Experimental Investigation of Dynamic Contact Angle and Capillary

Oct 16, 2014 - University of Wyoming, Laramie, Wyoming 82071-2000, United States. Langmuir , 2014, 30 (47), pp 14151–14162. DOI: 10.1021/la501724y...
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Experimental Investigation of Dynamic Contact Angle and Capillary Rise in Tubes with Circular and Noncircular Cross Sections Mohammad Heshmati* and Mohammad Piri University of Wyoming, Laramie, Wyoming 82071-2000, United States ABSTRACT: 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.

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INTRODUCTION Flow of fluids through porous media and the associated capillarity phenomenon have long been the focus of physicists, soil scientists, petroleum and environmental engineers, and researchers in many other areas of science, technology, and engineering. This along with the fact that a porous medium is a complicated system of connected and mostly rough-walled capillary pores and throats makes the study of fluid/fluid displacement mechanisms in capillary systems critically relevant. However, direct pore-level investigation of flow and transport in random porous mediums is very difficult due to scale and imaging challenges. Using glass micromodels and capillary tubes simplifies the study of such systems, thereby enabling investigation of complicated displacement physics and parameters such as dynamic contact angle, which are difficult to probe directly in a naturally-occurring random porous medium. Experimental and modeling studies of displacement processes in simplified capillary systems have long been the focus of authors in different research areas. Insights developed through these studies coupled with representative pore space topology maps obtained using, for instance, X-ray imaging technologies enable development of predictive, physically-based pore-scale models of flow and transport in porous media. It is therefore imperative to investigate subtle aspects of dynamic flow in capillary tubes and thereby enrich and improve the predictive capabilities of dynamic pore-scale flow models. Kinetics of liquid rise in single capillary tubes of circular cross section was formulated almost at the same time by Lucas,1 Washburn,2 and Rideal3 in the early 20th century. Later on, © XXXX American Chemical Society

many other scientists in different areas of science and engineering attempted to improve the formulation and associated analysis.7,9,11,12,14,15,17,20,23,26−28,37−39 Washburn2 modeled the fluid flow in circular capillary tubes using Poiseuille’s law. The author ignored the changes in contact angle of fluid meniscus during displacement. This assumption is one of the main reasons why the rise-versus-time curves predicted by the proposed equation do not match the experimental data. Hence, scientists have tried to improve the predictions through collection of more experimental data and improvements in the modeling of the displacement process. Quere26 performed experiments showing that the position of the meniscus versus time in the early stages of rise, can be described using a linear relationship. Furthermore, the oscillations around the equilibrium occur if the liquid viscosity is low enough. The author along with Hamraoui et al.27 and Siebold et al.,28 on liquid/air systems, and Mumley et al.,14 on liquid/liquid systems, emphasized on the importance of implementing dynamic contact angle in Washburn’s model in order to obtain a better agreement with the measured riseversus-time data. There have been several experimental investigations on the effects of velocity and capillary number on dynamic contact angle. Some of these studies have led to development of empirical correlations. Hoffman7 performed experiments in Received: May 7, 2014 Revised: October 7, 2014

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This is also the case in Lee’s35 and Hilpert’s semianalytical models.37,38 In Lee’s35 work, a simple geometrical model is utilized to solve the force balance equation on the liquid meniscus. While Hilpert37 applied a power law and a power series model for dynamic contact angle in order to generalize Washburn’s analytical solution for flow in horizontal capillary tubes. For flow in tilted capillary tubes, the author applied the power law model and a polynomial.38 The resulting semianalytical models compared well with the numerical solutions. The second approach does not include the velocity of contact line to correlate the dynamic contact angle. For instance, Deganello et al.39 proposed a numerical approach that does not explicitly include velocity of the contact line to model the dynamic contact angle. The authors combined an equilibrium of forces in the contact region near the solid boundary with a diffuse free fluid interface within a level-set finite volume numerical framework. The dynamic contact angle versus capillary number data derived from the model showed an excellent agreement with the empirical correlations proposed by Hoffman7 and Jiang et al.9 There have also been studies of capillary rise performed in microgravity systems.13,24,31−33 Utilizing available experimental data, van Mourik et al.33 tested some dynamic contact angle models in a numerical simulator. They found that Blake’s theoretical dynamic contact angle model23 gives the best agreement with two sets of available experimental data presented in their paper. Even though the majority of studies in this domain have focused on displacements in capillary tubes with circular cross section, there have been some limited investigations performed on tubes with angular cross sections as well. Ransohoff and Radke16 developed a model for the flow of fluids at low Reynolds numbers in tubes with angular cross section. They divided the problem into individual corner flows and solved it numerically. They defined a dimensionless flow resistance parameter, β, which depended on the corner half angle, degree of roundedness, surface shear viscosity, and contact angle. Tang and Tang25 analytically studied the dynamics of fluid flow in tubes with sharp grooves and proposed that when the diameter of the tube is smaller than the capillary length, the early-time rise-versus-time data for main terminal meniscus (MTM) and late-time data for arc meniscus (AM) follow t1/2 and t1/3 relationships, respectively. This is also reported in the studies performed by Ponomarenko et al.40 in which they found a universal relationship for the capillary rise in the corners. As discussed earlier, dynamic contact angle plays a critically important role in the description of the dynamic displacements in capillary tubes with varying cross-sectional geometries and wettabilities. Therefore, the extent and quality of the experimental data on this interfacial parameter control the predictive capabilities of the models that one can develop. To the best of our knowledge, the experimental data on dynamic contact angle in capillary rise experiments are scarce and have been generated under limited range of relevant conditions. In other words, there are very limited number of experimental data sets available in the literature that can be used to characterize the variation of dynamic contact angle for different capillary tube/fluid systems. For instance, majority of the experimental studies focused on measuring dynamic contact angles have been performed in horizontal capillary tubes using a piston to move the fluid phases, or as in the case of Bracke et al.,17 a continuous solid strip has been drawn into a large pool of liquid. In this work, we present, to the best of our knowledge, the first extensive, well-characterized experimental study of rise-

horizontal circular tubes and presented a general trend showing that dynamic contact angle correlates with capillary number. Using Hoffman’s experimental data, Jiang et al.9 suggested a correlation for the advancing dynamic contact angle measured through the liquid phase during liquid−gas interface displacement in circular glass capillary tubes. Rillaerts and Joos11 used mercury to perform displacements in circular glass capillary tubes and presented a correlation between the dynamic contact angle and the capillary number. Bracke et al.17 utilized a continuous solid strip drawn into a large pool of liquid and measured dynamic contact angle. This resulted in a correlation between the dynamic contact angle values and the speed by which the strip was drawn into the liquid. Another study by Girardo et al.19 focused on the effect of roughness of the walls of trapezoidal polydimethylsiloxane (PDMS) microchannels on the dynamics of imbibition by ethanol. The authors concluded that the roughness of the walls of a microcapillary plays an important role in the dynamics of advancement of the wetting front on a solid surface and the values of dynamic contact angle. It also results in the “stick−slip” motion of the wetting liquid at the edges of the microcapillary tubes with rough walls. Li et al.18 measured dynamic contact angle values in horizontal glass capillary tubes of 100−250 μm diameter using several liquids ranging from silicone oils with different viscosities to deionized water and crude oils. They reported the change of dynamic contact angle with the change of the velocity of the contact line at low capillary numbers for different fluid/tube sizes. They also derived a master curve which relates the dynamic contact angle variation at a specific contact line velocity with the Crispation number, Cr = (ηα)/(σl), in which Cr is the Crispation number, η is viscosity, α is thermal diffusivity, σ is the surface tension, and l is the length scale or the pore radius. Researchers have used hydrodynamic and the molecular kinetic theories to explain the reason for the variations in dynamic contact angle with changes in meniscus velocity. The hydrodynamic theory divides the distance from the surface of the solid to the bulk of the moving liquid into three regions of micro-, meso-, and macroscopic scales. It states that the bending of the liquid−gas interface due to viscous forces within the mesoscopic region is the main reason for the changes in the experimentally observed dynamic contact angle. On the other hand, the microscopic dynamic contact angle is governed by intermolecular short-range forces and is equal to the static contact angle.8,10,21,22 In the molecular kinetic theory, the dependence of dynamic contact angle to the velocity of contact line is studied at the molecular level and is related to the attachment and detachment of liquid molecules to and from the solid surface. Therefore, the microscopic dynamic contact angle is considered to be velocity dependent, and the same as the macroscopically measured dynamic contact angle.4,5,23,34 There are two main approaches that are generally used to take the effect of dynamic contact angle into account in the modeling of dynamics of rise in capillary tubes. The first approach is based on incorporation of the dependence of dynamic contact angle on velocity of contact line or capillary number. In other words, one can improve rise-versus-time models by integrating those relationships with rise equations. This approach has successfully been utilized in Hoffman’s molecular model12 and Cox’s theoretical approach.15 It is based on the solution of Stoke’s equation and the assumption of fluid slip in the vicinity of the three-phase contact line. This approach is also used in Joos’ model20 in which the correlation proposed by Bracke et al.17 is integrated with Poiseuille’s law. B

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Table 1. Properties of Liquids, Surface Tensions, and Tube Sizes and Tilt Angles Used in This Study at a Temperature of 25°C WP

ρ (gr/cm3)

η (cP)

σ (dyn/cm)

NWP

ID (mm)

θtilt

glycerol Soltrol 170 water

1.26 0.774 0.997

1011.1 2.6 1.1

63.47 24.83 72.8

air air air

0.5, 1.0, 2.0 0.75, 1.0, 1.3 0.75, 1.0, 1.3

90°, 45° 90° 90°

Figure 1. Schematic of the experimental setup.

versus-time and the change of dynamic contact angle with meniscus velocity in capillary rise experiments for different fluid systems and tilt angles using capillary tubes with circular and angular cross-sectional shapes. Furthermore, we present a new general correlation for the variation of dynamic contact angle versus meniscus rise that is developed based on our experimental measurements. The correlation can be used to (1) validate their theoretical counterparts and (2) predict, when coupled with Washburn’s equation, rise-versus-time trends for other systems for which measured dynamic contact angle data may not be available. The data can also be used for validation of numerical or theoretical solutions of rise as well as development of new correlation for dependence of dynamic contact angle on velocity of meniscus. Finally, the observed trends can be incorporated in, for instance, dynamic pore-scale network models used to predict multiphase flow functions (e.g., relative permeabilities and capillary pressures) in porous media. In this document, we first present the materials and the experimental setup and procedure used to perform the rise experiments. Our experimental results are first validated against dynamic contact angle data available in the literature. Measured rise data are then compared with those predicted by the Washburn’s equation with and without experimental values of dynamic contact angle. Four widely used semiempirical and theoretical correlations of dynamic contact angle versus velocity of contact line are validated against our experimental data. A new correlation is introduced for variation of dynamic contact angle versus rise for different fluid systems in tubes with varying internal diameters and tilt angles. We then present experimental data of rise in noncircular capillary tubes and study the rise of

both MTM and AMs. The paper is then concluded by a set of final remarks.

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EXPERIMENTAL SECTION

Materials and Properties. We used water, glycerol, and Soltrol 170 as the wetting fluids and the laboratory air as the nonwetting phase in the experiments performed under this study. Ultra clean distilled water was obtained from a water distiller made of glass, ensuring there were no contaminants present in the water. Certified ACS Fisher Scientific glycerol was used as received and Soltrol 170 was supplied by Chevron Phillips Chemical Company, The Woodlands, TX. Soltrol 170 was purified using a dual-packed column of silica gel and alumina. Glass capillary tubes of 0.5, 0.75, 1.0, 1.3, and 2 mm internal diameter with circular and square cross sections were obtained from Friedrich & Dimmock Inc., Millville, NJ. The tubes were 50 cm long, and therefore, they were cut, depending on the tube internal diameter, to a desired length for each experiment. The tips of the tubes were straightened using a rotary drill and a very fine sand paper. Each single tube was used only once to avoid any possible contaminations and to enhance the accuracy and reproducibility of the results. Every single capillary tube was thoroughly cleaned to establish strongly water-wet glass surfaces. Glass capillary tubes were first rinsed with isopropyl alcohol and then with 150 mL of distilled water. The tubes were immersed in a mixture of 0.5 L of sulfuric acid (95-98)% obtained from Sigma-Aldrich and 25 g of Nochromix from Godax Laboratories Inc., Cabin John, MD. The beaker containing the tubes in the acid solution was then placed in an ultrasonic bath for 15 min. They were then left in the same solution to soak overnight. The tubes were vacuum rinsed thoroughly by flowing 600 cm3 of distilled water through them using the laboratory’s vacuum line.41 Upon finishing the cleaning procedure, the tubes used in experiments with Soltrol 170 and glycerol were vacuum-dried for 1 min and then immediately used to perform the flow tests. For experiments with water, the tubes were C

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partially dried using a piece of paper filter and immediately used to perform the measurements. Dry tubes were not used in water experiments because clean dry borosilicate glass surfaces and distilled water are both very active6 and a small amount of contamination dramatically affects the values of contact angle and equilibrium rise. Fluid densities used in the experiments were measured using Anton Paar DMA 5000 M density meter. Viscosities were obtained using a Cambridge Viscosity viscometer, and surface tensions were determined with a state-of-the-art IFT measurement system using pendant drop technique. All measurements were made at ambient temperature and pressure conditions. The inner diameters of the tubes were measured optically with 0.01 mm precision using a camera and a highmagnification lens attached to a vertical positioning column. Table 1 lists the measured values as well as the fluid pairs and tube sizes for each group of rise experiments. In Table 1, WP stands for wetting phase, ρ is the density, η is the viscosity, σ is the surface tension, NWP is the nonwetting phase, ID is the inner diameter of the capillary tube, and θtilt is the tilt angle of the axis of the capillary tube (along the length) with respect to a horizontal plane. Experimental Setup. A precise positioning column was built to hold and move the capillary tubes in the vertical direction. The column could be tilted to perform rise experiments with different tilt angles. Two high-speed cameras, Sony SCD-V60CR and Phantom V310, were employed to record the position and shape of the moving meniscus. The cameras were also used to time-stamp the images. Two different types of lenses with different magnifications were utilized. The one with the lower magnification was used to detect the position of the meniscus and time-stamp the images during the rise experiments. The lens with the higher magnification captured the shape of the meniscus, providing high-quality, high-magnification images in order to measure dynamic contact angles during the rise. A Schott fiber optic flat backlight system with an active area of 20 × 20 cm2 was mounted to evenly illuminate the capillary tubes (Figure 1). Experimental Procedure. Each capillary tube was placed in the capillary tube holder attached to a precise vertical positioning column. The column’s position was controlled manually and accurately (0.01 mm increments) using an adjustment knob. The invading fluid was poured in a wide Petri dish, 9 cm in diameter, to eliminate any surface curvature caused by the edges of the dish that could affect the rise experiment. The tube was lowered slowly toward the surface of the fluid in the Petri dish. Recording of the images was started a few seconds before the tube tip touched the fluid surface and continued until the meniscus finally stopped at the equilibrium height. The procedure to measure the dynamic contact angle was different from the one used to measure the rise-versus-time values; that is, to measure the dynamic contact angle, it was necessary to have a magnified image of the interface. This in turn meant that the field of view of the camera/lens system had to be less than the whole height of rise. Thus, for glycerol with a high viscosity and a low rise velocity, the camera could be moved up along with the meniscus, while recording. However, for water and Soltrol 170, having low viscosities and very high speeds of rise, the tube length had to be divided into several intervals and imaged separately. For example, for a rise of 20 mm, if the field of view of the high magnification lens was 5 mm, one would need at least four separate measurements to cover the whole range of the dynamic contact angle for one tube/fluid combination. Furthermore, for reproducibility purposes, each experiment for a tube/fluid set was repeated at least three times, and if all the results compared well with each other within experimental error, the tests were called acceptable. During the contact angle measurement tests, we also recorded rise-versus-time data. These data agreed very well with those generated by the experiments mentioned in the first paragraph of this section. The dynamic contact angle is the angle that the meniscus formed by two fluids makes with a contacting solid surface through the denser phase. In the experiments presented here, one fluid was always air, while the other was glycerol, Soltrol 170, or water. The solid surface was the inner surface of the glass capillary tubes. In order to measure dynamic contact angle using the magnified images recorded during the

flow tests, we, similar to the approach used by Siebold et al.,28 assumed that the meniscus was part of a circle and used the following equation:

θ=

⎛ 2x ⎞ π − 2arctan⎜ m ⎟ ⎝ d ⎠ 2

(1)

where θ is the contact angle, xm is the height, and d is the diameter of the meniscus. To compare with the results obtained using the above-mentioned approach, the contact angles were also determined by drawing a tangent to the meniscus at the point of contact of the fluids with the solid surface, using ImageJ software. The results obtained using these two techniques were comparable within experimental error. We also examined the effect of gravity on the shape of the meniscus. To this end, we calculated the Bond number for our experiments: Bo = (ΔρgL2)/σ, in which Δρ is the difference between the density of the liquid and that of the air, g is acceleration due to gravity, and L is the characteristic length of the system which in this case is the radius of the capillary tube. It gives the ratio of gravity to capillary forces. For all the fluid systems and tube sizes we used in this study (see Table 1), the Bond number ranged between 0.012 and 0.076 except for glycerol experiment in 2 mm tube and Soltrol 170 test in 1.3 mm tube for which Bond numbers were 0.194 and 0.129, respectively. We believe gravity had negligible impact on the meniscus shape in our experiments, but one should note that for the cases of Bond numbers greater than 0.1 (i.e., tests with Glycerol in 2 mm diameter tubes and with Soltrol 170 in 1.3 mm diameter tubes) the shape of the interface might have been slightly affected by gravity. In order to eliminate the refraction of light on the curved surface of the tubes, which could result in a deformed meniscus image, tubes were mounted inside a square cross section cuvette made of glass. The cuvette was open at one end. Its closed end had a hole, made to the size of the outer diameter of the test capillary tube. The capillary tube was passed through the hole to make its tip available for contact with the test fluid surface. Figure 2 shows the difference between the cases

Figure 2. Meniscus image without the glycerol bath (left) and with glycerol bath (right). with and without a cuvette. In the former, the space between the capillary tube and the cuvette was filled with glycerol because it has the same refractive index as the glass. The above-mentioned setup eliminated any image distortion caused by the refraction of the light.

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RESULTS AND DISCUSSION In this section, we present and discuss all the results generated under this study. We first compare our dynamic contact angle data against those available in the literature. We then investigate the ability of rise equations in predicting our measurements with and without use of measured contact angles. Some of the correlations available in the literature for variation of dynamic contact angle with the velocity of moving meniscus are tested against our experimental data. We then present a general dynamic contact angle versus rise correlation that can be used for a wide range of applications. These are followed by the D

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Figure 3. Variation of dynamic contact angle-versus-capillary number for glycerol/air in 45° tilted and vertical glass capillary tubes and Soltrol 170/ air in vertical glass capillary tubes (top), and water/air in vertical glass capillary tubes (bottom). Hoffman’s7 experimental data are included for comparison.

effect of thickness of water film on dynamic contact angle by Hirasaki and Yang.30 The trend still compares relatively well with those published by Hoffman.7 It is noteworthy that the dynamic contact angle for a given capillary number shows, within experimental error, weak sensitivity to the type of fluid system and the tilt angle used. This may have important implications for development of modeling tools used to predict flow at the pore scale in porous media. Dynamics of Capillary Rise. In this section, we provide an extensive data set characterizing the dynamics of rise versus time in capillary tubes with different internal diameters and with different invading liquids. We compare the data with those predicted by Washburn equation using a fixed contact angle as well as our measured dynamic contact angles. We start with a brief discussion of Washburn’s equation.2 It was originally developed for a single capillary tube of uniform circular cross section. It was assumed that the velocity of fluid penetrating the tube would, after a short initial period, drop to a value at which the conditions of Poiseuille flow establish and persist. Poiseuille’s equation, neglecting any air resistance, is as follows:

study of displacements in capillary tubes with square cross section. Validation. The dynamic contact angle is dependent on the velocity of the moving meniscus, or capillary number. In Figure 3 (top), we show the variation of dynamic contact angle with capillary number for displacement of air by glycerol in circular capillary tubes with 0.5, 1.0, and 2.0 mm ID and at 45° and 90° tilt angles as well as those of air/Soltrol 170 in vertical tubes of 0.75, 1.0, and 1.3 mm ID. The results are compared against the experimental data reported by Hoffman.7 The agreement is encouraging and indicates the accuracy and reproducibility of our measurements. Furthermore, our results are consistent when tubes with various sizes and tilt angles are used. Figure 3 (bottom) presents similar measurements for the air/ water system in vertical circular glass capillary tubes with 0.75, 1.0, and 1.3 mm ID. The measured values pertaining to some of the rise tests with this fluid system show slight deviation from those presented by Hoffman.7 This uncertainty may have been introduced into the measured dynamic contact angle values due to presence of a thin water film on the inner walls of the capillary tubes, as discussed in a more detailed study on the E

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Figure 4. Experimentally measured values of rise versus time for glycerol/air in 45° tilted (left column) and vertical glass capillary tubes (right column) of different sizes. Predicted values of Washburn equation with fixed contact angle and with measured values of θd are included for comparison.

dV =

π ΣP 4 (r + 4εr 3) dt 8ηl

(

PA + ρg (h − ls sin ψ ) + dl = dt 8ηl

(2)

where dV is the volume of the liquid which flows during the time dt through any cross section of the capillary tube, l is the length of the column of liquid in the capillary at time t, η is the viscosity of the liquid, ε is the coefficient of slip, r is the radius of the capillary tube, and ΣP is the total effective pressure which acts to force the liquid along the capillary and is the sum of three separate pressures: the unbalanced atmospheric pressure, PA, the hydrostatic pressure, Ph, and the capillary pressure, Ps. The following ordinary differential equation for the penetration velocity was derived and integrated for the rise in single capillary tubes:2

2σ cos r

)

θ (r 2 + 4εr ) (3)

where g is the acceleration due to gravity, h is the height of liquid column, ls is the linear distance between the tip of the tube and any point along the tube, ψ is the angle that the straight line between the tip of the tube to any point along the tube makes with a horizontal surface, ρ is the fluid density, and θ is the contact angle. Washburn assumed that θ was constant. Findings of different investigations available in the literature (e.g., Hoffman,7 Jiang et al.,9 and Bracke et al.17) as well as our experimental results presented in the previous section show strong sensitivity of dynamic contact angle to variations in the velocity of the moving meniscus. Those findings along with the fact that the velocity of the meniscus changes as the liquid rises F

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Figure 5. Experimentally measured values of rise versus time for Soltrol 170 (left column) and water (right column) in vertical glass capillary tubes of different sizes. Predicted values of Washburn equation with fixed contact angle and with measured values of θd are included for comparison.

in a capillary tube indicate that Washburn’s assumption of fixed contact angle during the rise introduces uncertainty into the predicted rise-versus-time values. In order to mitigate this uncertainty, measured experimental values of dynamic contact angle must be incorporated. We have therefore used our measured values of this parameter in the original Washburn’s equation and compared the predicted rise-versus-time values against their experimental counterparts also reported in this study. For reference, we also include predictions using a fixed value of contact angle (i.e., zero). Two methods for implementing the experimental values of dynamic contact angle into Washburn’s equation were examined. The first approach involved fitting the contact angle values with a curve and the second approach was to implement the contact angle measurements on a point-by-point basis. The latter was used in this study as it led to less discrepancy between predicted results and the measured values. The coefficient of slip was assumed to be zero in our calculations.

The comparisons are shown in Figures 4 and 5. It is seen that in all cases the values predicted using measured dynamic contact angles expectedly agree much more closely with the experimental rise counterparts than those obtained with a fixed contact angle of zero. The agreements are encouraging, indicating the accuracy and reproducibility of the measurements. Interestingly, when measured dynamic contact angles are used with Washburn’s equation, the deviations between the measured and predicted rise-versus-time values for large capillary tubes become smaller, or comparable to, those of smaller tubes (see Figures 4 and 5). In the case of experiments with water, we observe a level of discrepancy that might be attributed to the water film present in the tubes at the start of each experiment, which could have impacted the measured contact angle values. Correlations for Velocity-Dependent Dynamic Contact Angle. Over the last several decades, researchers have introduced various semiempirical correlations to describe the G

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velocity-dependence of dynamic contact angle.15,20,22,42 Popescu et al.36 performed comparative analysis of the predictions made by some of these models for dynamic contact angle versus the velocity of the contact line for typical water and highviscosity silicone oil systems. The authors, however, did not compare any of the predicted trends against independent experimental counterparts mainly due to lack of such data in the literature. In this section, we use the experimental dynamic capillary rise data presented earlier to examine the predictive capabilities of four correlations presented by Cox,15 Joos et al.,20 Shikhmurzaev, 22 and Sheng.42 We discuss the details of the models followed by comparison of the predicted trends against our experimental measurements. Bracke et al.17 used two experimental methods to study the dependence of dynamic contact angle on the velocity of contact line. In the first set of experiments, they used polyethylene/ polyethylene terephthalate solid strips drawn into pools of different aqueous glycerol solutions, aqueous ethylenegelycol solutions, and ordinary corn oil. In the second method, they utilized a dry platina Wilhelmy plate. Joos et al.20 then used the experimental data to develop a semiempirical correlation for the dependence of dynamic contact angle to the velocity of contact line in a circular capillary tube. They replaced the dynamic contact angle term in Washburn’s equation2 with their correlation for the dynamic contact angle17 and showed that this semiempirical correlation leads to more accurate predictions of capillary rise than that of Washburn. The proposed empirical correlation is given by cos(θd) = cos(θe) − 2(1 + cos(θe))Ca1/2

divided by microscopic length scales.36 The parameters used with Cox’s model are also tabulated in Table 2. Utilizing numerical hydrodynamic calculations, Hoffman’s7 slipping function, and eq 6, Sheng and Zhou42 linked macroscopic flow behavior (e.g., dynamic contact angle) to the microscopic parameters governing the contact-line region. The authors also introduced parameters to take the effect of surface roughness into account. G(θ ) = G(θe) + Ca ln(K /ls)

where G(q) =

∫0

q

dϕ[f (ϕ)]−1

and f (ϕ) = ⎡⎣2sin ϕ{q2(ϕ2 − sin 2 ϕ) + 2q[ϕ(π − ϕ) + sin 2 ϕ] + (π − ϕ)2 − sin 2 ϕ}⎤⎦ ÷ {q(ϕ2 − sin 2 ϕ)[(π − ϕ) + sin ϕ cos ϕ] + (ϕ − sin ϕ cos ϕ)[(π − ϕ)2 − sin 2 ϕ]}

In the above equations, ls is the slipping length and K is a slipping model dependent constant. We also considered the mathematical model proposed by Shikhmurzaev.22 The author suggested that the surface tension gradient caused by the flow of a liquid flowing on a solid surface, influences the flow. This gradient, in the case of small capillaries, determines the dynamic contact angle. The shear stress singularity present in classical approaches is eliminated in this model. Blake and Shikhmurzaev29 and Popescu et al.36 simplified Shikhmurzaev’s original model into eq 7:

(4)

where θd is the dynamic contact angle in radians, θe is the equilibrium contact angle in radians, and Ca = (ηv)/σ is the capillary number, in which η is the fluid viscosity, σ is the surface tension, and v the velocity of contact line. Table 2 lists all the parameters needed to use this correlation with our fluid systems.

cos(θe) − cos(θd) =

2V (ρ2es* + ρ1es*u0) (1 − ρ1es*)[(ρ2es* + V 2)1/2 + V ] (7)

where

Table 2. Parameters Used with the Models Proposed by Joos,20 Cox,15 Shikhmurzaev,22 and Sheng42

u0(θd , 0) =

sin(θd) − θd cos(θd) sin(θd) cos(θd) − θd

fluid

surface tension (dyn/cm)

viscosity (cP)

θe (rad)

ρs1e*22

Sc

σ*SG22

V = Sc × Ca

glycerol Soltrol 170 water

63.47 24.83 72.8

1011.1 2.6 1.1

0 0 0

0.54 0.54 0.54

17.90 12.56 12.5

−0.07 −0.07 −0.07

*) ρ2es* = 1 + (1 − ρ1es*)(cos(θe) − σSG

In the above equations, Ca is the capillary number, Sc is a scaling factor that depends on the material properties, and ρs2e* and ρs1e* are two phenomenological coefficients (see Blake and Shikhmurzaev29 for more details). Table 2 lists values of the parameters used with the above-mentioned model (Sc for water is obtained from Popescu et al.36). We compare our experimental dynamic contact angle data for glycerol, Soltrol 170, and water with the trends predicted by these four correlations. For each of the models, we use our measured values of the physical parameters (i.e., viscosity and surface tension) and correlation coefficients available in the literature for the same fluid pair (see Table 2). Figure 6 compares our measured dynamic contact angles versus velocity of the triple contact line against the trends predicted by the above-mentioned correlations. This figure shows that the model proposed by Joos generates the best match against the Soltrol 170 data. It predicts the low velocity trend for glycerol well, but deviates at larger velocity

In a thermodynamics-based approach, Cox15 assumed a microscopic slip boundary condition for a moving fluid on a solid surface. This assumption helps removing the stress singularity at the triple contact line of the system. Cox’s analysis resulted in the following relationship for the dynamic contact angle versus velocity of the contact line: ηv G(θd) = G(θe) + χ (5) σ where G (θ ) =

∫0

θ

(6)

x − sin(x) cos(x) dx 2sin(x)

In this correlation, v is the contact line velocity and χ ≈ 16, which is defined as the natural logarithm of macroscopic H

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Figure 6. Experimentally measured variation of dynamic contact angle versus the velocity of the contact line for glycerol (top), Soltrol 170 (middle), and water (bottom). Predicted trends by the models of Joos,20 Cox,15 Shikhmurzaev,22 Sheng42are included for comparison.

values of 0.41, 2.9 × 10−3 and 4.1 × 10−4 for glycerol, Soltrol 170 and water respectively. Sheng’s model predicts the experimental data for glycerol and Soltrol 170 well; however, it slightly overpredicts the experimental data for water at almost all velocities. In case of Shikhmurzaev’s model, one needs to use parameters that can be found only for a very few fluid systems in the literature; see, for instance, Popescu et al.36 and Blake and Shikhmurzaev.29 When available, we have used parameters from literature (i.e., for distilled water). In other cases, that is, for Soltrol 170 and glycerol, we adjusted Sc to 12.56 and 17.90, respectively, to obtain the best agreement with the

values. In the case of water, it overpredicts the experimental data at both low and high velocities. Cox’s model produces a very good fit in the case of glycerol, while it overpredicts the Soltrol 170 experimental data particularly at higher velocities. This model shows more significant deviation from the experimental data in the case of water. It is important to note that there are no fitting parameters used in the calculations performed with these two models. This is, however, not the case for Sheng’s and Shikhmurzaev’s models. In Sheng’s model, the value of ls = 10−7 cm was obtained from molecular dynamic simulations,42 while K was the fitting parameter which has I

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Figure 7. General trend for variation of normalized rise, Ln, with changes in dynamic contact angle, θd, generated using the experimental data gathered under this study. The experimental data presented by Siebold et al.28 are included for comparison.

Figure 8. Experimentally measured values of rise versus time for the AMs (top) and MTMs (bottom) for glycerol/air in 0.5, 1.0, and 2.0 mm ID square capillary tubes. The solid line represents a slope of 1/3 for late-time AMs and 1/2 for early-time MTMs, following Ponomarenko et al.40 J

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versus-time for different fluid systems (i.e., water/air, Soltrol 170/air, and glycerol/air) and tube tilt angles. To the best of our knowledge, this is the first time that an extensive experimental data set for variation of dynamic contact angle with capillary number during rise with different fluid systems is reported. Measured dynamic contact angles were used to obtain rise-versus-time trends that agree more closely with the measured rise values (also reported in this study) than those predicted by the Washburn equation using a fixed contact angle. Four empirical models of velocity-dependence on dynamic contact angle were validated by comparing their predicted trends against our experimental data. The predictive capabilities of the models were discussed. A new general trend was introduced for the changes in normalized rise with variations in dynamic contact angle. The trend was compared successfully against the experimental data available in the literature. This relation can be used to find the dynamic contact angle values for a given rise, if the invading fluid density, surface tension, tube size, and tilt angle of the system are known. Combined with Washburn equation, one can then produce a rise-versus-time curve for the system. Finally, we presented our measurements of rise of glycerol in square cross section capillary tubes with different sizes. We reported rise-versus-time for both AMs and MTM. The AM late time scales and MTM early time scales data showed an encouraging agreement with the universal relationships proposed by Ponomarenko et al.40 for rise in capillary tubes with angular cross section. The experimental data and the trends presented here can be used in dynamic pore-scale models of flow in porous media to predict multiphase flow functions.

experimental measurements. All the other parameters were kept the same as the values used by Blake and Shikhmurzaev29 for silicon oil and different glycerol/water solutions, respectively. The model developed by Shikhmurzaev expectedly shows a good agreement with the experimental trends at low and moderate velocities in glycerol and water systems. The model always overpredicts the experimental values at higher velocities. Dynamic Contact Angle and Normalized Rise. In this section, we present, for the systems investigated in this study, a general trend for the changes in normalized rise (Ln = l/le, where l is the value of rise and le is the equilibrium rise), with variations in dynamic contact angle. We consolidated all the rise and dynamic contact angle data (except for those of the water/ air system) generated under this study to obtain the trend shown in Figure 7. The rise and dynamic contact angle data for all fluid/tube-size/tilt-angle combinations follow the same trend. Figure 7 also shows a curve fit for the data that is given by ⎛ −(θ − b)2 ⎞1/4r l d ⎟ = a exp⎜ le 2c 2 ⎝ ⎠

(8)

where r is the radius of the capillary tube in mm, a = 1.03, b = −2.47, and c = 27.7. In this figure, we also include the data presented by Siebold et al.,28 which follow the trend relatively well. There are very limited numbers of dynamic rise-versus-contact angle data sets available in the literature. This relation can be used to find the dynamic contact angle value for a given rise, if the invading fluid density, surface tension, tube size, and tilt angle of the system are known. Combined with the Washburn equation, one can then produce a rise-versus-time curve for the system. Capillary Tubes with Square Cross Section. Here we study the rise in capillary tubes with square cross section. The cross section of a typical square capillary tube used in these experiments is not a perfect square and has round corners. We investigate the rise-versus-time of both AMs and MTMs in these experiments. The data are presented in log−log plots in Figure 8. As shown in this figure, the rise-versus-time of AMs and MTM in 1 and 2 mm square tubes with glycerol/air fluid system follow the universal relationship proposed by Ponomarenko et al.40 As seen, the late time scales of AM rise-versus-time data follow a trend proportional to t1/3, while the early time scales of MTM rise-versus-time values follow a t1/2 profile. As expected, the AMs rise faster ahead of MTMs. Ransohoff and Radke16 presented a dimensionless flow resistance parameter (β), which, among other parameters, depends on the roundedness of the corner of the tube in which the fluid rises. Based on the calculations presented by the authors, the higher is the roundedness of the corner, the greater is the value of the dimensionless flow resistance (β). And the higher is the value of β, the lower is the average velocity of the moving meniscus. Therefore, the large roundedness values in our tubes makes the value of β much larger than it is in sharp cornered channels used by Ponomarenko et al.40 This may have contributed to the slight deviation between the slope of our AM rise-versus-time values from the universal relationships proposed by Ponomarenko et al.40

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support of the School of Energy Resources and the Enhanced Oil Recovery Institute at the University of Wyoming. We extend our gratitude to Professor George Hirasaki for his valuable comments and Henry Plancher and Soheil Saraji of Piri Research Group for their assistance with the capillary tube cleaning procedures and surface tension measurements.

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REFERENCES

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CONCLUSIONS An extensive set of capillary rise experiments were performed in circular and square cross section tubes with various internal diameters ranging from 0.5 to 2.0 mm. We measured riseK

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