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Experimental studies on the parametrization of liquid spreading and dynamic contact angles
COLLOIDS AND Colloids and Surfaces A: Physicochemical and Engineering Aspects 116 (1996) I15-124
ELSEVIER
A
SURFACES
Experimental studies on the parametrization of liquid spreading and dynamic contact angles Q. Chen, E. Ramr, S. Garoff * Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Received 15 September 1995; accepted 1 February 1996
Abstract The dynamics of a spreading liquid are controlled by the details of the fluid motion very near the moving contact line. Modeling this motion is not trivial. The classical hydrodynamic model has a singular stress field at the moving contact line. This singularity prevents the use of the contact angle in dynamic conditions and predicts that an infinite force would be needed to sink a solid into a fluid. A model, valid in the small capillary number (Ca) limit, describes the fluid motion and viscous interfacial deformation near the moving contact line. The model contains a single free parameter, too, which can be related to material parameters and must be determined experimentally. Experiments are reported that tested the range of validity of this asymptotic hydrodynamic model. The fluid-vapor interface shape and fluid velocity field produced by a glass tube entering a bath of polydimethylsiloxane at constant speed were measured near and far from the contact line. They were compared with the model using the free parameter too as a fitting constant. This procedure established the validity of the theory and provided a means of measuring too. The ranges (in capillary number and in space) of validity of the theory were established. The model fails near the contact line at Ca ~ 0.1. This failure starts near the contact line, propagates out and increases in magnitude as Ca increases. For Ca <<.0.1, the model with viscous deformation fails far from the contact line but describes the interface shape within ~ 400 ~tm from the contact line. The model begins to fail at distances where the interface shape ceases to be controlled by geometry-independent viscous forces but responds instead to a competition between viscous and gravitational forces. At even larger distances from the contact line, viscous forces become negligible and the interface looks staticlike. The experiments showed that the contact angle formed by the extrapolation to the solid surface of the static-like interface far from the contact line equals ~Oo as predicted by the theory. Comparisons of too and apparent dynamic contact angles based on meniscus height measurements, 0apv, are presented. Small but systematic errors were found which increase with Ca. In contrast to too, 0avp cannot be related to material parameters and hence cannot be used to generate archival modeling information for spreading dynamics.
Keywords: Dynamic contact angles; Liquid spreading; Parametrization
1. Introduction The displacement of one fluid by another immiscible fluid on a solid surface controls m a n y natural and technological processes. Such pro* Corresponding author. 0927-7757/96/$15.00 © 1996 Elsevier Science B,V. All rights reserved
PH 0927-7757(96)03581-9
cesses include spraying of paint and insecticides, two-phase flow in p o r o u s media, application of coatings, imbibition of inks in papers and formation or rewetting of dry patches in the eye or heat transfer equipment. In all these phenomena, surface tension forces are i m p o r t a n t and the details of the fluid m o t i o n very near the contact line control the
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Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
spreading. Investigations over the last 25 years have attempted to describe the fluid motion in this region. The reviews by Dussan V. [ 1], De Gennes [2] and Kistler [3] are invaluable discussions of the physics, modeling difficulties and approaches taken. The central difficulty is that the usual hydrodynamic model (this assumes Newtonian and incompressible fluids, non-deformable solids and continuity of velocity at the boundaries) predicts an infinite curvature of the fluid-vapor interface at the moving contact line. This singularity makes it impossible to apply the contact angle as a boundary condition to the differential equation that describes the fluid-vapor interface shape. The usual hydrodynamic model also predicts that the force required to sink a solid into a fluid bath is infinite [4]. This failure of the traditional model occurs only because of the presence of the moving contact line [1,5], suggesting that the classical model cannot apply all the way to the contact line. New, unique hydrodynamic mechanisms must operate in a very small neighborhood of the contact line characterized by a length scale Li. This region is usually called the "inner" region and the new mechanisms acting in it the "inner" mechanisms or models. Traditionally, the inner models relax one or more of the assumptions of the classical model and remove the contact line singularity [6-10]. To achieve predictive power, parametrizations of the macroscopic dynamics of the spreading process have been attempted which do not require knowledge of inner mechanisms [5,6,11-14]. These analyses take advantage of the observation that, when Ca<< 1 (Ca = U # / a is the capillary number and represents the relative magnitude of viscous to surface tension forces, where U is the spreading velocity, # the viscosity and a the surface tension), differing assumed inner models (all removing the no-slip condition at the solid) give rise to the same flow field at distances >>Li. At Ca << 1, a single parameter, o~o, carries the result of the inner hydrodynamics outward and controls the interface shape in the macroscopic region. The parameter coo is formed by a combination of inner region parameters, but the combination is independent of the particular inner model chosen. Hence, coo forms a universal boundary condition for the macro-
scopic dynamics which does not require knowledge of the specific inner model. ~o0 can be connected to a geometry-independent boundary condition for the macroscopic flow field (see Eq. (5)) [ 14]. This picture rests on the notion that, when Ca << 1, information about the state of deformation of the fluid interface travels from the inner region (where unique hydrodynamics are present, very near the contact line), through an intermediate geometry-free region controlled by classical hydrodynamics, out to the outer region where viscous forces are negligible and the system geometry is important. Specifically, the flow field near the contact line is controlled by the local geometry of the wedge-like region formed between the solid and the fluid-vapor interface near the contact line. The local flow is therefore independent of the macroscopic geometry. This local flow field generates viscous forces which cause a geometry-free viscous deformation. The region characterized by this geometry-free interfacial deformation includes the inner region and the intermediate region, controlled primarily by viscous and surface tension forces. Far from the contact line where viscous forces are negligible, the interface looks almost like a static capillary surface. This static-like portion defines the outer region, scaled with the macroscopic length of the system. The viscous, hydrodynamic deformation of the interface in the intermediate region causes a "dynamic contact angle" to be imposed on the macroscopic, staticlike interface [5,15]. These ideas yield a model which describes the interface shape up to but not including the inner region. This model uses classical hydrodynamics and assumes that the contact line singularity has been removed by some mechanism such as slip which produces a specific asymptotic form at distances from the contact line >>Li. Based on prior analyses [15,16], Dussan V. et al. [5] solved the interface shape when a tube of radius RT enters a fluid bath at constant speed U parallel to its axis and at an angle ~ from the horizontal. To order 1 [i.e. O( 1)] as Ca ~ 0 holding Ca ln(a/Li) fixed, this shape is 0 = g- 1[g(Ogo) + Ca ln(r/a)] + fo(r/a; COo, or, R T / a ) -- 09o
(1)
Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
where 0 is the angle between the local tangent to the interface and the solid (Fig. 1), r is the distance from the contact line to the interface, a is the capillary length ~ which acts as the macroscopic length scale in this geometry, p is the fluid density, g is the acceleration of gravity and I x - sin x cos x dx 2 sin x
geometry-independent parameter [9]: g(too) - Ca In a = g(A) - Ca In Li
The local flow near the contact line has radial and angular velocity components, u and v [15]:
(2) -
o
0ou t =fo(r/a; ¢Oo,at, RT/a)
(3)
is the solution in the outer region, and to O(1) as Ca ~ 0 is a static interface, o~o is related to inner region parameters by g(¢oo) = g(A) + Ul~/tr ln(a/Li)
(5)
U u = -~ [fl cos ~b+ ~bsin fl sin(~b - r)
0
g(O) =
117
/)
I
090 = fo(0; O~o,cx,RT/a ) Eq. (1) applies in the intermediate region and its overlap with the outer region. 090 must be measured macroscopically by fitting Eq. (1) to interface measurements, using O~o as a fitting parameter. This model is predictive because the parameter 090, which describes the wetting characteristics of the system, can be related back to the material,
Fig. 1. Geometry and coordinate system used in the experiments and analysis.
+
(6)
U r fl sin $ - $ sin a cos($ - r) / J
+ r
(4)
where A is a constant which for certain inner models corresponds to the actual contact angle, Oact. At the same time, 090 arises naturally as a constant of integration in the outer region. It is the contact angle of the static-like interface far from the contact line, i.e.
sin fl cos($ - fl)]
-
2 sin 2 fl(fl sin $ - $ sin fi cos($ - r) d2
sin ~b- ~ cos(2fl - ~b)~ dfl] d ,} d r ]
(7)
where ~b= fl(r) is the location of the interface in polar coordinates with origin at the contact line, with the solid at ~ = 0, and d _-__fi - sin fl cos ft. The theory on which Eqs. (6) and (7) are based is limited to situations where dfl/d(ln r)<< 1. When the boundary ~b= fl(r) is a fluid-vapor interface, this condition is met when Ca << 1 and Ca ln(r/a) = O i l ) [15-17]. In Eqs. (6) and (7), Ca enters indirectly through the interface curvature, dfi/dr, which is a function of Ca. In this paper, we summarize recent experiments characterizing the macroscopic motion of fluids near moving contact lines. In the experiments, the shape and velocity field near the contact line are compared with the theory [Eqs. (1), (3), (6) and (7)]. The objective was to investigate the range of validity of the model with respect to Ca and to r. We find that Eq.(1) holds for Ca<,O.1 and that the distance to which it holds increases with Ca. Although these experiments are primarily focused on the macroscopic aspects of the flow, we have found intriguing results consistent with the inner mechanisms extending out into the field of view of the experiment. We have also examined the global shape of the interface for Ca ~<0.1, where the model of Eq. (1) describes the interface shape near the contact line. We find that Eq. (3) describes the interface shape far from the contact line. Thus, the assumptions of the model are correct: (1) at Ca << 1,
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Q Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
the interface far from the contact line is static-like; and (2) coo is the contact angle of the extrapolation to the solid of the static-like shape far away. Finally, we measured the apparent contact angle based on the capillary rise height of the dynamic meniscus, 0app. We show that 0ap p differs systematically from coo with increasing Ca. Unlike coo, 0app cannot be related back to material modeling information.
2. Experimental In this section, we give a brief outline of the experimental method, materials, optics and image analysis. Details can be found in Refs. [18] and [ 19]. We immerse a cylindrical Pyrex tube, 2.54 cm outer diameter, in a bath of polydimethylsiloxane (PDMS) at constant speed. We record and analyze the shape of the fluid-vapor interface using Eq. (1) near the contact line and a static interface, Eq. (3), far from it. In these experiments we control the spreading velocity, an advantage over spontaneously spreading drops. A constant spreading velocity is necessary for testing models of fluid motion and interface shapes near moving contact lines without knowing a priori the dependence of coo on spreading velocity. We observe the PDMS meniscus against the edge of the Pyrex tube. PDMS forms a zero static contact angle against Pyrex. The static meniscus remains above the bulk fluid level, creating a meniscus "in rise". A motorized stage drives the tube into the fluid. As the tube enters the fluid, the height of the meniscus in rise decreases with increasing immersion speed. Above a certain speed, the meniscus is driven below the bulk fluid level, creating a meniscus "in depression". We observe menisci both in rise and in depression (Fig. 2). We use Koehler illumination [.20] to produce a shadow of the meniscus with a uniformly lit background. The edges of the meniscus and the glass tube are viewed with a long working distance microscope attached to a CCD camera. The image is digitally stored in a computer. For menisci in rise, we view the edges through the air using a beaker overfilled with PDMS. For menisci in
Fig. 2. Typicalimagesof menisciin (a) rise and (b) depression. depression, we view the edges through the fluid contained in a partially filled square glass cell. The data analysis of the interface shape consists in finding the location of the fluid interface and fitting it to the appropriate theoretical expression, Eqs. (1) or (3). We use the highest gradient of gray level to locate the interface. The interface shape is the angle between the local tangent to the interface and the surface of the solid, as a function of position. The image analysis produces about 1000 points for each interface shape. The data points form a cloud about 1-3 ° thick. Two successive images of a static interface have overlapping clouds but the individual data points do not overlap. Thus, pixel noise, not rapid spatial oscillations in the interface shape, causes the scatter in the data
Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
of each image. As a result, we assess a 1-3 ° standard deviation, #, for individual data points. We also perform particle image velocimetry (PIV) measurements using images of menisci in depression. Details of the technique used in the PIV study will be published elsewhere. The trajectories of small seed particles (mostly bubbles trapped in the fluid) about 5-10 gm in diameter track the local velocity field, which we compare with Eqs. (6) and (7). For general information on PIV, see Ref. [21]. We perform different analyses on interface data taken with high and low magnifications. With high magnification images, we fit data in the range 20gm
119
theory no longer fits the data as the location where the theory deviates from the data by #/3. We calibrate the optical and image analysis systems against a static capillary surface whose shape is known theoretically. Systematic optical distortions were present within 20 and 50 gm from the contact line in high and low magnification images, respectively, and beyond 1500 ~tm in low magnification images. We exclude data in the regions with distortions. The low magnification images exhibit an additional, minor systematic distortion at large distances due to the changing curvature of the interface in the direction of the light path, at the focal plane [19]. These distortions lead to an uncertainty of 0.4 ° in rise and 0.8 ° in depression in col) from fitting portions of the dynamic interface far from the contact line at low magnifications [ 19]. To attain the desired range of Ca, we use two PDMS fluids with nominal kinematic viscosities of 1000 cSt (Dow Corning 200 fluid) and 60 000 cSt (United Chemical). The surface tension is 20.8+0.5dyncm -1, the viscosities are 9.6 and 611 g cm -1 s -1, respectively, and the densities are 0.97 g cm -3. In our experiments we studied the range 0.001 ~
3. Results and discussion 1 The normalized chi-squared deviation for N points, defined as
)(2,
is
Z2 ==~ [0~ xp -- 0the°(gi)]2 ~ 2 ( N - I) where 0~xp is an individual experimental value and 0th*°(ri) is the corresponding theoretical value; 0 is the standard deviation of the data points from a smooth curve fit locally to the points [19].
We have shown that Eq. (1) describes the interface shape for 20 gm < r < 400 gm and Ca ~ 0.1 [5,14,18] (Fig. 3). The agreement with the data at Ca <<.0.1 validates the model and provides a way to determine coo. When the experiment is done at several Ca < 0.1, the dependence of coo on U constitutes material information for modeling the spreading dynamics of this system. The coo values together
Q. Chen et aL /Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
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with Eq. (1) provide the boundary condition for the macroscopic interface shape as r--*0. The asymptotic matching process used in the model requires that the inner model for the material system must come out of the inner region as Ca Inr. Thus, our experiments imply that our system must have an inner mechanism of the form Ca In r at low Ca and r/Li >> 1 [18]. We measured the local velocity field near the contact line for menisci in depression. The velocity components agree with the Eqs. (6) and (7) for Ca <~0.1. Fig. 4 shows the velocity data for Ca = 0.1. Deviations from the theory are random in direction and below experimental error in magnitude throughout the domain shown. At Ca > 0.1, Eq. (1) begins to fail to describe the interface shape data near the contact line [18],
but it can still fit portions of the interface in r b < r < 400 pm, where 20 pm < r b < 400 pm is the inner boundary of the fitted region. As seen in Fig. 5, for Ca > 0.1 the model fails near the contact line but continues to describe the data further from the contact line. Because the model begins to fail near contact line, the conclusions about the range of Ca where the model holds depend on how close to the contact line we can measure the interface. This failure extends out as Ca increases. At Ca ~ 0.45, the theory no longer fits any portion of the interface for r < 400 pm (Fig. 6). The velocity measurements also show systematic deviations from the model Eqs. (6) and (7) at Ca = 0.4 (Fig. 7). Nevertheless, and independent of the agreement with the present models, we have identified a region, located within 100 pm around the contact line at Ca ~ 0.45, where the interface and flow fields are independent of geometry changes caused by varying the tube immersion angle [18]. We have narrowed the possible causes of the failure of the model to (a) the O(1) analysis in the intermediate region is insufficient at C a > 0 . 1 and/or (b) inner region mechanisms (which Eq. (1) does not include) extend out and into the field of view. When the model fails, it overestimates the interface curvature. This overprediction is consistent with the inner region extension as stresses, and hence curvature, in the inner region relax and
Q. Chen et al./Colloids Surfaces A." Physicochem. Eng. Aspects 116 (1996) 115-124
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must be smaller than those predicted by Eq.(1) using classical hydrodynamics in the same region of space. Low magnification experiments test assumptions of the model about the state of the interface far from the contact line. These experiments were done at Ca <~0.1, where the model of Eq. (1) properly describes the hydrodynamics for small r. Through extensive fitting, we have confirmed that the interface behaves quasi-statically far from the contact line. In this region, viscous forces are negligible compared with surface tension and gravity. We determined, as a function of Ca, the inner boundary beyond which the interface is no longer static, i.e. viscous forces begin to play a role large enough to be detected• Using a similar method we found, as a function of Ca, the extent of the region near the
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Fig. 6. Plot of 0 vs. r at Ca = 0.46. (a) Dots, experimental data; solid line, extrapolated best theoretical fit of Eq. (1) to data 300 gm < r < 400 gm. (b) Dots, difference of the data and the best fit of Eq. (1). Note that no region of the data is described properly by Eq. ( 1 ).
contact line which is correctly described by Eq. ( 1). Fig. 8 shows a typical example of data with a region at small r fit to Eq. (1) and at large r fit to Eq. (3). Fig. 9 shows the regions described by Eqs. (1) and (3) for all Ca values tested. The boundary of the static-like region far from the contact line moves to larger r with increasing Ca, whereas the range of the region described by Eq. (1) increases with Ca. This is consistent with the growth of viscous deformation as Ca increases. These measurements are quantitatively different from previous measurements of dynamic menisci performed by Petrov and Sedev [22]. We also verified the prediction of the model that the contact angle of the extrapolation to the solid surface of the static-like
Q. Chen et aL/Colloids Surfaces A. Physicochem. Eng. Aspects 116 (1996) 115-124
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shape, co~, is equal to coo obtained by fitting Eq. ( 1) near the contact line [ 19]. Between the static-like region described by Eq.(3) and the region described by Eq.(l), a portion of interface remains which is neither staticlike nor described by Eq. (1). In this middle region (Fig. 9), viscous and gravity forces compete on an equal basis. As a result, viscous forces in this region are no longer purely local and are influenced by the system geometry at the capillary length scale. However, the middle region is not related to recirculation flows in the experimental cell because the
Viscous Theory Eq. [1]
300
Fig. 7. (a) Vector plot of the measured velocity field at Ca = 0.4. Scale marker of 150 p,m s 1 also shows velocity of tube. (b) Vector plot of the difference of the measured and theoretical velocity fields using Eqs. (6) and (7).
~.......--.--'~ ~
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cell radius is ,-~5 cm, whereas the middle region lies at 200 gm < r < 1200 ktm, depending on Ca. From our interface measurements, we determined 0app, the contact angle of the static meniscus having the same capillary rise height as the dynamic interface [19]. Most dynamic contact angle measurements are performed by measuring Oapp. Oapp is a property of the whole dynamic meniscus from the contact line to the bulk level at infinity. Relating Oappto material parameters would therefore require knowledge of the flow field gener-
Q. Chen et aL/Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
ated by an immersing solid, and valid in the range 0 < r < ~ . Since this analysis is not yet available, 0app cannot be connected to a geometry-free parameter in the same manner as coo. We have shown that the difference coo- Oappincreases systematically with Ca, although it remains below 4 ° for Ca<~O.1 [,19]. Our results also show that viscous deformation is detectable at distances as large as the capillary length scale at Ca >/0.1, violating the assumption that almost all the meniscus looks static used in 0app measurements. This viscous deformation is responsible for the measured difference coo- 0app. Qualitatively similar results appear for a meniscus moving down a capillary tube.
4. Conclusions We have studied the global shape of dynamic interfaces and the velocity field when a solid surface enters a liquid bath. We looked at the interface in two modes. At high magnifications, we imaged dynamic menisci and fluid velocities within distances ~ 400 gm of the contact line. We compared the measurements with Eqs. (1), (6) and (7) which are the result of an asymptotic hydrodynamic model describing the viscous deformation of the interface and the local flow field near the moving contact line [,5,15,16]. For PDMS spreading on clean glass, Eq. (1) begins to fail at Ca > 0.1. The failure of Eq. (1) is due to either (a) the inability of the O(1) contribution from the intermediate region to accurately describe the interfacial deformation due to viscous forces and/or (b) the possibility that, at Ca >0.1, aspects of the unique hydrodynamics acting in the inner region, not included in the model, project out and become visible in the imaged region. Our results are consistent with this second possibility since the theory overpredicts the measured interface curvature. The same model developed for the interface shape yields the velocity field in a domain shaped as a wedge with a slowly varying angle [,15]. Our velocity measurements agree with the theory [-Eqs. (6) and (7)] for Ca ~<0.1. They show systematic deviations above Ca = 0.1. These data should be useful in identifying the cause of breakdown of
123
Eq. (1). Even at Ca values where the model fails, our measurements provide vital archival information as boundary conditions for numerical simulations of the macroscopic flow in the form of geometry-independent interface shapes and velocity fields. For the range of Ca (~< 0.1 ) where Eq. (1) is valid near the contact line, we studied the condition of the interface shape far from the moving contact line. This tested assumptions made in deriving Eq. (1). Concerning the accuracy of our measurements, our results confirm that (a) the shape of the interface far from the contact line approaches that of a static capillary surface and (b) the contact angle of the extrapolation to the solid surface of that static-like shape, co~, equals coo obtained from fitting Eq. (1) to data near the contact line. We have detected viscous deformation for Ca as small as 0.001. The region with viscous deformation lies in the range r <300 ~tm at Ca=0.001 and expands out as Ca increases. This has implications for the ability to transport apparent contact angles based on meniscus heights to geometries different from that used in the measurement. In summary, we have presented up-to-date results of our experimental studies of the validity of asymptotic models of liquid spreading. These experiments tested the ability of the model to describe the macroscopic dynamics of spreading systems. We have established that, for our material system, the theory represented by Eqs. (1), (6) and (7) holds for Ca <<,0.1. We verified the assumptions of the theory. By demonstrating the use of the dynamic contact angle and velocity field as a boundary condition for the macroscopic spreading dynamics, these experiments will allow the development of new, predictive models of macroscopic spreading dynamics.
Acknowledgment This work was supported in part by NASA Grant No. NAG3-1390. References [-1] E.B. Dussan V., Annu. Rev. Fluid Mech., 11 (1979) 371. [2] P.G. de Gennes, Rev. Mod. Phys., 57 (1985) 827.
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Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
[3] S. Kistler, in J.C. Berg (Ed.), Wettability, Marcel Dekker, New York, 1993, Ch. 6. I-4] C. Huh and L.E. Scriven, J. Colloid Interface Sci., 35 (1971) 85. [5] E.B. Dussan V., E. Ram6 and S. Garoff, J. Fluid. Mech., 230 (1991) 97. 1,6] E.B. Dussan V., J. Fluid Mech., 77 (1976) 665. 1,7] S. Rosenblat and S.H. Davis, in S.H. Davis and J.L. Lumley (Eds.), Frontiers in Fluid Mechanics, a Collection of Papers in Commemoration of the 65th Birthday of Stanley Corrsin, Springer, Berlin, 1985, pp. 171-183. 1,8] E.O. Tuck and L.W. Schwartz, SIAM Rev., 32 (1990) 453. [9] L.M. Hocking, J. Fluid Mech., 77 (1977) 209. 1,10] C. Huh and S.G. Mason, J. Fluid Mech., 81 (1977) 401. [11] R.J. Hansen and T.Y. Toong, J. Colloid Interface Sci., 37 (1971) 196. [12] F.Y. Kafka and E.B. Dussan V., J. Fluid Mech., 95 (1979) 539.
[13] C.G. Ngan and E.B. Dussan V., J. Fluid Mech., 209 (1989) 191. [14] J.A. Marsh, S. Garoff and E.B. Dussan V., Phys. Rev. Lett., 70 (1993) 2778. [15] R.G. Cox, J. Fluid Mech., 168 (1986) 169. 1,16] L.M. Hocking and A.D. Rivers, J. Fluid Mech., 121 (1982) 425. [17] UM. Pismen and A. Nir, Phys. Fluids, 25 (1982) 3. 1-18] Q. Chen, E. Ram6 and S. Garoff, Phys. Fluids, 7 (1995) 2631. [19] E, Ram6 and S. Garoff, J. Colloid Interface Sci., 177 (1996) 234. [20] J.A. Marsh, PhD Thesis, Carnegie Mellon University, Pittsburgh, PA, 1992. [21] W. Mertzkirch, Flow Visualization, Academic Press, New York, 1974. 1,22] J.G. Petrov and R.V. Sedev, Colloids Surf. A, 74 (1993) 233.
ELSEVIER
A
SURFACES
Experimental studies on the parametrization of liquid spreading and dynamic contact angles Q. Chen, E. Ramr, S. Garoff * Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA Received 15 September 1995; accepted 1 February 1996
Abstract The dynamics of a spreading liquid are controlled by the details of the fluid motion very near the moving contact line. Modeling this motion is not trivial. The classical hydrodynamic model has a singular stress field at the moving contact line. This singularity prevents the use of the contact angle in dynamic conditions and predicts that an infinite force would be needed to sink a solid into a fluid. A model, valid in the small capillary number (Ca) limit, describes the fluid motion and viscous interfacial deformation near the moving contact line. The model contains a single free parameter, too, which can be related to material parameters and must be determined experimentally. Experiments are reported that tested the range of validity of this asymptotic hydrodynamic model. The fluid-vapor interface shape and fluid velocity field produced by a glass tube entering a bath of polydimethylsiloxane at constant speed were measured near and far from the contact line. They were compared with the model using the free parameter too as a fitting constant. This procedure established the validity of the theory and provided a means of measuring too. The ranges (in capillary number and in space) of validity of the theory were established. The model fails near the contact line at Ca ~ 0.1. This failure starts near the contact line, propagates out and increases in magnitude as Ca increases. For Ca <<.0.1, the model with viscous deformation fails far from the contact line but describes the interface shape within ~ 400 ~tm from the contact line. The model begins to fail at distances where the interface shape ceases to be controlled by geometry-independent viscous forces but responds instead to a competition between viscous and gravitational forces. At even larger distances from the contact line, viscous forces become negligible and the interface looks staticlike. The experiments showed that the contact angle formed by the extrapolation to the solid surface of the static-like interface far from the contact line equals ~Oo as predicted by the theory. Comparisons of too and apparent dynamic contact angles based on meniscus height measurements, 0apv, are presented. Small but systematic errors were found which increase with Ca. In contrast to too, 0avp cannot be related to material parameters and hence cannot be used to generate archival modeling information for spreading dynamics.
Keywords: Dynamic contact angles; Liquid spreading; Parametrization
1. Introduction The displacement of one fluid by another immiscible fluid on a solid surface controls m a n y natural and technological processes. Such pro* Corresponding author. 0927-7757/96/$15.00 © 1996 Elsevier Science B,V. All rights reserved
PH 0927-7757(96)03581-9
cesses include spraying of paint and insecticides, two-phase flow in p o r o u s media, application of coatings, imbibition of inks in papers and formation or rewetting of dry patches in the eye or heat transfer equipment. In all these phenomena, surface tension forces are i m p o r t a n t and the details of the fluid m o t i o n very near the contact line control the
116
Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
spreading. Investigations over the last 25 years have attempted to describe the fluid motion in this region. The reviews by Dussan V. [ 1], De Gennes [2] and Kistler [3] are invaluable discussions of the physics, modeling difficulties and approaches taken. The central difficulty is that the usual hydrodynamic model (this assumes Newtonian and incompressible fluids, non-deformable solids and continuity of velocity at the boundaries) predicts an infinite curvature of the fluid-vapor interface at the moving contact line. This singularity makes it impossible to apply the contact angle as a boundary condition to the differential equation that describes the fluid-vapor interface shape. The usual hydrodynamic model also predicts that the force required to sink a solid into a fluid bath is infinite [4]. This failure of the traditional model occurs only because of the presence of the moving contact line [1,5], suggesting that the classical model cannot apply all the way to the contact line. New, unique hydrodynamic mechanisms must operate in a very small neighborhood of the contact line characterized by a length scale Li. This region is usually called the "inner" region and the new mechanisms acting in it the "inner" mechanisms or models. Traditionally, the inner models relax one or more of the assumptions of the classical model and remove the contact line singularity [6-10]. To achieve predictive power, parametrizations of the macroscopic dynamics of the spreading process have been attempted which do not require knowledge of inner mechanisms [5,6,11-14]. These analyses take advantage of the observation that, when Ca<< 1 (Ca = U # / a is the capillary number and represents the relative magnitude of viscous to surface tension forces, where U is the spreading velocity, # the viscosity and a the surface tension), differing assumed inner models (all removing the no-slip condition at the solid) give rise to the same flow field at distances >>Li. At Ca << 1, a single parameter, o~o, carries the result of the inner hydrodynamics outward and controls the interface shape in the macroscopic region. The parameter coo is formed by a combination of inner region parameters, but the combination is independent of the particular inner model chosen. Hence, coo forms a universal boundary condition for the macro-
scopic dynamics which does not require knowledge of the specific inner model. ~o0 can be connected to a geometry-independent boundary condition for the macroscopic flow field (see Eq. (5)) [ 14]. This picture rests on the notion that, when Ca << 1, information about the state of deformation of the fluid interface travels from the inner region (where unique hydrodynamics are present, very near the contact line), through an intermediate geometry-free region controlled by classical hydrodynamics, out to the outer region where viscous forces are negligible and the system geometry is important. Specifically, the flow field near the contact line is controlled by the local geometry of the wedge-like region formed between the solid and the fluid-vapor interface near the contact line. The local flow is therefore independent of the macroscopic geometry. This local flow field generates viscous forces which cause a geometry-free viscous deformation. The region characterized by this geometry-free interfacial deformation includes the inner region and the intermediate region, controlled primarily by viscous and surface tension forces. Far from the contact line where viscous forces are negligible, the interface looks almost like a static capillary surface. This static-like portion defines the outer region, scaled with the macroscopic length of the system. The viscous, hydrodynamic deformation of the interface in the intermediate region causes a "dynamic contact angle" to be imposed on the macroscopic, staticlike interface [5,15]. These ideas yield a model which describes the interface shape up to but not including the inner region. This model uses classical hydrodynamics and assumes that the contact line singularity has been removed by some mechanism such as slip which produces a specific asymptotic form at distances from the contact line >>Li. Based on prior analyses [15,16], Dussan V. et al. [5] solved the interface shape when a tube of radius RT enters a fluid bath at constant speed U parallel to its axis and at an angle ~ from the horizontal. To order 1 [i.e. O( 1)] as Ca ~ 0 holding Ca ln(a/Li) fixed, this shape is 0 = g- 1[g(Ogo) + Ca ln(r/a)] + fo(r/a; COo, or, R T / a ) -- 09o
(1)
Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
where 0 is the angle between the local tangent to the interface and the solid (Fig. 1), r is the distance from the contact line to the interface, a is the capillary length ~ which acts as the macroscopic length scale in this geometry, p is the fluid density, g is the acceleration of gravity and I x - sin x cos x dx 2 sin x
geometry-independent parameter [9]: g(too) - Ca In a = g(A) - Ca In Li
The local flow near the contact line has radial and angular velocity components, u and v [15]:
(2) -
o
0ou t =fo(r/a; ¢Oo,at, RT/a)
(3)
is the solution in the outer region, and to O(1) as Ca ~ 0 is a static interface, o~o is related to inner region parameters by g(¢oo) = g(A) + Ul~/tr ln(a/Li)
(5)
U u = -~ [fl cos ~b+ ~bsin fl sin(~b - r)
0
g(O) =
117
/)
I
090 = fo(0; O~o,cx,RT/a ) Eq. (1) applies in the intermediate region and its overlap with the outer region. 090 must be measured macroscopically by fitting Eq. (1) to interface measurements, using O~o as a fitting parameter. This model is predictive because the parameter 090, which describes the wetting characteristics of the system, can be related back to the material,
Fig. 1. Geometry and coordinate system used in the experiments and analysis.
+
(6)
U r fl sin $ - $ sin a cos($ - r) / J
+ r
(4)
where A is a constant which for certain inner models corresponds to the actual contact angle, Oact. At the same time, 090 arises naturally as a constant of integration in the outer region. It is the contact angle of the static-like interface far from the contact line, i.e.
sin fl cos($ - fl)]
-
2 sin 2 fl(fl sin $ - $ sin fi cos($ - r) d2
sin ~b- ~ cos(2fl - ~b)~ dfl] d ,} d r ]
(7)
where ~b= fl(r) is the location of the interface in polar coordinates with origin at the contact line, with the solid at ~ = 0, and d _-__fi - sin fl cos ft. The theory on which Eqs. (6) and (7) are based is limited to situations where dfl/d(ln r)<< 1. When the boundary ~b= fl(r) is a fluid-vapor interface, this condition is met when Ca << 1 and Ca ln(r/a) = O i l ) [15-17]. In Eqs. (6) and (7), Ca enters indirectly through the interface curvature, dfi/dr, which is a function of Ca. In this paper, we summarize recent experiments characterizing the macroscopic motion of fluids near moving contact lines. In the experiments, the shape and velocity field near the contact line are compared with the theory [Eqs. (1), (3), (6) and (7)]. The objective was to investigate the range of validity of the model with respect to Ca and to r. We find that Eq.(1) holds for Ca<,O.1 and that the distance to which it holds increases with Ca. Although these experiments are primarily focused on the macroscopic aspects of the flow, we have found intriguing results consistent with the inner mechanisms extending out into the field of view of the experiment. We have also examined the global shape of the interface for Ca ~<0.1, where the model of Eq. (1) describes the interface shape near the contact line. We find that Eq. (3) describes the interface shape far from the contact line. Thus, the assumptions of the model are correct: (1) at Ca << 1,
118
Q Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
the interface far from the contact line is static-like; and (2) coo is the contact angle of the extrapolation to the solid of the static-like shape far away. Finally, we measured the apparent contact angle based on the capillary rise height of the dynamic meniscus, 0app. We show that 0ap p differs systematically from coo with increasing Ca. Unlike coo, 0app cannot be related back to material modeling information.
2. Experimental In this section, we give a brief outline of the experimental method, materials, optics and image analysis. Details can be found in Refs. [18] and [ 19]. We immerse a cylindrical Pyrex tube, 2.54 cm outer diameter, in a bath of polydimethylsiloxane (PDMS) at constant speed. We record and analyze the shape of the fluid-vapor interface using Eq. (1) near the contact line and a static interface, Eq. (3), far from it. In these experiments we control the spreading velocity, an advantage over spontaneously spreading drops. A constant spreading velocity is necessary for testing models of fluid motion and interface shapes near moving contact lines without knowing a priori the dependence of coo on spreading velocity. We observe the PDMS meniscus against the edge of the Pyrex tube. PDMS forms a zero static contact angle against Pyrex. The static meniscus remains above the bulk fluid level, creating a meniscus "in rise". A motorized stage drives the tube into the fluid. As the tube enters the fluid, the height of the meniscus in rise decreases with increasing immersion speed. Above a certain speed, the meniscus is driven below the bulk fluid level, creating a meniscus "in depression". We observe menisci both in rise and in depression (Fig. 2). We use Koehler illumination [.20] to produce a shadow of the meniscus with a uniformly lit background. The edges of the meniscus and the glass tube are viewed with a long working distance microscope attached to a CCD camera. The image is digitally stored in a computer. For menisci in rise, we view the edges through the air using a beaker overfilled with PDMS. For menisci in
Fig. 2. Typicalimagesof menisciin (a) rise and (b) depression. depression, we view the edges through the fluid contained in a partially filled square glass cell. The data analysis of the interface shape consists in finding the location of the fluid interface and fitting it to the appropriate theoretical expression, Eqs. (1) or (3). We use the highest gradient of gray level to locate the interface. The interface shape is the angle between the local tangent to the interface and the surface of the solid, as a function of position. The image analysis produces about 1000 points for each interface shape. The data points form a cloud about 1-3 ° thick. Two successive images of a static interface have overlapping clouds but the individual data points do not overlap. Thus, pixel noise, not rapid spatial oscillations in the interface shape, causes the scatter in the data
Q. Chen et al./Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
of each image. As a result, we assess a 1-3 ° standard deviation, #, for individual data points. We also perform particle image velocimetry (PIV) measurements using images of menisci in depression. Details of the technique used in the PIV study will be published elsewhere. The trajectories of small seed particles (mostly bubbles trapped in the fluid) about 5-10 gm in diameter track the local velocity field, which we compare with Eqs. (6) and (7). For general information on PIV, see Ref. [21]. We perform different analyses on interface data taken with high and low magnifications. With high magnification images, we fit data in the range 20gm
119
theory no longer fits the data as the location where the theory deviates from the data by #/3. We calibrate the optical and image analysis systems against a static capillary surface whose shape is known theoretically. Systematic optical distortions were present within 20 and 50 gm from the contact line in high and low magnification images, respectively, and beyond 1500 ~tm in low magnification images. We exclude data in the regions with distortions. The low magnification images exhibit an additional, minor systematic distortion at large distances due to the changing curvature of the interface in the direction of the light path, at the focal plane [19]. These distortions lead to an uncertainty of 0.4 ° in rise and 0.8 ° in depression in col) from fitting portions of the dynamic interface far from the contact line at low magnifications [ 19]. To attain the desired range of Ca, we use two PDMS fluids with nominal kinematic viscosities of 1000 cSt (Dow Corning 200 fluid) and 60 000 cSt (United Chemical). The surface tension is 20.8+0.5dyncm -1, the viscosities are 9.6 and 611 g cm -1 s -1, respectively, and the densities are 0.97 g cm -3. In our experiments we studied the range 0.001 ~
3. Results and discussion 1 The normalized chi-squared deviation for N points, defined as
)(2,
is
Z2 ==~ [0~ xp -- 0the°(gi)]2 ~ 2 ( N - I) where 0~xp is an individual experimental value and 0th*°(ri) is the corresponding theoretical value; 0 is the standard deviation of the data points from a smooth curve fit locally to the points [19].
We have shown that Eq. (1) describes the interface shape for 20 gm < r < 400 gm and Ca ~ 0.1 [5,14,18] (Fig. 3). The agreement with the data at Ca <<.0.1 validates the model and provides a way to determine coo. When the experiment is done at several Ca < 0.1, the dependence of coo on U constitutes material information for modeling the spreading dynamics of this system. The coo values together
Q. Chen et aL /Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
120 120
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with Eq. (1) provide the boundary condition for the macroscopic interface shape as r--*0. The asymptotic matching process used in the model requires that the inner model for the material system must come out of the inner region as Ca Inr. Thus, our experiments imply that our system must have an inner mechanism of the form Ca In r at low Ca and r/Li >> 1 [18]. We measured the local velocity field near the contact line for menisci in depression. The velocity components agree with the Eqs. (6) and (7) for Ca <~0.1. Fig. 4 shows the velocity data for Ca = 0.1. Deviations from the theory are random in direction and below experimental error in magnitude throughout the domain shown. At Ca > 0.1, Eq. (1) begins to fail to describe the interface shape data near the contact line [18],
but it can still fit portions of the interface in r b < r < 400 pm, where 20 pm < r b < 400 pm is the inner boundary of the fitted region. As seen in Fig. 5, for Ca > 0.1 the model fails near the contact line but continues to describe the data further from the contact line. Because the model begins to fail near contact line, the conclusions about the range of Ca where the model holds depend on how close to the contact line we can measure the interface. This failure extends out as Ca increases. At Ca ~ 0.45, the theory no longer fits any portion of the interface for r < 400 pm (Fig. 6). The velocity measurements also show systematic deviations from the model Eqs. (6) and (7) at Ca = 0.4 (Fig. 7). Nevertheless, and independent of the agreement with the present models, we have identified a region, located within 100 pm around the contact line at Ca ~ 0.45, where the interface and flow fields are independent of geometry changes caused by varying the tube immersion angle [18]. We have narrowed the possible causes of the failure of the model to (a) the O(1) analysis in the intermediate region is insufficient at C a > 0 . 1 and/or (b) inner region mechanisms (which Eq. (1) does not include) extend out and into the field of view. When the model fails, it overestimates the interface curvature. This overprediction is consistent with the inner region extension as stresses, and hence curvature, in the inner region relax and
Q. Chen et al./Colloids Surfaces A." Physicochem. Eng. Aspects 116 (1996) 115-124
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must be smaller than those predicted by Eq.(1) using classical hydrodynamics in the same region of space. Low magnification experiments test assumptions of the model about the state of the interface far from the contact line. These experiments were done at Ca <~0.1, where the model of Eq. (1) properly describes the hydrodynamics for small r. Through extensive fitting, we have confirmed that the interface behaves quasi-statically far from the contact line. In this region, viscous forces are negligible compared with surface tension and gravity. We determined, as a function of Ca, the inner boundary beyond which the interface is no longer static, i.e. viscous forces begin to play a role large enough to be detected• Using a similar method we found, as a function of Ca, the extent of the region near the
-5
0
r
i
i
i
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200 r (I.tm)
300
400
Fig. 6. Plot of 0 vs. r at Ca = 0.46. (a) Dots, experimental data; solid line, extrapolated best theoretical fit of Eq. (1) to data 300 gm < r < 400 gm. (b) Dots, difference of the data and the best fit of Eq. (1). Note that no region of the data is described properly by Eq. ( 1 ).
contact line which is correctly described by Eq. ( 1). Fig. 8 shows a typical example of data with a region at small r fit to Eq. (1) and at large r fit to Eq. (3). Fig. 9 shows the regions described by Eqs. (1) and (3) for all Ca values tested. The boundary of the static-like region far from the contact line moves to larger r with increasing Ca, whereas the range of the region described by Eq. (1) increases with Ca. This is consistent with the growth of viscous deformation as Ca increases. These measurements are quantitatively different from previous measurements of dynamic menisci performed by Petrov and Sedev [22]. We also verified the prediction of the model that the contact angle of the extrapolation to the solid surface of the static-like
Q. Chen et aL/Colloids Surfaces A. Physicochem. Eng. Aspects 116 (1996) 115-124
122 400
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Fig. 8. Plot of 0 vs. r at low magnification. Ca = 0.005. Dots, experimental measurement; solid line, best fit of Eq.(1) at 50 ~tm < r < 300/am from the contact line. (% = 45 :'. Dashed line, static line [Eq.(3)] with contact angle (u0=45 ~'. Inset shows detail of static fit far away from contact line.
~ " / . , , , ¢" I * 71,. .., r.,,,,
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shape, co~, is equal to coo obtained by fitting Eq. ( 1) near the contact line [ 19]. Between the static-like region described by Eq.(3) and the region described by Eq.(l), a portion of interface remains which is neither staticlike nor described by Eq. (1). In this middle region (Fig. 9), viscous and gravity forces compete on an equal basis. As a result, viscous forces in this region are no longer purely local and are influenced by the system geometry at the capillary length scale. However, the middle region is not related to recirculation flows in the experimental cell because the
Viscous Theory Eq. [1]
300
Fig. 7. (a) Vector plot of the measured velocity field at Ca = 0.4. Scale marker of 150 p,m s 1 also shows velocity of tube. (b) Vector plot of the difference of the measured and theoretical velocity fields using Eqs. (6) and (7).
~.......--.--'~ ~
0.001
~
~
~
~ 1
. . . . . . . .
0.01
I
0.1
Ca Fig. 9. Plot of the regions described by Eqs. (1) and (3) vs. Ca. The middle region is not described by either Eqs.(1) or (3). The arrows on two of the data points denote upper bounds.
cell radius is ,-~5 cm, whereas the middle region lies at 200 gm < r < 1200 ktm, depending on Ca. From our interface measurements, we determined 0app, the contact angle of the static meniscus having the same capillary rise height as the dynamic interface [19]. Most dynamic contact angle measurements are performed by measuring Oapp. Oapp is a property of the whole dynamic meniscus from the contact line to the bulk level at infinity. Relating Oappto material parameters would therefore require knowledge of the flow field gener-
Q. Chen et aL/Colloids Surfaces A: Physicochem. Eng. Aspects 116 (1996) 115-124
ated by an immersing solid, and valid in the range 0 < r < ~ . Since this analysis is not yet available, 0app cannot be connected to a geometry-free parameter in the same manner as coo. We have shown that the difference coo- Oappincreases systematically with Ca, although it remains below 4 ° for Ca<~O.1 [,19]. Our results also show that viscous deformation is detectable at distances as large as the capillary length scale at Ca >/0.1, violating the assumption that almost all the meniscus looks static used in 0app measurements. This viscous deformation is responsible for the measured difference coo- 0app. Qualitatively similar results appear for a meniscus moving down a capillary tube.
4. Conclusions We have studied the global shape of dynamic interfaces and the velocity field when a solid surface enters a liquid bath. We looked at the interface in two modes. At high magnifications, we imaged dynamic menisci and fluid velocities within distances ~ 400 gm of the contact line. We compared the measurements with Eqs. (1), (6) and (7) which are the result of an asymptotic hydrodynamic model describing the viscous deformation of the interface and the local flow field near the moving contact line [,5,15,16]. For PDMS spreading on clean glass, Eq. (1) begins to fail at Ca > 0.1. The failure of Eq. (1) is due to either (a) the inability of the O(1) contribution from the intermediate region to accurately describe the interfacial deformation due to viscous forces and/or (b) the possibility that, at Ca >0.1, aspects of the unique hydrodynamics acting in the inner region, not included in the model, project out and become visible in the imaged region. Our results are consistent with this second possibility since the theory overpredicts the measured interface curvature. The same model developed for the interface shape yields the velocity field in a domain shaped as a wedge with a slowly varying angle [,15]. Our velocity measurements agree with the theory [-Eqs. (6) and (7)] for Ca ~<0.1. They show systematic deviations above Ca = 0.1. These data should be useful in identifying the cause of breakdown of
123
Eq. (1). Even at Ca values where the model fails, our measurements provide vital archival information as boundary conditions for numerical simulations of the macroscopic flow in the form of geometry-independent interface shapes and velocity fields. For the range of Ca (~< 0.1 ) where Eq. (1) is valid near the contact line, we studied the condition of the interface shape far from the moving contact line. This tested assumptions made in deriving Eq. (1). Concerning the accuracy of our measurements, our results confirm that (a) the shape of the interface far from the contact line approaches that of a static capillary surface and (b) the contact angle of the extrapolation to the solid surface of that static-like shape, co~, equals coo obtained from fitting Eq. (1) to data near the contact line. We have detected viscous deformation for Ca as small as 0.001. The region with viscous deformation lies in the range r <300 ~tm at Ca=0.001 and expands out as Ca increases. This has implications for the ability to transport apparent contact angles based on meniscus heights to geometries different from that used in the measurement. In summary, we have presented up-to-date results of our experimental studies of the validity of asymptotic models of liquid spreading. These experiments tested the ability of the model to describe the macroscopic dynamics of spreading systems. We have established that, for our material system, the theory represented by Eqs. (1), (6) and (7) holds for Ca <<,0.1. We verified the assumptions of the theory. By demonstrating the use of the dynamic contact angle and velocity field as a boundary condition for the macroscopic spreading dynamics, these experiments will allow the development of new, predictive models of macroscopic spreading dynamics.
Acknowledgment This work was supported in part by NASA Grant No. NAG3-1390. References [-1] E.B. Dussan V., Annu. Rev. Fluid Mech., 11 (1979) 371. [2] P.G. de Gennes, Rev. Mod. Phys., 57 (1985) 827.
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