Spreading of liquid drops over dry porous layers: complete wetting case

Spreading of small liquid drops over thin dry porous layers is investigated from both theoretical and experimental points of view. Drop motion over a porous layer is caused by an interplay of two processes: (a) the spreading of the drop over already saturated parts of the porous layer, which results in an expanding of the drop base; (b) the imbibition of the liquid from the drop into the porous substrate, which results in a shrinkage of the drop base and an expanding of the wetted region inside the porous layer. As a result of these two competing processes, the radius of the drop goes through a maximum value over time. A system of two differential equations is derived to describe the evolution with time of radii of both the drop base and the wetted region inside the porous layer. This system includes two parameters: one accounts for the effective lubrication coefficient of the liquid over the wetted porous substrate and the other is a combination of permeability and effective capillary pressure inside the porous layer. Two additional experiments are used for an independent determination of these two parameters. The system of differential equations does not include any fitting parameter after these two parameters are determined. Experiments were carried out on the spreading of silicone oil drops over various dry microfiltration membranes (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region inside the porous layer are monitored. All experimental data fell on two universal curves if appropriate scales are used with a plot of the dimensionless radii of the drop base and of the wetted region inside the porous layer on dimensionless time. The predicted theoretical relationships are two universal curves accounting quite satisfactorily for the experimental data. According to our theory prediction, (i) the dynamic contact angle dependence on the same dimensionless time as before should be a universal function and (ii) the dynamic contact angle should change rapidly over an initial short stage of spreading and should remain a constant value over the duration of the rest of the spreading process. The constancy of the contact angle on this stage has nothing to do with hysteresis of the contact angle: there is no hysteresis in our system. These conclusions again are in good agreement with our experimental observations.     

Spontaneous rise of surfactant solutions into vertical hydrophobic capillaries

It has been found earlier (N.V. Churaev, G.A. Martynov, V.M. Starov, Z.M. Zorin, Colloid Polym. Sci. 259 (1981) 747) that aqueous surfactant solutions spontaneously rise in vertical hydrophobized quartz capillaries. A theory of this phenomenon is presented, which connects the experimental observations with the adsorption of surfactant molecules in front of the moving meniscus on the bare hydrophobic interface.     

Spreading of Surfactant Solutions over Hydrophobic Substrates

The spreading of surfactant solutions over hydrophobic surfaces is considered from both theoretical and experimental points of view. Water droplets do not wet a virgin solid hydrophobic substrate. It is shown that the transfer of surfactant molecules from the water droplet onto the hydrophobic surface changes the wetting characteristics in front of the drop on the three-phase contact line. The surfactant molecules increase the solid-vapor interfacial tension and hydrophilize the initially hydrophobic solid substrate just in front of the spreading drop. This process causes water drops to spread over time. The time of evolution of the spreading of a water droplet is predicted and compared with experimental observations. The assumption that surfactant transfer from the drop surface onto the solid hydrophobic substrate controls the rate of spreading is confirmed by our experimental observations. Copyright 2000 Academic Press.     

Viscosity of Milk: Influence of Cluster Formation

Recently, a new theory of viscosity of concentrated emulsions dependence on volume fraction of droplets (Starov, V. and Zhdanov, V., J. Colloid Interface Sci., 2003, vol. 258, p. 404) has been proposed that relates the viscosity of concentrated emulsions to the formation of clusters. Through experiments with milk at different fat concentrations, cluster formation has been validated using optical microscopy and their properties determined using the aforementioned theory. Viscometric studies have shown that, within the studied range of shear rates, both the packing density of fat droplets inside clusters and the relative viscosity of milk (viscosity over skim milk viscosity) are independent of shear rate but vary with volume fraction. Comparison of the experimental data with previous theories that assumed that the particles remained discrete shows wide variation. We attribute the discrepancy to cluster formation.     

Dual-frequency electrowetting: application to drop evaporation gauging within a digital microsystem

This paper addresses a method to estimate the size of a sessile drop and to measure its evaporation kinetics by making use of both Michelson interferometry and coplanar electrowetting. From a high-frequency electrowetting voltage, the contact angle of the sessile droplet is monitored to permanently obtain a half-liquid sphere, thus complying perfectly with the drop evaporation theory based on a constant contact angle (Bexon, R.; Picknett, R. J. Colloid Interface Sci. 1977, 61, 336-350). Low-frequency modulation of the electrowetting actuation is also applied to cause droplet shape oscillations and capillary resonance. Interferometry allows us to measure a time-dependent capillary spectrum and, in particular, the shift in natural frequencies induced by drop evaporation. Consequently, diffusive kinetics of drop evaporation can be properly estimated, as demonstrated. Because of coplanar electrode configuration, our methodology can be integrated in open and covered microsystems, such as digital lab-on-a-chip devices.     

Can we predict the spreading of a two-liquid system from the spreading of the corresponding liquid-air systems?

We present new data obtained from the spreading of a series of oil droplets, on top of a hydrophobic grafted silicon substrate, in air and immersed in water. We follow the contact angle and radius dynamics of hexane, dodecane, hexadecane, dibutyl phthalate, and squalane from the first milliseconds to approximately 1 s. Analysis of the images allows us to make several hundred contact angle and droplet radius measurements with great accuracy. The G-Dyna (Seveno et al. Langmuir 2010, 25, 13034) software is then used to fit the data with one of the wetting theories, the molecular-kinetic theory (MKT) (Blake et al. J. Colloid Interface Sci.1969, 30, 421), which takes into account the dissipation at the three-phase zone at the contact line. This theory allows us to extract the coefficient of friction of the contact line, which expresses the relationship between the driving force, that is, the unbalanced Young force, and the contact-line velocity V. It is first shown that the MKT is appropriate to describe the experimental data and then that the contact-line friction is a linear function of the viscosity as theoretically predicted. This is checked for oil-air and oil-water systems. A linear relation between the contact-line friction measured in oil-water systems and the contact-line frictions of the parent single liquid system seems plausible. To the best of our knowledge, this is the first trial to establish a link between the dynamics of wetting in liquid-liquid and in liquid-air systems.     

The Kinetics of Dewetting of Hydrophobic Surfaces during the Evaporation of Surfactant Solution Drops

The kinetics of dewetting (a decrease in contact angles and wetted surface area) during the evaporation of drops of cetyltrimethylammonium bromide (CTAB) solutions from paraffin and Teflon surfaces was studied in a wide concentration range. Three different stages of this process were found: (1) a monotonic decrease in the contact angle at a fixed position of the three-phase contact line, (2) contraction of the wetted surface area (the drop base) at a constant contact angle, and (3) simultaneous contraction of the drop base and a decrease in the contact angle. The CTAB distribution over a solid surface after the drop evaporation was studied by autoradiography. Depending on the surfactant concentration and the nature of a hydrophobic substrate, dewetting occurs by two mechanisms: slipping and carpet rolling.     

Molecular dynamics study of the influence of surfactant structure on surfactant-facilitated spreading of droplets on solid surfaces

The spreading of a partially wetting aqueous drop in air on a hydrophobic surface can be facilitated by the adsorption of surfactants from the drop phase onto the air/aqueous and aqueous/hydrophobic solid interfaces of the drop. At the contact line at which these interfaces meet, conventional surfactants with a linear alkyl hydrophobic chain attached to a polar group adsorb onto the surfaces, forming monolayers which remain distinct as they merge at the contact juncture. The adsorption causes a decrease in the interfacial tensions and reduction in the contact angle but the angle remains above zero so the drop is still nonwetting. Trisiloxane surfactants with a T-shaped geometry in which the hydrophobic group is composed of a trisiloxane oligomer with a polar group attached at the center of the chain can give rise to a zero contact angle at the contact line and complete wetting (superspreading). Experimental evidence suggests the adsorption of the T-shaped molecule, in addition to significantly decreasing the tensions of the interfaces (relative to the conventional surfactants), promotes the formation of a precursor film consisting of a surfactant bilayer at the contact line which facilitates the spreading. The aim of this study is to use molecular dynamics to examine if the T-shaped structure can promote spreading by the formation of a bilayer and to contrast this case with that of the linear chain surfactant where complex assembly does not occur. The simulation models the solvent as a monatomic liquid, the substrate as a particle lattice, and the surfactants as united atom structures, with all interactions given by Lennard-Jones potentials. We start with a base case in which the solvent partially wets a substrate comprised of a lattice of particles. We demonstrate that adsorbed T-shaped surfactant monolayers can, when the interaction between the solvent and the hydrophile particles is strong enough, assemble into a bilayer, allowing the drop to extend to a thin planar film. In the case of the flexible linear chain surfactant, there is no interaction between the monolayers on the two interfaces in the case of a strong hydrophile-solvent interaction and less coordination for a weaker interaction. In either case, the monolayers remain distinct, as the surfactant only marginally improves wetting.     

Simultaneous spreading and evaporation: Recent developments

The recent progress in theoretical and experimental studies of simultaneous spreading and evaporation of liquid droplets on solid substrates is discussed for pure liquids including nanodroplets, nanosuspensions of inorganic particles (nanofluids) and surfactant solutions. Evaporation of both complete wetting and partial wetting liquids into a nonsaturated vapour atmosphere are considered. However, the main attention is paid to the case of partial wetting when the hysteresis of static contact angle takes place. In the case of complete wetting the spreading/evaporation process proceeds in two stages. A theory was suggested for this case and a good agreement with available experimental data was achieved. In the case of partial wetting the spreading/evaporation of a sessile droplet of pure liquid goes through four subsequent stages: (i) the initial stage, spreading, is relatively short (1–2 min) and therefore evaporation can be neglected during this stage; during the initial stage the contact angle reaches the value of advancing contact angle and the radius of the droplet base reaches its maximum value, (ii) the first stage of evaporation is characterised by the constant value of the radius of the droplet base; the value of the contact angle during the first stage decreases from static advancing to static receding contact angle; (iii) during the second stage of evaporation the contact angle remains constant and equal to its receding value, while the radius of the droplet base decreases; and (iv) at the third stage of evaporation both the contact angle and the radius of the droplet base decrease until the drop completely disappears. It has been shown theoretically and confirmed experimentally that during the first and second stages of evaporation the volume of droplet to power 2/3 decreases linearly with time. The universal dependence of the contact angle during the first stage and of the radius of the droplet base during the second stage on the reduced time has been derived theoretically and confirmed experimentally. The theory developed for pure liquids is applicable also to nanofluids, where a good agreement with the available experimental data has been found. However, in the case of evaporation of surfactant solutions the process deviates from the theoretical predictions for pure liquids at concentration below critical wetting concentration and is in agreement with the theoretical predictions at concentrations above it.     

Liquid–paper interactions during liquid drop impact and recoil on paper surfaces

The spreading and recoiling of water drops on several flat and macroscopically smooth model surfaces and on sized paper surfaces were studied over a range of drop impaction velocities using a high-speed CCD camera. The water drop spreading and recoiling results on several model hydrophobic and hydrophilic surfaces were found to be in agreement with observations reported in the literature. The maximum drop spreading diameter for those model surfaces at impact was found to be dependent upon the initial drop kinetic energy and the degree of hydrophobicity/hydrophilicity of the surface. The extent of the maximum drop recoiling was found to be much weaker for hydrophilic substrates than for hydrophobic substrates. Sized papers, however, showed an interesting switch of behaviour in the process of water drop impaction. They behave like a hydrophobic substrate when a water drop impacts on it, but like a hydrophilic substrate when water drop recoils. Although the contact angle between water and hydrophilic or hydrophobic non-porous surfaces changes from advancing to receding as reported in literature, the change of contact angle during water impact on paper surface is unique in that the level of sizing was found to have a smaller than expected influence on the degree of recoil. Atomic force microscopy (AFM) was used to probe fibres on a sized filter paper surface under water. The AFM data showed that water interacted strongly with the fibre even though the paper was heavily sized. Implications of this phenomenon were discussed in the context of inkjet print quality and of the surface conditions of sized papers. Results of this study are very useful in the understanding of inkjet ink droplet impaction on paper surfaces which sets the initial condition for ink penetration into paper after impaction.     

Shape analysis based critical Eotvos numbers for buoyancy induced partial detachment of oil drops from hydrophilic surfaces

Removal of oil drops from solid surfaces immersed in an aqueous medium is of interest in many applications. It has been shown that drop shape analysis can be used to predict conditions at which the stability limit of a lighter than water oil drop on a solid surface immersed in an aqueous bath is reached (Adv. Colloid Interface Sci. 98 (2002) 265). However the above analysis is restricted to cases where the contact angle made by the drop is below 90degrees and when the surface conditions result in a 'pinned' contact line. In this paper, it is shown that drop shape analysis can be used to predict the critical conditions at which drop stability limit is reached for drop contact angles of 90degrees and above, which is encountered with 'hydrophilic' surfaces. This critical condition can predict the occurrence of partial oil drop detachment, before complete removal due to 'roll-up', which occurs when the hydrophilic surface is adequately smooth which prevents 'pinning' of the contact line. The critical conditions at which partial drop detachment occurs can also be approximately predicted from simple force balances. It has been shown (Adv. Colloid Interface Sci. 98 (2002) 265) that for contact angles less than 90degrees, the critical limit based on shape analysis appears to resolve the differences that arise due to alternate expressions for capillary retention force. This paper shows that even for contact angles above 90degrees, the critical conditions predicted from the shape analysis resolves the differences in the predictions from the alternate force balances. Drop shape analysis used in this paper is based on the 'Arc-length' form of Young-Laplace or 'drop shape' equation, which is different from the 'Y vs X' form of the above equation that is used in Adv. Colloid Interface Sci. 98 (2002) 265. The above drop shape equation is solved by a fourth order Runge-Kutta technique and it is shown that for angles less than 90degrees, the two forms of the drop shape equation, predict almost identical values of the critical Eotvos number. This paper highlights the competing effects of interfacial tension lowering induced drop instability and 'roll-up', a term that is used to describe the retraction of the contact line of an oil drop on a surface, in being the primary c ause for drop detachment.     

Use of a mesoporous material for organic synthesis

A common problem in synthetic organic chemistry is attaining proper contact between lipophilic organic compounds and inorganic salts. Various strategies, for example, phase transfer catalysis (Starks, C. M.; Liotta, C. L.; Halpern, M. Phase Transfer Catalysis: Fundamentals, Applications and Industrial Perspectives; Chapman & Hall: New York, 1994) or use of a microheterogeneous medium such as a microemulsion (Hager, M.; Currie, F.; Holmberg, K. Organic Reactions in Microemulsions. In Colloid Chemistry II; Antonietti, M., Ed.; Topics in Current Chemistry 227; Springer-Verlag: Heidelberg, 2003; p 53) have been worked out to tackle the issue. Here, we report that mesoporous solid materials made from surfactant self-assembly can be used as medium for such reactions. The material is made from silica, and the pore size is large, relatively uniform, and can be controlled with a high degree of precision by the choice of surfactant that is being used as template (Palmqvist, A. E. C. Curr. Opin. Colloid Interface Sci. 2003, 8, 145). The pores are hydrophilic and are filled with an aqueous solution containing the inorganic salt. The porous material is dispersed in the lipophilic organic substrate, that is, 4-tert-butylbenzyl bromide, or in a hydrocarbon solution of this substrate. The reaction occurs at the hydrophilic/lipophilic interface, and, because the interface is large, the reaction is fast. A considerable advantage with this new reaction medium is that the workup procedure is extremely facile. After the reaction is completed, the solid is simply removed by filtering or centrifugation.     

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.     

纳米结构表面上冷凝液滴的生长模式及部分润湿液滴的形成机制

分析并计算了纳米结构表面上冷凝液滴按照不同途径长大的过程中液滴能量的增加速率, 并以能量增加最小为判据来确定液滴的生长途径. 结果表明, 纳米结构内形成的冷凝液斑在初期按接触角(CA)增加的模式生长时, 其能量增加速率远低于其它模式, 于是, 初始液斑先按增大接触角、并保持底面积不变的模式生长, 直至液滴达到前进角状态. 此后, 沿接触角增加的模式长大所导致的能量增加速率开始远高于其它生长模式, 于是液滴三相线开始移动, 底面积开始增加, 但接触角保持不变. 液滴所增加的底面积可以呈润湿或复合两种状态, 分别形成Wenzel 液滴及部分润湿液滴, 前者的表观接触角一般小于160°, 而后者则明显大于160°. 液滴的生长模式及其润湿状态均与纳米结构参数密切相关, 仅当纳米柱具有一定高度、且间距较小时, 冷凝液滴才能呈现部分润湿状态. 最后, 本模型对纳米结构表面上冷凝液滴润湿状态的计算结果与绝大部分实测结果相一致, 准确率达到91.9%, 明显高于已有公式的计算准确率.     

The validity of Cassie's law: a simple exercise using a simplified model

The contact angle of a macroscopic droplet on a heterogeneous but flat substrate is studied using the interface displacement model which can lead to the augmented Young-Laplace equation. Droplets under the condition of constant volume as well as constant vapor pressure are considered. By assuming a cylindrical liquid-vapor surface (meniscus) and minimizing the total free energy of the interface displacement model, we derive an equation which is similar but different from the well-known Cassie's law. Our modified Cassie's law is essentially the same as the formula obtained previously by Marmur [J. Colloid Interface Sci. 168 (1994) 40]. A few consequences from this modified Cassie's law are briefly described in the following sections of this paper. Several sets of recent experimental results seem to support the validity of our modified Cassie's law.     

On autophobing in surfactant-driven thin films

Recent experiments (Afsar-Siddiqui, A. B.; Luckham, P. F.; Matar, O. K. Langmuir 2004, 20, 7575-7582) on the spreading of aqueous droplets containing cationic surfactants over thin aqueous films supported by negatively charged substrates demonstrated trends in the spreading behavior with either increasing surfactant concentration or increasing film thickness. Although the substrate is initially hydrophilic and the droplet spreads, surfactant adsorption at the substrate renders it hydrophobic leading to droplet retraction. We generate a model here using lubrication theory that allows the effect of the surfactant on the wettability to be taken into account. Our numerical results show that due to basal adsorption of surfactant at the interface, the initially hydrophilic solid substrate is rendered hydrophobic. This then drives droplet retraction and dewetting, which is in agreement with the experimentally observed trends.     

On the role of energy barriers in determining contact angle hysteresis

The thermodynamic model of contact angles on rough, heterogeneous surfaces developed by Long et al. [J. Long, M.N. Hyder, R.Y.M. Huang and P. Chen, Adv. Colloid Interface Sci. 118 (2005) 173] was employed to study the role of energy barriers in determining contact angle hysteresis. Major energy barriers corresponding to metastable states and minor energy barriers corresponding to secondary metastable states were defined. Distributions of major and/or minor energy barriers as a function of apparent contact angle for various surfaces were obtained. The reproducibility of contact angle measurement, the effect of vibrational energy on contact angle hysteresis and the "stick-slip" phenomenon were discussed. Quantitative relations between contact angles and vibrational energy were obtained. It was found that receding contact angles are normally poorly reproducible for hydrophilic surfaces, but for extremely hydrophobic surfaces, advancing contact angles may have a poor reproducibility. When the vibrational energy available to a system increases, the measured advancing contact angle will decrease while the receding angle will increase until both reach a common value: the system equilibrium angle. This finding not only agrees well with the experimental observations in system equilibrium contact angle measurements, but also lays a theoretical foundation for such measurements. A small vibrational energy may result in a "stick-slip" phenomenon.     

A numerical study of the effect of insoluble surfactants on the stability of a viscous drop translating in a Hele-Shaw cell

A circular drop is a linearly stable solution for the buoyancy-driven motion of drops in a Hele-Shaw cell [Gupta et al. J. Colloid Interface Sci.218(1), 338 (1999)]. In the absence of surface-active agents, an initially prolate drop always goes to a steady circular shape while initially oblate drops exhibit complex dynamics [Gupta et al. J. Colloid Interface Sci.222, 107 (2000)]. In this study, the effect of insoluble surfactant impurities on the critical conditions for drop breakup is explored by using the Langmuir adsorption framework in conjunction with a physically based expression for the depth-averaged tangential stress exerted on a two-phase interface in a Hele-Shaw cell. It is shown that the presence of surfactants can have both a stabilizing and a destabilizing effect on the shape of the drop, depending on the Bond number, the magnitude of the initial perturbation, and the strength of surface convection. Similar to the clean drop dynamics, two marginally stable branches are found. Increasing the surface Peclet number results in the stabilization of the main branch while the secondary branch shifts to higher Bond numbers. The mode of breakup is also found to be strongly influenced by the strength of surface convection.     

Surfactant-assisted spreading of a liquid drop on a smooth solid surface

Axisymmetric spreading of a liquid drop covered with an insoluble surfactant monolayer on a smooth solid substrate is numerically investigated. As the drop spreads, the adsorbed surfactant molecules are constantly redistributed along the air-liquid interface by convection and diffusion, leading to nonuniformities in surface tension along the interface. The resulting Marangoni stresses affect the spreading rate by altering the surface flow and the drop shape. In addition, surfactant accumulation in the vicinity of the moving contact line affects the spreading rate by altering the balance of line forces. Two different models for the constitutive relation at the moving contact line are used, in conjunction with a surface equation of state based on the Frumkin adsorption framework, to probe the surfactant influence. The coupled evolution equations for the drop shape and monolayer concentration profile are integrated using a pseudospectral method to determine the rate of surfactant-assisted spreading over a wide range of the dimensionless parameters governing the spreading process. The insoluble monolayer enhances spreading through two mechanisms; a reduction in the equilibrium contact angle, and an increase in the magnitude of the radial pressure gradient within the drop due to the formation of positive surface curvature near the moving contact line. Both mechanisms are driven by the accumulation of surfactant at the contact line due to surface convection. Although the Marangoni stresses induced at the air-liquid interface reduce the rate of spreading during the initial stages of spreading, their retarding effect is overwhelmed by the favorable effects of the aforementioned mechanisms to lead to an overall enhancement in the rate of spreading in most cases. The spreading rate is found to be higher for bulkier surfactants with stronger repulsive interactions. With the exception of monolayers with strong cohesive interactions which tend to retard the spreading process, the overall effect of an insoluble monolayer is to increase the rate of drop spreading. Simulation results for small Bond numbers indicate the existence of a power-law region for the time-dependence of the basal radius of the drop, consistent with experimental measurements.