shapes in the intermediate viscous sublayer, although the inviscid sublayer can only be well presented when capillary-type waves are outside the contact-line region. However, $O(1)$ parameters taken to be unity by Cox must be specified and terms be corrected to ${\mathit{Ca}}^{+ 1} $ in order to achieve good agreement between the theory and the simulation, both of which are undertaken here. We also find that the apparent angle from numerical simulation, obtained by extrapolating the interface shape from the macro region to the contact line, agrees reasonably well with the modified theory of Cox. A simplified version of the inertial theory is proposed in the limit of negligible viscosity of the external fluid. Building on these results, weinvestigate the flow structure near the contact line, the shear stress and pressure along the wall, and the use of the analysis for droplet impact and rapid dewetting. Finally, we compare the modified theory of Cox with a recent experiment for rapid droplet spreading, the results of which suggest a spreading-velocity-dependent dynamic contact angle in the experiments. The paper is closed with a discussion of the outlook regarding the potential of using the present results in large-scale simulations wherein the contact-line region is not resolved down to the slip length, especially for inertial spreading.

Development and application of micromechanical techniques for characterising interfacial shear strength in fibre-thermoplastic composites

The development of single fibre pull-out and microbond tests for characterising interfacial strength in thermoplastic composites is reviewed in detail. Manufacture of an experimental jig and sample preparation regimes for both tests are described. The challenges addressed in the sample preparation include the measurement of embedded fibre length for pull-out samples and the low yield rate of axisymmetric resin droplets obtained during sample preparation under nitrogen. The applications of these laboratory developed techniques are demonstrated by characterisation of the interfacial shear strength (IFSS) of glass fibre-polypropylene (GF-PP) and natural fibre-polylactic acid (NF-PLA). The comparison of the IFSS between neat and modified GF-PP showed that both methods were sensitive to the interfacial performance change despite the poor agreement between them for the absolute IFSS values from the same composite. The effect of the material modification was also reflected in load-displacement curves with different behaviour of the frictional motion after complete debonding. When a high level of fibre-matrix adhesion was realised in the composites with weak fibres, the microbond test showed higher feasibility for characterising the IFSS. This was clearly shown in its application to NF-PLA.

Numerical simulation of natural convection in a sessile liquid droplet

Heat transfer in a sessile liquid droplet was studied with numerical methods. A computer code was developed for solving the problem of convection in an axisymmetric hemispherical droplet and in a spherical layer as well. The problem of establishing an equilibrium state in a droplet was solved using several variables: temperature, stream function, and vorticity. Simulation was performed for droplets of water, ethyl alcohol, and model liquids. Variable parameters: intensity of heat transfer from droplet surface, Rayleigh and Marangoni dimensionless criteria, and the characteristic temperature difference. It was revealed that the curve of convective flow intensity versus heat transfer intensity at droplet surface has a maximum. A dual-vortex structure was obtained in a stationary hemispherical profile of liquid droplet for the case of close values for thermocapillary and thermogravitational forces. Either thermocapillary or thermogravitational vortex might be dominating phenomena in the flow structure.

The effect of surfactant convection and diffusion on the evolution of an axisymmetric pendant droplet

In this work, we consider the evolution of a single axisymmetric droplet as it grows from an initially uniform thin liquid film on the underside of a solid, horizontal substrate. We consider the effects of an insoluble surfactant on the free surface, the concentration of which changes due to convection and diffusion. Employing the lubrication approximations, we derive a linear solution valid in the initial stages of drop growth, and a full nonlinear solution, which we solve numerically using finite differences. Both a linear analysis and a numerical solution show that for sufficiently thin films, corresponding to low Bond numbers, diffusion of surfactant can effectively negate the effects of surfactant convection, and the drop evolves as if there is no surfactant on the free surface. For sufficiently thick films, corresponding to high Bond numbers, gravitational forces are stronger than surface tension gradient forces, and again the droplet evolves as if there is no surfactant present. For intermediate Bond numbers, the convection of surfactant can significantly slow the growth rate and must be included in the analysis to accurately model the time evolution of a typical pendant droplet. The interactions between coating height, flux, and surface velocity are used to explain the physics of this behavior.

Plateau Rayleigh Instability Simulation

The well-known phenomena of Plateau-Rayleigh instability has been simulated using computational fluid dynamics (CFD). The breakup of a liquid film into an array of droplets on a cylindrical element was simulated using a volume-of-fluid (VOF) solver and compared to experimental observations and existing theory. It is demonstrated that the VOF method can correctly predict the breakup of thins films into an array of either axisymmetric droplets or clam-shell droplets, depending on the surface energy. The existence of unrealistically large films is precluded. Droplet spacing was found to show reasonable agreement with theory. Droplet motion and displacement under fluid flow was also examined and compared to that in previous studies. It was found that the presence of air flow around the droplet does not influence the stable film thickness; however, it reduces the time required for droplet formation. Novel relationships for droplet displacement were derived from the results.

Non‐conforming finite elements for axisymmetric charged droplet deformation dynamics and Coulomb explosions

SUMMARYThe deformations of both electrically charged and uncharged incompressible axisymmetric Newtonian viscous liquid droplets acting under the effects of surface tension (but not gravity) are studied using a non‐conforming, discontinuous Galerkin finite element procedure with moving meshes.The full Navier–Stokes equations are discretized and solved in an Eulerian manner with a simple predictor–corrector Lagrangian updating of the free boundary location coincident with the droplet surface at each solution iteration.By using linear Crouzeix–Raviart basis functions for the velocity and piecewise constant pressures, results are presented both for the simple (oscillatory) relaxation of elongated electrically charged/uncharged droplets to a sphere and for the deformation to steady state/Coulomb explosions of initially slightly oblate/prolate spheroidal droplets charged beyond/to the Rayleigh limit. Copyright © 2012 John Wiley & Sons, Ltd.

Fast Evaporation of Spreading Droplets of Colloidal Suspensions

When a coffee droplet dries on a countertop, a dark ring of coffee solute is left behind, a phenomenon often referred to as the coffee-ring effect. A closely related yet less-well-explored phenomenon is the formation of a layer of particles, or skin, at the surface of the droplet during drying. In this work, we explore the behavior of a mathematical model that can qualitatively describe both phenomena. We consider a thin axisymmetric droplet of a colloidal suspension on a horizontal substrate undergoing spreading and evaporation. In contrast to prior work, precursor films (rather than pinned contact lines) are present at the droplet edge, and evaporation is assumed to be limited by how quickly molecules can transfer out of the liquid phase (rather than by how quickly they can diffuse through the gas phase). The lubrication approximation is applied to simplify the mass and momentum conservation equations, and the colloidal particles are allowed to influence the droplet rheology through their effect on the viscosity. By describing the transport of the colloidal particles with the full convection-diffusion equation, we are able to capture depthwise gradients in particle concentration and thus describe skin formation, a feature neglected in prior models of droplet evaporation. The highly coupled model equations are solved for a range of problem parameters using a finite-difference scheme based on a moving overset grid. The presence of evaporation and a large particle Peclet number leads to the accumulation of particles at the liquid-air interface. Whereas capillarity creates a flow that drives particles to the droplet edge to produce a coffee ring, Marangoni flows can compete with this and promote skin formation. Increases in viscosity due to particle concentration slow down droplet dynamics and can lead to a reduction in the spreading rate.

Nanofluid Surface Wettability Through Asymptotic Contact Angle

This investigation introduces the asymptotic contact angle as a criterion to quantify the surface wettability of nanofluids and determines the variation of solid surface tensions with nanofluid concentration and nanoparticle size. The asymptotic contact angle, which is only a function of gas-liquid-solid physical properties, is independent of droplet size for ideal surfaces and can be obtained by equating the normal component of interfacial force on an axisymmetric droplet to that of a spherical droplet. The technique is illustrated for a series of bismuth telluride nanofluids where the variation of surface wettability is measured and evaluated by asymptotic contact angles as a function of nanoparticle size, concentration, and substrate material. It is found that the variation of nanofluid concentration, nanoparticle size, and substrate modifies both the gas-liquid and solid surface tensions, which consequently affects the force balance at the triple line, the contact angle, and surface wettability.

Jet printing of colloidal solutions – Numerical modeling and experimental verification of the influence of ink and surface parameters on droplet spreading

Ink jet printing of functional materials promises an efficient route for the manufacturing of future low cost and large-area electronics applications. The effect of capillary flow of thin liquid films, the control of droplet spreading by suitably influencing the wetting properties of surfaces, the rheology of the ink and the process design play a relevant role in improvement of ink jet printed patterns. This work presents the experimentally based numerical study of the shape of single ink jetted droplets controlled by homogeneous contact angle distributions. The dynamics of the fluid on the substrate surface is treated in the frame of the lubrication theory using the concept of a precursor film and modeling the equilibrium contact angle by a disjoining pressure. The model describes the spreading of axisymmetric droplets considering different material and process parameter configurations. It is shown that the spreading process can be modeled separately from the drying process within a certain range of contact angles.

Sessile-Water-Droplet Contact Angle Dependence on Adsorption at the Solid−Liquid Interface

We investigate both analytically and experimentally the possible role of line tension in determining the contact angle of sessile water droplets on a polished Cu substrate. In a closed system with constraints that make the Helmholtz function the thermodynamic potential, the curvature of the three-phase line and the height of an axisymmetric droplet on its center line could be measured. The adsorption on each of the surfaces used to construct the experimental chamber was taken into account, and the value of the total number of water moles in the system was determined from the minimum in the Helmholtz function. The number of water moles was then changed to a new value and the system allowed to come to equilibrium again. The contact angle in the second state could be both measured and predicted with the adsorption at the solid−liquid and solid−vapor interfaces fully taken into account but with line tension completely neglected. The predicted values of the contact angle compared closely with those measured, i...

Simulation of droplet impact on a solid surface using the level contour reconstruction method

We simulate the three-dimensional impact of a droplet onto a solid surface using the level contour reconstruction method (LCRM). A Navier-slip dynamic contact line model is implemented in this method and contact angle hysteresis is accounted for by fixing the contact angle limits to prescribed advancing or receding angles. Computation of a distance function directly from the tracked interface enables a straightforward implementation of the contact line dynamic model in the LCRM. More general and sophisticated contact line models are readily applicable in this front tracking approach with few modifications, since complete knowledge of the geometrical information of the interface in the vicinity of the wall contact region is available. Several validation tests are performed including 2D planar droplet, 2D axisymmetric droplet, and full three-dimensional droplet splashing problems. The results show good agreement compared with existing numerical and experimental solutions.

A mathematical model of the Leidenfrost effect on an axisymmetric droplet

A simple mathematical model is developed for the Leidenfrost effect on an axisymmetric droplet. We first examine the practice of describing large droplets as cylindrical and small droplets as almost spherical, by solving the Young–Laplace equation. Best fit formulas are presented relating the droplet surface area to volume. The full model for Leidenfrost is then discussed. This shows that the calculation may be reduced to the solution of three first order ordinary differential equations involving the volume, evaporation rate, and vapor film thickness. Comparison with experimental results shows good agreement provided the model incorporates two components neglected in previous theoretical studies. First, a finite heat transfer coefficient between the vapor and substrate is required. Second evaporation from the upper surface of the droplet must be included, even for large droplets.