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Microscopic and Macroscopic Dynamic Interface Shapes and the Interpretation of Dynamic Contact Angles
JOURNAL OF COLLOID AND INTERFACE SCIENCE
177, 234–244 (1996)
Article No. 0026
Microscopic and Macroscopic Dynamic Interface Shapes and the Interpretation of Dynamic Contact Angles ENRIQUE RAME´ 1
AND
STEPHEN GAROFF
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 U.S.A. Received March 17, 1995; accepted June 7, 1995
We have studied shapes of dynamic fluid interfaces at distances £1700 mm from the moving contact line at capillary numbers (Ca) ranging from 10 03 to 10 01 . Near the moving contact line where viscous deformation is important, an analysis valid to O(1) in Ca describes the shape of the fluid interface. Static capillarity should describe the interface shape far from the contact line. We have quantitatively determined the extent of the regions described by the analysis with viscous deformation and by a static shape as a function of Ca. We observe a third portion of the interface between the two regions cited above, which is not described by either the analysis with viscous deformation or a static shape. In this third region the interface shape is controlled by viscous and gravitational forces of comparable magnitude. We detect significant viscous deformation even far from the contact line at Ca ) 0.01. Our measured dynamic contact angle parameter extracted by fitting the analysis with viscous deformation to the shape near the moving contact line coincides with the contact angle of the static-like shape far from the contact line. We measure and explain the discrepancy between this dynamic contact angle parameter and the apparent contact angles based on meniscus or apex heights. Our observations of viscous effects at large distances from the contact line have implications for dynamic contact angle measurements in capillary tubes. q 1996 Academic Press, Inc. Key Words: wetting; dynamic wetting; moving contact line; dynamic contact angle; spreading.
1. INTRODUCTION
The dynamics of liquids spreading on solid surfaces controls natural and technologically important processes such as two-phase flow in porous media, the spraying of paint droplets, the deposition of coatings, and the imbibition of inks in papers. In each of these processes, surface tension forces are important and details of the fluid motion in the immediate vicinity of the moving contact line control the spreading dynamics. Unfortunately, the motion of the fluids in this region has proven hard to describe because the classi1
To whom correspondence should be sent. Email: er2n/@andrew. cmu.edu. 0021-9797/96 $12.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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cal hydrodynamic model 2 is singular at the moving contact line. If this classical model held, an infinite force would be required to sink a spoon into a coffee cup. The classical hydrodynamic model cannot be used up to and including the contact line. Dussan V., Rame´, and Garoff (abbreviated DRG) (1) developed a model for describing interface shapes near moving contact lines in the limit of low capillary number, Ca ( å Um/ s, where U is a characteristic velocity, m the fluid viscosity, and s the surface tension). This model, discussed in Section 2, describes the shape of the fluid interface when a solid enters a fluid at constant velocity and immersion angle. The region described by the model is characterized by a balance of viscous and surface tension forces. In this region, viscous forces deform the interface significantly. The model introduces a single adjustable parameter, v0 , and specifies the functional form describing the interface shape near the moving contact line. v0 depends on the material properties of the system, must be determined experimentally, and plays the role of a dynamic contact angle. Experiments have shown that this model accurately predicts the interface shape within 300 mm from the contact line for Ca õ 0.1 (1–4). These experiments also confirmed the model’s prediction that the viscous deformation in this region is independent of the macroscopic geometry. They established the functional form of the interface shape near the moving contact line as the boundary condition for the shape in the macroscopic region. Central to the analysis in DRG is the assumption that, when Ca ! 1, the interface shape far from the contact line is well described by static capillarity. This static-like shape far from the contact line should meet the solid surface with contact angle v *0 if it were continued as a static shape all the way to the solid. In the model, v *0 Å v0 , where v0 is extracted by fitting the model which contains the geometryindependent viscous deformation to the interface shape close to the moving contact line. 2 These assumptions are: Newtonian, incompressible fluid, undeformable solid, and the continuity of velocity at bounding surfaces. The latter amounts to the no-slip condition at a fluid–solid interface.
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In this study, we verify the assumptions of the present model about the condition of the fluid interface far from the moving contact line. We study the dynamic interface shape up to 1700 mm from the contact line, a length scale much larger than studied in previous experiments. This region is q on the order of the capillary length, s / rg (where r is the fluid density and g is the acceleration of gravity), the length scale where the interface is primarily controlled by gravity and surface tension forces. As a function of Ca, we determine the part of the interface near the contact line described by the analysis of DRG containing the viscous deformation. We also identify the region far from the contact line behaving as a static-like interface. In this region, viscous effects are undetectable by our optics and image analysis system. Our measurements in the staticlike region are consistent with the interpretation given in the theory, that v *0 Å v0 . We show that, within our detection limits, viscous effects reach 1400 mm at Ca Å 0.01. However, only the viscous deformation nearest the contact line is independent of the overall geometry. Other measurements of dynamic interface shapes near a moving contact line have been reported (5, 6). We also compare v0 to apparent contact angles obtained from ‘‘two-point measurements.’’ Typical two-point measurements are based on the rise of a dynamic meniscus against a moving plate or cylinder, or the apex heights of menisci in capillaries. The apparent contact angle corresponds to the contact angle of a static interface with the same rise or apex height as the dynamic one. We show that, in our geometry, the difference between the apparent contact angle and the parameter v0 changes systematically with Ca. We discuss the errors that may arise when using two-point measurements to characterize dynamic wettability in geometries other than the one used in our measurements. In Section 2 we present the theoretical background for our experiments. In Section 3 we describe the experiment. In Section 4, we present and discuss the results; and in Section 5, we state the conclusions.
devoid of moving contact lines, this singularity has made evident the inadequacy of the classical model to describe systems with moving contact lines. The difficulty cited above does not play the same role under all conditions. The importance of the contact angle in determining the interface shape increases with surface tension both in static and in dynamic conditions. Hence, studies aimed at properly modeling moving contact lines concern themselves with conditions where surface tension forces dominate over viscous and gravitational forces. These conditions require that the capillary number, Ca, and the Bond number, Bd ( år gD 2 / s, where D is the characteristic length of the system; Bd measures the ratio of gravitational to surface tension forces) be !1. The motion of a contact line has been the object of numerous studies in the last 20 years (7). In order to describe systems with moving contact lines, the classical model needs to be modified by including a new region surrounding the moving contact line where the classical hydrodynamic assumptions are replaced by a new set of assumptions. This region has characteristic length scale Li and is commonly known as the inner region. Very near the moving contact line, but not so close that the classical model breaks down, models postulate the existence of a region where the fluid motion obeys the classical model and is independent of the macroscopic geometry of the system. This region, called the intermediate region, lies between the inner and the macroscopic—or outer—regions. In the intermediate region, the fluid interface can sustain significant viscous deformation even to lowest order as Ca r 0 (8–10). Based on these ideas, DRG performed an analysis which describes the interface shape near the moving contact line generated by a cylindrical tube of radius RT vertically entering a liquid bath. Their main result, asymptotically correct to O(1) as Ca r 0, holding Ca ln(a/Li ) fixed, is
2. THEORETICAL BACKGROUND
where u is the local slope of the interface with respect to the solid, r is the distance from the contact line to the point on the interface (see Fig. 1), a is the capillary length q s / rg, and g 01 is the inverse function of g, i.e., x å g 01 (g(x)) with
The spreading of liquids is controlled by the motion of a contact line across a solid surface, a feature that puts spreading systems in a special class of fluid mechanics problems. The difficulty in this class of problems is that the classical hydrodynamic assumptions give rise to a singularity in the stress field at the contact line. The singularity prevents the use of the contact angle as a boundary condition for the interface shape. This marks a significant departure from capillary statics, where the contact angle is the classical material property needed to wholly determine the shape of the fluid interface on the macroscopic scale. Even though the same classical model works well for all other dynamic situations
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u Ç g 01 (g( v0 ) / Ca ln(r/a)) / f0 (r/a; v0 , RT /a) 0 v0 ,
g(x) å
*
x
0
y 0 sin(y)cos(y) dy. 2 sin(y)
[1]
[2]
The first term in Eq. [1] represents the interface shape in the intermediate region, i.e., uint Ç g 01 (Ca ln(r/L)),
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FIG. 1. Coordinate notation used in equations and graphs. (A) observed contact line. (B) extrapolation of the static-like interface shape to the solid. (---) Static continuation of the static-like interface shape far from the contact line. (AB) contact line shift. The x-axis coincides with the bulk fluid level.
where the parameter L ( å a exp[ 0Ca/g( v0 )]) depends only on the physics of the inner region (2). The second term is the solution in the outer region, correct to O(1). It is the same as for a static interface rising on a vertical cylinder of radius RT (11): uout Ç f0 (r/a; v0 , RT /a).
sible outer length scale. The outer qlength scale in this geometry was the capillary length a å s / rg. Given the range of Ca studied, menisci formed both in rise and in depression. For this reason, two kinds of liquid containers were used. For menisci in rise, we used a slightly overfilled cylindrical Teflon container about 8 cm in diameter and 12 cm deep. The meniscus rose above the rim of the container. For menisci in depression, we used a square glass container 10 cm on a side and filled to about 4/5 of its total volume. The PDMS wetted the walls of this container. The Pyrex tube was driven by a motorized stage. Two types of motors were used: for menisci in rise, a computercontrolled Newport Klinger (Model UE30CC) mounted on a Newport Klinger translation stage (UT100); and for menisci in depression, a screw motor (Newport 860SC Motorizer) mounted on a stage (Newport 435-2). We produced menisci in rise using a 1000 cSt fluid at three different immersion speeds, 2.1 1 10 03 cm/s, 1 1 10 02 cm/s, and 2.1 1 10 02 cm/s, and in depression, with a 60000 cSt fluid at 3.26 1 10 03 cm/s. These conditions correspond to capillary numbers of 9.7 { 0.1 1 10 04 , 4.62 { 0.01 1 10 03 , 9.70 { 0.01 1 10 03 , and 9.580 { 0.002 1 10 02 , respectively. The capillary length was different for each of the fluids: a Å 0.146 cm for the 1000 cSt fluid, and a Å 0.148 cm for the 60000 cSt fluid.
[4] 3.2. Optics and Image Analysis
Equation [1] represents the interface shape in the intermediate region up to its overlap with the outer region. In Eq. [1] the single parameter v0 must be determined experimentally. Within the model, v0 is interpreted as the contact angle of the extrapolation back to the solid of the static-like interface far away (described by Eq. [4]). 3. MATERIALS AND PROCEDURE
3.1. Experimental Apparatus We immersed a vertical, cylindrical Pyrex tube at constant velocity into a container of polydimethylsiloxane (PDMS). Since PDMS completely wetted the glass, the static meniscus rose up the tube and attained a zero static contact angle.3 When the tube moved into the fluid, viscous forces distorted the fluid interface and the meniscus appeared flatter. Below some speed, the meniscus remained above the general fluid level, a ‘‘meniscus in rise.’’ At higher speeds, the meniscus was bent below the general fluid level, a ‘‘meniscus in depression.’’ This experimental geometry gave the largest pos3 Work being done in our laboratory addresses PDMS spreading on glass coated with a perfluorinated surfactant, where the static advancing contact angle uA x 0. This allows study of receding contact lines. We do not anticipate qualitative differences between uA x 0 and this work, where uA Å 0.
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The meniscus was backlit using Koehler illumination (3). The silhouette of the meniscus and of the tube were viewed using a long working distance microscope attached to a CCD camera with 640 1 480 pixel array. The fluid container and the glass tube were placed between the Koehler illumination and the microscope. The image was sent from the camera to a computer for digital storage and analysis. Two camera/ microscope systems were used, Pulnix TM-740 camera/ Bausch & Lomb Monozoom 7 microscope for depression images, and Cohu 4110 Solid State Camera/Infinity K2 microscope for rise images. In rise, we slightly overfilled the Teflon container and viewed the meniscus above the bulk fluid; in depression we partly filled the square glass cell and viewed through the bulk liquid. Figure 2 shows typical images. Great care must be taken to align the optics (3). Since the light that forms images of menisci in depression travels through the liquid, the faces of the square glass container must be set normal to the light beam. The focus must be changed between rise and depression images to account for the fact that the fluid has an index of refraction different than air (4). Finally, we recorded the vertical direction before each experiment using an image of a plumb bob. This was necessary to compensate for misalignments between the pixel columns in the camera and the true vertical. Five images, at 5-s intervals, were taken at each immersion velocity.
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FIG. 2. Typical images of a meniscus. The straight vertical edge is the tube edge; the curved boundary is the liquid–air interface. (a) In rise, the liquid and the solid are dark; the air is light. (b) In depression, the liquid is light; the solid and the air are dark.
The data analysis consists of identifying the location and slope of the meniscus and fitting sections of the interface slope data using Eq. [1] or [4]. The air–liquid interface is
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located by the highest gradient of gray levels in the digital image. The interface location and slope are found using discrete gray-level algorithms adapted by Marsh (3). A typi-
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for the region of data used in the fit. To assess the full extent of regions of agreement of the data and the theory, we examine the systematic deviations outside the fitted region. In order to identify the point where either Eq. [1] or [4] no longer fits appropriate sections of data, we set the criterion that a smooth line fit to the deviation between data and theory must depart from zero by sV /3. 3.3. Materials
FIG. 3. (a) Data and best fit for the static interface in capillary rise used as calibration. (b) Difference of the data and the best theoretical fit.
cal data set of the slope versus distance from the contact line consists of about 1000 data points. The data appear as a cloud of points about 1 to 3 degrees wide. See Fig. 3 for a typical example. For several images of the same meniscus taken in rapid succession, the data clouds overlap but the slope values at each spatial location do not. Thus the cloud cannot be due to real variation in the interface shape. Rather, it is due to pixel noise in the camera. Therefore the thickness of the data cloud sets the standard deviation assigned to the slope values. Due to the density of points, our detection limit of systematic deviations of the data from a theoretical form is far better than the 17 to 37 standard deviation in the individual data points. We calculate the normalized chi-squared deviation, x 2 , of a set of N data points as x2 å
( ( u iexp 0 u itheo ) 2 , sV 2 (N 0 1)
where sV is the standard deviation of each point. Since we treat each point as independent, we compute sV as the standard deviation of the data cloud about a smooth line passing through the cloud’s center. A second- or third-degree polynomial is normally sufficient. The criteria for a good fit are that x 2 Ç 1 and that the plot of the deviation of the data from the best-fit theoretical line scatter uniformly about zero
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The solid on which the liquid spreads is a 2.54 cm outer diameter Pyrex tube. Its outer surface is fire-polished and carefully cleaned to ensure a smooth contact line all around the tube. The Pyrex tube was cleaned by soaking it in sulfochromic acid solution for 40 min, rinsing it with pure water (from a Barnstead system, model ROPure LP, fitted with NANOPure II filtration cartridges), soaking it in 15% HCl solution for 40 min to remove Cr ions, and rinsing it again with pure water. The water layer wetting the glass after the rinse was blown off with dry nitrogen. If the water did not sheet off uniformly from the glass we repeated the cleaning. The Teflon container was cleaned with ethanol, then rinsed with pure water and dried in a vacuum oven. The glass container was washed with dish detergent and rinsed thoroughly with pure water. Both containers were filled with the PDMS fluid after cleaning and remained filled for the duration of the experiments. Two PDMS fluids were used, with nominal kinematic viscosities of 1000 cSt (Dow Corning 200 fluid) and 60000 cSt (United Chemical Inc.). The fluid properties for the duration of the experiments were: density r Å 0.97 g/cm3 , surface tensions s1 Å 20.3 dyn/cm and s2 Å 20.8 dyn/cm, and viscosities m1 Å 9.6 g/cm s and m2 Å 611 g/cm s, respectively. 3.4. Calibration The interface measurement system, which consists of the combined optical setup and digital image analysis algorithm, was calibrated using a static meniscus as reference. The calibration consists of finding the best fit between static theory and a static interface image using the contact angle as the fitting parameter. Since the theory describing static menisci is known exactly, systematic deviations between static data and the best static fit indicate the distortions in the system. Because the curvature of the interface along the optic axis and the angle at the contact line are crucial to the image formation ( 3 ) , calibration must be done on static interfaces similar to the ones obtained in the dynamic experiments. We produced a static interface with contact angle near 357 by contacting PDMS with a Pyrex tube coated with a monolayer of fluorinated surfactant. Figure 3a shows the data and the best fit. Data for x õ 50 mm and x ú 1500 mm exhibited systematic deviations. Thus, we
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FIG. 4. x 2 vs xc , the inner edge of the fitted region, for the best fits of the static interface used as calibration in capillary rise.
restricted the data analysis of menisci in rise to 50 mm õ x õ 1500 mm. Figure 3b shows that the difference between data and the best fit is uniformly distributed about zero for 50 mm õ x õ 1500 mm. x 2 is statistically constant near 1.03 using fits to subsets of data xc õ x õ 1500 mm, where 100 mm õ xc õ 1200 mm, suggesting that the system could be distortion-free (see Fig. 4). However, a test of the sensitivity of the fitting parameter (the contact angle) to xc , as xc moves progressively out reveals the presence of systematic distortions. Figure 5 shows that the value of the contact angle is rather stable up to about 600 mm from the solid, at which point it begins to show a systematic upward drift. The magnitude of this drift is 0.47 in the field of view of our images. The contact angle continues to change with xc while x 2 is statistically constant because each portion of the interface being analyzed looks almost static, with the same x 2 but slightly ‘‘different’’ contact angles. The contact angle changes slowly but systematically as we fit portions of the interface farther and farther from the solid. The ‘‘different’’ contact angle can be rationalized as a progressive tilting of the image relative to the real interface as we move away from the solid. The interaction of the light waves with the curved surface of the fluid interface depends on the edge’s
FIG. 5. Contact angle vs xc , the inner edge of the fitted region, for the static interface used as calibration in capillary rise.
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FIG. 6. (a) Data and best static fit for the static interface in capillary depression used as calibration. (b) Difference of the data and the best theoretical fit.
curvature in the direction of the optic axis. This curvature is highest near the solid and gets progressively smaller moving away from the solid. The position of the maximum gradient in the intensity of the transmitted image relative to the physical edge of the interface may drift across the image. As a reference, we estimated that in order to cause a 0.47 variation in the contact angle for 600 mm õ xc õ 1200 mm, the data of the local interface slope angle should have a linearly varying error of only about 0.047 /100 mm. The static interface in depression was created by immersing a 2.54 cm outer diameter brass rod with a sharp edge machined on its circumference. The meniscus pins on the sharp edge as it is immersed into the fluid. Images of menisci in depression exhibit a distortion qualitatively similar to that of images in rise (Fig. 6), except that the contact angle decreases uniformly as xc moves from 100 mm to 1000 mm, with a total variation of about 0.87 (Fig. 7). In depression, the distortion-free region (deviations uniformly distributed about zero) was 50 mm õ x õ 2000 mm. Thus we restricted our analysis of depression images to data in this region. The different sign of the systematic distortion in rise and in depression further supports the notion that the image formation process is responsible for the systematic distortions. Given the result of our static calibration, we must assign an uncertainty of 0.47 (for rise) and 0.87 (for depression) to
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FIG. 7. Contact angle vs inner edge of the fitted region, xc , for the static interface used as calibration in capillary depression.
contact angles from fittings to portions of the dynamic meniscus far from the contact line. 4. RESULTS AND DISCUSSION
In the first part of this section, we provide direct measurements of the regions where geometry-free viscous deformations dominate the interface shape and where the interface shape is essentially static. We show how the extent of interfacial regions in agreement with the viscous deformation described by Eq. [1] or with a static shape (Eq. [4]) changes with Ca. In the next subsection, we test whether the interpretation of v0 is correct, i.e., whether v0 as determined by Eq. [1] is the contact angle of the static-like interface far from the contact line when it is extrapolated statically to the solid (see Fig. 1). We test this by independently measuring v0 from fittings of Eq. [1] to data near the contact line and of Eq. [4] to data far from it. We then discuss two related measures of viscous deformation: (a) the distance between the dynamic contact line and the point where the static extrapolation of the static-like shape far away intersects the solid (contact line shift, AB in Fig. 1), and (b) the difference between v0 and the apparent dynamic contact angle derived from a measurement of the meniscus height in our experimental geometry (AO in Fig. 1). We conclude by discussing these effects for a dynamic meniscus in a capillary tube.
even closer to the contact line, viscous forces increase even more. From a certain point approaching the contact line, the flow becomes locally determined. The influence of gravity and of the overall system geometry is lost. The interface shape in the region controlled by the local flow is geometryindependent and results from a balance between viscous and capillary forces. In this region the interface shape is described by Eq. [1]. The extent of the interface deformed by viscous forces increases with increasing Ca. Consequently both the inner boundary of the region described by static capillarity (Eq. [4]) and the outer boundary of the region with geometryfree viscous deformation described by Eq. [1] should increase with increasing Ca. To identify these regions, we analyze data from our dynamic experiments at each Ca in two different ways. First, we fit the data in the region 50 mm õ r õ 300 mm using Eq. [1]. We then use the deviation of the data from the best fit to probe the full extent of the region described by Eq. [1]. Next, we fit various segments of the data xc õ x õ 1500 mm (for 100 mm £ xc £ 1200 mm) to Eq. [4]. This determines the extent of the static-like region of the interface. Figure 8 shows how Eq. [1] describes a portion of the interface shape at two of the Ca’s tested. Figure 9 shows
4.1. Regions of the Interface Shape All regions of the fluid interface are not controlled by the same dominant balance of forces. For a given Ca, the fluid interface sufficiently far from the contact line is controlled by surface tension and, depending on Bd, gravity. In this region viscous forces are negligible, and the interface shape looks static. This is the region where the interface shape is well described by Eq. [4]. But this static-like situation can not be valid everywhere because viscous forces grow compared to surface tension and gravity as the contact line is approached, distorting the shape away from static. Moving
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FIG. 8. Data and fits versus distance from the contact line. (a) Ca Å 0.005. (b) Ca Å 0.1. ( —) Best fit of Eq. [1] in 50 mm õ x õ 300 mm. ( – – – ) Static interface with contact angle equal to v 0* from the fit of Eq. [4] for (a) and to v0 from Eq. [1] for (b). The insets are blow-ups showing static fit far from the contact line.
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FIG. 9. Difference of the data from the best fit of Eq. [1] in 50 mm õ x õ 300 mm at Ca Å 0.005. ( – - – ) 300 mm boundary of fitted region. ( —) Third degree polynomial fit of data cloud in region where departure from zero crosses sV /3.
the deviation of the data at Ca Å 0.005 from the best fit of Eq. [1] to data in 50 mm õ r õ 300 mm. Similar behaviors appear at all other Ca’s tested. From graphs like that of Fig. 9 we locate the region described by Eq. [1] within our detection limits. To find this region we determined the point where the deviation cloud departs from zero by sV /3. We found the deviation’s departure from zero using a thirddegree polynomial fit of the relevant part of the cloud (solid line in Fig. 9). The region described by Eq. [1] moves out monotonically with Ca (lower line in Fig. 10). The model producing Eq. [1] postulates that the interface far from the contact line is static-like. To establish whether a static-like region is present in our field of view, we fit a static interface to different sections of the data in xc õ x õ 1500 mm, with 100 £ xc £ 1200 mm. The fitting parameter is v *0 (the contact angle of the static shape). To simplify the fitting, we used x/a as the independent variable because it is an absolute distance, unlike rV /a, which depends on the unknown location of the putative contact line (point B, Fig. 1). The best fit static line at Ca Å 0.005 for x ú 1200 mm is shown by the dashed line in Fig. 8a. In order to determine the region where the interface becomes static-like, we ana-
FIG. 10. The boundary of the regions fit by Eq. [1] and by Eq. [4] as a function of Ca. Points marked f are upper bounds on the region boundary.
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FIG. 11. x 2 as a function of xc , for fits of static shapes to static and dynamic menisci. ( xv ) static. ( s ) Ca Å 0.001. ( h ) Ca Å 0.005. ( n ) Ca Å 0.01.
lyzed x 2 as a function of xc for each image. Figure 11 shows the x 2 vs xc for the static interface and for one typical interface at each Ca Å 0.001, 0.005, and 0.01. x 2 for the static interface stays flat near 1 and serves as a baseline for comparisons to x 2 from dynamic measurements. At Ca Å 0.001, x 2 becomes stable at about xc ú 200 mm, while at Ca Å 0.005, x 2 stabilizes at xc ú 800 mm. At Ca § 0.01, x 2 approaches 1 but does not stabilize within our field of view. The region of stable x 2 near 1 represents the static-like region of the interface. We did not find a static-like region at Ca § 0.01. Therefore in Fig. 8b, we plotted the static shape using Eq. [4] with v0 found from the good fit of Eq. [1] to data within 300 mm from the contact line. The boundary of the static-like region fitted by Eq. [4] is shown by the upper line in Fig. 10. This distance was estimated using the same method described above for finding the region fit by Eq. [1]. Since we did not find a static-like region within our field of view at Ca Å 0.01 and 0.1, we estimated the region fitted by Eq. [4] using v0 from the fit of Eq. [1] near the contact line. As shown in Fig. 10, we can detect significant viscous deformation at distances £1400 mm from the contact line, at Ca as low as 0.01. This distance is well within the outer region (given here by the capillary length a Ç 1.5 mm), and has implications for similar measurements in capillary tubes (see Section 4.4). Our findings contrast other reports (6) stating that static-like shapes prevail all the way in, to about 100 mm from the moving contact line at Ca Ç 0.1. For our experimental conditions, the regions well described by either Eq. [1] or Eq. [4] do not overlap. The ‘‘middle region’’ in Fig. 10 is neither static-like nor described by Eq. [1], which contains the geometry-free viscous deformation. In this middle region, viscous and gravitational forces are of the same order and both together balance surface tension. This region must exist since the viscous deformation described by Eq. [1] can not be valid for all r. The viscous effects in Eq. [1] arise from balancing only viscous and surface tension forces, without accounting for gravity.
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But, for each Ca, there exists some r* such that for r ú r* gravity becomes comparable to viscous forces before the latter become negligible. Hence in the region where viscous and gravitational forces compete, the viscous deformation is perceived as being affected by the geometry of the outer region. Since r* increases with Ca, so does the size of the geometry-free viscous region. Note that the ‘‘middle region’’ discussed here has no relation to recirculation flows in the fluid container. 4.2. Interpretation of v0 In the DRG model (1), v0 is determined by the interface shape near the contact line. Yet v0 physically represents the contact angle of the static limit of the interface shape far from the contact line. We have proven this supposition by comparing v0 obtained by fitting Eq. [1] to data near the contact line to v *0 obtained by fitting Eq. [4] to data in the static-like region far from the contact line. This static-like region has been identified for Ca Å 0.001 and 0.005. At Ca Å 0.001, in the region of stable x 2 , v *0 0 v0 Å 00.26 { 0.27. This difference lies well within the uncertainty identified in the calibration. At Ca Å 0.005, v *0 0 v0 Å 01.07 { 0.037. This relatively large magnitude may be due to residual viscous deformation coupled with the measurement distortions. These results are consistent with the identification of v *0 with v0 and strongly suggest that v0 must be the correct boundary condition for the macroscopic shape. 4.3. Apparent Contact Angle and Contact Line Shift Apparent contact angles, uapp , have been a popular way of quantifying dynamic wettability in the past, both with immersing cylinder and capillary tube geometries (12, 13). In experiments where meniscus heights are measured, uapp is calculated as the contact angle of a static interface rising to the same meniscus height (or to the same apex height, for a capillary tube) as measured from the dynamic interface. uapp measured in this manner cannot be transferred to another geometry nor is it the correct boundary condition for the static-like interface shape far from the contact line. uapp is also different from apparent dynamic contact angles obtained from Wilhelmy plate force measurements. Force measurements on dynamic Wilhelmy plates that do not account for viscous forces exerted on the plate will produce an apparent contact angle, u *app x uapp . Further, the relation between u *app and v0 can only be known when a viscous analysis of the Wilhelmy plate flow is available. Here, we examine the efficacy of these relatively simple measurements by comparing uapp with v0 , the proper dynamic contact angle. At the same time we consider a related measurable quantity, the distance between the observed contact line and the point where the extrapolation of the static-like interface far away intersects the solid (AB in Fig. 1). We call this distance the
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TABLE 1
Ca
Contact line shift (mm)
v0 0 uapp (degrees)
0.001 0.005 0.01 0.1
40 50 60 73
1.93 2.38 2.2 3.2
contact line shift. v0 0 uapp and the contact line shift are not independent. They both are zero in the absence of viscous deformation and vary together in the presence of viscous forces. Usually, uapp is determined by measuring the vertical distance from the contact line to the bulk fluid level (or the apex of the meniscus, in a capillary tube geometry). Unfortunately, we cannot measure this distance from our images because our field of view does not show the bulk fluid level. Measurements with a cathetometer have insufficient resolution for our purposes. To find the bulk fluid level from our images we take the interface slope, u, at x Å 1200 mm and continue the interface to infinity as a static shape. To mimic experiments, uapp is calculated from the height between this bulk level and the observed contact line using the relation for a static interface shape. The continuation of the same static shape to the solid locates point B in Fig. 1 and thus determines the contact line shift. For Ca Å 0.001 and 0.005, this method is straightforward. Results appear in Table 1. At Ca Å 0.01, a static-like region did not appear within our field of view. However, uapp was stable at 58.767 { 0.037 when calculated using the slope, u, at 800, 1000, and 1200 mm. At Ca Å 0.1, uapp calculated as outlined above did not stabilize. By noting the effects of residual viscous deformation in our measurements of u at 1200 mm, we can bound the values at 71 mm õ shift õ 74 mm, 117.47 õ uapp õ 117.67 at Ca Å 0.1. Table 1 shows the contact line shift and v0 0 uapp vs Ca for our experimental geometry. Our measurements show that uapp õ v0 . As predicted by Eq. [1], the viscous deformation near the contact line makes the interface more curved than the static baseline shape, hence pulling the meniscus above the extrapolation of the static-like shape (see Fig. 1). v0 0 uapp and the contact line shift increase modestly with Ca. The explanation for their moderate growth is that viscous effects on interface curvature are controlled by two competing factors. Increasing Ca causes the viscous deformation to increase at any fixed r and u. However, as Ca increases the interface angle, u, also increases at any given r. This decreases the velocity gradients responsible for the viscous deformation. Though these results are valid for all material systems having inner mechanisms equivalent to that of our system, they are limited to the geometry used in our experiments.
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Because parameters such as v0 or uapp depend on the system geometry, their usefulness from a modeling standpoint hinges on the possibility of relating them to a material property of the system. Even though the value of uapp is approximately correct, it can not be related to any material property because there exists no analysis of the flow field valid for 0 õ x õ ` . Such analysis would be necessary to relate uapp to a geometry-independent feature such as the interface shape in the intermediate region. On the other hand, v0 measured using Eq. [1] can be easily transformed into the material parameter L (see Eq. [3]). This parameter can then be exported to other geometries to be used as the needed material specific boundary condition. The contact line shift discussed here (AB in Fig. 1) is not equal to the shift in contact line position between static (i.e., no motion) and dynamic conditions (14). In fact, the two have opposite signs. A static contact line on a vertical plate is higher than the dynamic contact line when the plate is advancing into the fluid. The fluid flow in the intermediate region causes v0 to increase as Ca increases (9, 14). Over the macroscopic extent of the fluid interface, the motion of the surface lowers the dynamic interface height compared to the static height at each x. Unmagnified, the meniscus appears to have a larger apparent contact angle. In contrast, near the contact line where viscous forces strongly deform the interface, the dynamic contact line rises above the extrapolation of the static-like region of the actual dynamic interface (AB in Fig. 1). Without a good microscopic view showing the actual dynamic interface shape, the observer is unlikely to sense this upturn of the meniscus but will sense the position of the contact line which is slightly higher than the extrapolation of the static-like region. Thus the fluid flow has two effects on the contact line position; it lowers it compared to the completely static shape but raises it relative to the static-like component of the macroscopic dynamic interface. The magnitude of the former is larger than that of the latter. 4.4. Capillary Tube Geometry Many measurements of apparent dynamic contact angles are done in capillary tubes. The experimenter measures the dynamic apex height and computes uapp as the contact angle of a static meniscus having the same apex height as the dynamic meniscus. Our measurements in the immersing cylinder geometry show that uapp systematically differs from the correct dynamic contact angle v0 . However, it would be improper to assess potential errors in capillary tube measurements by merely applying the results of our experiments to the capillary tube geometry because the dominant force balance is different in a capillary tube. In a thin capillary tube, surface tension forces—not gravity—dominate in the outer region at low Ca. This change in the dominant force balance has the potential of altering the extent of the region where the interface is static-like.
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Since interface shape measurements in capillaries have not been performed, we analyze Lowndes’ (15) numerical simulation of a meniscus rising up a cylindrical tube of radius a. From Lowndes’ results we can compute uapp and v0 . uapp appears tabulated or can be roughly obtained from plots of the interface shapes (15). We calculate v0 from g( v0 ) Å g( U ) / Ca ln(a/Li ),
[5]
where U is the actual contact angle applied to the interface in the inner region and Li is the length scale of the inner region (1). Both U and Li are inputs to Lowndes’ calculations. At Ca Å 0.075, v0 0 uapp remains of the same order as in our geometry and material system at 3.27. A better measure of the extent of viscous effects would be to find the static-like portion of the meniscus by fitting the interface near the centerline to a static shape and analyzing the deviations as we did in Section 4.1. Unfortunately Lowndes’ data are only in a plot whose quality does not allow us to make this comparison. However, our measurements in the immersing tube geometry and the example from Lowndes indicate a strong possibility that in the outer region of the capillary tube geometry, i.e., near the center line, significant viscous deformation exists at Ca Ç 0.1. 5. CONCLUSIONS
In this study, the global analysis of a fluid interface under dynamic conditions, including the viscous deformation near the contact line, has been documented for the first time. The work assessed the appropriateness of assumptions made in deriving Eq. [1], a model for the shape of the fluid interface valid near the moving contact line (i.e., r/a ! 1) in the limit of Ca r 0. The assumptions tested include: (a) Near the dynamic contact line, the interface shape shows viscous deformation; but far from the contact line the interface shape becomes static-like. (b) The parameter of the model, v0 , corresponds to the contact angle of the static extrapolation back to the solid of the static-like region of the dynamic interface far from the contact line. We investigated the regions of validity of Eq. [1] and of the static theory (Eq. [4]) by analyzing each image of a dynamic interface near and far from the contact line. The descriptions provided by Eq. [1] and Eq. [4] are not guaranteed to be found in the field of view (a little larger than the capillary length a). Specifically, we studied the shape of a dynamic fluid interface at capillary numbers ranging from 0.001 to 0.1 by imaging dynamic interface shapes out to about 1700 mm from the moving contact line, about five times farther than in previous studies (1–3). We have shown that the region of the fluid interface described by the static shape with contact angle v0 moves out as Ca increases. In fact, at Ca as low as 0.01, we detected significant viscous deformation out to about 1400 mm from
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the contact line, a distance well embedded in the outer region of our geometry. We also found that the region well described by Eq. [1] extends farther out as Ca increases but it does not overlap with that described by static theory. The ‘‘middle region’’ in Fig. 10 lies between the two regions cited above. It is neither static-like nor described by Eq. [1], which accounts for the geometry-free viscous deformation. Therefore, the viscous deformation in the ‘‘middle region’’ must depend on geometry. This geometry dependence comes from a competition between viscous and gravity forces in a region where viscous forces have not yet become negligible. However, the shape in this region is not related to recirculation flows which must prevail at a length scale on the order of the fluid container size. Our interface measurements demonstrate that the second assumption of the model is correct. The model parameter, v0 , obtained from fitting data near the contact line where viscous deformation is significant, is the boundary condition for the static-like shape far from the contact line. From our experimental data we determined uapp (the static contact angle for the observed meniscus height) and the contact line shift for the Ca’s studied. We showed that v0 0 uapp and the contact line shift have a similar dependence on Ca. They grow modestly with Ca because of the competing effects of increasing Ca; it increases viscous deformation but the simultaneous opening of the angle between the solid and the fluid interface decreases the velocity gradients responsible for viscous deformation. The contact line shift studied here is only the smaller component of the contact line displacement between a static and a dynamic situation. The larger component is perceived as the contact angle of the macroscopic interface increases to v0 from its static advancing value. Our measurements show that errors made in measuring uapp in an immersing tube geometry or in a capillary tube are a few degrees and systematic with Ca for Ca £ 0.1. Even at lower Ca, the viscous deformations in capillary tubes
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can potentially extend well into the outer region near the tube center, leading to errors made in assuming a static shape for the center of the meniscus. The extent of the static-like region near the tube center should be determined by accurate measurements or careful numerical simulations for the capillary geometry. Even though uapp is approximately correct, it cannot be related back to a material property of the system in the same way as v0 . Thus, simpler two-point measurements cannot be used in different geometries as a predictive measure of dynamic wettability nor for careful material studies of the wetting dynamics. ACKNOWLEDGMENTS This work was supported by NASA Grant NAG3-1390. The authors acknowledge Ms. Qun Chen and Mr. Keith R. Willson, Department of Physics of Carnegie Mellon University, for their invaluable assistance with the experiments and many helpful suggestions.
REFERENCES 1. Dussan, V., E. B., Rame´, E., and Garoff, S., J. Fluid Mech. 230, 97 (1991). 2. Marsh, J. A., Garoff, S., and Dussan V., E. B., Phys. Rev. Lett. 70, 2778 (1993). 3. Marsh, J. A., ‘‘Dynamic Contact Angles and Hydrodynamics Near a Moving Contact Line,’’ Ph.D. thesis, Carnegie Mellon University, 1992. 4. Chen, Q., Rame´, E., and Garoff, S., Phys. Fluids (1995), in press. 5. Chen, J., and Wada, N., Phys. Rev. Lett. 62, 3050 (1989). 6. Petrov, J. G., and Sedev, R. V., Colloids Surf. A 74, 233 (1993). 7. Kistler, S. F., in ‘‘Wettability’’ (J. C. Berg, Ed.), p. 311. Dekker, New York, 1993. 8. Hocking, L. M., and Rivers, A. D., J. Fluid Mech. 121, 425 (1982). 9. Cox, R. G., J. Fluid Mech. 168, 169 (1986). 10. Boender, W., Chester, A. K., and van der Zanden, A. J. J., Int. J. Multiphase Flow 17, 661 (1991). 11. Huh, C., and Scriven, L. E., J. Colloid Interface Sci. 30, 323 (1969). 12. Seeberg, J. E., and Berg, J. C., Chem. Eng. Sci. 47, 4455 (1992). 13. Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975). 14. De Gennes, P. G., Hua, X., and Levinson, P., J. Fluid Mech. 212, 55 (1990). 15. Lowndes, J., J. Fluid Mech. 101, 631 (1980).
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Article No. 0026
Microscopic and Macroscopic Dynamic Interface Shapes and the Interpretation of Dynamic Contact Angles ENRIQUE RAME´ 1
AND
STEPHEN GAROFF
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 U.S.A. Received March 17, 1995; accepted June 7, 1995
We have studied shapes of dynamic fluid interfaces at distances £1700 mm from the moving contact line at capillary numbers (Ca) ranging from 10 03 to 10 01 . Near the moving contact line where viscous deformation is important, an analysis valid to O(1) in Ca describes the shape of the fluid interface. Static capillarity should describe the interface shape far from the contact line. We have quantitatively determined the extent of the regions described by the analysis with viscous deformation and by a static shape as a function of Ca. We observe a third portion of the interface between the two regions cited above, which is not described by either the analysis with viscous deformation or a static shape. In this third region the interface shape is controlled by viscous and gravitational forces of comparable magnitude. We detect significant viscous deformation even far from the contact line at Ca ) 0.01. Our measured dynamic contact angle parameter extracted by fitting the analysis with viscous deformation to the shape near the moving contact line coincides with the contact angle of the static-like shape far from the contact line. We measure and explain the discrepancy between this dynamic contact angle parameter and the apparent contact angles based on meniscus or apex heights. Our observations of viscous effects at large distances from the contact line have implications for dynamic contact angle measurements in capillary tubes. q 1996 Academic Press, Inc. Key Words: wetting; dynamic wetting; moving contact line; dynamic contact angle; spreading.
1. INTRODUCTION
The dynamics of liquids spreading on solid surfaces controls natural and technologically important processes such as two-phase flow in porous media, the spraying of paint droplets, the deposition of coatings, and the imbibition of inks in papers. In each of these processes, surface tension forces are important and details of the fluid motion in the immediate vicinity of the moving contact line control the spreading dynamics. Unfortunately, the motion of the fluids in this region has proven hard to describe because the classi1
To whom correspondence should be sent. Email: er2n/@andrew. cmu.edu. 0021-9797/96 $12.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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cal hydrodynamic model 2 is singular at the moving contact line. If this classical model held, an infinite force would be required to sink a spoon into a coffee cup. The classical hydrodynamic model cannot be used up to and including the contact line. Dussan V., Rame´, and Garoff (abbreviated DRG) (1) developed a model for describing interface shapes near moving contact lines in the limit of low capillary number, Ca ( å Um/ s, where U is a characteristic velocity, m the fluid viscosity, and s the surface tension). This model, discussed in Section 2, describes the shape of the fluid interface when a solid enters a fluid at constant velocity and immersion angle. The region described by the model is characterized by a balance of viscous and surface tension forces. In this region, viscous forces deform the interface significantly. The model introduces a single adjustable parameter, v0 , and specifies the functional form describing the interface shape near the moving contact line. v0 depends on the material properties of the system, must be determined experimentally, and plays the role of a dynamic contact angle. Experiments have shown that this model accurately predicts the interface shape within 300 mm from the contact line for Ca õ 0.1 (1–4). These experiments also confirmed the model’s prediction that the viscous deformation in this region is independent of the macroscopic geometry. They established the functional form of the interface shape near the moving contact line as the boundary condition for the shape in the macroscopic region. Central to the analysis in DRG is the assumption that, when Ca ! 1, the interface shape far from the contact line is well described by static capillarity. This static-like shape far from the contact line should meet the solid surface with contact angle v *0 if it were continued as a static shape all the way to the solid. In the model, v *0 Å v0 , where v0 is extracted by fitting the model which contains the geometryindependent viscous deformation to the interface shape close to the moving contact line. 2 These assumptions are: Newtonian, incompressible fluid, undeformable solid, and the continuity of velocity at bounding surfaces. The latter amounts to the no-slip condition at a fluid–solid interface.
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In this study, we verify the assumptions of the present model about the condition of the fluid interface far from the moving contact line. We study the dynamic interface shape up to 1700 mm from the contact line, a length scale much larger than studied in previous experiments. This region is q on the order of the capillary length, s / rg (where r is the fluid density and g is the acceleration of gravity), the length scale where the interface is primarily controlled by gravity and surface tension forces. As a function of Ca, we determine the part of the interface near the contact line described by the analysis of DRG containing the viscous deformation. We also identify the region far from the contact line behaving as a static-like interface. In this region, viscous effects are undetectable by our optics and image analysis system. Our measurements in the staticlike region are consistent with the interpretation given in the theory, that v *0 Å v0 . We show that, within our detection limits, viscous effects reach 1400 mm at Ca Å 0.01. However, only the viscous deformation nearest the contact line is independent of the overall geometry. Other measurements of dynamic interface shapes near a moving contact line have been reported (5, 6). We also compare v0 to apparent contact angles obtained from ‘‘two-point measurements.’’ Typical two-point measurements are based on the rise of a dynamic meniscus against a moving plate or cylinder, or the apex heights of menisci in capillaries. The apparent contact angle corresponds to the contact angle of a static interface with the same rise or apex height as the dynamic one. We show that, in our geometry, the difference between the apparent contact angle and the parameter v0 changes systematically with Ca. We discuss the errors that may arise when using two-point measurements to characterize dynamic wettability in geometries other than the one used in our measurements. In Section 2 we present the theoretical background for our experiments. In Section 3 we describe the experiment. In Section 4, we present and discuss the results; and in Section 5, we state the conclusions.
devoid of moving contact lines, this singularity has made evident the inadequacy of the classical model to describe systems with moving contact lines. The difficulty cited above does not play the same role under all conditions. The importance of the contact angle in determining the interface shape increases with surface tension both in static and in dynamic conditions. Hence, studies aimed at properly modeling moving contact lines concern themselves with conditions where surface tension forces dominate over viscous and gravitational forces. These conditions require that the capillary number, Ca, and the Bond number, Bd ( år gD 2 / s, where D is the characteristic length of the system; Bd measures the ratio of gravitational to surface tension forces) be !1. The motion of a contact line has been the object of numerous studies in the last 20 years (7). In order to describe systems with moving contact lines, the classical model needs to be modified by including a new region surrounding the moving contact line where the classical hydrodynamic assumptions are replaced by a new set of assumptions. This region has characteristic length scale Li and is commonly known as the inner region. Very near the moving contact line, but not so close that the classical model breaks down, models postulate the existence of a region where the fluid motion obeys the classical model and is independent of the macroscopic geometry of the system. This region, called the intermediate region, lies between the inner and the macroscopic—or outer—regions. In the intermediate region, the fluid interface can sustain significant viscous deformation even to lowest order as Ca r 0 (8–10). Based on these ideas, DRG performed an analysis which describes the interface shape near the moving contact line generated by a cylindrical tube of radius RT vertically entering a liquid bath. Their main result, asymptotically correct to O(1) as Ca r 0, holding Ca ln(a/Li ) fixed, is
2. THEORETICAL BACKGROUND
where u is the local slope of the interface with respect to the solid, r is the distance from the contact line to the point on the interface (see Fig. 1), a is the capillary length q s / rg, and g 01 is the inverse function of g, i.e., x å g 01 (g(x)) with
The spreading of liquids is controlled by the motion of a contact line across a solid surface, a feature that puts spreading systems in a special class of fluid mechanics problems. The difficulty in this class of problems is that the classical hydrodynamic assumptions give rise to a singularity in the stress field at the contact line. The singularity prevents the use of the contact angle as a boundary condition for the interface shape. This marks a significant departure from capillary statics, where the contact angle is the classical material property needed to wholly determine the shape of the fluid interface on the macroscopic scale. Even though the same classical model works well for all other dynamic situations
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u Ç g 01 (g( v0 ) / Ca ln(r/a)) / f0 (r/a; v0 , RT /a) 0 v0 ,
g(x) å
*
x
0
y 0 sin(y)cos(y) dy. 2 sin(y)
[1]
[2]
The first term in Eq. [1] represents the interface shape in the intermediate region, i.e., uint Ç g 01 (Ca ln(r/L)),
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FIG. 1. Coordinate notation used in equations and graphs. (A) observed contact line. (B) extrapolation of the static-like interface shape to the solid. (---) Static continuation of the static-like interface shape far from the contact line. (AB) contact line shift. The x-axis coincides with the bulk fluid level.
where the parameter L ( å a exp[ 0Ca/g( v0 )]) depends only on the physics of the inner region (2). The second term is the solution in the outer region, correct to O(1). It is the same as for a static interface rising on a vertical cylinder of radius RT (11): uout Ç f0 (r/a; v0 , RT /a).
sible outer length scale. The outer qlength scale in this geometry was the capillary length a å s / rg. Given the range of Ca studied, menisci formed both in rise and in depression. For this reason, two kinds of liquid containers were used. For menisci in rise, we used a slightly overfilled cylindrical Teflon container about 8 cm in diameter and 12 cm deep. The meniscus rose above the rim of the container. For menisci in depression, we used a square glass container 10 cm on a side and filled to about 4/5 of its total volume. The PDMS wetted the walls of this container. The Pyrex tube was driven by a motorized stage. Two types of motors were used: for menisci in rise, a computercontrolled Newport Klinger (Model UE30CC) mounted on a Newport Klinger translation stage (UT100); and for menisci in depression, a screw motor (Newport 860SC Motorizer) mounted on a stage (Newport 435-2). We produced menisci in rise using a 1000 cSt fluid at three different immersion speeds, 2.1 1 10 03 cm/s, 1 1 10 02 cm/s, and 2.1 1 10 02 cm/s, and in depression, with a 60000 cSt fluid at 3.26 1 10 03 cm/s. These conditions correspond to capillary numbers of 9.7 { 0.1 1 10 04 , 4.62 { 0.01 1 10 03 , 9.70 { 0.01 1 10 03 , and 9.580 { 0.002 1 10 02 , respectively. The capillary length was different for each of the fluids: a Å 0.146 cm for the 1000 cSt fluid, and a Å 0.148 cm for the 60000 cSt fluid.
[4] 3.2. Optics and Image Analysis
Equation [1] represents the interface shape in the intermediate region up to its overlap with the outer region. In Eq. [1] the single parameter v0 must be determined experimentally. Within the model, v0 is interpreted as the contact angle of the extrapolation back to the solid of the static-like interface far away (described by Eq. [4]). 3. MATERIALS AND PROCEDURE
3.1. Experimental Apparatus We immersed a vertical, cylindrical Pyrex tube at constant velocity into a container of polydimethylsiloxane (PDMS). Since PDMS completely wetted the glass, the static meniscus rose up the tube and attained a zero static contact angle.3 When the tube moved into the fluid, viscous forces distorted the fluid interface and the meniscus appeared flatter. Below some speed, the meniscus remained above the general fluid level, a ‘‘meniscus in rise.’’ At higher speeds, the meniscus was bent below the general fluid level, a ‘‘meniscus in depression.’’ This experimental geometry gave the largest pos3 Work being done in our laboratory addresses PDMS spreading on glass coated with a perfluorinated surfactant, where the static advancing contact angle uA x 0. This allows study of receding contact lines. We do not anticipate qualitative differences between uA x 0 and this work, where uA Å 0.
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The meniscus was backlit using Koehler illumination (3). The silhouette of the meniscus and of the tube were viewed using a long working distance microscope attached to a CCD camera with 640 1 480 pixel array. The fluid container and the glass tube were placed between the Koehler illumination and the microscope. The image was sent from the camera to a computer for digital storage and analysis. Two camera/ microscope systems were used, Pulnix TM-740 camera/ Bausch & Lomb Monozoom 7 microscope for depression images, and Cohu 4110 Solid State Camera/Infinity K2 microscope for rise images. In rise, we slightly overfilled the Teflon container and viewed the meniscus above the bulk fluid; in depression we partly filled the square glass cell and viewed through the bulk liquid. Figure 2 shows typical images. Great care must be taken to align the optics (3). Since the light that forms images of menisci in depression travels through the liquid, the faces of the square glass container must be set normal to the light beam. The focus must be changed between rise and depression images to account for the fact that the fluid has an index of refraction different than air (4). Finally, we recorded the vertical direction before each experiment using an image of a plumb bob. This was necessary to compensate for misalignments between the pixel columns in the camera and the true vertical. Five images, at 5-s intervals, were taken at each immersion velocity.
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FIG. 2. Typical images of a meniscus. The straight vertical edge is the tube edge; the curved boundary is the liquid–air interface. (a) In rise, the liquid and the solid are dark; the air is light. (b) In depression, the liquid is light; the solid and the air are dark.
The data analysis consists of identifying the location and slope of the meniscus and fitting sections of the interface slope data using Eq. [1] or [4]. The air–liquid interface is
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located by the highest gradient of gray levels in the digital image. The interface location and slope are found using discrete gray-level algorithms adapted by Marsh (3). A typi-
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for the region of data used in the fit. To assess the full extent of regions of agreement of the data and the theory, we examine the systematic deviations outside the fitted region. In order to identify the point where either Eq. [1] or [4] no longer fits appropriate sections of data, we set the criterion that a smooth line fit to the deviation between data and theory must depart from zero by sV /3. 3.3. Materials
FIG. 3. (a) Data and best fit for the static interface in capillary rise used as calibration. (b) Difference of the data and the best theoretical fit.
cal data set of the slope versus distance from the contact line consists of about 1000 data points. The data appear as a cloud of points about 1 to 3 degrees wide. See Fig. 3 for a typical example. For several images of the same meniscus taken in rapid succession, the data clouds overlap but the slope values at each spatial location do not. Thus the cloud cannot be due to real variation in the interface shape. Rather, it is due to pixel noise in the camera. Therefore the thickness of the data cloud sets the standard deviation assigned to the slope values. Due to the density of points, our detection limit of systematic deviations of the data from a theoretical form is far better than the 17 to 37 standard deviation in the individual data points. We calculate the normalized chi-squared deviation, x 2 , of a set of N data points as x2 å
( ( u iexp 0 u itheo ) 2 , sV 2 (N 0 1)
where sV is the standard deviation of each point. Since we treat each point as independent, we compute sV as the standard deviation of the data cloud about a smooth line passing through the cloud’s center. A second- or third-degree polynomial is normally sufficient. The criteria for a good fit are that x 2 Ç 1 and that the plot of the deviation of the data from the best-fit theoretical line scatter uniformly about zero
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The solid on which the liquid spreads is a 2.54 cm outer diameter Pyrex tube. Its outer surface is fire-polished and carefully cleaned to ensure a smooth contact line all around the tube. The Pyrex tube was cleaned by soaking it in sulfochromic acid solution for 40 min, rinsing it with pure water (from a Barnstead system, model ROPure LP, fitted with NANOPure II filtration cartridges), soaking it in 15% HCl solution for 40 min to remove Cr ions, and rinsing it again with pure water. The water layer wetting the glass after the rinse was blown off with dry nitrogen. If the water did not sheet off uniformly from the glass we repeated the cleaning. The Teflon container was cleaned with ethanol, then rinsed with pure water and dried in a vacuum oven. The glass container was washed with dish detergent and rinsed thoroughly with pure water. Both containers were filled with the PDMS fluid after cleaning and remained filled for the duration of the experiments. Two PDMS fluids were used, with nominal kinematic viscosities of 1000 cSt (Dow Corning 200 fluid) and 60000 cSt (United Chemical Inc.). The fluid properties for the duration of the experiments were: density r Å 0.97 g/cm3 , surface tensions s1 Å 20.3 dyn/cm and s2 Å 20.8 dyn/cm, and viscosities m1 Å 9.6 g/cm s and m2 Å 611 g/cm s, respectively. 3.4. Calibration The interface measurement system, which consists of the combined optical setup and digital image analysis algorithm, was calibrated using a static meniscus as reference. The calibration consists of finding the best fit between static theory and a static interface image using the contact angle as the fitting parameter. Since the theory describing static menisci is known exactly, systematic deviations between static data and the best static fit indicate the distortions in the system. Because the curvature of the interface along the optic axis and the angle at the contact line are crucial to the image formation ( 3 ) , calibration must be done on static interfaces similar to the ones obtained in the dynamic experiments. We produced a static interface with contact angle near 357 by contacting PDMS with a Pyrex tube coated with a monolayer of fluorinated surfactant. Figure 3a shows the data and the best fit. Data for x õ 50 mm and x ú 1500 mm exhibited systematic deviations. Thus, we
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FIG. 4. x 2 vs xc , the inner edge of the fitted region, for the best fits of the static interface used as calibration in capillary rise.
restricted the data analysis of menisci in rise to 50 mm õ x õ 1500 mm. Figure 3b shows that the difference between data and the best fit is uniformly distributed about zero for 50 mm õ x õ 1500 mm. x 2 is statistically constant near 1.03 using fits to subsets of data xc õ x õ 1500 mm, where 100 mm õ xc õ 1200 mm, suggesting that the system could be distortion-free (see Fig. 4). However, a test of the sensitivity of the fitting parameter (the contact angle) to xc , as xc moves progressively out reveals the presence of systematic distortions. Figure 5 shows that the value of the contact angle is rather stable up to about 600 mm from the solid, at which point it begins to show a systematic upward drift. The magnitude of this drift is 0.47 in the field of view of our images. The contact angle continues to change with xc while x 2 is statistically constant because each portion of the interface being analyzed looks almost static, with the same x 2 but slightly ‘‘different’’ contact angles. The contact angle changes slowly but systematically as we fit portions of the interface farther and farther from the solid. The ‘‘different’’ contact angle can be rationalized as a progressive tilting of the image relative to the real interface as we move away from the solid. The interaction of the light waves with the curved surface of the fluid interface depends on the edge’s
FIG. 5. Contact angle vs xc , the inner edge of the fitted region, for the static interface used as calibration in capillary rise.
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FIG. 6. (a) Data and best static fit for the static interface in capillary depression used as calibration. (b) Difference of the data and the best theoretical fit.
curvature in the direction of the optic axis. This curvature is highest near the solid and gets progressively smaller moving away from the solid. The position of the maximum gradient in the intensity of the transmitted image relative to the physical edge of the interface may drift across the image. As a reference, we estimated that in order to cause a 0.47 variation in the contact angle for 600 mm õ xc õ 1200 mm, the data of the local interface slope angle should have a linearly varying error of only about 0.047 /100 mm. The static interface in depression was created by immersing a 2.54 cm outer diameter brass rod with a sharp edge machined on its circumference. The meniscus pins on the sharp edge as it is immersed into the fluid. Images of menisci in depression exhibit a distortion qualitatively similar to that of images in rise (Fig. 6), except that the contact angle decreases uniformly as xc moves from 100 mm to 1000 mm, with a total variation of about 0.87 (Fig. 7). In depression, the distortion-free region (deviations uniformly distributed about zero) was 50 mm õ x õ 2000 mm. Thus we restricted our analysis of depression images to data in this region. The different sign of the systematic distortion in rise and in depression further supports the notion that the image formation process is responsible for the systematic distortions. Given the result of our static calibration, we must assign an uncertainty of 0.47 (for rise) and 0.87 (for depression) to
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FIG. 7. Contact angle vs inner edge of the fitted region, xc , for the static interface used as calibration in capillary depression.
contact angles from fittings to portions of the dynamic meniscus far from the contact line. 4. RESULTS AND DISCUSSION
In the first part of this section, we provide direct measurements of the regions where geometry-free viscous deformations dominate the interface shape and where the interface shape is essentially static. We show how the extent of interfacial regions in agreement with the viscous deformation described by Eq. [1] or with a static shape (Eq. [4]) changes with Ca. In the next subsection, we test whether the interpretation of v0 is correct, i.e., whether v0 as determined by Eq. [1] is the contact angle of the static-like interface far from the contact line when it is extrapolated statically to the solid (see Fig. 1). We test this by independently measuring v0 from fittings of Eq. [1] to data near the contact line and of Eq. [4] to data far from it. We then discuss two related measures of viscous deformation: (a) the distance between the dynamic contact line and the point where the static extrapolation of the static-like shape far away intersects the solid (contact line shift, AB in Fig. 1), and (b) the difference between v0 and the apparent dynamic contact angle derived from a measurement of the meniscus height in our experimental geometry (AO in Fig. 1). We conclude by discussing these effects for a dynamic meniscus in a capillary tube.
even closer to the contact line, viscous forces increase even more. From a certain point approaching the contact line, the flow becomes locally determined. The influence of gravity and of the overall system geometry is lost. The interface shape in the region controlled by the local flow is geometryindependent and results from a balance between viscous and capillary forces. In this region the interface shape is described by Eq. [1]. The extent of the interface deformed by viscous forces increases with increasing Ca. Consequently both the inner boundary of the region described by static capillarity (Eq. [4]) and the outer boundary of the region with geometryfree viscous deformation described by Eq. [1] should increase with increasing Ca. To identify these regions, we analyze data from our dynamic experiments at each Ca in two different ways. First, we fit the data in the region 50 mm õ r õ 300 mm using Eq. [1]. We then use the deviation of the data from the best fit to probe the full extent of the region described by Eq. [1]. Next, we fit various segments of the data xc õ x õ 1500 mm (for 100 mm £ xc £ 1200 mm) to Eq. [4]. This determines the extent of the static-like region of the interface. Figure 8 shows how Eq. [1] describes a portion of the interface shape at two of the Ca’s tested. Figure 9 shows
4.1. Regions of the Interface Shape All regions of the fluid interface are not controlled by the same dominant balance of forces. For a given Ca, the fluid interface sufficiently far from the contact line is controlled by surface tension and, depending on Bd, gravity. In this region viscous forces are negligible, and the interface shape looks static. This is the region where the interface shape is well described by Eq. [4]. But this static-like situation can not be valid everywhere because viscous forces grow compared to surface tension and gravity as the contact line is approached, distorting the shape away from static. Moving
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FIG. 8. Data and fits versus distance from the contact line. (a) Ca Å 0.005. (b) Ca Å 0.1. ( —) Best fit of Eq. [1] in 50 mm õ x õ 300 mm. ( – – – ) Static interface with contact angle equal to v 0* from the fit of Eq. [4] for (a) and to v0 from Eq. [1] for (b). The insets are blow-ups showing static fit far from the contact line.
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FIG. 9. Difference of the data from the best fit of Eq. [1] in 50 mm õ x õ 300 mm at Ca Å 0.005. ( – - – ) 300 mm boundary of fitted region. ( —) Third degree polynomial fit of data cloud in region where departure from zero crosses sV /3.
the deviation of the data at Ca Å 0.005 from the best fit of Eq. [1] to data in 50 mm õ r õ 300 mm. Similar behaviors appear at all other Ca’s tested. From graphs like that of Fig. 9 we locate the region described by Eq. [1] within our detection limits. To find this region we determined the point where the deviation cloud departs from zero by sV /3. We found the deviation’s departure from zero using a thirddegree polynomial fit of the relevant part of the cloud (solid line in Fig. 9). The region described by Eq. [1] moves out monotonically with Ca (lower line in Fig. 10). The model producing Eq. [1] postulates that the interface far from the contact line is static-like. To establish whether a static-like region is present in our field of view, we fit a static interface to different sections of the data in xc õ x õ 1500 mm, with 100 £ xc £ 1200 mm. The fitting parameter is v *0 (the contact angle of the static shape). To simplify the fitting, we used x/a as the independent variable because it is an absolute distance, unlike rV /a, which depends on the unknown location of the putative contact line (point B, Fig. 1). The best fit static line at Ca Å 0.005 for x ú 1200 mm is shown by the dashed line in Fig. 8a. In order to determine the region where the interface becomes static-like, we ana-
FIG. 10. The boundary of the regions fit by Eq. [1] and by Eq. [4] as a function of Ca. Points marked f are upper bounds on the region boundary.
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FIG. 11. x 2 as a function of xc , for fits of static shapes to static and dynamic menisci. ( xv ) static. ( s ) Ca Å 0.001. ( h ) Ca Å 0.005. ( n ) Ca Å 0.01.
lyzed x 2 as a function of xc for each image. Figure 11 shows the x 2 vs xc for the static interface and for one typical interface at each Ca Å 0.001, 0.005, and 0.01. x 2 for the static interface stays flat near 1 and serves as a baseline for comparisons to x 2 from dynamic measurements. At Ca Å 0.001, x 2 becomes stable at about xc ú 200 mm, while at Ca Å 0.005, x 2 stabilizes at xc ú 800 mm. At Ca § 0.01, x 2 approaches 1 but does not stabilize within our field of view. The region of stable x 2 near 1 represents the static-like region of the interface. We did not find a static-like region at Ca § 0.01. Therefore in Fig. 8b, we plotted the static shape using Eq. [4] with v0 found from the good fit of Eq. [1] to data within 300 mm from the contact line. The boundary of the static-like region fitted by Eq. [4] is shown by the upper line in Fig. 10. This distance was estimated using the same method described above for finding the region fit by Eq. [1]. Since we did not find a static-like region within our field of view at Ca Å 0.01 and 0.1, we estimated the region fitted by Eq. [4] using v0 from the fit of Eq. [1] near the contact line. As shown in Fig. 10, we can detect significant viscous deformation at distances £1400 mm from the contact line, at Ca as low as 0.01. This distance is well within the outer region (given here by the capillary length a Ç 1.5 mm), and has implications for similar measurements in capillary tubes (see Section 4.4). Our findings contrast other reports (6) stating that static-like shapes prevail all the way in, to about 100 mm from the moving contact line at Ca Ç 0.1. For our experimental conditions, the regions well described by either Eq. [1] or Eq. [4] do not overlap. The ‘‘middle region’’ in Fig. 10 is neither static-like nor described by Eq. [1], which contains the geometry-free viscous deformation. In this middle region, viscous and gravitational forces are of the same order and both together balance surface tension. This region must exist since the viscous deformation described by Eq. [1] can not be valid for all r. The viscous effects in Eq. [1] arise from balancing only viscous and surface tension forces, without accounting for gravity.
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But, for each Ca, there exists some r* such that for r ú r* gravity becomes comparable to viscous forces before the latter become negligible. Hence in the region where viscous and gravitational forces compete, the viscous deformation is perceived as being affected by the geometry of the outer region. Since r* increases with Ca, so does the size of the geometry-free viscous region. Note that the ‘‘middle region’’ discussed here has no relation to recirculation flows in the fluid container. 4.2. Interpretation of v0 In the DRG model (1), v0 is determined by the interface shape near the contact line. Yet v0 physically represents the contact angle of the static limit of the interface shape far from the contact line. We have proven this supposition by comparing v0 obtained by fitting Eq. [1] to data near the contact line to v *0 obtained by fitting Eq. [4] to data in the static-like region far from the contact line. This static-like region has been identified for Ca Å 0.001 and 0.005. At Ca Å 0.001, in the region of stable x 2 , v *0 0 v0 Å 00.26 { 0.27. This difference lies well within the uncertainty identified in the calibration. At Ca Å 0.005, v *0 0 v0 Å 01.07 { 0.037. This relatively large magnitude may be due to residual viscous deformation coupled with the measurement distortions. These results are consistent with the identification of v *0 with v0 and strongly suggest that v0 must be the correct boundary condition for the macroscopic shape. 4.3. Apparent Contact Angle and Contact Line Shift Apparent contact angles, uapp , have been a popular way of quantifying dynamic wettability in the past, both with immersing cylinder and capillary tube geometries (12, 13). In experiments where meniscus heights are measured, uapp is calculated as the contact angle of a static interface rising to the same meniscus height (or to the same apex height, for a capillary tube) as measured from the dynamic interface. uapp measured in this manner cannot be transferred to another geometry nor is it the correct boundary condition for the static-like interface shape far from the contact line. uapp is also different from apparent dynamic contact angles obtained from Wilhelmy plate force measurements. Force measurements on dynamic Wilhelmy plates that do not account for viscous forces exerted on the plate will produce an apparent contact angle, u *app x uapp . Further, the relation between u *app and v0 can only be known when a viscous analysis of the Wilhelmy plate flow is available. Here, we examine the efficacy of these relatively simple measurements by comparing uapp with v0 , the proper dynamic contact angle. At the same time we consider a related measurable quantity, the distance between the observed contact line and the point where the extrapolation of the static-like interface far away intersects the solid (AB in Fig. 1). We call this distance the
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TABLE 1
Ca
Contact line shift (mm)
v0 0 uapp (degrees)
0.001 0.005 0.01 0.1
40 50 60 73
1.93 2.38 2.2 3.2
contact line shift. v0 0 uapp and the contact line shift are not independent. They both are zero in the absence of viscous deformation and vary together in the presence of viscous forces. Usually, uapp is determined by measuring the vertical distance from the contact line to the bulk fluid level (or the apex of the meniscus, in a capillary tube geometry). Unfortunately, we cannot measure this distance from our images because our field of view does not show the bulk fluid level. Measurements with a cathetometer have insufficient resolution for our purposes. To find the bulk fluid level from our images we take the interface slope, u, at x Å 1200 mm and continue the interface to infinity as a static shape. To mimic experiments, uapp is calculated from the height between this bulk level and the observed contact line using the relation for a static interface shape. The continuation of the same static shape to the solid locates point B in Fig. 1 and thus determines the contact line shift. For Ca Å 0.001 and 0.005, this method is straightforward. Results appear in Table 1. At Ca Å 0.01, a static-like region did not appear within our field of view. However, uapp was stable at 58.767 { 0.037 when calculated using the slope, u, at 800, 1000, and 1200 mm. At Ca Å 0.1, uapp calculated as outlined above did not stabilize. By noting the effects of residual viscous deformation in our measurements of u at 1200 mm, we can bound the values at 71 mm õ shift õ 74 mm, 117.47 õ uapp õ 117.67 at Ca Å 0.1. Table 1 shows the contact line shift and v0 0 uapp vs Ca for our experimental geometry. Our measurements show that uapp õ v0 . As predicted by Eq. [1], the viscous deformation near the contact line makes the interface more curved than the static baseline shape, hence pulling the meniscus above the extrapolation of the static-like shape (see Fig. 1). v0 0 uapp and the contact line shift increase modestly with Ca. The explanation for their moderate growth is that viscous effects on interface curvature are controlled by two competing factors. Increasing Ca causes the viscous deformation to increase at any fixed r and u. However, as Ca increases the interface angle, u, also increases at any given r. This decreases the velocity gradients responsible for the viscous deformation. Though these results are valid for all material systems having inner mechanisms equivalent to that of our system, they are limited to the geometry used in our experiments.
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Because parameters such as v0 or uapp depend on the system geometry, their usefulness from a modeling standpoint hinges on the possibility of relating them to a material property of the system. Even though the value of uapp is approximately correct, it can not be related to any material property because there exists no analysis of the flow field valid for 0 õ x õ ` . Such analysis would be necessary to relate uapp to a geometry-independent feature such as the interface shape in the intermediate region. On the other hand, v0 measured using Eq. [1] can be easily transformed into the material parameter L (see Eq. [3]). This parameter can then be exported to other geometries to be used as the needed material specific boundary condition. The contact line shift discussed here (AB in Fig. 1) is not equal to the shift in contact line position between static (i.e., no motion) and dynamic conditions (14). In fact, the two have opposite signs. A static contact line on a vertical plate is higher than the dynamic contact line when the plate is advancing into the fluid. The fluid flow in the intermediate region causes v0 to increase as Ca increases (9, 14). Over the macroscopic extent of the fluid interface, the motion of the surface lowers the dynamic interface height compared to the static height at each x. Unmagnified, the meniscus appears to have a larger apparent contact angle. In contrast, near the contact line where viscous forces strongly deform the interface, the dynamic contact line rises above the extrapolation of the static-like region of the actual dynamic interface (AB in Fig. 1). Without a good microscopic view showing the actual dynamic interface shape, the observer is unlikely to sense this upturn of the meniscus but will sense the position of the contact line which is slightly higher than the extrapolation of the static-like region. Thus the fluid flow has two effects on the contact line position; it lowers it compared to the completely static shape but raises it relative to the static-like component of the macroscopic dynamic interface. The magnitude of the former is larger than that of the latter. 4.4. Capillary Tube Geometry Many measurements of apparent dynamic contact angles are done in capillary tubes. The experimenter measures the dynamic apex height and computes uapp as the contact angle of a static meniscus having the same apex height as the dynamic meniscus. Our measurements in the immersing cylinder geometry show that uapp systematically differs from the correct dynamic contact angle v0 . However, it would be improper to assess potential errors in capillary tube measurements by merely applying the results of our experiments to the capillary tube geometry because the dominant force balance is different in a capillary tube. In a thin capillary tube, surface tension forces—not gravity—dominate in the outer region at low Ca. This change in the dominant force balance has the potential of altering the extent of the region where the interface is static-like.
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Since interface shape measurements in capillaries have not been performed, we analyze Lowndes’ (15) numerical simulation of a meniscus rising up a cylindrical tube of radius a. From Lowndes’ results we can compute uapp and v0 . uapp appears tabulated or can be roughly obtained from plots of the interface shapes (15). We calculate v0 from g( v0 ) Å g( U ) / Ca ln(a/Li ),
[5]
where U is the actual contact angle applied to the interface in the inner region and Li is the length scale of the inner region (1). Both U and Li are inputs to Lowndes’ calculations. At Ca Å 0.075, v0 0 uapp remains of the same order as in our geometry and material system at 3.27. A better measure of the extent of viscous effects would be to find the static-like portion of the meniscus by fitting the interface near the centerline to a static shape and analyzing the deviations as we did in Section 4.1. Unfortunately Lowndes’ data are only in a plot whose quality does not allow us to make this comparison. However, our measurements in the immersing tube geometry and the example from Lowndes indicate a strong possibility that in the outer region of the capillary tube geometry, i.e., near the center line, significant viscous deformation exists at Ca Ç 0.1. 5. CONCLUSIONS
In this study, the global analysis of a fluid interface under dynamic conditions, including the viscous deformation near the contact line, has been documented for the first time. The work assessed the appropriateness of assumptions made in deriving Eq. [1], a model for the shape of the fluid interface valid near the moving contact line (i.e., r/a ! 1) in the limit of Ca r 0. The assumptions tested include: (a) Near the dynamic contact line, the interface shape shows viscous deformation; but far from the contact line the interface shape becomes static-like. (b) The parameter of the model, v0 , corresponds to the contact angle of the static extrapolation back to the solid of the static-like region of the dynamic interface far from the contact line. We investigated the regions of validity of Eq. [1] and of the static theory (Eq. [4]) by analyzing each image of a dynamic interface near and far from the contact line. The descriptions provided by Eq. [1] and Eq. [4] are not guaranteed to be found in the field of view (a little larger than the capillary length a). Specifically, we studied the shape of a dynamic fluid interface at capillary numbers ranging from 0.001 to 0.1 by imaging dynamic interface shapes out to about 1700 mm from the moving contact line, about five times farther than in previous studies (1–3). We have shown that the region of the fluid interface described by the static shape with contact angle v0 moves out as Ca increases. In fact, at Ca as low as 0.01, we detected significant viscous deformation out to about 1400 mm from
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the contact line, a distance well embedded in the outer region of our geometry. We also found that the region well described by Eq. [1] extends farther out as Ca increases but it does not overlap with that described by static theory. The ‘‘middle region’’ in Fig. 10 lies between the two regions cited above. It is neither static-like nor described by Eq. [1], which accounts for the geometry-free viscous deformation. Therefore, the viscous deformation in the ‘‘middle region’’ must depend on geometry. This geometry dependence comes from a competition between viscous and gravity forces in a region where viscous forces have not yet become negligible. However, the shape in this region is not related to recirculation flows which must prevail at a length scale on the order of the fluid container size. Our interface measurements demonstrate that the second assumption of the model is correct. The model parameter, v0 , obtained from fitting data near the contact line where viscous deformation is significant, is the boundary condition for the static-like shape far from the contact line. From our experimental data we determined uapp (the static contact angle for the observed meniscus height) and the contact line shift for the Ca’s studied. We showed that v0 0 uapp and the contact line shift have a similar dependence on Ca. They grow modestly with Ca because of the competing effects of increasing Ca; it increases viscous deformation but the simultaneous opening of the angle between the solid and the fluid interface decreases the velocity gradients responsible for viscous deformation. The contact line shift studied here is only the smaller component of the contact line displacement between a static and a dynamic situation. The larger component is perceived as the contact angle of the macroscopic interface increases to v0 from its static advancing value. Our measurements show that errors made in measuring uapp in an immersing tube geometry or in a capillary tube are a few degrees and systematic with Ca for Ca £ 0.1. Even at lower Ca, the viscous deformations in capillary tubes
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can potentially extend well into the outer region near the tube center, leading to errors made in assuming a static shape for the center of the meniscus. The extent of the static-like region near the tube center should be determined by accurate measurements or careful numerical simulations for the capillary geometry. Even though uapp is approximately correct, it cannot be related back to a material property of the system in the same way as v0 . Thus, simpler two-point measurements cannot be used in different geometries as a predictive measure of dynamic wettability nor for careful material studies of the wetting dynamics. ACKNOWLEDGMENTS This work was supported by NASA Grant NAG3-1390. The authors acknowledge Ms. Qun Chen and Mr. Keith R. Willson, Department of Physics of Carnegie Mellon University, for their invaluable assistance with the experiments and many helpful suggestions.
REFERENCES 1. Dussan, V., E. B., Rame´, E., and Garoff, S., J. Fluid Mech. 230, 97 (1991). 2. Marsh, J. A., Garoff, S., and Dussan V., E. B., Phys. Rev. Lett. 70, 2778 (1993). 3. Marsh, J. A., ‘‘Dynamic Contact Angles and Hydrodynamics Near a Moving Contact Line,’’ Ph.D. thesis, Carnegie Mellon University, 1992. 4. Chen, Q., Rame´, E., and Garoff, S., Phys. Fluids (1995), in press. 5. Chen, J., and Wada, N., Phys. Rev. Lett. 62, 3050 (1989). 6. Petrov, J. G., and Sedev, R. V., Colloids Surf. A 74, 233 (1993). 7. Kistler, S. F., in ‘‘Wettability’’ (J. C. Berg, Ed.), p. 311. Dekker, New York, 1993. 8. Hocking, L. M., and Rivers, A. D., J. Fluid Mech. 121, 425 (1982). 9. Cox, R. G., J. Fluid Mech. 168, 169 (1986). 10. Boender, W., Chester, A. K., and van der Zanden, A. J. J., Int. J. Multiphase Flow 17, 661 (1991). 11. Huh, C., and Scriven, L. E., J. Colloid Interface Sci. 30, 323 (1969). 12. Seeberg, J. E., and Berg, J. C., Chem. Eng. Sci. 47, 4455 (1992). 13. Hoffman, R. L., J. Colloid Interface Sci. 50, 228 (1975). 14. De Gennes, P. G., Hua, X., and Levinson, P., J. Fluid Mech. 212, 55 (1990). 15. Lowndes, J., J. Fluid Mech. 101, 631 (1980).
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