Capillary rise with velocity-dependent dynamic contact angle

Capillary imbibition of confined monodisperse emulsions in microfluidic channels

We study imbibition of a monodisperse emulsion into high-aspect ratio microfluidic channels with the height h comparable to the droplet diameter d. Two distinct regimes are identified in the imbibition dynamics. In a strongly confined system (the confinement ratio d/h = 1.2 in our experiments), the droplets are flattened between the channel walls and move more slowly compared to the average suspension velocity. As a result, a droplet-free region forms behind the meniscus (separated from the suspension region by a sharp concentration front) and the suspension exhibits strong droplet-density and velocity fluctuations. In a weaker confinement, d/h = 0.65, approximately spherical droplets move faster than the average suspension flow, causing development of a dynamically unstable high-concentration region near the meniscus. This instability results in the formation of dense droplet clusters, which migrate downstream relative to the average suspension flow, thus affecting the entire suspension dynamics. We explain the observed phenomena using linear transport equations for the particle-phase and suspension fluxes driven by the local pressure gradient. We also use a dipolar particle interaction model to numerically simulate the imbibition dynamics. The observed large velocity fluctuations in strongly confined systems are elucidated in terms of migration of self-assembled particle chains with highly anisotropic mobility.

A machine learning-based framework to design capillary-driven networks

We present a novel approach for the design of capillary-driven microfluidic networks using a machine learning genetic algorithm (ML-GA). This strategy relies on a user-friendly 1D numerical tool specifically developed to generate the necessary data to train the ML-GA. This 1D model was validated using analytical results issued from a Y-shaped capillary network and experimental data. For a given microfluidic network, we defined the objective of the ML-GA to obtain the set of geometric parameters that produces the closest matching results against two prescribed curves of delivered volume against time. We performed more than 20 generations of 10 000 simulations to train the ML-GA and achieved the optimal solution of the inverse design problem. The optimisation took less than 6 hours, and the results were successfully validated using experimental data. This work establishes the utility of the presented method for the fast and reliable design of complex capillary-driven devices, enabling users to optimise their designs via an easy-to-use 1D numerical tool and machine learning technique.

Capillary Flow Dynamics in Composite Rectangular Microchannels with Rough Walls

In this article, we consider rectangular microchannels composed of glass and thin polymeric walls with different roughness in which opposed walls are of the same material but adjacent walls are not. We propose a model for fluid capillary transport into these rectangular microchannels when horizontally positioned and focus our research on how the microchannel aspect ratio modifies the motion during the initial viscous regimes. The model relies on an effective static contact angle and an effective friction coefficient that averages local magnitudes in the cross section. An extensive experimental investigation with different microchannels enabled us to obtain these coefficients for different aspect ratios. While for low aspect ratios, the effective contact angle presents the smallest values, the effective friction coefficient shows the larger ones. With rough surfaces, the spontaneous occurrence of pinning and depinning events associated with sharp wall defects notably reduces the effective static contact angle even when high aspect ratios are used. The obtained values of the effective friction coefficient show good agreement with previous literature investigations for rough and smooth lateral wall surfaces. Finally, we propose a nondimensional time to establish when contact angle effects dominate the dynamics. We found that for the materials and fluid properties used in this work, these effects become negligible for times larger than t ∼ 1 s.

Meniscus Formation in a Vertical Capillary Tube

In this theoretical work, the energies associated with the formation of a meniscus in a small diameter capillary tube are analyzed. A mechanism for meniscus creation and an associated energy balance are proposed. Equations for work of wetting, surface energy, gravitational energy, and dissipation are derived. The relative magnitude of these quantities is compared, first to each other and then to energies from capillary rise. In capillary rise, the energy released as work of wetting is evenly split between gravitational energy stored in the liquid column and heat dissipated there. The analysis performed here suggests that meniscus formation is energetically distinct and more complex than capillary rise. In meniscus formation, most of the energy released as work of wetting is stored in the stretched air-liquid interface or dissipated in the bulk liquid; their relative distribution depends on the properties of the liquid and the tube.

Dynamic Effects during the Capillary Rise of Fluids in Cylindrical Tubes

The mathematical models for the capillary-driven flow of fluids in tubes typically assume a static contact angle at the fluid-air contact line on the tube walls. However, the dynamic evolution of the fluid-air interface is an important feature during capillary rise. Furthermore, inertial effects are relevant at early times and may lead to oscillations. To incorporate and quantify the different effects, a fundamental description of the physical processes within the tube is used to derive an upscaled model of capillary-driven flow in circular cylindrical tubes. The upscaled model extends the classical Lucas-Washburn model by incorporating a dynamic contact angle and slip. It is then further extended to account for inertial effects. Finally, the solutions of the different models are compared to experimental data. In contrast to the Lucas-Washburn model, the models with dynamic contact angle match well the experimental data, both the rise height and the contact angle, even at early times. The models have a free parameter through the dynamic contact angle description, which is fitted using the experimental data. The findings here suggest that this parameter depends only on the properties of the fluid but is independent of geometrical features, such as the tube radius. Therefore, the presented models can predict the capillary-driven flow in tubular systems upon knowledge of the underlying dynamic contact-angle relation.

Spontaneous Imbibition of Capillaries under the End Effect and Wetting Hysteresis

The phenomenon of spontaneous imbibition is widely present in the development process of tight oil/gas reservoirs. To further explore the spontaneous imbibition behavior of capillary tubes to provide theoretical and methodological references for the study of microscopic porous media imbibition phenomena, the capillaries that can be observed with the naked eye on the order of 10-100 μm were selected as research objects. Based on the theory of interface chemistry, the capillary end effect, and wetting hysteresis, the influence of the additional pressures generated by the two-phase interface on the spontaneous absorption of the horizontal capillary was studied. Some of the capillaries were processed for wettability, and then the water wettability of different capillaries was measured by the introduced concept, which is the conversion height of the self-absorption phase in the capillary. The capillaries were horizontally placed in the liquid for a spontaneous imbibition experiment, and the air-liquid two-phase menisci behavior was observed at the same time, and then the influence of water wettability, surfactant, and capillary diameter on spontaneous imbibition was discussed. It was found that in the equal diameter capillaries, the spontaneous air-liquid imbibition behavior of capillary tubes with different water wetting properties is different in sensitivity to surfactants and tube diameters; when surfactants are used to improve capillary water wettability to increase spontaneous imbibition efficiency, the initial water wettability of the capillary and the comprehensive changes in the capillary pressure caused by interfacial tension should be considered.

Capillary flow of liquids in open microchannels: overview and recent advances

Capillary flow is the spontaneous wicking of liquids in narrow spaces without the assistance of external forces. Examples of capillary flow can be found in numerous applications ranging from controlling and transporting fuel in spacecrafts to printed electronics manufacturing. Open rectangular microchannels often appear in these applications, with the lack of a top resulting in a complex free-surface morphology and evaporation. Here, we present a brief overview of this topic and discuss some recent advances.

Microscopic imbibition characterization of sandstone reservoirs and theoretical model optimization

Traditional porous media imbibition models deviate from the actual imbibition process in oil and gas reservoirs. Experimental studies on gas-water imbibition in reservoirs were carried out to describe the dynamic profile variation process of wet phase saturation in reservoirs and to further reveal the variation of the imbibition front and the imbibition amount. Optimization and correction methods were established, and experimental verifications were performed. Studies have shown the following: (1) Unlike homogeneous porous media, the water phase imbibition process in oil and gas reservoirs is more complicated, and it is impossible for the maximum saturation of imbibition to reach 100%. (2) Contrary to the theoretical hypothesis, the imbibition of water is not piston-like, and there is a clear transition zone at the imbibition front. This transition zone is the main cause of water saturation variations in the imbibition zone; with the expansion of the imbibition zone, the influence of the transition zone on water saturation weakens. (3) Traditional theoretical models predict a positive correlation between the imbibition amount and the measurements; however, there is a large deviation in the numerical values, which must be corrected. (4) The L-W model was optimized and the parameter group fluid factor F and the reservoir factor R were proposed to characterize the properties of the fluid and the reservoir, respectively. These two parameters have a clear physical significance and are easy to accurately test. After experimental correction, the optimized model is favourably suitable for oil and gas reservoirs.

Evolution of Thin-Liquid Films Surrounding Bubbles in Microfluidics and Their Impact on the Pressure Drop and Fluid Movement

The evolution of thin-liquid films in a microchannel is one of the most critical and intricate phenomena to understand two-phase movement, evaporation, micromixing, heat transfer, chemical synthesis, biological processes, and efficient energy devices. In this paper, we demonstrate experimentally the effect of a liquid film on the removal of an initially dry and lodged bubble in laser-etched poly(methyl methacrylate) microfluidic networks and discuss the evolution of the liquid film in accordance with the bubble superficial velocity and the effect of liquid properties and branch angle on the evolution of the liquid film and the pressure drop. During the removal of a dry bubble, four stages have been observed in the bubble velocity profile and they directly relate to the evolution of the liquid film. The correlation of maximum bubble velocity has been derived as a function of bubble length, fluid viscosity, surface tension, geometry of the cross-sectional area, and dimensions of the microchannel and agrees with the experimental results. The bubble moving distance required for the full deposition of a continuous and stable thin-liquid film is affected by the liquid viscosity and network branch angle. The liquid with a higher viscosity will increase the pressure drop for removing dry bubbles from microfluidic networks, while this effect will be hampered by increasing the microfluidic network complexity. The deposition of the thin-liquid film surrounding bubbles significantly decreases the pressure drop required to remove bubbles from microfluidics. Compared with deionized water, the glycerol solution is prone to acting as the lubricating liquid due to its strong H-bond interaction with the channel wall and the reduction in interfacial energy of the gas-water interface.

Direct Measurement of Contact Angle Change in Capillary Rise

Capillary rise is important in many aspects of physical phenomena from transport in porous media to biotechnology. It is typically described by the Lucas-Washburn-Rideal equation (LWRE), but discrepancy between some experiments and the model still remains elusive. In this paper, we show that the discrepancy is simply from the contact angle change during the capillary rise with no help of any specific models, such as dynamic contact angle (DCA) models. To demonstrate this, we directly measure the contact angle change in the capillary rise for glycerol and carboxymethyl cellulose solutions as examples of Newtonian and non-Newtonian liquids. Unlike previous studies that used DCA models to explain the discrepancy, when the contact angle change is directly applied to the LWRE for all four tested fluids, the model agrees well with experimental data. The estimated contact angle from the capillary rise as a function of time is in good agreement with the directly measured contact angle within a narrow margin of error. To pinpoint the conditions for the discrepancy, we propose a new time scale when contact angle dynamics dominates. The contact angle dynamics that can be obtained from the macroscopic capillary rise may provide useful information for capillary flow in a more complicated geometry such as porous media.

Analysis of Capillary Flow in a Parallel Microchannel-Based Wick Structure with Circular and Noncircular Geometries

Capillary flow in porous media is of great significance to many different applications including microfluidics, chromatography, and passive thermal management. For example, heat pipe has been widely used in the thermal management of electronic system due to its high flexibility and low thermal resistance. However, the critical heat flux of heat pipe is often limited by the maximum capillary-driven liquid transport rate through the wicking material. A significant number of novel porous material with complex structures have been proposed in past studies to provide enhanced capillary-driven flow without substantial reduction in pore size and porosity. However, the increasing level of structural complexity often leads to a more tortuous flow path, which deprives the merits of enhanced capillarity. In this study, we examined the capillary performance of a porous material with simple geometric structures both analytically and numerically. Specifically, the capillary rate of rise of water in parallel hollow microchannels with different cross-sectional shapes is derived by solving the momentum transport equation. The relationships between the capillary flow rate and wicking height are further validated by two-phase flow simulation based on the conservative level-set method. The results demonstrate that parallel microchannel configuration, despite its geometric simplicity, provides superior capillary performance than most existing porous media in terms of both capillary flow rate and ultimate wicking height. In addition, design of noncircular cross section reduces the viscous drag and increases the packing density of the microchannels in the bulk solid without affecting the capillary pumping pressure. These features contribute to a further enhancement in the capillary performance by up to 32%. These results provide important guidance to the rational design of porous material with enhanced fluid transport property in a variety of microfluidic systems.

Pressure Drop with Moving Contact Lines and Dynamic Contact Angles in a Hydrophobic Round Minichannel: Visualization via Synchrotron X-ray Imaging and Verification of Experimental Correlations

In hydrophobic mini- and microchannels, slug flow with moving contact lines is typically generated under various two-phase flow conditions. There is a significant pressure drop in this flow pattern with moving contact lines, which is closely related to the dynamic contact angles. Researchers have investigated dynamic contact angles experimentally for decades, but due to the limitations of visualization techniques, these experiments have typically been conducted in low Weber number regions (We < 10^-3). In this study, we clearly visualized the dynamic contact angles of a liquid slug in high Weber number regions (10^-3 < We <1) via synchrotron X-ray imaging with high temporal (∼1000 fps) and spatial (∼2 μm/pixel) resolutions. We precisely measured the pressure drop with moving contact lines in a hydrophobic minichannel (inner diameter = 1.018 mm). On the basis of our experimental data, we verified previous correlations for dynamic contact angles and explored the relationship between pressure drop with moving contact lines and dynamic contact angles.

Inertial capillary uptake of drops

Uptake of liquid drops into capillary tubes has been experimentally studied and quantitatively analyzed. In experiments, drops of water and aqueous glycerol (≤50 wt %) were drawn into cylindrical borosilicate glass and quartz tubes with an inner diameter of 0.50-0.75 mm. The meniscus height rise was measured using high-speed images captured at 4000 frames per second, and results within a conservatively defined inertial regime indicate constant uptake velocity. An increase in the inertial velocity with drop curvature was observed due to increasing Laplace pressure in the drop, as drop sizes were comparable to the width of the capillary tubes. Measured velocities were slower than predicted by a purely inertial-capillary model and best described by introducing a contact line friction, consistent with the observed variability and viscosity dependence of the results. Mean friction coefficients in borosilicate capillaries ranged from 169±1 for 50 wt % glycerol drops to 218±1 for water drops. Peaks in the instantaneous Laplace pressure caused by surface oscillations were also measured. Correlations with uptake velocity were qualitatively apparent, with a delay between peaks of similar magnitude to the inertial-capillary oscillation time.

Capillary Flow with Evaporation in Open Rectangular Microchannels

Numerous applications rely upon capillary flow in microchannels for successful operation including lab-on-a-chip devices, porous media flows, and printed electronics manufacturing. Open microchannels often appear in these applications, and evaporation of the liquid can significantly affect its flow. In this work, we develop a Lucas-Washburn-type one-dimensional model that incorporates the effects of concentration-dependent viscosity and uniform evaporation on capillary flow in channels of a rectangular cross section. The model yields predictions of the time evolution of the liquid front down the length of the microchannel. For the case where evaporation is absent, prior studies have demonstrated better agreement between model predictions and experimental observations in low-viscosity liquids when using a no-slip rather than a no-stress boundary condition at the upper liquid-air interface. However, flow visualization experiments conducted in this work suggest the absence of a rigidified liquid-air interface. The use of the no-stress condition results in overestimation of the time evolution of the liquid front, which appears to be due to underestimation of the viscous forces from (i) the upper and front meniscus morphology, (ii) dynamic contact angle effects, and (iii) surface roughness, none of which are accounted for in the model. When high-viscosity liquids are considered, the large bulk viscosity is found to suppress these factors, resulting in better agreement between model predictions using the no-stress condition and experiments. Model predictions are also compared to prior experiments involving poly(vinyl alcohol) in the presence of evaporation by using the evaporation rate as a fitting parameter. Scaling relationships obtained from the model for the dependence of the final liquid-front position and total flow time on the channel dimensions and rate of uniform evaporation are found to be in good agreement with experimental observations.

Effect of Nanoscale Surface Textures on Multiphase Flow Dynamics in Capillaries

Multiphase flow through porous media is important in a wide range of environmental applications such as enhanced oil recovery and geologic storage of CO₂. Recent in situ observations of the three-phase contact line between immiscible fluid phases and solid surfaces suggest that existing models may not fully capture the effects of nanoscale surface textures, impacting flow prediction. To better characterize the role of surface roughness in these systems, spontaneous and forced imbibition experiments were carried out using glass capillaries with modified surface roughness or wettability. Dynamic contact angle and interfacial speed deviation, both resulting from stick-slip flow conditions, were measured to understand the impact these microscale dynamics would have on macroscale flow processes. A 2 ^k factorial experimental design was used to test the ways in which the dynamic contact angle was impacted by the solid surface properties (e.g., wettability, roughness), ionic strength in the aqueous phase, nonaqueous fluid type (water/Fluorinert and water/dodecane), and the presence/absence of a wetting film prior to the imbibition of the wetting phase. The analysis of variance of spontaneous imbibition results suggests that surface roughness and ionic strength play important roles in controlling dynamic contact angle in porous media, more than other factors tested here. The presence of a water film alone does not affect dynamic contact angle, but its interactions with surface roughness and aqueous chemistry have a statistically significant effect. Both forced imbibition and spontaneous imbibition experiments suggest that nanoscale textures can have a larger impact on flow dynamics than chemical wettability. These experimental results are used to extend the Joos and Wenzel equations relating apparent static and dynamic contact angles to roughness, presence of a water film, and water chemistry. The new empirical equation improves prediction accuracy by taking water film and aqueous chemistry into account, reducing error by up to 50%.

Spontaneous rise in open rectangular channels under gravity

Fluid movement in microfluidic devices, porous media, and textured surfaces involves coupled flows over the faces and corners of the media. Spontaneous wetting of simple grooved surfaces provides a model system to probe these flows. This numerical study investigates the spontaneous rise of a liquid in an array of open rectangular channels under gravity, using the Volume-of-Fluid method with adaptive mesh refinement. The rise is characterized by the meniscus height at the channel center, outer face and the interior and exterior corners. At lower contact angles and higher channel aspect ratios, the statics and dynamics of the rise in the channel center show little deviation with the classical model for capillarity, which ignores the existence of corners. For contact angles smaller than 45°, rivulets are formed in the interior corners and a cusp at the exterior corner. The rivulets at long times obey the one-third power law in time, with a weak dependence on the geometry. The cusp behaviour at the exterior corner transforms into a smooth meniscus when the capillary force is higher in the channel, even for contact angles smaller than 45°. The width of the outer face does not influence the capillary rise inside the channel, and the channel size does not influence the rise on the outer face.

Capillary Coatings: Flow and Drying Dynamics in Open Microchannels

Capillary flow and drying of polymer solutions in open microchannels are explored over time scales spanning seven orders of magnitude: from capillary filling (10^-3-10 s) to the formation of a dry thin film (a "capillary coating"; 10²-10³ s). During capillary filling, drying-induced changes (increased solids content and viscosity) generate microscale pinning events that impede contact line motion. Three unique types of pinning are identified and characterized, each defined by the specific location(s) along the contact line at which pinning is induced. Drying is shown to ultimately pin the contact line permanently, and the associated total flow distances and times are revealed to be strong functions of channel width and drying rate. In general, lower drying rates coupled with intermediate channel widths are found to be most conducive to longer flow distances and times. After the advancing contact line permanently pins, internal flows driven by uneven evaporation rates continue to drive polymer to the contact line. This phenomenon promotes a local accumulation of solids and persists until all motion is arrested by drying. The effects of channel width and drying rate are investigated at each stage of this capillary coating process. These results are then applied to case studies of two functional inks commonly used in printed electronics fabrication: a PEDOT:PSS (poly(3,4-ethylenedioxythiophene)-poly(styrenesulfonate)) ink and a graphene ink. Although drying is shown to permanently arrest flow in both inks, both systems exhibit an increased resistance to pinning unexplained by mechanisms identified in aqueous polymer systems. Instead, arguments based on chemistry, particle size, and rheology are used to explain their novel behavior. These case studies provide insight into how functional inks can be better designed to optimize flow distances and maximize overall dry film uniformity in capillary coatings.

Capillary Rise: Validity of the Dynamic Contact Angle Models

The classical Lucas-Washburn-Rideal (LWR) equation, using the equilibrium contact angle, predicts a faster capillary rise process than experiments in many cases. The major contributor to the faster prediction is believed to be the velocity dependent dynamic contact angle. In this work, we investigated the dynamic contact angle models for their ability to correct the dynamic contact angle effect in the capillary rise process. We conducted capillary rise experiments of various wetting liquids in borosilicate glass capillaries and compared the model predictions with our experimental data. The results show that the LWR equations modified by the molecular kinetic theory and hydrodynamic model provide good predictions on the capillary rise of all the testing liquids with fitting parameters, while the one modified by Joos' empirical equation works for specific liquids, such as silicone oils. The LWR equation modified by molecular self-layering model predicts well the capillary rise of carbon tetrachloride, octamethylcyclotetrasiloxane, and n-alkanes with the molecular diameter or measured solvation force data. The molecular self-layering model modified LWR equation also has good predictions on the capillary rise of silicone oils covering a wide range of bulk viscosities with the same key parameter W(0), which results from the molecular self-layering. The advantage of the molecular self-layering model over the other models reveals the importance of the layered molecularly thin wetting film ahead of the main meniscus in the energy dissipation associated with dynamic contact angle. The analysis of the capillary rise of silicone oils with a wide range of bulk viscosities provides new insights into the capillary dynamics of polymer melts.

Dynamics of capillary transport in semi-solid channels

Capillary action has been described by Lucas and Washburn and extensively studied experimentally in hard materials, but few studies have examined capillary action in soft materials such as hydrogels. In tissue engineering, cells or dispersions must be often distributed within a hydrogel via microporous paths. Capillary action is one way to disperse such substances. Here, we examine the dynamics of capillary action in a model system of straight capillaries in two hydrogels. The channels had a circular cross-section in the micrometer size range (180-630 μm). The distance travelled over time was recorded and compared with the predictions of Lucas and Washburn. Besides water, we used a sucrose solution and a hydroxyethyl cellulose solution, both with viscosities slightly higher than that of water. The results showed that the distance travelled is proportional to the square root of time, , and that larger capillaries and lower viscosities result, as expected, in faster speeds. However, the absolute experimental values display large discrepancies from the predictions. We demonstrate that several possible reasons for these discrepancies can be ruled out and we describe a novel hypothesis for the cause of the retarded meniscus movement.

Dynamics of Capillary-Driven Flow in 3D Printed Open Microchannels

Microchannels have applications in microfluidic devices, patterns for micromolding, and even flexible electronic devices. Three-dimensional (3D) printing presents a promising alternative manufacturing route for these microchannels due to the technology's relative speed and the design freedom it affords its users. However, the roughness of 3D printed surfaces can significantly influence flow dynamics inside of a microchannel. In this work, open microchannels are fabricated using four different 3D printing techniques: fused deposition modeling (FDM), stereolithography (SLA), selective laser sintering, and multi jet modeling. Microchannels printed with each technology are evaluated with respect to their surface roughness, morphology, and how conducive they are to spontaneous capillary filling. Based on this initial assessment, microchannels printed with FDM and SLA are chosen as models to study spontaneous, capillary-driven flow dynamics in 3D printed microchannels. Flow dynamics are investigated over short (∼10^-3 s), intermediate (∼1 s), and long (∼10² s) time scales. Surface roughness causes a start-stop motion down the channel due to contact line pinning, while the cross-sectional shape imparted onto the channels during the printing process is shown to reduce the expected filling velocity. A significant delay in the onset of Lucas-Washburn dynamics (a long-time equilibrium state where meniscus position advances proportionally to the square root of time) is also observed. Flow dynamics are assessed as a function of printing technology, print orientation, channel dimensions, and liquid properties. This study provides the first in-depth investigation of the effect of 3D printing on microchannel flow dynamics as well as a set of rules on how to account for these effects in practice. The extension of these effects to closed microchannels and microchannels fabricated with other 3D printing technologies is also discussed.

Investigation of the Dynamic Contact Angle Using a Direct Numerical Simulation Method

A large amount of residual oil, which exists as isolated oil slugs, remains trapped in reservoirs after water flooding. Numerous numerical studies are performed to investigate the fundamental flow mechanism of oil slugs to improve flooding efficiency. Dynamic contact angle models are usually introduced to simulate an accurate contact angle and meniscus displacement of oil slugs under a high capillary number. Nevertheless, in the oil slug flow simulation process, it is unnecessary to introduce the dynamic contact angle model because of a negligible change in the meniscus displacement after using the dynamic contact angle model when the capillary number is small. Therefore, a critical capillary number should be introduced to judge whether the dynamic contact model should be incorporated into simulations. In this study, a direct numerical simulation method is employed to simulate the oil slug flow in a capillary tube at the pore scale. The position of the interface between water and the oil slug is determined using the phase-field method. The capacity and accuracy of the model are validated using a classical benchmark: a dynamic capillary filling process. Then, different dynamic contact angle models and the factors that affect the dynamic contact angle are analyzed. The meniscus displacements of oil slugs with a dynamic contact angle and a static contact angle (SCA) are obtained during simulations, and the relative error between them is calculated automatically. The relative error limit has been defined to be 5%, beyond which the dynamic contact angle model needs to be incorporated into the simulation to approach the realistic displacement. Thus, the desired critical capillary number can be determined. A three-dimensional universal chart of critical capillary number, which functions as static contact angle and viscosity ratio, is given to provide a guideline for oil slug simulation. Also, a fitting formula is presented for ease of use.

Capillary driven flow of polydimethylsiloxane in open rectangular microchannels

The flow of liquid polydimethylsiloxane (PDMS, Dow Corning Sylgard 184, 10 : 1 base to cross-linker ratio) in open, rectangular silicon microchannels, with and without a coating (100 nm) of poly-tetra-fluoro-ethylene (PTFE), was studied. Photolithographic patterning and etching of silicon wafers was used to create microchannels with a range of widths (∼5-50 μm) and depths (5-20 μm). Experimental PDMS flow rates in both PTFE-coated and uncoated channels were compared to an analytical model based on the work of Lucas and Washburn. The experimental flow rates matched the predicted flow rates reasonably well when the channel aspect ratio (width to depth), p, was less than 2. For channels with p > 2, the observed flow rates progressively lagged model predictions with increasing p. The experimental data, including zero flow rates in certain high aspect ratio PTFE-coated channels, can largely be explained by changes in the front and upper meniscus morphology of the flow as the channel aspect ratio is varied. The results strongly suggest that meniscus morphology needs to be taken into account to accurately model capillary flow in microchannels, especially those with large aspect ratios.

Capillary Penetration into Inclined Circular Glass Tubes

The spontaneous penetration of a wetting liquid into a vertical tube against the force of gravity and the imbibition of the same liquid into a horizontal tube (or channel) are both driven by capillary forces and described by the same fundamental equations. However, there have been few experimental studies of the transition from one orientation to the other. We report systematic measurements of capillary penetration of polydimethylsiloxane oils of viscosities 9.6, 19.2, and 48.0 mPa·s into glass capillary tubes. We first report the effect of tube radii R between 140 and 675 μm on the dynamics of spontaneous imbibition. We show that the data can be fitted using the exact numerical solution to the governing equations and that these are similar to fits using the analytical viscogravitational approximation. However, larger diameter tubes show a rate of penetration slower than expected using an equilibrium contact angle and the known value of liquid viscosity. To account for the slowness, an increase in viscosity by a factor (η/ρ)(scaling) is needed. We show full agreement with theory requires the ratio R/κ(-1) ∼ 0.1 or less, where κ(-1) is the capillary length. In addition, we propose an experimental method that enables the determination of the dynamic contact angle during imbibition, which gives values that agree with the literature values. We then report measurements of dynamic penetration into the tubes of R = 190 and 650 μm for a range of inclination angles to the horizontal, φ, from 5 to 90°. We show that capillary penetration can still be fitted using the viscogravitational solution, rather than the Bosanquet solution which describes imbibition without gravity, even for inclination angles as low as 10°. Moreover, at these low angles, the effect of the tube radius is found to diminish and this appears to relate to an effective capillary length, κ(-1)(φ) = (γ(LV)/ρg sin φ)(1/2).

Early regimes of water capillary flow in slit silica nanochannels

Molecular simulation of the capillary filling of water in a silica nanoslit. An atomistic description of the capillary filling process allows us to conduct a detailed study of the validity of the Bosanquet equation at the nanoscale.

Experimental investigation of dynamic contact angle and capillary rise in tubes with circular and noncircular cross sections

An extensive experimental study of the kinetics of capillary rise in borosilicate glass tubes of different sizes and cross-sectional shapes using various fluid systems and tube tilt angles is presented. The investigation is focused on the direct measurement of dynamic contact angle and its variation with the velocity of the moving meniscus (or capillary number) in capillary rise experiments. We investigated this relationship for different invading fluid densities, viscosities, and surface tensions. For circular tubes, the measured dynamic contact angles were used to obtain rise-versus-time values that agree more closely with their experimental counterparts (also reported in this study) than those predicted by Washburn equation using a fixed value of contact angle. We study the predictive capabilities of four empirical correlations available in the literature for velocity-dependence of dynamic contact angle by comparing their predicted trends against our measured values. We also present measurements of rise in noncircular capillary tubes where rapid advancement of arc menisci in the corners ahead of main terminal meniscus impacts the dynamics of rise. Using the extensive set of experimental data generated in this study, a new general empirical trend is presented for variation of normalized rise with dynamic contact angle that can be used in, for instance, dynamic pore-scale models of flow in porous media to predict multiphase flow behavior.

An algorithm for selecting the most accurate protocol for contact angle measurement by drop shape analysis

In this study, an error analysis is performed to study real water drop images and the corresponding numerically generated water drop profiles for three widely used static contact angle algorithms: the circle- and ellipse-fitting algorithms and the axisymmetric drop shape analysis-profile (ADSA-P) algorithm. The results demonstrate the accuracy of the numerically generated drop profiles based on the Laplace equation. A significant number of water drop profiles with different volumes, contact angles, and noise levels are generated, and the influences of the three factors on the accuracies of the three algorithms are systematically investigated. The results reveal that the above-mentioned three algorithms are complementary. In fact, the circle- and ellipse-fitting algorithms show low errors and are highly resistant to noise for water drops with small/medium volumes and contact angles, while for water drop with large volumes and contact angles just the ADSA-P algorithm can meet accuracy requirement. However, this algorithm introduces significant errors in the case of small volumes and contact angles because of its high sensitivity to noise. The critical water drop volumes of the circle- and ellipse-fitting algorithms corresponding to a certain contact angle error are obtained through a significant amount of computation. To improve the precision of the static contact angle measurement, a more accurate algorithm based on a combination of the three algorithms is proposed. Following a systematic investigation, the algorithm selection rule is described in detail, while maintaining the advantages of the three algorithms and overcoming their deficiencies. In general, static contact angles over the entire hydrophobicity range can be accurately evaluated using the proposed algorithm. The ease of erroneous judgment in static contact angle measurements is avoided. The proposed algorithm is validated by a static contact angle evaluation of real and numerically generated water drop images with different hydrophobicity values and volumes.

Generalized modeling of spontaneous imbibition based on Hagen-Poiseuille flow in tortuous capillaries with variably shaped apertures

Spontaneous imbibition of wetting liquids in porous media is a ubiquitous natural phenomenon which has received much attention in a wide variety of fields over several decades. Many traditional and recently presented capillary-driven flow models are derived based on Hagen-Poiseuille (H-P) flow in cylindrical capillaries. However, some limitations of these models have motivated modifications by taking into account different geometrical factors. In this work, a more generalized spontaneous imbibition model is developed by considering the different sizes and shapes of pores, the tortuosity of imbibition streamlines in random porous media, and the initial wetting-phase saturation. The interrelationships of accumulated imbibition weight, imbibition rate and gas recovery and the properties of the porous media, wetting liquids, and their interactions are derived analytically. A theoretical analysis and comparison denote that the presented equations can generalize several traditional and newly developed models from the literature. The proposed model was evaluated using previously published data for spontaneous imbibition measured in various natural and engineered materials including different rock types, fibrous materials, and silica glass. The test results show that the generalized model can be used to characterize the spontaneous imbibition behavior of many different porous media and that pore shape cannot always be assumed to be cylindrical.

Inverse problem of capillary filling

The inverse problem of capillary filling, as defined in this work, consists in determining the capillary radius profile from experimental data of the meniscus position l as a function of time t. This problem is central in diverse applications, such as the characterization of nanopore arrays or the design of passive transport in microfluidics; it is mathematically ill posed and has multiple solutions; i.e., capillaries with different geometries may produce the same imbibition kinematics. Here a suitable approach is proposed to solve this problem, which is based on measuring the imbibition kinematics in both tube directions. Capillary filling experiments to validate the calculation were made in a wide range of length scales: glass capillaries with a radius of around 150  μm and anodized alumina membranes with a pores radius of around 30  nm were used. The proposed method was successful in identifying the radius profile in both systems. Fundamental aspects also emerge in this study, notably the fact that the l(t)∝t1/2 kinematics (Lucas-Washburn relation) is not exclusive of uniform cross-sectional capillaries.

The effect of contact angles and capillary dimensions on the burst frequency of super hydrophilic and hydrophilic centrifugal microfluidic platforms, a CFD study

This paper employs the volume of fluid (VOF) method to numerically investigate the effect of the width, height, and contact angles on burst frequencies of super hydrophilic and hydrophilic capillary valves in centrifugal microfluidic systems. Existing experimental results in the literature have been used to validate the implementation of the numerical method. The performance of capillary valves in the rectangular and the circular microfluidic structures on super hydrophilic centrifugal microfluidic platforms is studied. The numerical results are also compared with the existing theoretical models and the differences are discussed. Our experimental and computed results show a minimum burst frequency occurring at square capillaries and this result is useful for designing and developing more sophisticated networks of capillary valves. It also predicts that in super hydrophilic microfluidics, the fluid leaks consistently from the capillary valve at low pressures which can disrupt the biomedical procedures in centrifugal microfluidic platforms.

Capillary rise dynamics of aqueous glycerol solutions in glass capillaries: a critical examination of the Washburn equation

The classic description of capillary rise given by the Washburn equation was recently questioned in the light of experimental evidence for a velocity dependent dynamic contact angle at a moving contact line. We present a systematic investigation of the capillary rise dynamics of glycerol and aqueous glycerol solutions in vertical glass capillaries of various radii. For pure glycerol, the results of our experiments are in almost perfect agreement with the predictions of the Washburn equation using independently measured values for the liquid and capillary parameters. For aqueous glycerol solutions we observe discrepancies between the theoretical expectations and the experimental results, which are increasing with the water content of the solution. A thorough analysis, combined with scaling arguments, allows us to conclude that dynamic contact angle effects alone cannot provide a consistent explanation for these discrepancies. Rather, they can be perfectly accounted for if the mixture flowing in the capillary would have an effective, increased viscosity (in respect to the nominal value). We suggest and briefly discuss various mechanisms that could contribute to this observed behavior.

Experimental investigation of droplet acceleration and collision in the gas phase in a microchannel

We developed a novel microfluidic system, termed a micro-droplet collider, by utilizing the spatial-temporal localized liquid energy to realize chemical processes, which achieved rapid mixing between droplets having a large volume ratio by collision. In this paper, in order to clarify the characteristics of the micro-droplet collider, dynamics of droplet acceleration, stationary motion and collision in the gas phase in a microchannel were experimentally investigated with visualized images using a microscope equipped with a high-speed camera. The maximum velocity of 450 mm s(-1) and acceleration of 1500 m s(-2) of a 1.6 nL water droplet were achieved at an air pressure of 100 kPa. Measurement results of dynamic contact angles of droplets indicated that wettability of the surface played an important role in the stability of droplet acceleration and collision. We found that the bullet droplet penetrated into the target droplet at collision, which differed from bulk scale. The deformation of the droplet was strongly suppressed by the channel structure, thus stable collision and efficient utilization of the droplet energy were possible. These results are useful for estimating the localized energy, for improving the system in order to realize extreme performance, and for extending the applications of microfluidic devices.

Capillary filling in nanostructured porous silicon

An experimental study on the capillary filling of nanoporous silicon with different fluids is presented. Thin nanoporous membranes were obtained by electrochemical anodization, and the filling dynamics was measured by laser interferometry, taking advantage of the optical properties of the system, related with the small pore radius in comparison to light wavelength. This optical technique is relatively simple to implement and yields highly reproducible data. A fluid dynamic model for the filling process is also proposed including the main characteristics of the porous matrix (tortuosity, average hydraulic radius). The model was tested for different ambient pressures, porous layer morphology, and fluid properties. It was found that the model reproduces well the experimental data according to the different conditions. The predicted pore radii quantitatively agree with the image information from scanning electron microscopy. This technique can be readily used as nanofluidic sensor to determine fluid properties such as viscosity and surface tension of a small sample of liquid. Besides, the whole method can be suitable to characterize a porous matrix.

An investigation on the capillary wetting of glass fiber tow and fabric structures with nanoclay-enriched reactive epoxy and silicone oil mixtures

Capillary wetting of a biaxially-oriented glass fabric and the tow comprising the fabric was examined via the Washburn equation with silicone oil and a reactive epoxy-curative system containing 0-4 wt % nanoclay reinforcement. Capillary wetting of silicone oil was used to measure the hydraulic constant of the fabric and tow. The wetting rates for fabric were found to be greater than those for tow and this was based on the larger pore radii of fabric compared to tow. Reynolds and capillary numbers calculated from wetting rate data indicated that the flow is dominated by interfacial tension. The presence of nanoclay offered a significant perturbation to capillary wetting behavior. Wetting rates indicate that the effect of nanoclay is 2-fold: blocking of tow pores and increasing the wetted area. This suggests that nanoclay particles are aggregating and the flow field is such that the particles are not dispersed additionally under the present conditions. Capillary wetting rates of fabric and tow samples were measured with reactive epoxy-amine mixtures at various nanoclay loadings. The presence of nanoclay offers another level of complexity to the Washburn equation. In addition to the liquid/fabric or tow interfacial tension, the interfacial tension of the liquid/nanoclay interface has to be reconciled. As a consequence, hydraulic constant, surface tension, and contact angle are convoluted in such a binary system.