Experimental Investigation of Dynamic Contact Angle and

Experimental Investigation of Dynamic Contact Angle and CapillaryRise in Tubes with Circular and Noncircular Cross SectionsMohammad Heshmati* and Mohammad Piri

University of Wyoming, Laramie, Wyoming 82071-2000, United States

ABSTRACT: An extensive experimental study of the kineticsof capillary rise in borosilicate glass tubes of different sizes andcross-sectional shapes using various fluid systems and tube tiltangles is presented. The investigation is focused on the directmeasurement of dynamic contact angle and its variation withthe velocity of the moving meniscus (or capillary number) incapillary rise experiments. We investigated this relationship fordifferent invading fluid densities, viscosities, and surfacetensions. For circular tubes, the measured dynamic contactangles were used to obtain rise-versus-time values that agreemore closely with their experimental counterparts (alsoreported in this study) than those predicted by Washburn equation using a fixed value of contact angle. We study thepredictive capabilities of four empirical correlations available in the literature for velocity-dependence of dynamic contact angleby comparing their predicted trends against our measured values. We also present measurements of rise in noncircular capillarytubes 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 ofnormalized rise with dynamic contact angle that can be used in, for instance, dynamic pore-scale models of flow in porous mediato predict multiphase flow behavior.

■ INTRODUCTIONFlow of fluids through porous media and the associatedcapillarity phenomenon have long been the focus of physicists,soil scientists, petroleum and environmental engineers, andresearchers in many other areas of science, technology, andengineering. This along with the fact that a porous medium is acomplicated system of connected and mostly rough-walledcapillary pores and throats makes the study of fluid/fluiddisplacement mechanisms in capillary systems criticallyrelevant. However, direct pore-level investigation of flow andtransport in random porous mediums is very difficult due toscale and imaging challenges. Using glass micromodels andcapillary tubes simplifies the study of such systems, therebyenabling investigation of complicated displacement physics andparameters such as dynamic contact angle, which are difficult toprobe directly in a naturally-occurring random porous medium.Experimental and modeling studies of displacement processesin simplified capillary systems have long been the focus ofauthors in different research areas. Insights developed throughthese studies coupled with representative pore space topologymaps obtained using, for instance, X-ray imaging technologiesenable development of predictive, physically-based pore-scalemodels of flow and transport in porous media. It is thereforeimperative to investigate subtle aspects of dynamic flow incapillary tubes and thereby enrich and improve the predictivecapabilities 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,

many other scientists in different areas of science andengineering attempted to improve the formulation andassociated analysis.7,9,11,12,14,15,17,20,23,26−28,37−39 Washburn2

modeled the fluid flow in circular capillary tubes usingPoiseuille’s law. The author ignored the changes in contactangle of fluid meniscus during displacement. This assumption isone of the main reasons why the rise-versus-time curvespredicted by the proposed equation do not match theexperimental data. Hence, scientists have tried to improve thepredictions through collection of more experimental data andimprovements in the modeling of the displacement process.Quere26 performed experiments showing that the position ofthe meniscus versus time in the early stages of rise, can bedescribed using a linear relationship. Furthermore, theoscillations around the equilibrium occur if the liquid viscosityis low enough. The author along with Hamraoui et al.27 andSiebold et al.,28 on liquid/air systems, and Mumley et al.,14 onliquid/liquid systems, emphasized on the importance ofimplementing dynamic contact angle in Washburn’s model inorder to obtain a better agreement with the measured rise-versus-time data.There have been several experimental investigations on the

effects of velocity and capillary number on dynamic contactangle. Some of these studies have led to development ofempirical correlations. Hoffman7 performed experiments in

Received: May 7, 2014Revised: October 7, 2014Published: October 16, 2014

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horizontal circular tubes and presented a general trend showingthat dynamic contact angle correlates with capillary number.Using Hoffman’s experimental data, Jiang et al.9 suggested acorrelation for the advancing dynamic contact angle measuredthrough the liquid phase during liquid−gas interface displace-ment in circular glass capillary tubes. Rillaerts and Joos11 usedmercury to perform displacements in circular glass capillarytubes and presented a correlation between the dynamic contactangle and the capillary number. Bracke et al.17 utilized acontinuous solid strip drawn into a large pool of liquid andmeasured dynamic contact angle. This resulted in a correlationbetween the dynamic contact angle values and the speed bywhich the strip was drawn into the liquid. Another study byGirardo et al.19 focused on the effect of roughness of the wallsof trapezoidal polydimethylsiloxane (PDMS) microchannels onthe dynamics of imbibition by ethanol. The authors concludedthat the roughness of the walls of a microcapillary plays animportant role in the dynamics of advancement of the wettingfront on a solid surface and the values of dynamic contact angle.It also results in the “stick−slip” motion of the wetting liquid atthe edges of the microcapillary tubes with rough walls. Li etal.18 measured dynamic contact angle values in horizontal glasscapillary tubes of 100−250 μm diameter using several liquidsranging from silicone oils with different viscosities to deionizedwater and crude oils. They reported the change of dynamiccontact angle with the change of the velocity of the contact lineat low capillary numbers for different fluid/tube sizes. They alsoderived a master curve which relates the dynamic contact anglevariation at a specific contact line velocity with the Crispationnumber, 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 indynamic contact angle with changes in meniscus velocity. Thehydrodynamic theory divides the distance from the surface ofthe solid to the bulk of the moving liquid into three regions ofmicro-, meso-, and macroscopic scales. It states that thebending of the liquid−gas interface due to viscous forces withinthe mesoscopic region is the main reason for the changes in theexperimentally observed dynamic contact angle. On the otherhand, the microscopic dynamic contact angle is governed byintermolecular short-range forces and is equal to the staticcontact angle.8,10,21,22 In the molecular kinetic theory, thedependence of dynamic contact angle to the velocity of contactline is studied at the molecular level and is related to theattachment and detachment of liquid molecules to and from thesolid surface. Therefore, the microscopic dynamic contact angleis considered to be velocity dependent, and the same as themacroscopically measured dynamic contact angle.4,5,23,34

There are two main approaches that are generally used totake the effect of dynamic contact angle into account in themodeling of dynamics of rise in capillary tubes. The firstapproach is based on incorporation of the dependence ofdynamic contact angle on velocity of contact line or capillarynumber. In other words, one can improve rise-versus-timemodels by integrating those relationships with rise equations.This approach has successfully been utilized in Hoffman’smolecular model12 and Cox’s theoretical approach.15 It is basedon the solution of Stoke’s equation and the assumption of fluidslip in the vicinity of the three-phase contact line. Thisapproach is also used in Joos’ model20 in which the correlationproposed by Bracke et al.17 is integrated with Poiseuille’s law.

This is also the case in Lee’s35 and Hilpert’s semianalyticalmodels.37,38 In Lee’s35 work, a simple geometrical model isutilized to solve the force balance equation on the liquidmeniscus. While Hilpert37 applied a power law and a powerseries model for dynamic contact angle in order to generalizeWashburn’s analytical solution for flow in horizontal capillarytubes. For flow in tilted capillary tubes, the author applied thepower law model and a polynomial.38 The resulting semi-analytical models compared well with the numerical solutions.The second approach does not include the velocity of

contact line to correlate the dynamic contact angle. Forinstance, Deganello et al.39 proposed a numerical approach thatdoes not explicitly include velocity of the contact line to modelthe dynamic contact angle. The authors combined anequilibrium of forces in the contact region near the solidboundary with a diffuse free fluid interface within a level-setfinite volume numerical framework. The dynamic contact angleversus capillary number data derived from the model showed anexcellent agreement with the empirical correlations proposedby Hoffman7 and Jiang et al.9 There have also been studies ofcapillary rise performed in microgravity systems.13,24,31−33

Utilizing available experimental data, van Mourik et al.33 testedsome dynamic contact angle models in a numerical simulator.They found that Blake’s theoretical dynamic contact anglemodel23 gives the best agreement with two sets of availableexperimental data presented in their paper.Even though the majority of studies in this domain have

focused on displacements in capillary tubes with circular crosssection, there have been some limited investigations performedon tubes with angular cross sections as well. Ransohoff andRadke16 developed a model for the flow of fluids at lowReynolds numbers in tubes with angular cross section. Theydivided the problem into individual corner flows and solved itnumerically. They defined a dimensionless flow resistanceparameter, β, which depended on the corner half angle, degreeof roundedness, surface shear viscosity, and contact angle. Tangand Tang25 analytically studied the dynamics of fluid flow intubes with sharp grooves and proposed that when the diameterof the tube is smaller than the capillary length, the early-timerise-versus-time data for main terminal meniscus (MTM) andlate-time data for arc meniscus (AM) follow t1/2 and t1/3

relationships, respectively. This is also reported in the studiesperformed by Ponomarenko et al.40 in which they found auniversal 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 displacementsin capillary tubes with varying cross-sectional geometries andwettabilities. Therefore, the extent and quality of theexperimental data on this interfacial parameter control thepredictive capabilities of the models that one can develop. Tothe best of our knowledge, the experimental data on dynamiccontact angle in capillary rise experiments are scarce and havebeen generated under limited range of relevant conditions. Inother words, there are very limited number of experimentaldata sets available in the literature that can be used tocharacterize the variation of dynamic contact angle for differentcapillary tube/fluid systems. For instance, majority of theexperimental studies focused on measuring dynamic contactangles have been performed in horizontal capillary tubes using apiston to move the fluid phases, or as in the case of Bracke etal.,17 a continuous solid strip has been drawn into a large poolof liquid. In this work, we present, to the best of our knowledge,the first extensive, well-characterized experimental study of rise-

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dx.doi.org/10.1021/la501724y | Langmuir 2014, 30, 14151−1416214152

versus-time and the change of dynamic contact angle withmeniscus velocity in capillary rise experiments for different fluidsystems and tilt angles using capillary tubes with circular andangular cross-sectional shapes. Furthermore, we present a newgeneral correlation for the variation of dynamic contact angleversus meniscus rise that is developed based on ourexperimental measurements. The correlation can be used to(1) validate their theoretical counterparts and (2) predict, whencoupled with Washburn’s equation, rise-versus-time trends forother systems for which measured dynamic contact angle datamay not be available. The data can also be used for validation ofnumerical or theoretical solutions of rise as well as developmentof new correlation for dependence of dynamic contact angle onvelocity of meniscus. Finally, the observed trends can beincorporated in, for instance, dynamic pore-scale networkmodels used to predict multiphase flow functions (e.g., relativepermeabilities and capillary pressures) in porous media.In this document, we first present the materials and the

experimental setup and procedure used to perform the riseexperiments. Our experimental results are first validated againstdynamic contact angle data available in the literature. Measuredrise data are then compared with those predicted by theWashburn’s equation with and without experimental values ofdynamic contact angle. Four widely used semiempirical andtheoretical correlations of dynamic contact angle versus velocityof contact line are validated against our experimental data. Anew correlation is introduced for variation of dynamic contactangle versus rise for different fluid systems in tubes with varyinginternal diameters and tilt angles. We then present experimentaldata of rise in noncircular capillary tubes and study the rise of

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

■ EXPERIMENTAL SECTIONMaterials and Properties. We used water, glycerol, and Soltrol

170 as the wetting fluids and the laboratory air as the nonwettingphase in the experiments performed under this study. Ultra cleandistilled water was obtained from a water distiller made of glass,ensuring there were no contaminants present in the water. CertifiedACS Fisher Scientific glycerol was used as received and Soltrol 170 wassupplied by Chevron Phillips Chemical Company, The Woodlands,TX. Soltrol 170 was purified using a dual-packed column of silica geland alumina.

Glass capillary tubes of 0.5, 0.75, 1.0, 1.3, and 2 mm internaldiameter with circular and square cross sections were obtained fromFriedrich & 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 werestraightened using a rotary drill and a very fine sand paper. Each singletube was used only once to avoid any possible contaminations and toenhance the accuracy and reproducibility of the results.

Every single capillary tube was thoroughly cleaned to establishstrongly water-wet glass surfaces. Glass capillary tubes were first rinsedwith isopropyl alcohol and then with 150 mL of distilled water. Thetubes were immersed in a mixture of 0.5 L of sulfuric acid (95-98)%obtained from Sigma-Aldrich and 25 g of Nochromix from GodaxLaboratories Inc., Cabin John, MD. The beaker containing the tubes inthe acid solution was then placed in an ultrasonic bath for 15 min.They were then left in the same solution to soak overnight. The tubeswere vacuum rinsed thoroughly by flowing 600 cm3 of distilled waterthrough them using the laboratory’s vacuum line.41 Upon finishing thecleaning procedure, the tubes used in experiments with Soltrol 170 andglycerol were vacuum-dried for 1 min and then immediately used toperform the flow tests. For experiments with water, the tubes were

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 1.26 1011.1 63.47 air 0.5, 1.0, 2.0 90°, 45°Soltrol 170 0.774 2.6 24.83 air 0.75, 1.0, 1.3 90°water 0.997 1.1 72.8 air 0.75, 1.0, 1.3 90°

Figure 1. Schematic of the experimental setup.

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partially dried using a piece of paper filter and immediately used toperform the measurements. Dry tubes were not used in waterexperiments because clean dry borosilicate glass surfaces and distilledwater are both very active6 and a small amount of contaminationdramatically 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 aCambridge Viscosity viscometer, and surface tensions were deter-mined with a state-of-the-art IFT measurement system using pendantdrop technique. All measurements were made at ambient temperatureand pressure conditions. The inner diameters of the tubes weremeasured optically with 0.01 mm precision using a camera and a high-magnification lens attached to a vertical positioning column. Table 1lists the measured values as well as the fluid pairs and tube sizes foreach 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 isthe inner diameter of the capillary tube, and θtilt is the tilt angle of theaxis of the capillary tube (along the length) with respect to a horizontalplane.Experimental Setup. A precise positioning column was built to

hold and move the capillary tubes in the vertical direction. The columncould be tilted to perform rise experiments with different tilt angles.Two high-speed cameras, Sony SCD-V60CR and Phantom V310, wereemployed to record the position and shape of the moving meniscus.The cameras were also used to time-stamp the images. Two differenttypes of lenses with different magnifications were utilized. The onewith the lower magnification was used to detect the position of themeniscus and time-stamp the images during the rise experiments. Thelens with the higher magnification captured the shape of the meniscus,providing high-quality, high-magnification images in order to measuredynamic contact angles during the rise. A Schott fiber optic flat back-light system with an active area of 20 × 20 cm2 was mounted to evenlyilluminate 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 (0.01mm increments) using an adjustment knob. The invading fluid waspoured in a wide Petri dish, 9 cm in diameter, to eliminate any surfacecurvature caused by the edges of the dish that could affect the riseexperiment. The tube was lowered slowly toward the surface of thefluid in the Petri dish. Recording of the images was started a fewseconds before the tube tip touched the fluid surface and continueduntil 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, tomeasure the dynamic contact angle, it was necessary to have amagnified image of the interface. This in turn meant that the field ofview of the camera/lens system had to be less than the whole height ofrise. Thus, for glycerol with a high viscosity and a low rise velocity, thecamera could be moved up along with the meniscus, while recording.However, for water and Soltrol 170, having low viscosities and veryhigh speeds of rise, the tube length had to be divided into severalintervals and imaged separately. For example, for a rise of 20 mm, ifthe field of view of the high magnification lens was 5 mm, one wouldneed at least four separate measurements to cover the whole range ofthe dynamic contact angle for one tube/fluid combination.Furthermore, for reproducibility purposes, each experiment for atube/fluid set was repeated at least three times, and if all the resultscompared well with each other within experimental error, the testswere called acceptable. During the contact angle measurement tests,we also recorded rise-versus-time data. These data agreed very wellwith those generated by the experiments mentioned in the firstparagraph 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 denserphase. In the experiments presented here, one fluid was always air,while the other was glycerol, Soltrol 170, or water. The solid surfacewas the inner surface of the glass capillary tubes. In order to measuredynamic contact angle using the magnified images recorded during the

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

θ π= − ⎜ ⎟⎛⎝

⎞⎠

xd2

2arctan2 m

(1)

where θ is the contact angle, xm is the height, and d is the diameter ofthe meniscus.

To compare with the results obtained using the above-mentionedapproach, the contact angles were also determined by drawing atangent to the meniscus at the point of contact of the fluids with thesolid surface, using ImageJ software. The results obtained using thesetwo techniques were comparable within experimental error. We alsoexamined the effect of gravity on the shape of the meniscus. To thisend, we calculated the Bond number for our experiments: Bo =(ΔρgL2)/σ, in which Δρ is the difference between the density of theliquid and that of the air, g is acceleration due to gravity, and L is thecharacteristic length of the system which in this case is the radius ofthe capillary tube. It gives the ratio of gravity to capillary forces. For allthe 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 glycerolexperiment in 2 mm tube and Soltrol 170 test in 1.3 mm tube forwhich Bond numbers were 0.194 and 0.129, respectively. We believegravity had negligible impact on the meniscus shape in ourexperiments, but one should note that for the cases of Bond numbersgreater than 0.1 (i.e., tests with Glycerol in 2 mm diameter tubes andwith Soltrol 170 in 1.3 mm diameter tubes) the shape of the interfacemight have been slightly affected by gravity.

In order to eliminate the refraction of light on the curved surface ofthe tubes, which could result in a deformed meniscus image, tubeswere mounted inside a square cross section cuvette made of glass. Thecuvette was open at one end. Its closed end had a hole, made to thesize of the outer diameter of the test capillary tube. The capillary tubewas passed through the hole to make its tip available for contact withthe test fluid surface. Figure 2 shows the difference between the cases

with and without a cuvette. In the former, the space between thecapillary tube and the cuvette was filled with glycerol because it has thesame refractive index as the glass. The above-mentioned setupeliminated any image distortion caused by the refraction of the light.

■ RESULTS AND DISCUSSIONIn this section, we present and discuss all the results generatedunder this study. We first compare our dynamic contact angledata against those available in the literature. We then investigatethe ability of rise equations in predicting our measurementswith and without use of measured contact angles. Some of thecorrelations available in the literature for variation of dynamiccontact angle with the velocity of moving meniscus are testedagainst our experimental data. We then present a generaldynamic contact angle versus rise correlation that can be usedfor a wide range of applications. These are followed by the

Figure 2. Meniscus image without the glycerol bath (left) and withglycerol bath (right).

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study of displacements in capillary tubes with square crosssection.Validation. The dynamic contact angle is dependent on the

velocity of the moving meniscus, or capillary number. In Figure3 (top), we show the variation of dynamic contact angle withcapillary number for displacement of air by glycerol in circularcapillary 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 of0.75, 1.0, and 1.3 mm ID. The results are compared against theexperimental data reported by Hoffman.7 The agreement isencouraging and indicates the accuracy and reproducibility ofour measurements. Furthermore, our results are consistentwhen 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 ofthe rise tests with this fluid system show slight deviation fromthose presented by Hoffman.7 This uncertainty may have beenintroduced into the measured dynamic contact angle values dueto presence of a thin water film on the inner walls of thecapillary tubes, as discussed in a more detailed study on the

effect of thickness of water film on dynamic contact angle byHirasaki and Yang.30 The trend still compares relatively wellwith those published by Hoffman.7 It is noteworthy that thedynamic contact angle for a given capillary number shows,within experimental error, weak sensitivity to the type of fluidsystem and the tilt angle used. This may have importantimplications for development of modeling tools used to predictflow at the pore scale in porous media.

Dynamics of Capillary Rise. In this section, we provide anextensive data set characterizing the dynamics of rise versustime in capillary tubes with different internal diameters andwith different invading liquids. We compare the data with thosepredicted by Washburn equation using a fixed contact angle aswell as our measured dynamic contact angles. We start with abrief discussion of Washburn’s equation.2 It was originallydeveloped for a single capillary tube of uniform circular crosssection. It was assumed that the velocity of fluid penetrating thetube would, after a short initial period, drop to a value at whichthe conditions of Poiseuille flow establish and persist.Poiseuille’s equation, neglecting any air resistance, is as follows:

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 forcomparison.

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πη

ε= Σ +VPl

r r td8

( 4 ) d4 3

(2)

where dV is the volume of the liquid which flows during thetime dt through any cross section of the capillary tube, l is thelength of the column of liquid in the capillary at time t, η is theviscosity of the liquid, ε is the coefficient of slip, r is the radiusof the capillary tube, and ΣP is the total effective pressure whichacts to force the liquid along the capillary and is the sum ofthree 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 insingle capillary tubes:2

ρ ψ θ ε

η=

+ − + +σ( )lt

P g h l r r

ldd

( sin ) cos ( 4 )

8rA s

2 2

(3)

where g is the acceleration due to gravity, h is the height ofliquid column, ls is the linear distance between the tip of thetube and any point along the tube, ψ is the angle that thestraight line between the tip of the tube to any point along thetube makes with a horizontal surface, ρ is the fluid density, andθ 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 ourexperimental results presented in the previous section showstrong sensitivity of dynamic contact angle to variations in thevelocity of the moving meniscus. Those findings along with thefact that the velocity of the meniscus changes as the liquid rises

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

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in a capillary tube indicate that Washburn’s assumption of fixedcontact angle during the rise introduces uncertainty into thepredicted rise-versus-time values. In order to mitigate thisuncertainty, measured experimental values of dynamic contactangle must be incorporated. We have therefore used ourmeasured values of this parameter in the original Washburn’sequation and compared the predicted rise-versus-time valuesagainst their experimental counterparts also reported in thisstudy. For reference, we also include predictions using a fixedvalue of contact angle (i.e., zero). Two methods forimplementing the experimental values of dynamic contactangle into Washburn’s equation were examined. The firstapproach involved fitting the contact angle values with a curveand the second approach was to implement the contact anglemeasurements on a point-by-point basis. The latter was used inthis study as it led to less discrepancy between predicted resultsand the measured values. The coefficient of slip was assumed tobe zero in our calculations.

The comparisons are shown in Figures 4 and 5. It is seen thatin all cases the values predicted using measured dynamiccontact angles expectedly agree much more closely with theexperimental rise counterparts than those obtained with a fixedcontact angle of zero. The agreements are encouraging,indicating the accuracy and reproducibility of the measure-ments. Interestingly, when measured dynamic contact anglesare used with Washburn’s equation, the deviations between themeasured and predicted rise-versus-time values for largecapillary tubes become smaller, or comparable to, those ofsmaller tubes (see Figures 4 and 5). In the case of experimentswith water, we observe a level of discrepancy that might beattributed to the water film present in the tubes at the start ofeach experiment, which could have impacted the measuredcontact angle values.

Correlations for Velocity-Dependent Dynamic Con-tact Angle. Over the last several decades, researchers haveintroduced various semiempirical correlations to describe the

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

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velocity-dependence of dynamic contact angle.15,20,22,42 Popes-cu et al.36 performed comparative analysis of the predictionsmade by some of these models for dynamic contact angleversus the velocity of the contact line for typical water and high-viscosity silicone oil systems. The authors, however, did notcompare any of the predicted trends against independentexperimental counterparts mainly due to lack of such data inthe literature. In this section, we use the experimental dynamiccapillary rise data presented earlier to examine the predictivecapabilities of four correlations presented by Cox,15 Joos etal.,20 Shikhmurzaev, 22 and Sheng.42 We discuss the details ofthe models followed by comparison of the predicted trendsagainst our experimental measurements.Bracke et al.17 used two experimental methods to study the

dependence of dynamic contact angle on the velocity of contactline. In the first set of experiments, they used polyethylene/polyethylene terephthalate solid strips drawn into pools ofdifferent aqueous glycerol solutions, aqueous ethylenegelycolsolutions, and ordinary corn oil. In the second method, theyutilized a dry platina Wilhelmy plate. Joos et al.20 then used theexperimental data to develop a semiempirical correlation for thedependence of dynamic contact angle to the velocity of contactline in a circular capillary tube. They replaced the dynamiccontact angle term in Washburn’s equation2 with theircorrelation for the dynamic contact angle17 and showed thatthis semiempirical correlation leads to more accuratepredictions of capillary rise than that of Washburn. Theproposed empirical correlation is given by

θ θ θ= − + Cacos( ) cos( ) 2(1 cos( ))d e e1/2

(4)

where θd is the dynamic contact angle in radians, θe is theequilibrium contact angle in radians, and Ca = (ηv)/σ is thecapillary number, in which η is the fluid viscosity, σ is thesurface tension, and v the velocity of contact line. Table 2 listsall the parameters needed to use this correlation with our fluidsystems.

In a thermodynamics-based approach, Cox15 assumed amicroscopic slip boundary condition for a moving fluid on asolid surface. This assumption helps removing the stresssingularity at the triple contact line of the system. Cox’s analysisresulted in the following relationship for the dynamic contactangle versus velocity of the contact line:

θ θ χ ησ

= +G Gv

( ) ( )d e (5)

where

∫θ = −θG

x x xx

x( )sin( ) cos( )2sin( )

d0

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

divided by microscopic length scales.36 The parameters usedwith Cox’s model are also tabulated in Table 2.Utilizing numerical hydrodynamic calculations, Hoffman’s7

slipping function, and eq 6, Sheng and Zhou42 linkedmacroscopic flow behavior (e.g., dynamic contact angle) tothe microscopic parameters governing the contact-line region.The authors also introduced parameters to take the effect ofsurface roughness into account.

θ θ= +G G Ca K l( ) ( ) ln( / )e s (6)

where

∫ ϕ ϕ= −G q f( ) d [ ( )]q

0

1

and

ϕ ϕ ϕ ϕ ϕ π ϕ

ϕ π ϕ ϕ

ϕ ϕ π ϕ ϕ ϕ

ϕ ϕ ϕ π ϕ ϕ

= − + −

+ + − −

÷ − − +

+ − − −

⎡⎣⎤⎦

f q q

q

( ) 2sin { ( sin ) 2 [ ( )

sin ] ( ) sin }

{ ( sin )[( ) sin cos ]

( sin cos )[( ) sin ]}

2 2 2

2 2 2

2 2

2 2

In the above equations, ls is the slipping length and K is aslipping model dependent constant.We also considered the mathematical model proposed by

Shikhmurzaev.22 The author suggested that the surface tensiongradient caused by the flow of a liquid flowing on a solidsurface, influences the flow. This gradient, in the case of smallcapillaries, determines the dynamic contact angle. The shearstress singularity present in classical approaches is eliminated inthis model. Blake and Shikhmurzaev29 and Popescu et al.36

simplified Shikhmurzaev’s original model into eq 7:

θ θρ ρ

ρ ρ− =

+

− + +

* *

* *

V u

V Vcos( ) cos( )

2 ( )

(1 )[( ) ]e d2es

1es

0

1es

2es 2 1/2

(7)

where

θθ θ θθ θ θ

=−

−u ( , 0)

sin( ) cos( )sin( ) cos( )0 d

d d d

d d d

= ×V Sc Ca

ρ ρ θ σ= + − − ** *1 (1 )(cos( ) )2es

1es

e SG

In the above equations, Ca is the capillary number, Sc is ascaling factor that depends on the material properties, and ρ2e

s*and ρ1e

s* are two phenomenological coefficients (see Blake andShikhmurzaev29 for more details). Table 2 lists values of theparameters used with the above-mentioned model (Sc for wateris obtained from Popescu et al.36).We compare our experimental dynamic contact angle data

for glycerol, Soltrol 170, and water with the trends predicted bythese four correlations. For each of the models, we use ourmeasured values of the physical parameters (i.e., viscosity andsurface tension) and correlation coefficients available in theliterature for the same fluid pair (see Table 2). Figure 6compares our measured dynamic contact angles versus velocityof the triple contact line against the trends predicted by theabove-mentioned correlations.This figure shows that the model proposed by Joos generates

the best match against the Soltrol 170 data. It predicts the lowvelocity trend for glycerol well, but deviates at larger velocity

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

fluid

surfacetension

(dyn/cm)viscosity(cP)

θe(rad) ρ1e

s*22 Sc σSG*22

glycerol 63.47 1011.1 0 0.54 17.90 −0.07Soltrol 170 24.83 2.6 0 0.54 12.56 −0.07water 72.8 1.1 0 0.54 12.5 −0.07

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values. In the case of water, it overpredicts the experimentaldata at both low and high velocities. Cox’s model produces avery good fit in the case of glycerol, while it overpredicts theSoltrol 170 experimental data particularly at higher velocities.This model shows more significant deviation from theexperimental data in the case of water. It is important to notethat there are no fitting parameters used in the calculationsperformed with these two models. This is, however, not thecase for Sheng’s and Shikhmurzaev’s models. In Sheng’s model,the value of ls = 10−7 cm was obtained from molecular dynamicsimulations,42 while K was the fitting parameter which has

values of 0.41, 2.9 × 10−3 and 4.1 × 10−4 for glycerol, Soltrol170 and water respectively. Sheng’s model predicts theexperimental data for glycerol and Soltrol 170 well; however,it slightly overpredicts the experimental data for water at almostall velocities. In case of Shikhmurzaev’s model, one needs to useparameters that can be found only for a very few fluid systemsin the literature; see, for instance, Popescu et al.36 and Blakeand Shikhmurzaev.29 When available, we have used parametersfrom 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

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.

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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 datagathered 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 IDsquare 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

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experimental measurements. All the other parameters were keptthe same as the values used by Blake and Shikhmurzaev29 forsilicon oil and different glycerol/water solutions, respectively.The model developed by Shikhmurzaev expectedly shows agood agreement with the experimental trends at low andmoderate velocities in glycerol and water systems. The modelalways 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, ageneral trend for the changes in normalized rise (Ln = l/le,where l is the value of rise and le is the equilibrium rise), withvariations in dynamic contact angle. We consolidated all the riseand dynamic contact angle data (except for those of the water/air system) generated under this study to obtain the trendshown in Figure 7. The rise and dynamic contact angle data forall fluid/tube-size/tilt-angle combinations follow the sametrend. Figure 7 also shows a curve fit for the data that isgiven by

θ=

− −⎛⎝⎜

⎞⎠⎟

ll

ab

cexp

( )2

r

e

d2

2

1/4

(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 verylimited numbers of dynamic rise-versus-contact angle data setsavailable in the literature. This relation can be used to find thedynamic contact angle value for a given rise, if the invading fluiddensity, surface tension, tube size, and tilt angle of the systemare known. Combined with the Washburn equation, one canthen 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. Thecross section of a typical square capillary tube used in theseexperiments is not a perfect square and has round corners. Weinvestigate the rise-versus-time of both AMs and MTMs inthese experiments. The data are presented in log−log plots inFigure 8. As shown in this figure, the rise-versus-time of AMsand MTM in 1 and 2 mm square tubes with glycerol/air fluidsystem follow the universal relationship proposed byPonomarenko et al.40 As seen, the late time scales of AMrise-versus-time data follow a trend proportional to t1/3, whilethe early time scales of MTM rise-versus-time values follow at1/2 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 whichthe fluid rises. Based on the calculations presented by theauthors, the higher is the roundedness of the corner, the greateris the value of the dimensionless flow resistance (β). And thehigher is the value of β, the lower is the average velocity of themoving meniscus. Therefore, the large roundedness values inour tubes makes the value of β much larger than it is in sharpcornered channels used by Ponomarenko et al.40 This may havecontributed to the slight deviation between the slope of our AMrise-versus-time values from the universal relationshipsproposed by Ponomarenko et al.40

■ CONCLUSIONSAn extensive set of capillary rise experiments were performed incircular and square cross section tubes with various internaldiameters ranging from 0.5 to 2.0 mm. We measured rise-

versus-time for different fluid systems (i.e., water/air, Soltrol170/air, and glycerol/air) and tube tilt angles. To the best ofour knowledge, this is the first time that an extensiveexperimental data set for variation of dynamic contact anglewith capillary number during rise with different fluid systems isreported. Measured dynamic contact angles were used to obtainrise-versus-time trends that agree more closely with themeasured rise values (also reported in this study) than thosepredicted by the Washburn equation using a fixed contactangle. Four empirical models of velocity-dependence ondynamic contact angle were validated by comparing theirpredicted trends against our experimental data. The predictivecapabilities of the models were discussed. A new general trendwas introduced for the changes in normalized rise withvariations in dynamic contact angle. The trend was comparedsuccessfully against the experimental data available in theliterature. This relation can be used to find the dynamic contactangle values for a given rise, if the invading fluid density, surfacetension, tube size, and tilt angle of the system are known.Combined with Washburn equation, one can then produce arise-versus-time curve for the system. Finally, we presented ourmeasurements of rise of glycerol in square cross sectioncapillary tubes with different sizes. We reported rise-versus-timefor both AMs and MTM. The AM late time scales and MTMearly time scales data showed an encouraging agreement withthe universal relationships proposed by Ponomarenko et al.40

for rise in capillary tubes with angular cross section. Theexperimental data and the trends presented here can be used indynamic pore-scale models of flow in porous media to predictmultiphase flow functions.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected].

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe gratefully acknowledge the financial support of the Schoolof Energy Resources and the Enhanced Oil Recovery Instituteat the University of Wyoming. We extend our gratitude toProfessor George Hirasaki for his valuable comments andHenry Plancher and Soheil Saraji of Piri Research Group fortheir assistance with the capillary tube cleaning procedures andsurface tension measurements.

■ REFERENCES(1) Lucas, R. The Time Law of the Capillary Rise of Liquids. Kolloid-Z. 1918, 23, 15−22.(2) Washburn, E. W. The Dynamics of Capillary Flow. Phys. Rev.1921, 17, 273−283.(3) Rideal, E. K. On the Flow of Liquids under Capillary Flow. Philos.Mag. 1922, 44, 1152−1159.(4) Cherry, B. W.; Holmes, C. M. Kinetics of Wetting of Surfaces byPolymers. J. Colloid Interface Sci. 1969, 29, 174−176.(5) Blake, T. D.; Haynes, J. M. Kinetics of Liquid/LiquidDisplacement. J. Colloid Interface Sci. 1969, 30, 421−423.(6) Goldfinger, G. Clean Surfaces: Their Preparation and Character-ization for Interfacial Studies; Marcel Dekker Incorporation: New York,1970; pp 269−284.(7) Hoffman, R. L. A Study of the Advancing Interface I. InterfaceShape in Liquid-Gas Systems. J. Colloid Interface Sci. 1975, 50, 228−241.

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(8) Huh, C.; Mason, S. G. The Steady Movement of a LiquidMeniscus in a Capillary Tube. J. Fluid Mech. 1977, 81, 401−419.(9) Jiang, T.-S.; Oh, S.-G.; Slattery, J. C. Correlation for DynamicContact Angle. J. Colloid Interface Sci. 1979, 69, 74−77.(10) Dussan V, E. B. On the Spreading of Liquids on Solid Surfaces:Static and Dynamic Contact Lines. Annu. Rev. Fluid Mech. 1979, 11,371−400.(11) Rillaerts, E.; Joos, P. The Dynamic Contact Angle. Chem. Eng.Sci. 1980, 35, 883−887.(12) Hoffman, R. L. A Study of the Advancing Interface II.Theoretical Prediction of the Dynamic Contact Angle in Liquid-GasSystems. J. Colloid Interface Sci. 1983, 94, 470−486.(13) Sell, P.; Maisch, E. Fluid Transport in Capillary Systems UnderMicrogravity. Acta Astronaut. 1984, 11, 577−583.(14) Mumley, T. E.; Radke, C. J.; Williams, M. C. Kinetics of Liquid/Liquid Capillary Rise II Development and Test of Theory. J. ColloidInterface Sci. 1986, 109, 398−412.(15) Cox, R. G. The Dynamics of the Spreading of the Liquids on aSolid Surface. Part 1. Viscous Flow. J. Fluid Mech. 1986, 168, 169−194.(16) Ransohoff, T. C.; Radke, C. J. Laminar Flow of a Wetting Liquidalong the Corners of a Predominantly Gas-Occupied NoncircularPore. Chem. Eng. Sci. 1988, 121, 392−401.(17) Bracke, M.; Voeght, F. D.; Joos, P. The Kinetics of Wetting: TheDynamic Contact Angle. Prog. Colloid Polym. Sci. 1989, 79, 142−149.(18) Li, X.; Fan, X.; Askounis, A.; Wu, K.; Sefiane, K.; Koutsos, V. AnExperimental Study on Dynamic Pore Wettability. Chem. Eng. Sci.2013, 104, 988−997.(19) Girardo, S.; Palpacelli, S.; De Maio, A.; Cingolani, R.; Succi, S.;Pisignano, D. Interplay between Shape and Roughness in Early-StageMicrocapillary Imbibition. Langmuir 2012, 28, 2596−2603.(20) Joos, P.; Remoortere, P. V.; Bracke, M. The Kinetics of Wettingin a Capillary. J. Colloid Interface Sci. 1989, 136, 189−197.(21) Dussan V, E. B.; Rame, E.; Garoff, S. On Identifying theAppropriate Boundary Conditions at a Moving Contact Line: AnExperimental Investigation. J. Fluid Mech. 1991, 230, 97−116.(22) Shikhmurzaev, Y. D. The Moving Contact Line on a SmoothSolid Surface. Int. J. Multiphase Flow 1993, 19, 589−610.(23) Blake, T. D. Surfactant science series, Wettability. Surfactant Sci.Ser. 1993, 49, 251−309.(24) Dreyer, M.; Delgado, A.; Rath, H.-J. Capillary Rise of Liquidbetween Parallel Plates under Microgravity. J. Colloid Interface Sci.1994, 163, 158−168.(25) Tang, L.-H.; Tang, Y. Capillary Rise in Tubes with SharpGrooves. J. Phys. II 1994, 4, 881−890.(26) Quere, D. Inertial Capillarity. Europhys. Lett. 1997, 39, 533−538.(27) Hamraoui, A.; Thuresson, K.; Nylander, T.; Yaminsky, V. Can aDynamic Contact Angle Be Understood in Terms of a FrictionCoefficient? J. Colloid Interface Sci. 2000, 226, 199−204.(28) Siebold, A.; Nardin, M.; Schultz, J.; Walliser, A.; Oppliger, M.Effect of Dynamic Contact Angle on Capillary Rise Phenomena.Colloids Surf., A 2000, 161, 81−87.(29) Blake, T. D.; Shikhmurzaev, Y. D. Dynamic Wetting by Liquidsof Different Viscosity. J. Colloid Interface Sci. 2002, 253, 196−202.(30) Hirasaki, G. J.; Yang, S. Y. Dynamic Contact Line withDisjoining Pressure, Large Capillary Numbers, Large Angles and Pre-wetted, Precursor, or Entrained Films. Contact Angle, Wettability Adhes.2002, 2, 1−30.(31) Stange, M.; Dreyer, M. E.; Rath, H. J. Capillary Driven Flow inCircular Cylindrical Tubes. Phys. Fluids 2003, 15, 2587−2601.(32) Chen, Y.; Collicott, S. H. Experimental Study on Capillary Flowin a Vane-Wall Gap Geometry. AIAA J. 2005, 43, 2395−2403.(33) van Mourik, S.; Veldman, A. E. P.; Dreyer, M. E. Simulation ofCapillary Flow with a Dynamic Contact Angle. Microgravity Sci.Technol. 2005, 17, 87−94.(34) Blake, T. D. The Physics of Moving Wetting Lines. J. ColloidInterface Sci. 2006, 299, 1−13.(35) Lee, S.-L.; Lee, H.-D. Evolution of Liquid Meniscus Shape in aCapillary Tube. J. Fluids Eng. 2007, 129, 957−965.

(36) Popescu, M. N.; Ralston, J.; Sedev, R. Capillary Rise withVelocity-Dependent Dynamic Contact Angle. Langmuir 2008, 24,12710−12716.(37) Hilpert, M. Effects of Dynamic Contact Angle on LiquidInfiltration into Horizontal Capillary Tubes: (Semi)-AnalyticalSolutions. J. Colloid Interface Sci. 2009, 337, 131−137.(38) Hilpert, M. Effects of Dynamic Contact Angle on LiquidInfiltration into Inclined Capillary Tubes: (Semi)-Analytical Solutions.J. Colloid Interface Sci. 2009, 337, 138−144.(39) Deganello, D.; Croft, T. N.; Williams, A. J.; Lubansky, A. S.;Gethin, D. T.; Claypole, T. C. Numerical Simulation of DynamicContact Angle Using a Force Based Formulation. J. Non-NewtonianFluid Mech. 2011, 166, 900−907.(40) Ponomarenko, A.; Quere, D.; Clanet, C. A Universal Law forCapillary Rise in Corners. J. Fluid Mech. 2011, 666, 146−154.(41) Saraji, S.; Goual, L.; Piri, M.; Plancher, H. Wettability of sc-CO2/Water/Quartz Systems: Simultaneous Measurement of ContactAngle and Interfacial Tension at Reservoir Conditions. Langmuir2013, 29, 6856−6866.(42) Sheng, P.; Zhou, M. Immiscible-Fluid Displacement: Contact-Line Dynamics and the Velocity-Dependent Capillary Pressure. Phys.Rev. A 1992, 45, 5694−5708.

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