Capillary rise of a non-Newtonian power law liquid: impact of the fluid rheology and dynamic contact angle

The impact of non-Newtonian behavior and the dynamic contact angle on the rise dynamics of a power law liquid in a vertical capillary is studied theoretically and experimentally for quasi-steady-state flow. An analytical solution for the time evolution of the meniscus height is obtained in terms of a Gaussian hypergeometric function, which in the case of a Newtonian liquid reduces to the Lucas-Washburn equation modified by the dynamic contact angle correction. The validity of the solution is checked against experimental data on the rise dynamics of a shear-thinning cmc solution in a glass microcapillary, and excellent agreement is found.     

Spreading dynamics and dynamic contact angle of non-Newtonian fluids

The spreading dynamics of power-law fluids, both shear-thinning and shear-thickening fluids, that completely or partially wet solid substrate was investigated theoretically and experimentally. An evolution equation for liquid-film thickness was derived using a lubrication approximation, from which the dynamic contact angle versus the contact line moving velocity relationship was evaluated. In the capillary spreading regime, film thickness h is proportional to xi3/(n+2) (xi is the distance from the contact line), whereas in the gravitational regime, h is proportional to xi1/(n+2), relating to the rheological power exponent n. The derived model fit the experimental data well for a shear-thinning fluid (0.2% w/w xanthan solution) or a shear-thickening fluid (7.5% w/w 10 nm silica in polypropylene glycol) on a completely wetted substrate. The derived model was extended using Hoffmann's proposal for partially wetting fluids. Good agreement was also attained between model predictions and the shear-thinning fluid (1% w/w cmc solution) and shear-thickening fluid (10% w/w 15 nm silica) on partially wetted surfaces.     

Deformation of advancing gas-liquid interfaces in capillary tubes

Deformation of an advancing gas-liquid meniscus is considered in two cases: prewetted and dry capillary tubes. The shape, slope, and curvature of the gas-liquid interface are determined assuming small Weber and Bond numbers, i.e., in the case of negligible inertia and gravity terms. For the prewetted capillary case, the dynamic contact angle rate-dependency is found to depend on both the capillary number and the ratio of the macroscopic prewetting film thickness to the capillary radius. Results are found intermediate between rate-dependency relations available in the literature. In the case of dry capillaries, the relative magnitudes of the viscous, capillary, and disjoining pressure effects are determined. The actual location of the three-phase contact line is analyzed in relation to the spreading coefficient. Results for the dynamic contact angle rate-dependency are found to agree well with published experimental data. In both cases, prewetted and dry capillaries, results are compared with Tanner's relationship and previous theoretical investigations.     

The exponential power law: partial wetting kinetics and dynamic contact angles

An equation for the kinetics of partial drop spreading is proposed. This equation was empirically derived from experimental data for the spreading kinetics of partially wetting liquids in terms of the wet area versus time. The equation has the form of an exponential power law (EPL), and transforms into the well-known power law for complete wetting, when the equilibrium contact angle approaches zero. The EPL fits very well available experimental data. To lend additional support to the validity of this generalized equation, it will be demonstrated that when it is transformed to present the dynamic contact angle (DCA), it fits very well DCA experimental data for other wetting processes, such as capillary flow and tape coating.     

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.     

Contact angle hysteresis of microbead suspensions

Microbead suspensions are often used in microfluidic devices for transporting biomolecules. An experimental investigation on the wettability of microbead suspension is presented in this study. The variation in the surface tension and the equilibrium contact angle with the change in the volume fraction of the microbead is presented here. The surface tension of the microbead suspension is measured with the pendant drop technique, whereas the dynamic contact angle measurements, i.e., advancing and receding contact angles, are measured with the sessile drop technique. An equilibrium contact angle of a suspension with particular volume fraction is determined by computing an average over the measured advancing and receding contact angles. It is observed that the surface tension and the equilibrium contact angle determined from advancing and receding contact angles vary with the magnitude of the microbeads volume fraction in the suspension. A decrease in the surface tension with an increase in the volume fraction of the microbead suspension is observed. The advancement and the recession in contact line for dynamic contact angle measurements are achieved with the motorized dosing mechanism. For microbead suspensions, the advancement of the contact line is faster as compared to the recession of the contact line for the same flow rate. The presence of microbeads assists in the advancement and the recession of the contact line of the suspension. A decrease in the equilibrium contact angles with an increase in the microbead suspension volume fraction is observed. Inclusion of microbeads in the suspension increases the wetting capability for the considered combination of the microbead suspension and substrate. Finally, empirical correlations for the surface tension and the contact angle of the suspension as a function of microbead volume fraction are proposed. Such correlations can readily be used to develop mechanistic models for the capillary transport of microbead suspensions related to LOC applications.     

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.     

Replacing the solid needle by a liquid one when measuring static and advancing contact angles

In general, the optical determination of static and advancing contact angle is made on drops applied or extended, respectively, onto a substrate through the use of thin solid needles. Although this method is used extensively, this way of dosing can be time consuming, cumbersome and if not performed meticulously can lead to erroneous results. Herein, we present an alternative way of applying drops onto substrates using a liquid jet produced by a liquid pressure dosing system acting as a “liquid needle”. We performed a comparative static contact angle study on 14 different surfaces with two different liquids (water and diiodomethane) utilizing two different ways of dosing: the conventional solid and a novel liquid needle-based technique. We found, for all but one sample, that the obtained results on μl size drops were comparable within the experimental error bars provided the liquid needle is thin enough. Observed differences are explained by the special characteristics of either way of dosing. In addition, we demonstrate how the liquid pressure-based dosing system facilitates high-speed optical advancing contact angle measurement by expanding a drop from 0.1 to 22 μl within less than 1.2 s but yet providing constant contact angle versus drop base diameter curves. The obtained results were compared with data from tensiometric dynamic Wilhelmy contact angle measurements. These data, in conjunction with sequences of live images of the dosing process of the liquid pressure dosing system, illustrate how this system can replace the solid needle by a liquid needle.     

Interfacial phenomena and dynamic contact angle modulation in microcapillary flows subjected to electroosmotic actuation

The dynamic evolution of an incompressible liquid meniscus inside a microcapillary is investigated, under the combined influences of viscous, capillary, intermolecular, pondermotive, and electroosmotic effects. In the limit of small capillary numbers, an advancing meniscus shape is shown to merge smoothly with the precursor film, using matched asymptotic analysis. A scaling relationship is also established for the dynamic contact angle as a nondimensional function of the capillary number and the applied electrical voltage. The analysis is further generalized by invoking a kinetic slip model for overcoming the constraints of meniscus tip singularity. The kinetic slip model is subsequently utilized to analyze the interfacial dynamics from the perspective of the results obtained from the matched asymptotic analysis. A generalization is achieved in this regard, which may provide a sound basis for controlling the topographical features of a dynamically evolving meniscus in a microcapillary subjected to electrokinetic effects. These results are also in excellent agreement with the experimental findings over a wide range of capillary number values.     

Viscoelastic properties of insoluble amphiphiles at the air/water interface

The effects of the presence of a molecular monolayer on the dilatational properties of the air/water interface have been investigated. Two water insoluble amphiphiles, dipalmitoyl phosphatidyl choline and quercetin 3-O-palmitate, were spread onto a pendant drop and the dynamic surface pressure was measured by means of drop shape analysis. The surface dilatational elasticity and viscosity of the spread monolayers were also determined by the oscillating drop technique. Constraints on the range of measuring conditions were investigated and we demonstrated that the pressure-area isotherms derived from oscillatory dynamic measurements display phase behaviour similar to that found in equilibrium measurements, albeit at reduced resolution. Both the amphiphiles formed purely elastic films that were characterised by a dilatational modulus that depended on the surface concentration and obeyed a power scaling law. The exponent of the relationship could be related to the thermodynamic conditions prevailing at the interface. The phospholipid monolayer scaling exponent was 2.8 in a temperature range of 20-26 degrees C indicates a favourable solvency of molecules in the bidimensional matrix. A very high scaling exponent (11.8 at 7 degrees C) for quercetin palmitate was interpreted assuming that molecules self-organise in fibre-like structures. This interface structure and the phase behaviour was found consistent with observations of the surface film obtained by Brewster angle microscopy. The structured quercetin 3-O-palmitate monolayers are disrupted by temperature increase or by adding a 0.2 molar fraction of the immiscible dipalmitoyl phosphatidyl choline.     

Static contact angle hysteresis on smooth, homogeneous solid substrates

A theory of contact angle hysteresis on smooth, homogeneous solid substrates is developed in terms of shape of disjoining/conjoining pressure isotherm and quasi-equilibrium phenomena. It is shown that all contact angles, θ, in the range θ _r?<?θ?<?θ _a, which are different from the unique equilibrium contact angle θ?≠?θ _e, correspond to the state of slow “microscopic” advancing or receding motion of the liquid if θ _e ?<?θ?<?θ _a or θ _r?<?θ?<?θ _e, respectively. This “microscopic” motion almost abruptly becomes fast “macroscopic” advancing or receding motion after the contact angle reaches the critical values θ?=?θ _a or θ _r?=?θ, correspondingly. The values of the static receding, θ _r, and static advancing, θ _a, contact angles in cylindrical capillaries were calculated earlier, based on the shape of disjoining pressure isotherm. It is shown that an advancing contact angle of a droplet on a solid substrate depends on the drop volume and is not a unique characteristic of the liquid–solid system. The suggested mechanism of contact angle hysteresis has direct experimental confirmation.     

An analytical solution for determination of small contact angles from sessile drops of arbitrary size

An analytical solution to the capillary equation of Young and Laplace is derived that allows determination of the static contact angle based on the volume of a sessile drop and the wetted area of the substrate. This solution does not require numerical integration to determine the drop profile and accounts for surface deformation due to gravitational effects. Calculation of the static contact angle by this method is remarkably simple and accurate when the contact angle is less than 30 degrees. A natural scaling arises in the solution, which provides indication of when a drop is small enough so as to neglect gravitational influences on the surface shape which, for small contact angles, is generally less than 1 microl. The technique described has the simplicity of the spherical cap approximation but remains accurate for any size of sessile drop.     

As-placed contact angles for sessile drops

As-placed contact angle is the contact angle a drop adapts as a result of its placement on a surface. As expected, the as-placed contact angle, thetaAP, of a sessile drop on a horizontal surface decreases with the drop size due to the increase in hydrostatic pressure. We present a theoretical prediction for thetaAP which shows that it is a unique function of the advancing contact angle, thetaA, drop size, and material properties (surface tensions and densities). We test our prediction with published and new data. The theory agrees with the experiments. From the relation of the as-placed contact angle to drop size the thermodynamic equilibrium contact angle is also calculated.     

Spreading of completely wetting or partially wetting power-law fluid on solid surface

This study investigated the drop-spreading dynamics of pseudo-plastic and dilatant fluids. Experimental results indicated that the spreading law for both fluids is related to rheological characteristics or power exponent n. For the completely wetting system, the evolution of the wetting radius over time can be expressed by the power law R = atm, where the spreading exponent m of the dilatant fluids is >0.1 and the spreading exponent m of pseudo-plastic fluids is <0.1. The strength of non-Newtonian effects is positively correlated to the extent of deviation from the theoretical value 0.1 of m for Newtonian fluids. For the partially wetting system, the power law on the time dependence of the wetting radius no longer holds; therefore, an exponential power law, R = Req(1-exp(-at(m)/Req)), is proposed, where Req denotes the equilibrium radius of drop and a is a coefficient. Comparing experimental data with the exponential power law revealed that both are in good agreement.     

Contact angle hysteresis of cylindrical drops on chemically heterogeneous striped surfaces

Contact angle hysteresis of a macroscopic droplet on a heterogeneous but flat substrate is studied using the interface displacement model. First, the apparent contact angle of a droplet on a heterogeneous surface under the condition of constant volume is considered. By assuming a cylindrical liquid-vapor surface (meniscus) and minimizing the total free energy, we derive an equation for the apparent contact angle, which is similar but different from the well-known Cassie's law. Next, using this modified Cassie's law as a guide to predict the behavior of a droplet on a heterogeneous striped surface, we examine several scenarios of contact angle hysteresis using a periodically striped surface model. By changing the volume of the droplet, we predict a sudden jump of the droplet edge, and a continuous change of the apparent contact angle at the edge of two stripes. Our results suggest that as drop volume is increased (advancing contact lines), the predominant drop configuration observed is the one whose contact angle is large; whereas, decreasing drop volume from a large value (receding contact lines) yields drop configuration that predominantly exhibit the smaller contact angle.     

Theory of slope-dependent disjoining pressure with application to Lennard-Jones liquid films

A liquid film of thickness h<100 nm is subject to additional intermolecular forces, which are collectively called disjoining pressure Pi. Since Pi dominates at small film thicknesses, it determines the stability and wettability of thin films. Current theory derived for uniform films gives Pi=Pi(h). This solution has been applied recently to non-uniform films and becomes unbounded near a contact line as h-->0. Consequently, many different effects have been considered to eliminate or circumvent this singularity. We present a mean-field theory of Pi that depends on the slope h(x) as well as the height h of the film. When this theory is implemented for Lennard-Jones liquid films, the new Pi=Pi(h,h(x)) is bounded near a contact line as h-->0. Thus, the singularity in Pi(h) is artificial because it results from extending a theory beyond its range of validity. We also show that the new Pi can capture all three regimes of drop behavior (complete wetting, partial wetting, and pseudo-partial wetting) without altering the signs of the long and short-range interactions. We find that a drop with a precursor film is linearly stable.     

Sliding behavior of liquid droplets on tilted Langmuir-Blodgett surfaces

The sliding behavior of liquid droplets on inclined Langmuir-Blodgett surfaces was investigated. The critical sliding angle defined as the tilt angle of the surface at which the drop slides down as well as the advancing and receding contact angles was measured for five different liquids on five surfaces. In addition, the contact line geometry was analyzed at critical sliding angle. The experimental relationship between the surface tension forces resulting from contact angle hysteresis and the weight of the drop was compared to theoretical predictions. Even though the shape of the drop bases was found as skewed ellipses, a model assuming parallel-sided elongated drops is shown to describe reasonably the experimental values. This result probably indicates the main influence of the capillary forces at the rear and front edges of the drop with respect to that exerted on the lateral sides.     

Drop manipulation and surgery using electric fields

We study the dynamics of a slender drop sandwiched between two electrodes using lubrication theory. A coupled system of evolution equations for the film thickness and interfacial charge density is derived and simplified for the case of a highly conducting fluid. The contact line singularity is relieved by postulating the existence of a wetting precursor film, which is stabilised by intermolecular forces. We examine the motion of the drop as a function of system parameters: the electrode separation, beta, an electric capillary number, C, and a spatio-temporally varying bottom electrode potential. The possibility of drop manipulation and surgery, which include drop spreading, translation, splitting and recombination, is demonstrated using appropriate tuning of the properties of the bottom potential; these results could have potential implications for drop manipulation schemes in various microfluidic applications. For relatively small beta and/or large C values, the drop assumes cone-like structures as it approaches the top electrode; the latter stages of this approach are found to be self-similar and a power-law exponent has been extracted for this case.     

Electrowetting-induced capillary flow in a parallel-plate channel

This paper investigated, theoretically and experimentally, the electrowetting-induced capillary rise in a parallel-plate channel. The measured equilibrium height of the meniscus was proportional to the square of the applied potential. A model, based on the kinetic equation of capillary flow with the consideration of an electrowetting dynamic contact angle, was established to simulate the capillary rise. The effects of the electrostatic charge and the contact-line friction were linearly added to describe the electrowetting dynamic contact angle. The model was found to be able to adequately describe the experimental data under different initial heights and applied voltages. The non-Poseuille flow effect had little influence in the meniscus rising phenomenon studied in this work.     

Nanodrop on a nanorough solid surface: density functional theory considerations

The density distributions and contact angles of liquid nanodrops on nanorough solid surfaces are determined on the basis of a nonlocal density functional theory. Two kinds of roughness, chemical and physical, are examined. The former considers the substrate as a sequence of two kinds of semi-infinite vertical plates of equal thicknesses but of different natures with different strengths for the liquid-solid interactions. The physical roughness involves an ordered set of pillars on a flat homogeneous surface. Both hydrophobic and hydrophilic surfaces were considered. For the chemical roughness, the contact angle which the drop makes with the flat surface increases when the strength of the liquid-solid interaction for one kind of plates decreases with respect to the fixed value of the other kind of plates. Such a behavior is in agreement with the Cassie-Baxter expression derived from macroscopic considerations. For the physical roughness on a hydrophobic surface, the contact angle which a drop makes with the plane containing the tops of the pillars increases with increasing roughness. Such a behavior is consistent with the Wenzel formula developed for macroscopic drops. For hydrophilic surfaces, as the roughness increases the contact angle first increases, in contradiction with the Wenzel formula, which predicts for hydrophilic surfaces a decrease of the contact angle with increasing roughness. However, a further increase in roughness changes nonmonotonously the contact angle, and at some roughness, the drop disappears and only a liquid film is present on the surface. It was also found that the contact angle has a periodic dependence on the volume of the drop.