Three-phase capillary entry conditions in pores of noncircular cross-section

In this work we challenge the assumption that the capillary entry pressures for displacements in three-phase flow are the same as those in two-phase flow. Using an energy balance, as derived by R.P. Mayer and R.A. Stowe (J. Colloid Interface Sci. 20 (1965) 893-911) and H.M. Princen (J. Colloid Interface Sci. 30 (1969) 69-75; 30 (1969) 359-371; 34 (1970) 171-184) for two-phase flow, we derive a general formula for determination of the capillary entry pressures for piston-like displacement of two bulk phases in a pore where a third phase may also be present. The method applies to capillaries of angular cross-section and uniform but arbitrary wettability. To use this method we have determined all possible underlying phase occupancies in cross-sections on either side of the main terminal meniscus, in particular the presence of corner arc menisci (AMs). Indeed, the capillary entry pressures for piston-like displacements depend on the pressure in the remaining third phase if the cross-sectional fluid configurations contain this phase. This dependence only vanishes when layers of the intermediate-wetting phase completely separate the wetting and the non-wetting phases. The complexity of the corresponding equations and the quantitative effects are studied using two different geometries, the equilateral triangle and the rhombus. The main difference is that the latter geometry has unequal corners, which may carry different AMs. We have carried out a limited sensitivity study with respect to the effect of wettability, the spreading coefficient of the intermediate-wetting phase, and the aspect ratio of the principal radii of the rhombus.     

Simulations of novel nanostructures formed by capillary effects in lithography

High aspect ratio three-dimensional nanostructures are of tremendous interest to a wide range of fields such as photonics, plasmonics, fluid mechanics, and biology. Recent developments in capillary force lithography (CFL) have focused on taking advantage of the formation of menisci to enhance the functionality of small size-scale structures. In this study, simulations of the three-dimensional shapes of equilibrium menisci formed in capillaries with various cross-section geometries are studied. The capillary cross sections include regular polygons and equilateral star-shapes with sharp and rounded corners. The characteristic dimension of the physical lithography systems which are simulated is on the order of 100nm. At such size-scale, surface-tension-effects are predominant, and as a consequence, our simulations demonstrate that nanometer-sized structures with great application potentials can be fabricated. Specifically, this study demonstrates that surfaces with three-dimensional nanoscale structures can be fabricated from templates with micron or sub-micron features through the development of cusps in the corners of the polygonal capillaries. Quantitatively, the effects of contact angle, corner angle, meniscus confinement, and corner rounding radius are examined and scaling analyses are presented to describe the dependencies of the height variation across the meniscus on these parameters. These simulations serve as useful guides for extending the development and implementation of capillary force lithography.     

Free energy balance for three fluid phases in a capillary of arbitrarily shaped cross-section: capillary entry pressures and layers of the intermediate-wetting phase

In this work we derive rigorously the free energy balance for three fluid phases in a straight capillary of arbitrarily shaped cross-section. This balance is then used to derive the general equation for the capillary entry pressures of all possible two-phase and three-phase displacements. Moreover, the equation provides the criterion determining the existence of layers of the intermediate-wetting phase separating the wetting and non-wetting phases in the corners or cavities of a capillary, by also treating the spreading of such layers as a capillary displacement. For a number of combinations of interfacial tensions and contact angles, illustrating all the different relevant situations, we calculate the criteria for spreading of such a layer in the corner of a capillary with polygonal cross-section. In a capillary with a cross-section in the shape of an isosceles triangle of varying corner size, these criteria are used to determine the unique capillary entry pressures for piston-like displacement from alternative solutions of the general equation. These solutions relate to displacements in the presence or absence of layers in the various differently sized corners.     

Some mechanisms of immiscible fluid displacement in small networks

We report experimental observations on immiscible displacement in two small networks using three different pairs of fluids, air-oil, air-water, and oil-water, to vary the wettability. The experiments were run for a wide range of capillary number, from 10⁻⁷ to 10⁻³. Various mechanisms are observed. These are film spreading and drainage, Haines' jump, free slip and stick-slip meniscus motion, contact angle hysteresis, snap-off, coalescence, and blocking of film and bubble. For the air-oil case, oil is perfectly wetting in the network. In imbibition, the displacement occurs first via thin film spreading, followed by snap-off of menisci, and then by piston-like displacement at low flow rates. As the flow rate increases, piston-like displacement dominates because film spreading is comparatively slow. Snap-off of menisci in the throats is a necessary condition for air trapping. In drainage, meniscus snap-off and coalescence are observed in one network. For both imbibition and drainage, during each snap-off or piston-like displacement event, all menisci move freely along the channels to adjust their curvatures, due to the lubrication of the wetting film. For the other two fluid pairs at low flow rates, this curvature readjustment through free slipping of meniscus is not observed, presumably due to the absence of wetting film during the displacement. At high flow rate, oscillation of menisci due to volumetric competition is observed. Neither wetting film spreading nor throat snap-off is observed. Stick and slip motion of meniscus is observed, probably due to the roughness and/or heterogeneous wettability of the solid surface. For the oil-water system the wettability seems to be time dependent. Coalescence between two menisci can occur in the throat, in the pore, or at the pore-throat boundary during displacement. Trapping of the displaced phase is due to its being bypassed or snapped off in the throat.     

Surface-directed boundary flow in microfluidic channels

Channel geometry combined with surface chemistry enables a stable liquid boundary flow to be attained along the surfaces of a 12 microm diameter hydrophilic glass fiber in a closed semi-elliptical channel. Surface free energies and triangular corners formed by PDMS/glass fiber or OTS/glass fiber surfaces are shown to be responsible for the experimentally observed wetting phenomena and formation of liquid boundary layers that are 20-50 microm wide and 12 microm high. Viewing this stream through a 20 microm slit results in a virtual optical window with a 5 pL liquid volume suitable for cell counting and pathogen detection. The geometry that leads to the boundary layer is a closed channel that forms triangular corners where glass fiber and the OTS coated glass slide or PDMS touch. The contact angles and surfaces direct positioning of the fluid next to the fiber. Preferential wetting of corner regions initiates the boundary flow, while the elliptical cross-section of the channel stabilizes the microfluidic flow. The Young-Laplace equation, solved using fluid dynamic simulation software, shows contact angles that exceed 105 degrees will direct the aqueous fluid to a boundary layer next to a hydrophilic fiber with a contact angle of 5 degrees. We believe this is the first time that an explanation has been offered for the case of a boundary layer formation in a closed channel directed by a triangular geometry with two hydrophobic wetting edges adjacent to a hydrophilic surface.     

Co- and counter-current spontaneous imbibition into groups of capillary tubes with lateral connections permitting cross-flow

A model for co- and counter-current imbibition through independent capillaries has already been developed and experiments conducted to verify the theory [E. Unsal, G. Mason, N.R. Morrow, D.W. Ruth, J. Colloid Interface Sci. 306 (2007) 105]. In this paper, the work is extended to capillaries which are connected laterally and in which cross-flow can take place. The fundamental pore geometry is a rod in an angled round-bottomed slot with a gap between the rod and a capping glass plate. The surfaces of the slot, rod and plate form capillaries and interconnecting passages which have non-axisymmetric cross-sections. Depending on the gap size either (i) a large single meniscus, (ii) two menisci one on each side of the rod, or (iii) three menisci, one between the rod and the glass additional to the ones on each side can be formed. A viscous refined oil was applied to one end of the capillaries and co-current and counter-current spontaneous imbibition experiments were performed. The opposite end was left open to the atmosphere for co-current experiments. When the gap between the rod and the plate was large, the imbibing oil advanced into the tubes with the meniscus in the largest capillary always lagging behind the two menisci in the other two smaller capillaries. For counter-current imbibition experiments the open end was sealed and connected to a sensitive pressure transducer. In some experiments, the oil imbibed into the smaller capillaries and expelled air as a series of bubbles from the end of the largest capillary. In other experiments, the oil was allowed to imbibe part way into the tubes before counter-current imbibition was started. The meniscus curvatures of the capillaries have been calculated using the Mayer and Stowe-Princen method for different cell slot angles and gap sizes using a value of zero for the contact angle. These values have been compared with actual values by measuring the capillary rise in the tubes; agreement was very close. A model for co-current and counter-current imbibition has also been developed. The significance of this model is that some hydraulic/capillary properties are common for both co-current and counter-current imbibition. The experiments give an illustration of behavior expected in a real porous material and verify the importance of the 'perfect cross-flow' modification to the 'bundle of parallel tubes' model.     

Capillary-driven flow in corner geometries

Wetting of corner-containing geometries is ubiquitous, since the man-made surfaces and natural surfaces are usually not atomically smooth and contain pores, grooves, and cracks. In spite of the very long history of the research of capillary phenomena, the most attention was paid to capillary rise in cylindrical capillaries leaving the rich physics of the capillary transport of the liquids in the corner geometries unravelled. The present work aims to review the progress in studying of wetting of corner-containing geometries: isolated corners, rectangular channels, and confined angular geometries. The review is believed to be of interest for readers from fields such as oil and gas industry, space science, biophysics, and microfluidics.     

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.     

Source-like solution for radial imbibition into a homogeneous semi-infinite porous medium

We describe the imbibition process from a point source into a homogeneous semi-infinite porous material. When body forces are negligible, the advance of the wetting front is driven by capillary pressure and resisted by viscous forces. With the assumption that the wetting front assumes a hemispherical shape, our analytical results show that the absorbed volume flow rate is approximately constant with respect to time, and that the radius of the wetting evolves in time as r ≈ t(1/3). This cube-root law for the long-time dynamics is confirmed by experiments using a packed cell of glass microspheres with average diameter of 42 μm. This result complements the classical one-dimensional imbibition result where the imbibition length l ≈ t(1/2), and studies in axisymmetric porous cones with small opening angles where l ≈ t(1/4) at long times.     

Recovery of oil by spontaneous imbibition

Spontaneous imbibition is of particular importance to oil recovery from fractured reservoirs. There has been a surge in the growth of technical literature over the past 5 years. This review is centered on developments in the scaling of laboratory imbibition data. Results for variation in interfacial tension, wetting and non-wetting phase viscosity, sample size, shape and boundary conditions, and initial wetting phase saturation have been correlated for a variety of strongly water-wet rocks as plots of normalized oil recovery vs. dimensionless time. Correlations have been tested for weakly water-wet conditions induced by adsorption from crude oil. In situ fluid saturation measurements have been used to distinguish between modes of imbibition that range from frontal to global displacement. Research on surfactant-enhanced imbibition has advanced from laboratory to field tests.     

Induced detachment of coalescing droplets on superhydrophobic surfaces

Coalescence of a falling droplet with a stationary sessile droplet on a superhydrophobic surface is investigated by a combined experimental and numerical study. In the experiments, the droplet diameter, the impact velocity, and the distance between the impacting droplets were controlled. The evolution of surface shape during the coalescence of two droplets on the superhydrophobic surface is captured using high speed imaging and compared with numerical results. A two-phase volume of fluid (VOF) method is used to determine the dynamics of droplet coalescence, shape evaluation, and contact line movement. The spread length of two coalesced droplets along their original center is also predicted by the model and compared well with the experimental results. The effect of different parameters such as impact velocity, center to center distance, and droplet size on contact time and restitution coefficient are studied and compared to the experimental results. Finally, the wetting and the self-cleaning properties of superhydrophobic surfaces have been investigated. It has been found that impinging water drops with very small amount of kinetic impact energy were able to thoroughly clean these surfaces.     

Electroosmotic flow mixing in zigzag microchannels

In this study we performed numerical and experimental investigations into the mixing of EOFs in zigzag microchannels with two different corner geometries, namely sharp corners and flat corners. In the zigzag microchannel with sharp corners, the flow travels more rapidly near the inner wall of the corner than near the outer wall as a result of the higher electric potential drop. The resulting velocity gradient induces a racetrack effect, which enhances diffusion within the fluid and hence improves the mixing performance. The simulation results reveal that the mixing index is approximately 88.83%. However, the sharp-corner geometry causes residual liquid or bubbles to become trapped in the channel at the point where the flow is almost stationary, when the channel is in the process of cleaning. Accordingly, a zigzag microchannel with flat-corner geometry is developed. The flat-corner geometry forms a convergent-divergent type nozzle which not only enhances the mixing performance in the channel, but also prevents the accumulation of residual liquid or bubbles. Scaling analysis reveals that this corner geometry leads to an effective increase in the mixing length. The experimental results reveal that the mixing index is increased to 94.30% in the flat-corner zigzag channel. Hence, the results demonstrate that the mixing index of the flat-corner zigzag channel is better than that of the conventional sharp-corner microchannel. Finally, the results of Taguchi analysis indicate that the attainable mixing index is determined primarily by the number of corners in the microchannel and by the flow passing height at each corner.     

Spreading, evaporation, and contact line dynamics of surfactant-laden microdrops

An optical technique based on the reflectivity measurements of a thin film was used to experimentally study the spreading, evaporation, contact line motion, and thin film characteristics of drops consisting of a water-surfactant (polyalkyleneoxide-modified heptamethyltrisiloxane, called superspreader) solution on a fused silica surface. On the basis of the experimental observations, we concluded that the surfactant adsorbs primarily at the solid-liquid and liquid-vapor interfaces near the contact line region. At equilibrium, the completely wetting corner meniscus was associated with a flat adsorbed film having a thickness of approximately 31 nm. The calculated Hamaker constant, A = -4.47 x 10(-)(20) J, shows that this thin film was stable under equilibrium conditions. During a subsequent evaporation/condensation phase-change process, the thin film of the surfactant solution was unstable, and it broke into microdrops having a finite contact angle. The thickness of the adsorbed film associated with the drops was lower than that of the equilibrium meniscus. The drop profiles were experimentally measured and analyzed during the phase-change process as the contact line advanced and receded. The apparent contact angle, the maximum concave curvature near the contact line region, and the convex curvature of the drop increased as the drop grew during condensation, whereas these quantities decreased during evaporation. The position of the maximum concave curvature of the drop moved toward the center of the drop during condensation, whereas it moved away from the center during evaporation. The contact line velocity was correlated to the observed experimental results and was compared with the results of the drops of a pure alcohol. The experimentally obtained thickness profiles, contact angle profiles, and curvature profiles of the drops explain how the surfactant adsorption affects the contact line motion. We found that there was an abrupt change in the velocity of the contact line when the adsorbed film of the surfactant solution was just hydrated or desiccated during the phase-change processes. This result shows the effect of vesicles and aggregates of the surfactant on the shape evolution of the drops. For these surfactant-laden water drops, we found that the apparent contact angle increased during condensation and decreased during evaporation. However, for the drop of a pure liquid (n-butanol and 2-propanol) the apparent contact angle remained constant at a constant velocity during condensation and evaporation. The contact line was pinned during the evaporation and spreading of the surfactant-laden water drops, but it was not pinned for a drop of a pure alcohol (self-similar shape evolution).     

Co-current and counter-current imbibition in independent tubes of non-axisymmetric geometry

Experiments that illustrate and quantify the basics of co- and counter-current spontaneous imbibition have been conducted in a series of simple model pore systems. The fundamental pore geometry is a rod in an angled round-bottomed slot with the rod touching a capping glass plate. The capillaries thus formed by the surfaces of the slot, rod and plate do not have circular cross-sections but more complicated geometric structures with angular corners. The tubes formed at each side of the rod connect at both ends. A viscous, refined oil was applied from one end. For co-current experiments, the opposite end was left open to the atmosphere and oil imbibed into both tubes. For counter-current experiments the opposite end was sealed and connected to a sensitive pressure transducer. Oil imbibed into the smaller capillary and expelled air as a series of bubbles from the end of the larger capillary. Bubble snap-off was observed to be rate-dependent and occurred at a lower curvature than that of the cylindrical meniscus that just fits inside the tube. Only the corners of the larger capillary filled with oil during counter-current imbibition. Meniscus curvatures were calculated using the Mayer and Stowe-Princen method and were compared with actual values by measuring the capillary rise in the tubes; agreement was close. A simple model for co-current and counter-current imbibition has also been developed and the predictions compared with the experimental results. The model results were in agreement with the experiments. The experiments demonstrate that the capillary back pressure generated by the interfaces and bubbles in counter-current imbibition can slow the process significantly.     

Spontaneous Penetration of Liquids into Capillaries and Porous Membranes Revisited

A critical review of the problem of spontaneous penetration of a wetting liquid into pore channels shows that no theory exists to quantitatively predict the initial stage of imbibition. Since C. H. Bosanquet (1923, Phil. Mag. 45, 525), the theory operates with an universal velocity U(Bosanquet)=(2gammacosstraight thetarhor(1/2), with gamma being the surface tension, straight theta the contact angle, r the capillary/pore radius, and rho the fluid density. It is assumed that the initial impulse of the liquid entering the pore is insignificant for the penetration dynamics. Though the importance of the outside flow pattern has been noted in many papers, a thorough mathematical analysis of this effect is lacking in the literature. We derived a generalized equation of the fluid front motion by averaging the Euler equations of flow inside and outside the pore space. This analysis shows the significance of the flow patterns at the pore entrance. The initial stage of liquid imbibition is studied in the inviscid approximation using the methods of dynamic systems. The phase portrait of the dynamic system reveals a multiplicity of penetration regimes. Remarkably, the Bosanquet solution represents a particular regime, with the apparent mass being set zero. The Bosanquet trajectory refers to a separatrix of the phase portrait. It is shown that the initial conditions affect the rate of uptake significantly. The initial conditions stem from the prehistory of the fluid motion outside the pores prior to the liquid-solid contact. The phase portrait method allows us to distinguish two groups of solutions for the capillary rise dynamics of an inviscid fluid. The first group of trajectories corresponds to the liquid front rebound; the second group includes cyclic trajectories which correspond to the periodic regimes with liquid front oscillations at the equilibrium position. The upper estimate of the oscillation amplitude is found. Copyright 2001 Academic Press.     

The possibility of different time scales in the dynamics of pore imbibition

To model the imbibition of liquids into porous solids, use is often made of the Lucas-Washburn equation, which relates the distance of penetration of a liquid at a given time to the pore radius, the viscosity and surface tension of the liquid, and the effective contact angle between the liquid and the solid. In this paper, we extend previous large-scale molecular dynamics simulations to show how this tool can be used to study the details of liquid imbibition, including the impact of the contact angle on the dynamics of penetration and the evolution of the internal flow field. In particular, we show that the asymptotic behavior of the contact angle versus time for a completely wetting liquid is given by approximately t(-1/4).     

Groovy drops: effect of groove curvature on spontaneous capillary flow

Spontaneous capillary flow (SCF) of a drop in a groove with an ideally sharp corner is possible when the Concus-Fin (CF) condition is fulfilled. However, since ideally sharp corners do not exist in reality, it is important to understand the effect of finite corner curvature on SCF. This effect is analytically studied for long drops in a V-shaped groove with a curved corner, leading to a generalization of the CF condition for such drops. The generalized condition implies that SCF depends on the geometry of the corner as well as on the dimensionless length of the drop, in addition to its dependence on the opening angle and contact angle that is covered by the CF condition. Specific calculations are presented for rounded corners. In addition, this effect is numerically calculated for short drops in V-shaped grooves with rounded corners, using the Surface Evolver software. The results of both types of calculations show that even a relatively small corner radius strongly affects the possibility of SCF: when the corner is not ideally sharp, SCF requires conditions that are more difficult to achieve than predicted by the CF condition; also, the spreading of the drop stops at a finite length and does not proceed indefinitely.