Hydrodynamic interactions of evaporating droplet with droplet of nonvolatile substance

The hydrodynamic interactions of freely evaporating or growing droplet (suspended in gaseous medium) in the supersaturated vapor with the droplet of nonvolatile substance or spherical solid particle are theoretically studied with allowance for effects that are linear with respect to the Knudsen number. The process of interaction between the volatile droplet and the infinite plane surface of nonvolatile liquid or solid is considered as a limiting case. Numerical estimates of the velocities of the steady motion of evaporating droplets of water and castor oil are reported. For the droplet of water and spherical solid particle, the effect of the heat conductivity of the latter on the velocity of particle motion is considered. Analogous estimates are obtained for a water droplet that evaporates near the infinite solid surface of castor oil or solid. The effects of the droplet size and the heat conductivity of wall on the rate of the evaporation of water droplet are analyzed.     

Hydrodynamic interactions of evaporating droplet with plane liquid surface

The hydrodynamic interactions between freely evaporating or growing droplet suspended in gas and the infinite plane liquid layer is theoretically studied with allowance for effects that are linear with respect to the Knudsen number. It is assumed that substances comprising droplet and liquid layer are identical, and it is taken into account that these substances can contain dissolved nonvolatile component whose concentration is so low that its presence should necessarily be accounted for only in the calculation of the concentration of saturated vapor. Vapor pressure at large distance from the droplet was assumed to be equal to the pressure of saturated vapor above the plane liquid surface. Results of numerical calculations of the rate of steady motion of water droplet evaporating or growing in the air are reported.     

Electrostatic interaction between a cylinder and a planar surface

 An exact analytical expression for the potential energy of the electrostatic interaction between a plate-like particle 1 and a cylindrical particle 2 of radius a ₂ immersed in an electrolyte solution of Debye–Hückel parameter κ is derived on the basis of the linearized Poisson–Boltzmann equation without recourse to Derjaguin's approximation. Both particles may have either constant surface potential or constant surface charge density. In the limit of κa ₂→0, in particular, the interaction between a plate with zero surface charge density and a cylinder having constant surface charge density becomes identical to the usual image interaction between a line charge (a charged rod of infinitesimal thickness) and an uncharged plate. Received: 22 September 1998  Accepted in revised form: 27 January 1999     

Hydrodynamic interaction between spheres coated with deformable thin liquid films

In this article, we considered the hydrodynamic interaction between two unequal spheres coated with thin deformable liquids in the asymptotic lubrication regime. This problem is a prototype model for drop coalescence through the so-called "film drainage" mechanism, in which the hydrodynamic contribution comes dominantly from the lubrication region apart from the van der Waals interaction force. First, a general formulation was derived for two unequal coated spheres that experienced a head-to-head collision at a very close proximity. The resulting set of the evolution equations for the deforming film shapes and stress distributions was solved numerically. The film shapes and hydrodynamic interaction forces were determined as functions of the separation distance, film thickness, viscosity ratios, and capillary numbers. The results show that as the two spheres approach each other, the films begin to flatten and eventually to form negative curvature (or a broad dimple) at their forehead areas in which high lubrication pressure is formed. The dimple formation occurs earlier as the capillary number increases. For large capillary numbers, the film liquids are drained out from their forehead areas and the coated liquid films rupture before the two films "touch" each other. Meanwhile, for small capillary numbers, the gap liquid is drained out first and the two liquid films eventually coalesce.     

On the solidification of a supercooled liquid droplet lying on a surface

We model the solidification and subsequent cooling of a supercooled liquid droplet that is lying on a cold solid substrate after impact. It is assumed that solidification occurs for a given fixed droplet shape. The shapes used by the model are a sphere, truncated spheres, and an experimentally registered droplet shape. The freezing process is conduction-dominant and is modeled as a one-phase Stefan problem. This moving boundary problem is reformulated with the enthalpy method and then solved numerically with an implicit finite-difference technique. The numerical results for the simple case of a spherical droplet touching a surface are similar to those of a freely freezing spherical droplet and are well confirmed by the 1D asymptotic analytical model of Feuillebois et al. (J. Colloid Interface Sci. 169 (1995) 90). A freezing water droplet is considered as an example. The numerical results for full freezing time, subsequent cooling time, and last freezing point coordinate for the various droplets shapes are fitted by analytical functions depending on supercooling, thermal resistance of the target surface (expressed by Biot number), and spreading parameter. These functions are proposed for direct application, thus avoiding the need to solve the full freezing and cooling problem.     

Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal

We use dynamic simulations to explore the pairwise interaction and multiparticle assembly of droplets suspended in a nematic liquid crystal. The computation is based on a regularized Leslie-Ericksen theory that allows orientational defects. The homeotropic anchoring on the drop surface is of sufficient strength as to produce a satellite point defect near the droplet. Based on the position of the defects relative to the host droplet and the far-field molecular orientation, we have identified five types of pairwise attractive and repulsive forces. In particular, long-range attraction between two droplets with their line of centers along the far-field orientation decays as R-4, with R being the center-to-center separation. This agrees with prior static calculations and a phenomenological model that treats the attraction as that between two dipoles. For interaction in shorter ranges, our simulations agree qualitatively with experimental measurements and static calculations. However, there is considerable quantitative discrepancy among the few existing studies and our simulation. We suggest that this is partly due to the dynamic nature of the process, which has never been taken into account in prior calculations. Multidrop simulations show the formation of linear chains through pairwise interactions between nearby droplets. Parallel chains repel or attract each other depending on the relative orientation of the drop-to-defect vector. These are consistent with experimental observations of chain formation and two-dimensional self-assembly in bulk nematics and smectic-C films.     

Self-assembly of spherical particles on an evaporating sessile droplet

Particles adsorbed on the surface of a droplet form three-dimensional packings when the droplet evaporates. We study the final packings when the liquid droplet is attached to a solid substrate. In contrast to a droplet evaporating away from a substrate, here the final packings are highly dependent on both the number of particles and the contact angle between the droplet and the surface. Simple geometrical constraints quantitatively determine the parameter regions that particular packings can form.     

Evaporation of two solution droplets and a droplet located near a flat liquid surface

The mutual influence of two moderate-sized droplets of a dilute nonvolatile substance solution on the processes of their evaporation or condensation is theoretically analyzed under the assumption of a uniform concentration distribution inside the droplets. The conditions for the applicability of this approach are revealed. The evaporation or condensation of a droplet near a flat liquid surface is considered as a limiting case. The fluxes of water molecules to and from the surface of aqueous glycerol solution droplets occurring in air are numerically estimated depending on the droplet radii, distances between their surfaces, and air humidity. Analogous estimates are obtained for an aqueous glycerol solution droplet growing near a flat water surface.     

On the evaporation and motion of a volatile droplet near a volatile liquid surface

A problem concerning the free evaporation or condensation growth of a droplet near an infinite planar surface of the same liquid is solved. The behavior of the droplet is considered at vapor temperature and concentration gradients preset at an infinite distance from it. The boundary conditions take into account effects that are linear with respect to the Knudsen number. Equations are derived for the rate of variations in the radius of the droplet and the velocity of its steady motion induced by nonuniform temperature and concentration of the vapor. Dependences of the rate of variations in the radius and the velocity of the steady motion of the droplet on the distance from the planar surface are presented for a droplet 1 ??m in radius suspended in air.     

Evidence of oscillatory convection inside an evaporating multicomponent droplet in a closed chamber

Visualization of an evaporating binary (ethanol-water) droplet reveals presence of oscillatory internal circulation. The visualization is done by using a laser scattering technique. The oscillatory circulation possibly results from the opposing effect of solutal and thermal Marangoni convection as proposed in some earlier theoretical works. The frequency of this oscillation is measured and the variation of this frequency with the initial concentration of the volatile component (ethanol) is reported.     

Analysis of the microfluid flow in an evaporating sessile droplet

The axisymmetric time-dependent flow field in an evaporating sessile droplet whose contact line is pinned is studied numerically and using an analytical lubrication theory with a zero-shear-stress boundary condition on the free surface of the droplet at low capillary and Reynolds numbers. A finite element algorithm is developed to solve simultaneously the vapor concentration and flow field in the droplet under conditions of slow evaporation. The finite element solution confirms the accuracy of the lubrication solution, especially when terms of higher order in the droplet flatness ratio (the ratio of droplet height to radius, h/R) are included in the lubrication theory to account more accurately for the singular flow near the contact line.     

Modeling contact angle hysteresis of a liquid droplet sitting on a cosine wave-like pattern surface

A liquid droplet sitting on a hydrophobic surface with a cosine wave-like square-array pattern in the Wenzel state is simulated by using the Surface Evolver to determine the contact angle. For a fixed drop volume, multiple metastable states are obtained at two different surface roughnesses. Unusual and non-circular shape of the three-phase contact line of a liquid droplet sitting on the model surface is observed due to corrugation and distortion of the contact line by structure of the roughness. The contact angle varies along the contact line for each metastable state. The maximum and minimum contact angles among the multiple metastable states at a fixed viewing angle correspond to the advancing and the receding contact angles, respectively. It is interesting to observe that the advancing/receding contact angles (and contact angle hysteresis) are a function of viewing angle. In addition, the receding (or advancing) contact angles at different viewing angles are determined at different metastable states. The contact angle of minimum energy among the multiple metastable states is defined as the most stable (equilibrium) contact angle. The Wenzel model is not able to describe the contact angle along the three-phase contact line. The contact angle hysteresis at different drop volumes is determined. The number of the metastable states increases with increasing drop volume. Drop volume effect on the contact angles is also discussed.     

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.     

Impact of a heterogeneous liquid droplet on a dry surface: Application to the pharmaceutical industry

Droplet impact has been studied for over a hundred years dating back to the pioneering work of Worthington [1]. In fact, much of his ingenuity contributed to modern day high speed photography. Over the past 40 years significant contributions in theoretical, numerical, and experimental work have been made. Droplet impact is a problem of fundamental importance due to the wealth of applications involved, namely, spray coating, spray painting, delivery of agricultural chemicals, spray cooling, inkjet printing, soil erosion due to rain drop impact, and turbine wear. Here we highlight one specific application, spray coating. Although most studies have focused their efforts on low viscosity Newtonian fluids, many industrial applications such as spray coating utilize more viscous and complex rheology liquids. Determining dominant effects and quantifying their behavior for colloidal suspensions and polymer solutions remains a challenge and thus has eluded much effort. In the last decade, it has been shown that introducing polymers to Newtonian solutions inhibits the rebounding of a drop upon impact, Bergeron et al. [2]. Furthermore Bartolo et al. [3] concluded that the normal stress component of the elongational viscosity was responsible for the rebounding inhibition of polymer based non-Newtonian solutions. We aim to uncover the drop impact dynamics of highly viscous Newtonian and complex rheology liquids used in pharmaceutical coating processes. The generation and impact of drops of mm and μm size drops of coating liquids and glycerol/water mixtures on tablet surfaces are systematically studied over a range of We ∼ O(1-300), Oh ∼ O(10^− 2-1), and Re ∼ O(1-700). We extend the range of Oh to values above 1, which are not available to previous studies of droplet impacts. Outcomes reveal that splashing and rebounding are completely inhibited and the role of wettability is negligible in the early stages of impact. The maximum spreading diameter of the drop is compared with three models demonstrating reasonable agreement.     

A cell model of a liquid droplet

An expression for the free energy of a droplet composed of attracting hard spheres is found using a simple cell model. The hard-sphere repulsion is assumed to act only between molecules in the same cell, whereas attraction extends over many cells. A maximum term analysis gives rise to a mean-field free energy which includes terms proportional to the first and second power of the droplet radius R with coefficients which can be related to the planar surface tension and Tolman length. Certain Gaussian fluctuations about the maximum term are also considered, corresponding to droplet translation and capillary wave fluctuations. Inclusion of these fluctuations is necessary to ensure that the nucleation rate is proportional to the system volume. They also reduce the planar surface tension and introduce a logarithmic term, -4/3 ln R, into the free energy. The inclusion of other fluctuations and the relationship between these equations and those arising in density-functional theories of nucleation is discussed.     

Cost-effective electrodeposition of platinum nanoparticles with ionic liquid droplet confined onto electrode surface as micro-media

We report here for the first time on the use of a droplet of water-immiscible ionic liquid (IL) containing metallic precursor confined onto electrode surface as new micro-media for cost-effective electrodeposition of platinum nanoparticles. 1-n-Butyl-3-methylimidazolium hexafluorophosphate (BmimPF₆), a typical water-immiscible IL, is found to be able to form a stable droplet onto electrode surface in which the metallic precursor (i.e., chloroplatinic acid hexahydrate (H₂PtCl₆)) for electrodeposition of Pt nanoparticles can be stably dissolved when the prepared electrode is used in aqueous solutions. The electrodeposition of Pt nanoparticles is carried out in the aqueous solution of 0.1 M KPF₆ with the H₂PtCl₆-containing IL droplet-confined glassy carbon electrode as working electrode at −1.5 V vs. Ag/AgCl. The Pt nanoparticles electrodeposited from the IL droplet micro-medium are characterized to have a uniform morphology and to possess an excellent electrocatalytic activity toward the oxidation of methanol. Compared with the existing methods for the electrodeposition of metals with ILs as the solvents, the method demonstrated here requires a less amount of ILs and metallic precursors and is thus anticipated to provide a new and cost-effective approach to the deposition of metallic nanoparticles onto conducting substrate.     

Dynamics of a droplet imbibing on a rough surface

We consider the imbibition of a liquid droplet of finite size on a rough surface and theoretically show that the imbibition dynamics is significantly slower than the familiar Washburn diffusive dynamics, ～t(0.5). The imbibition does not follow a simple power law. The droplet starts to imbibe with ～t(0.5) dynamics but progressively becomes slower with time. The slower imbibition is mainly attributed to the finite size of the droplet, resulting in a limited capillary driving force as compared to a steady capillary driving force in the case of imbibition from a steady source.     

Analysis of droplet evaporation on a superhydrophobic surface

The evaporation process for small, 1-2-mm-diameter droplets of water from patterned polymer surfaces is followed and characterized. The surfaces consist of circular pillars (5-15 microm diameter) of SU-8 photoresist arranged in square lattice patterns such that the center-to-center separation between pillars is 20-30 microm. These types of surface provide superhydrophobic systems with theoretical initial Cassie-Baxter contact angles for water droplets of up to 140-167 degrees, which are significantly larger than can be achieved by smooth hydrophobic surfaces. Experiments show that on these SU-8 textured surfaces water droplets initially evaporate in a pinned contact line mode, before the contact line recedes in a stepwise fashion jumping from pillar to pillar. Provided the droplets of water are deposited without too much pressure from the needle, the initial state appears to correspond to a Cassie-Baxter one with the droplet sitting upon the tops of the pillars. In some cases, but not all, a collapse of the droplet into the pillar structure occurs abruptly. For these collapsed droplets, further evaporation occurs with a completely pinned contact area consistent with a Wenzel-type state. It is shown that a simple quantitative analysis based on the diffusion of water vapor into the surrounding atmosphere can be performed, and estimates of the product of the diffusion coefficient and the concentration difference (saturation minus ambient) are obtained.     

Stationary concentration establishment in the growing or evaporating droplet of ideal binary solution

The establishment of stationary solution concentration in a growing or evaporating droplet of an ideal binary solution (binary droplet) placed in a vapor mixture of constituting substances and passive gas is described analytically. Relations defining time dependences of solution concentration in a droplet, the number of molecules of each constituting component, and droplet radius are derived at known parameters of the vapor-gas mixture and the initial composition of a binary droplet. The results of calculations of time dependences of aforementioned values are reported for several variants of the initial composition of a droplet and a vapor mixture.