The leading edge of evaporating droplets

New experiments on drops evaporating in normal atmosphere from smooth substrates in the situation of complete wetting are reported and compared with the available theoretical model. They are the continuation of previous work with alkane or water sessile drops, which is first briefly summarized. The model accounts very well for the dynamics of the drop radius, but the predictions are only qualitative for the contact angle, especially for small angles. Experiments with hanging drops allow us first to discard any influence of convection in the gas phase on the drops dynamics. Then the main part of the paper concerns new experiments with polydimethylsiloxane oligomers. These silicone oils are similar to alkanes as far as evaporation rate is concerned, but have lower surface tensions, and therefore smaller dynamic contact angles. The purity of the oils appears to be critical for the experiments, and requires a preliminary investigation. Then a systematic study of the drops dynamics is presented, as a basis for forthcoming theoretical work.     

Spreading Dynamics of Polydimethylsiloxane Drops: Crossover from Laplace to Van der Waals Spreading

The spreading dynamics of small polydimethylsiloxane (PDMS) drops was studied on substrates with varying surface energies. For experimental parameters near the wetting transition, we observed small PDMS drops of different drop volumes as a function of time using interference video microscopy. While for large drops the contact angle θ decreases with the well-established power-law relation θ approximately t(-0.3) (Tanner's law), the effect of dispersive van der Waals (VW) interactions must be taken into account when interpreting the evolution of small drops. Two signatures of the VW forces are observed. For a positive Hamaker constant, the disjoining pressure acts as an additional driving force, leading to an acceleration of droplet spreading as soon as the drop height becomes comparable to the range of the VW interactions. In addition, a precursor film forms ahead of the contact line, leading to an apparent volume loss, particularly noticeable for very small drops. Contact line pinning may be a problem and we describe its effect on our experimental results. We present a theory that discusses the interplay of surface tension and VW forces in the case of a spreading drop. This model predicts a new spreading regime for very thin drops, in agreement with our experimental results. Copyright 2001 Academic Press.     

Application of Young’s equation to two immiscible liquids resting on a solid substrate

The geometry of two immiscible liquid drops (water and mercury) resting on a solid substrate is reported. The problem under investigation is a very simple situation of a non-miscible two-phase liquid system which is properly documented. The simple experimental observation of water micro drop on top of a mercury drop on polypropylene substrate is discussed. The static situation of the drops is explored, and a new equation is derived for such situation by applying Young’s equation to the contact angles. The obtained equation simply represents the additive summation of Young equations for two independent drops, and may be applied for the case when the interfaces of two drops are close to each other and hence interact. The forces at the contact lines of each drop are treated separately as well as in correlation. This paper shows a schematic representation of two-liquid phase contact angle system. The value of this work may be appreciated from a pedagogical point of view.     

Wetting and evaporation of binary mixture drops

Experimental results on the wetting behavior of water, methanol, and binary mixture sessile drops on a smooth, polymer-coated substrate are reported. The wetting behavior of evaporating water/methanol drops was also studied in a water-saturated environment. Drop parameters (contact angle, shape, and volume) were monitored in time. The effects of the initial relative concentrations on subsequent evaporation and wetting dynamics were investigated. Physical mechanisms responsible for the various types of wetting behavior during different stages are proposed and discussed. Competition between evaporation and hydrodynamic flow are evoked. Using an environment saturated with water vapor allowed further exploration of the controlling mechanisms and underlying processes. Wetting stages attributed to differential evaporation of methanol were identified. Methanol, the more volatile component, evaporates predominantly in the initial stage. The data, however, suggest that a small proportion of methanol remained in the drop after the first stage of evaporation. This residual methanol within the drop seems to influence subsequent wetting behavior strongly.     

Contact line dynamics in drop coalescence and spreading

The dynamics of coalescence of two water sessile drops is investigated and compared with the spreading dynamics of a single drop in partially wetting regime. The composite drop formed due to coalescence relaxes exponentially toward equilibrium with a typical relaxation time that decreases with contact angle. The relaxation time can reach a few tenths of seconds and depends also on the drop size, initial conditions, and surface properties (contact angle, roughness). The relaxation dynamics is larger by 5 to 6 orders of magnitude than the bulk hydrodynamics predicts, due to the high dissipation in the contact line vicinity. The coalescence is initiated at a contact of the drops growing in a condensation chamber or by depositing a small drop at the top of neighboring drops with a syringe, a method also used for the studies of the spreading. The dynamics is systematically faster by an order of magnitude when comparing the syringe deposition with condensation. We explain this faster dynamics by the influence of the unavoidable drop oscillations observed with fast camera filming. Right after the syringe deposition, the drop is vigorously excited by deformation modes, favoring the contact line motion. This excitation is also observed in spreading experiments while it is absent during the condensation-induced coalescence.     

Evaporation of pure liquid sessile and spherical suspended drops: a review

A sessile drop is an isolated drop which has been deposited on a solid substrate where the wetted area is limited by a contact line and characterized by contact angle, contact radius and drop height. Diffusion-controlled evaporation of a sessile drop in an ambient gas is an important topic of interest because it plays a crucial role in many scientific applications such as controlling the deposition of particles on solid surfaces, in ink-jet printing, spraying of pesticides, micro/nano material fabrication, thin film coatings, biochemical assays, drop wise cooling, deposition of DNA/RNA micro-arrays, and manufacture of novel optical and electronic materials in the last decades. This paper presents a review of the published articles for a period of approximately 120 years related to the evaporation of both sessile drops and nearly spherical droplets suspended from thin fibers. After presenting a brief history of the subject, we discuss the basic theory comprising evaporation of micrometer and millimeter sized spherical drops, self cooling on the drop surface and evaporation rate of sessile drops on solids. The effects of drop cooling, resultant lateral evaporative flux and Marangoni flows on evaporation rate are also discussed. This review also has some special topics such as drop evaporation on superhydrophobic surfaces, determination of the receding contact angle from drop evaporation, substrate thermal conductivity effect on drop evaporation and the rate evaporation of water in liquid marbles.     

Contact line dynamics in the late-stage coalescence of diethylene glycol drops

We investigated the contact line dynamics of a composite drop formed as a result of the coalescence during the condensation of two diethylene glycol (DEG) drops at -4 degrees C on a silicon surface. The composite drop relaxes exponentially toward equilibrium with a typical relaxation time, tc, which depends on the equilibrium radius, R, of the composite drop. The value of tc is found to be in the range of 10-100 s for R approximately 1-4 microm. The relaxation dynamics is found to be larger by 6 orders of magnitude than that predicted by bulk hydrodynamics because of high dissipation in the contact line vicinity. Similar to low viscous liquids (water), this high dissipation can be attributed to an Arrhenius factor resulting from the phase change in the contact line vicinity and to the influence of surface defects that pin the contact line.     

High internal phase emulsions: catastrophic phase inversion, stability, and triggered destabilization

We have investigated the formation, drop sizes, and stability of emulsions prepared by hand shaking in a closed vessel in which the emulsion is in contact with a single type of surface during its formation. The emulsions undergo catastrophic phase inversion from oil-in-water (o/w) to water-in-oil (w/o) as the oil volume fraction is increased. We find that the oil volume fraction required for catastrophic inversion exhibits a linear correlation with the oil-water-solid surface contact angle. W/o high internal phase emulsions (HIPEs) prepared in this way contain water drops of diameters in the range 10-100 μm; emulsion drop size depends on the surfactant concentration and method of preparation. W/o HIPEs with large water drops show water separation but w/o HIPEs with small water drops are stable with respect to water separation for more than 100 days. The destabilization of the w/o HIPEs can be triggered by either evaporation of the oil continuous phase or by contact the emulsion with a solid surface of the "wrong" wettability.     

Motion of liquid drops on surfaces induced by asymmetric vibration: role of contact angle hysteresis

Hysteresis of wetting, like the Coulombic friction at solid/solid interface, impedes the motion of a liquid drop on a surface when subjected to an external field. Here, we present a counterintuitive example, where some amount of hysteresis enables a drop to move on a surface when it is subjected to a periodic but asymmetric vibration. Experiments show that a surface either with a negligible or high hysteresis is not conducive to any drop motion. Some finite hysteresis of contact angle is needed to break the periodic symmetry of the forcing function for the drift to occur. These experimental results are consistent with simulations, in which a drop is approximated as a linear harmonic oscillator. The experiment also sheds light on the effect of the drop size on flow reversal, where drops of different sizes move in opposite directions due to the difference in the phase of the oscillation of their center of mass.     

Unconventional multiple ring structure formation from evaporation-induced self-assembly of polymers

The formation of multiring deposits of poly(2-vinylpyridine) (P2VP) from the evaporation of a P2VP-(2,6-lutidine + water) drop on a glass substrate does not conform to the conventional pinning-depinning mechanism. Instead, ringlike deposits are formed when the droplet undergoes several cycles of spreading and receding where, for each spreading event, a P2VP ridge is formed at the contact line when the polymer flows toward the outward advancing edge. The complex interplay between an outward solutal-Marangoni flow due to a higher concentration of the polymer at the contact line and an inward solvent-Marangoni flow arising from the differences in volatilities and surface tensions of the pure solvent components plays an important role in enhancing the droplet spreading rate. The newly discovered surface patterning mechanism has important implications in the development of novel techniques for inducing self-assembly of functional materials from evaporating drops.     

Dynamic Photocontrol of the Coffee‐Ring Effect with Optically Tunable Particle Stickiness

When a colloidal drop dries on a surface, most of the particles accumulate at the drop periphery, yielding a characteristic ring‐shaped pattern. This so‐called coffee‐ring effect (CRE) is observed in any pinned evaporating drop containing non‐volatile solutes. Here, the CRE is dynamically controlled for the first time by using light, and an unprecedented reconfigurability of the deposit profile is demonstrated. This is achieved through a new mechanism where particle stickiness is optically tuned on demand, thus offering reliable modulation of the deposition pattern. The system consists of anionic nanoparticles and photosensitive cationic surfactants dispersed in water. It is shown that light‐dependent modulation of surfactant–particle interactions dictates particle attraction and trapping at the liquid–gas interface, which allows us to direct particle deposition into a wide range of patterns from rings to homogeneous disks. Patterning from single drops is photoreversible upon changing the wavelength whereas spatial control in multiple drop arrays is achieved using a photomask.     

Microdroplet growth mechanism during water condensation on superhydrophobic surfaces

By promoting dropwise condensation of water, nanostructured superhydrophobic coatings have the potential to dramatically increase the heat transfer rate during this phase change process. As a consequence, these coatings may be a facile method of enhancing the efficiency of power generation and water desalination systems. However, the microdroplet growth mechanism on surfaces which evince superhydrophobic characteristics during condensation is not well understood. In this work, the sub-10 μm dynamics of droplet formation on nanostructured superhydrophobic surfaces are studied experimentally and theoretically. A quantitative model for droplet growth in the constant base (CB) area mode is developed. The model is validated using optimized environmental scanning electron microscopy (ESEM) imaging of microdroplet growth on a superhydrophobic surface consisting of immobilized alumina nanoparticles modified with a hydrophobic promoter. The optimized ESEM imaging procedure increases the image acquisition rate by a factor of 10-50 as compared to previous research. With the improved imaging temporal resolution, it is demonstrated that nucleating nanodroplets coalesce to create a wetted flat spot with a diameter of a few micrometers from which the microdroplet emerges in purely CB mode. After the droplet reaches a contact angle of 130-150°, its base diameter increases in a discrete steplike fashion. The droplet height does not change appreciably during this steplike base diameter increase, leading to a small decrease of the contact angle. Subsequently, the drop grows in CB mode until it again reaches the maximum contact angle and increases its base diameter in a steplike fashion. This microscopic stick-and-slip motion can occur up to four times prior to the droplet coalescence with neighboring drops. Lastly, the constant contact angle (CCA) and the CB growth models are used to show that modeling formation of a droplet with a 150° contact angle in the CCA mode rather than in the CB mode severely underpredicts both the drop formation time and the average heat transfer rate through the drop.     

Roughness effects of cellulose and paper substrates on water drop impact and recoil

The impact dynamics of water drops on sized and unsized smooth cellulose films and paper surfaces with controlled roughness levels were studied. The objective was to better understand the effect of roughness on the liquid drop impact dynamics on paper surfaces, isolating from the effect chemical heterogeneity. Drop impact in the first few milliseconds were recorded using high-speed CCD camera and the three-phase contact line movement of the water drop was analyzed. Smooth cellulose film surface and rough paper surface showed similar impact dynamics, suggesting that the surface energy plays a more dominant role than surface roughness. Significantly different dynamic contact angles of water drop on the sized and unsized surfaces were observed during drop impact. The Laplace pressure of the curved spreading front pointing to the centre of a spreading drop on these sized cellulose and paper surfaces reduces the three-phase contact line movement, and leads to smaller maximum spreading diameter. Our results suggest that the water drop spreads on the rough surface is most likely via a “roll-over” action rather than “stick and jump” movements.     

Gibbs free energy of liquid drops on conical fibers

Small drops can move spontaneously on conical fibers. As a drop moves along the cone, it must change shape to maintain a constant volume, and thus, it must change its surface energy. Simultaneously, the exposed surface area of the underlying cone must also change. The associated surface energies should balance each other, and the drop should stop moving when it reaches a location where the free energy is a minimum. In this paper, a minimum Gibbs free energy analysis has been performed to predict where a drop will stop on a conical fiber. To obtain the Gibbs free energies of a drop at different locations of a conical fiber, the theoretical expressions for the shape of a droplet on a conical fiber are derived by extending Carroll's equations for a drop on a cylindrical fiber. The predicted Gibbs free energy exhibits a minimum along the length of the cone. For a constant cone angle, as the contact angle between the liquid and the cone increases, the drop will move toward the apex of the cone. Likewise, for a constant contact angle, as the cone angle increases, the drop moves toward the apex. Experiments in which water and dodecane were placed on glass cones verify these dependencies. Thus, the final location of a drop on a conical fiber can be predicted on the basis of the geometry and surface energy of the cone, the surface tension and volume of the liquid, and the original location where the drop was deposited.     

Assessing the accuracy of contact angle measurements for sessile drops on liquid-repellent surfaces

Gravity-induced sagging can amplify variations in goniometric measurements of the contact angles of sessile drops on super-liquid-repellent surfaces. The very large value of the effective contact angle leads to increased optical noise in the drop profile near the solid-liquid free surface and the progressive failure of simple geometric approximations. We demonstrate a systematic approach to determining the effective contact angle of drops on super-repellent surfaces. We use a perturbation solution of the Bashforth-Adams equation to estimate the contact angles of sessile drops of water, ethylene glycol, and diiodomethane on an omniphobic surface using direct measurements of the maximum drop width and height. The results and analysis can be represented in terms of a dimensionless Bond number that depends on the maximum drop width and the capillary length of the liquid to quantify the extent of gravity-induced sagging. Finally, we illustrate the inherent sensitivity of goniometric contact angle measurement techniques to drop dimensions as the apparent contact angle approaches 180°.     

Wetting and absorption of water drops on Nafion films

Water drops on Nafion films caused the surface to switch from being hydrophobic to being hydrophilic. Contact angle hysteresis of >70 degrees between advancing and receding values were obtained by the Wilhelmy plate technique. Sessile drop measurements were consistent with the advancing contact angle; the sessile drop contact angle was 108 degrees . Water drop adhesion, as measured by the detachment angle on an inclined plane, showed much stronger water adhesion on Nafion than Teflon. Sessile water and methanol drops caused dry Nafion films to deflect. The flexure went through a maximum with time. Flexure increased with contact area of the drop, but was insensitive to the film thickness. Methanol drops spread more on Nafion and caused larger film flexure than water. The results suggest that the Nafion surface was initially hydrophobic but water and methanol drops caused hydrophilic sulfonic acid domains to be drawn to the Nafion surface. Local swelling of the film beneath the water drop caused the film to buckle. The maximum flexure is suggested to result from motion of a water swelling front through the Nafion film.     

Limiting conditions for applying the spherical section assumption in contact angle estimation

The shape of liquid drops on solid surfaces deviates from the spherical as tension decreases and gravity effects start affecting the drop shape. This paper attempts to define this deviation and estimates the dimensionless Eotvos number limits above which the deviation becomes "significant." The use of these limiting values can facilitate estimation of contact angle in the following manner. It is well known that the equilibrium contact angle made by a liquid drop on a solid surface can be estimated from measurements of two drop parameters. These parameters can be any two chosen from the drop volume, height, and wetted radius. In case the effect of gravity on the drop shape is negligible, simple algebraic relations derived from the spherical section assumption exist, from which the contact angle can be estimated. In systems where the "spherical section" assumption is invalid, the Laplace equation for the drop shape has been solved numerically with any two of the above parameters as the constraints, to obtain the contact angle. In this paper, Eotvos numbers at which the deviation of the drop profile from the spherical is significant enough to result in contact angle deviation of 1 degrees are estimated. The limiting values of Eotvos number, expressed as a function of the original contact angle made by the spherical profile, are obtained by solving the Laplace equation for the drop shape with the drop volume and wetted radius constraints for decreasing values of Interfacial tension. These limiting values are also estimated for different drop sizes and for cases where the drop phase is heavier (sessile) and lighter (buoyant) than the surrounding fluid. The independence of the Eotvos number estimates from the sign of the density difference as well as the drop size is shown. These Eotvos number limits can be used to check if the spherical section assumption, with the resulting simple algebraic relations, can be used for contact angle estimation and other shape-related analysis for a system.     

Non-aqueous phase liquid drop formation within a water saturated fracture

This work focuses on the mechanisms of non-aqueous phase liquid (NAPL) drop formation within a single fracture fed from a NAPL reservoir by way of a circular orifice, such as a pore. The fracture is assumed to be fully saturated, the relative wettability of the system is assumed water-wet, and the water velocity profile within the fracture is described by a Poiseuille flow. The size of the NAPL drops is investigated for various water flow velocities and NAPL entrance diameters. A force balancing method was used to determine the radii of detached drops. The drop sizes calculated from the model developed here are shown to be in agreement with available experimental drop size data. It is shown that at low Reynolds numbers the buoyancy force is the dominant force acting on the drop during the formation process and at high Reynolds numbers the viscous forces dominate. A simplified expression relating the geometry of the fractured system to the drop radii is developed from the model equations, and it is shown to predict drop radii that match well with both the model simulations and the available 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 (gamma) and contact angle (theta(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 gamma and theta(c) values obtained from the curve-fitting algorithm. The error in the experimental values for gamma from sessile drops of water on acrylic and Mazola corn oil on acrylic falls within the predicted range of errors obtained for gamma values from simulated sessile drop profiles with randomized errors that are comparable in magnitude to the resolution of the experimental setup.     

Liquid drops on vertical and inclined surfaces; I. An experimental study of drop geometry

Experiments have been conducted to investigate the geometric parameters necessary to describe the shapes of liquid drops on vertical and inclined plane surfaces. Two liquids and eight surfaces have been used to study contact angles, contact lines, profiles, and volumes of drops of different sizes for a range of surface conditions. The results show the contact-angle variation along the circumference of a drop to be best fit by a third-degree polynomial in the azimuthal angle. This contact-angle function is expressed in terms of the maximum and minimum contact angles of the drop, which are determined for various conditions. The maximum contact angle, thetamax, is approximately equal to the advancing contact angle, thetaA, of the liquid on the surface. As the Bond number, Bo, increases from 0 to a maximum, the minimum contact angle, thetamin, decreases almost linearly from the advancing to the receding angle. A general relation is found between thetamin/thetaA and Bo for different liquid-surface combinations. The drop contour can be described by an ellipse, with the aspect ratio increasing with Bo. These experimental results are valuable in modeling drop shape, as presented in Part II of this work.