Numerical simulations of capillary-driven flows in nonuniform cross-sectional capillaries

In this study the wetting behavior of converging-diverging and diverging-converging capillaries is investigated numerically using an in-house written, finite-element code. An interface tracking procedure based on the predicted change in the total liquid volume, to update the interface location, and Cox's formulation, to determine the dynamic contact angle and the interface shape, is proposed and used. Flow simulations revealed that both converging-diverging and diverging-converging capillaries exhibit significantly slower wetting behavior than straight capillaries and that any deviation in the capillary diameter necessarily tends to slow the overall wetting speed. This behavior was attributed to local regions of very low capillary pressure and high viscous retardation force when the capillary diameter at the interface was significantly larger than the capillary diameter over the upstream fluid. Though the local wetting velocities were different, when equivalent capillaries were compared it was found that both converging-diverging and diverging-converging capillaries had the same total fill time independent of the number of irregular regions, suggesting that the simple model is sufficient for predicting the overall effect. The influence of surface tension and contact angle on the total wetting time was found to be similar for both straight and irregularly shaped capillaries.     

Analytical approach for the Lucas-Washburn equation

Porous media can be characterized by studying the kinetics of liquid rise within the pore spaces. Although porous media generally have a complex structure, they can be modeled as a single, vertical capillary or as an assembly of such capillaries. The main difficulties lie in separately estimating the effective mean radius of the capillaries and the contact angle between the liquid and the pore. In this paper we circumvent these obstacles by exploring another approach and suggest an analytical approach of the classical Lucas-Washburn equation (LWE). Specifically, we consider that the contact angle between the liquid meniscus and the inner surface of the capillary becomes a dynamic contact angle when the liquid front is in movement. It has previously been demonstrated that the resulting time dependence is due to frictional dissipation at the moving wetting front.     

Thickness, stability and contact angle of liquid films on and inside nanofibres, nanotubes and nanochannels

While the stability of liquid films on substrates is a classical topic of colloidal science, the availability of nanostructured materials, such as nanotubes, nanofibres and nanochannels, has raised the question of how the stability of liquid films and their wetting behaviour is affected by nanoscale confinement. This paper will present the conditions for the stability of liquid films on and inside cylindrical solid substrates with nanometre scale characteristic dimensions. It is shown that the stability is determined by an effective disjoining/conjoining pressure isotherm which differs from the corresponding disjoining/conjoining pressure isotherm of flat liquid films on flat solid substrates. From the former, the equilibrium contact angles of drops on an outer or inner surface of a cylindrical capillary have been calculated as a function of surface curvature, showing that the expressions for equilibrium contact angles vary for different geometries, in view of the difference in thickness of the film of uniform thickness with which the bulk liquid (drops or menisci) is at equilibrium. These calculations have been extended to the case of glass nanocapillaries and carbon nanotubes, finding good agreement with experimental results in the literature.     

Wetting and spreading of nanofluids on solid surfaces driven by the structural disjoining pressure: statics analysis and experiments

The wetting and spreading of nanofluids composed of liquid suspensions of nanoparticles have significant technological applications. Recent studies have revealed that, compared to the spreading of base liquids without nanoparticles, the spreading of wetting nanofluids on solid surfaces is enhanced by the structural disjoining pressure. Here, we present our experimental observations and the results of the statics analysis based on the augmented Laplace equation (which takes into account the contribution of the structural disjoining pressure) on the effects of the nanoparticle concentration, nanoparticle size, contact angle, and drop size (i.e., the capillary and hydrostatic pressure); we examined the effects on the displacement of the drop-meniscus profile and spontaneous spreading of a nanofluid as a film on a solid surface. Our analyses indicate that a suitable combination of the nanoparticle concentration, nanoparticle size, contact angle, and capillary pressure can result not only in the displacement of the three-phase contact line but also in the spontaneous spreading of the nanofluid as a film on a solid surface. We show here, for the first time, that the complete wetting and spontaneous spreading of the nanofluid as a film driven by the structural disjoining pressure gradient (arising due to the nanoparticle ordering in the confined wedge film) is possible by decreasing the nanoparticle size and the interfacial tension, even at a nonzero equilibrium contact angle. Experiments were conducted on the spreading of a nanofluid composed of 5, 10, 12.5, and 20 vol % silica suspensions of 20 nm (geometric diameter) particles. A drop of canola oil was placed underneath the glass surface surrounded by the nanofluid, and the spreading of the nanofluid was monitored using an advanced optical technique. The effect of an electrolyte, such as sodium chloride, on the nanofluid spreading phenomena was also explored. On the basis of the experimental results, we can conclude that a nanofluid with an effective particle size (including the electrical double layer) of about 40 nm, a low equilibrium contact angle (<3°), and a high effective volume concentration (>30 vol %) is desirable for the dynamic spreading of a nanofluid system with an interfacial tension of 0.5 mN/m. Our experimental observations also validate the major predications of our theoretical analysis.     

Effect of hydrophobicity on the stability of the wetting films of water formed on gold surfaces

We have developed a methodology that can be used to determine disjoining pressures (Π) in both stable and unstable wetting films from the spatial and temporal profiles of dynamic wetting films. The results show that wetting films drain initially by the capillary pressure created by the changes in curvature at the air/water interface and subsequently by the disjoining pressure created by surface forces. The drainage rate of the film formed on a gold surface with a receding contact angle (θ(r)) of 17° decreases with film thickness due to a corresponding increase in positive Π, resulting in the formation of a stable film. The wetting film formed on a hydrophobic gold with θ(r)=81° drains much faster due to the presence of negative Π in the film, resulting in film rupture. Analysis of the experimental data using the Frumkin-Derjaguin isotherm suggests that short-range hydrophobic forces are responsible for film rupture and long-range hydrophobic forces accelerate film thinning.     

Spreading of liquid drops over dry surfaces

Spreading of thin, axisymmetric, non-volatile, Newtonian liquid drops over a dry, smooth, flat solid surface is considered both theoretically and experimentally in the case of complete wetting. The drop profile is solved analytically by matching the “outer” solution for large film thicknesses, where only the capillary effects are important, with the “inner” solution for small film thicknesses, where the viscous and disjoining pressure effects are comparable to capillary effects. It is shown that the apparent radius of the wetted spot, the apex height of the drop, and the apparent advancing dynamic contact angle follow different power laws in time and the advancing dynamic contact angle follows a power law in capillary number. Both the prefactor and the exponent of each power law are derived theoretically. Good agreement between the theory predictions and experimental measurements is shown for both the prefactor and exponent of each power law. It is necessary to emphasize that the theory suggested does not include any fitting parameters.     

Isotherms of Capillary Condensation Influenced by Formation of Adsorption Films

Isotherms of capillary condensation are often used to determine the vapor sorption capacity of porous adsorbents as well as the pore size distribution by radii. In this paper, for calculating the volume of capillary condensate and of adsorption films in a porous body, an approach based on the theory of surface forces is used. Adsorption isotherms and disjoining pressure isotherms of wetting films are presented here in an exponential form discussed earlier. The calculations were made for straight cylindrical capillaries of different radii and slit pores of different width. The mechanisms of capillary condensation differ in cylindrical and slit pores. In cylindrical pores capillary condensation occurs due to capillary instability of curved wetting films on a capillary surface, when film thickness grows. In the case of slit pores, coalescence of wetting films formed on opposite slit surfaces proceeds under the action of attractive dispersion forces. Partial volumes of liquid in the state of both capillary condensate and adsorbed films are calculated dependent on the relative vapor pressure in a surrounding media. Copyright 2000 Academic Press.     

A Novel Method for Surface Free-Energy Determination of Powdered Solids

Interfacial solid/liquid interactions play a crucial role in wetting, spreading, and adhesion processes. In the case of a flat solid surface, contact angle measurements are commonly utilized for the determination of the solid surface free energy and its components. However, if such a surface cannot be obtained, then the contact angle can not be measured directly. Usually methods based on imbibition of probe liquids into a thin porous layer or column are applied. In this paper a novel method, also based on the capillary rise, is proposed for the solid surface free-energy components determination. Actually, it is a modification of the thin column wicking method; similar theoretical background can be applied together with that appropriate for the capillary rise method of liquid surface tension determination. The proposed theoretical approach and procedure are verified by using single glass capillaries, and then alumina and ground glass powders were used for the method testing. Thus obtained surface free-energy components for these solids, for both glass and alumina, agree well with the literature values.     

Nonflat equilibrium liquid shapes on flat surfaces

The hydrostatic pressure in thin liquid layers differs from the pressure in the ambient air. This difference is caused by the actions of surface forces and capillary pressure. The manifestation of the surface force action is the disjoining pressure, which has a very special S-shaped form in the case of partial wetting (aqueous thin films and thin films of aqueous electrolyte and surfactant solutions, both free films and films on solid substrates). In thin flat liquid films the disjoining pressure acts alone and determines their thickness. However, if the film surface is curved then both the disjoining and the capillary pressures act simultaneously. In the case of partial wetting their simultaneous action results in the existence of nonflat equilibrium liquid shapes. It is shown that in the case of S-shaped disjoining pressure isotherm microdrops, microdepressions, and equilibrium periodic films exist on flat solid substrates. Criteria are found for both the existence and the stability of these nonflat equilibrium liquid shapes. It is shown that a transition from thick films to thinner films can go via intermediate nonflat states, microdepressions and periodic films, which both can be more stable than flat films within some range of hydrostatic pressure. Experimental investigations of shapes of the predicted nonflat layers can open new possibilities of determination of disjoining pressure in the range of thickness in which flat films are unstable.     

Shape Factor Correlations of Hydraulic Conductance in Noncircular Capillaries

In Part I of this paper, we introduced the Mason-Morrow shape factor and the corner half-angles to capture the part of geometry of angular capillaries essential in pore network calculations of single- and two-phase flow in drainage and imbibition. We then used this shape factor to obtain simple expressions for the hydraulic conductance in single-phase flow through triangular, rectangular, and oval capillaries. In Part II, we study two-phase fluid flow along angular capillaries. The nonwetting fluid occupies the central part of the capillary, whereas the wetting liquid fills the corners. First, we verify the numerical solution obtained by Ransohoff-Radke for concave corner menisci by using a high-resolution finite element method with zero and infinite surface shear viscosity. We present new numerical results for corner flow domains bounded by convex menisci, i.e., for pinned contact lines and forced imbibition. We also present numerical solutions for two-phase flow with momentum transfer across the interface. We introduce a dimensionless hydraulic conductance of wetting fluid in the corners and correlate it with the corner filament shape factor, corner half-angle, and contact angle. By appropriate scaling, we obtain an accurate universal curve for flow conductance in the corners of an arbitrary angular capillary and for arbitrary contact angles. We give error estimates of the Ransohoff-Radke flow resistance factors, of the Zhou et al. analytical expressions for the resistance factors, and of our universal curves for the hydraulic conductance with no-slip and perfect-slip boundary conditions at the interface. Our expressions for the hydraulic conductance in corner flow of wetting liquid not only are valid for both concave and convex fluid interfaces but also are more accurate than any other published correlation. Copyright 2001 Academic Press.     

Wetting and surface forces

In this review we discuss the fundamental role of surface forces, with a particular emphasis on the effect of the disjoining pressure, in establishing the wetting regime in the three phase systems with both plane and curved geometry. The special attention is given to the conditions of the formation of wetting/adsorption liquid films on the surface of poorly wetted substrate and the possibility of their thermodynamic equilibrium with bulk liquid. The calculations of contact angles on the basis of the isotherms of disjoining pressure and the difference in wettability of flat and highly curved surfaces are discussed. Mechanisms of wetting hysteresis, related to the action of surface forces, are considered.     

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.     

Why surfaces modified by flexible polymers often have a finite contact angle for good solvents

Polymers adsorbing from a dilute solution onto the solvent-vapor interface generate a nonzero surface pressure. When the same polymers are end-grafted onto a surface such that a so-called polymer brush is formed, one will find that the solvent wets this compound interface partially. The partial wetting and the finite surface pressure are intimately linked properties of the polymer-solvent-vapor combination. It is shown that the spreading parameter in the wetting problem is proportional to the surface pressure in the adsorption case. Complete wetting is only possible when this surface pressure is nonpositive. The wetting characteristics are hardly influenced by the grafting density and chain length characterizing the brush. We argue that the grafted polymer chains can bridge to the solvent-vapor interface, thereby preventing the wetting film to become macroscopically thick. We present experimental data underpinning our self-consistent field analysis. Indeed, finite contact angles should be expected in various systems in which bridging attraction contributes to the disjoining pressure in wetting films.     

When tethered chains meet free ones; the stability of polymer wetting films on polymer brushes

We present a combined experimental and theoretical self-consistent field (SCF) investigation of the wetting behavior of a polystyrene melt (composed of chains with degree of polymerization P) on top of a polystyrene brush (composed of chains with length N) grafted onto a silica surface. The control variables are the grafting density σ of the brush chains and the length of mobile chains P. Experiments show in agreement with the theory that there is a window of complete wetting. Both at very low and at high grafting densities the system remains partial wet. At large degree of polymerization P, there is a difference between the experimental and theoretical results. Theory predicts partial wetting only, whereas the window of complete wetting persists in the experiments even when P >> N. This difference is attributed to the double-well structure of the disjoining pressure as revealed by the SCF theory. With this type of disjoining pressure it is conceivable that a metastable zero contact angle remains present for very long times.     

Geometry-driven wetting transition

Wetting states are quantitatively described by the number of inflection points on the liquid-vapor interface and by the macroscopic contact angle. The number of inflection points required for complete, partial, and pseudopartial wetting is determined for geometries with positive, zero, and negative capillary pressures. The wetting state of a material system is not always independent of the magnitude of the capillary pressure; for example, the wetting state of a fluid inside a capillary tube may depend on the capillary radius. In particular, a fluid that pseudopartially wets the inside of a tube exhibits a transition to partial wetting (or complete wetting) as the capillary radius is decreased.     

Influence of the work of adhesion on the dynamic wetting of chemically heterogeneous surfaces

The velocity dependence of the dynamic contact angle for a glycerol-water mixture wetting two different chemically heterogeneous surfaces (mixed thiols on gold and partially methylated titania, 16 samples in all) was studied. The molecular kinetic theory (MKT) of wetting was used to interpret the dynamic contact angle data. The equilibrium displacement frequency ( K 0) was predominantly determined by the viscous contribution from the bulk liquid, with a minor contribution from the surface. The mean distance between surface sites (lambda) decreased with increasing work of adhesion. The contact line friction coefficient zeta 0 was found to vary exponentially with the work of adhesion, enabling the unit flow volume of the liquid to be obtained.     

Truncated versus extended microfilms at a vapor-liquid contact line on a heated substrate

The microstructure of a contact line formed by a liquid and its pure vapor on a perfectly wetted superheated smooth substrate, with the disjoining pressure most often in the form of a positive inverse cubic law (nonpolar case), is routinely considered to end up in a microfilm extended over adjacent "dry" parts of the solid surface. Invoking the spreading coefficient as an additional independent parameter within this framework, we argue however that a regime with a truncated microfilm is chosen instead if the spreading coefficient is decreased below a positive (still perfect wetting) critical value dependent upon the superheat, in which case the extended-microfilm thickness is surpassed by that of the "pancake" introduced by de Gennes and co-workers. Conversely, for a given positive spreading coefficient, there is a critical superheat above which the microfilm gets truncated, whereas for a negative one (partial wetting) the truncated regime should be preferred at any superheat. A parametric study of the apparent contact angle (a nonlinear eigenvalue of the steady microstructure problem) versus the spreading coefficient is carried out. When the latter is negative, Young's law is asymptotically recovered. Microfilm fronts on a bare surface are shown to be advancing or receding in accordance with the selected regime. A slightly more general class of disjoining pressures is also touched upon. The analysis is based in part upon thermodynamic considerations and in part upon a standard one-sided model of an evaporating liquid layer in the lubrication approximation.     

Equilibrium contact angles of liquid droplets on ideal rough solids

This work proposes a theoretical model for predicting the apparent equilibrium contact angle of a liquid on an ideal rough surface that is homogeneous and has a negligible body force, line tension, or contact angle hysteresis between solid and liquid. The model is derived from the conservation equations and the free-energy minimization theory for the changes of state of liquid droplets. The work of adhesion is expressed as the contact angles in the wetting process of the liquid droplets. Equilibrium contact angles of liquid droplets for rough surfaces are expressed as functions of the area ratios for the solid, liquid, and surrounding gas and the roughness ratio and wetting ratio of the liquid on the solid for the partially and fully wet states. It is found that the ideal critical angle for accentuating the contact angles by the surface roughness is 48°. The present model is compared with existing experimental data and the classical Wenzel and Cassie-Baxter models and agrees with most of the experimental data for various surfaces and liquids better than does the Wenzel model and accounts for trends that the Wenzel model cannot explain.     

Surface Tension and Dynamic Contact Angle of Water in Thin Quartz Capillaries

The surface tension of water has been measured in quartz capillaries with radii from 200 down to 40 nm. It appears that the surface tension does not differ from the known (bulk) values in the temperature range from 8 to 70 degrees C, within 1% experimental error. The dynamic contact angle, theta(d), vanishes when the capillary surface is covered with a wetting film left behind the receding meniscus. In the case of a dry surface, theta(d) depends on the velocity of the meniscus motion. The results obtained do not agree with presently available theoretical predictions from hydrodynamic theories of dynamic contact angles. Rather the kinetics of water vapor adsorption ahead of the moving meniscus seems to be the major controlling agent of the dynamic contact angle. Copyright 2000 Academic Press.     

Kinetics of Gas Bubble Motion in a Capillary

Two flow regimes were discovered by measuring flow rate vof water through thin capillaries containing small gas bubble. The first regime is realized at a low pressure drop P, when the main resistance to flow is created by the wetting film. The estimate of its thickness makes it possible to determine the parameters of the isotherm of the electrostatic component of the disjoining pressure corresponding to the constant charge of the film–gas interface and potential of the quartz surface, which requires the application of a new calculation procedure. As Pincreases, an advancing meniscus forms contact angle of 50°–60° due to the film rupture when it reaches critical pressure in the thinnest part near the meniscus. In this case, the rate vis controlled by the flow in the filled part of a capillary, although some additional viscous drag is also created by the film moving in front of advancing meniscus. The longer the bubble, the greater this contribution.