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Can a Dynamic Contact Angle Be Understood in Terms of a Friction Coefficient?
Journal of Colloid and Interface Science 226, 199–204 (2000) doi:10.1006/jcis.2000.6830, available online at http://www.idealibrary.com on
Can a Dynamic Contact Angle Be Understood in Terms of a Friction Coefficient? Ahmed Hamraoui,1 Krister Thuresson, Tommy Nylander, and Vassili Yaminsky∗ Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden; and ∗ Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Institute of Advanced Studies, Australian National University, Canberra, ACT 0200, Australia Received May 10, 1999; accepted March 13, 2000
The kinetics of capillary rise of pure water and ethanol as well as mixtures thereof that under static conditions wet glass capillaries has been studied by a high-speed imaging technique. To adequately describe the experimental data, a rate-dependent dynamic contact angle must be added to the Washburn-Lucas equation. This result is discussed in terms of a molecular friction coefficient at the front of the liquid flowing over the substrate. Dynamic inertia effects are significant at the initial stages of spreading and may have further importance for capillaries of larger radii. In addition, it was found that the preparation and cleaning of glass capillaries, which are essential for the liquid substrate interaction, have a significant effect on the initial rate of rise of the liquid. °C 2000 Academic Press Key Words: capillary rise; wetting; friction; dynamic contact angle.
INTRODUCTION
The wetting of a surface by a liquid can be described by classic thermodynamics using Young’s equation. Here the contact angle is related to the interfacial energy of the solid-liquid, liquid-vapor, and solid-vapor interfaces. However, in many real situations this equilibrium state is not reached, at least not within time frames of interest. Instead, the wetting process is controlled by rates of spreading of a liquid over the substrate. Since the contact angle depends on the speed and direction of movement of the contact line, an adequate description of the wetting process becomes quite complex. The influence of the hydrodynamic forces (1) on the dynamic contact angle in the microscopic region near the moving line of the three-phase contact is not yet understood. Two different models have been presented: the hydrodynamic theory of Cox and Voinov (2), and the molecular kinetic approach proposed by Blake (3). The main difference between the two models is how the frictional force contribution to the velocity dependence of the dynamic contact angle is calculated. The molecular kinetics model takes into account
the adsorption/desorption rates at the moving three-phase line. In the hydrodynamic approach the moving liquid is regarded as a macroscopic continuum that at the wedge forms a film on which classic hydrodynamic theory, like Navier-Stokes equation, is applied. The wetting front of a pure liquid may change the properties of the substrate. If this process takes place on the same time scale as the wetting front advances, the contact angle is no longer constant, but will vary with the velocity of the front. In this paper we have focused on the factors controlling such behavior. For this purpose the wetting of well-defined substrates under different conditions has been studied with the capillary rise method using high-speed imaging. To interpret the data, we have introduced a modification to the well-known Washburn-Lucas equation. Corrections for the Washburn-Lucas equation have been considered earlier (4), but these do not apply over the whole experimental time scale. The observed contact angle dynamics is discussed in terms of the friction factor of the liquid advancing over the substrate. THEORETICAL
The main forces that act on a liquid rising up a capillary tube are due to the surface tension, 2πr γ cos(θ ), the gravity, mg = πr 2 ρgh(t), and the viscosity of the liquid, 8π ηh(t)(∂h(t)/∂t). Here r is the radius of the capillary, γ is the surface tension of the liquid, θ is the contact angle, g is the acceleration due to gravity, h(t) is the height of the liquid within the capillary at time t, and, finally, ρ and η are the density and the viscosity of the liquid, respectively. The Washburn-Lucas (5) equation is obtained by balancing these three forces, in analogy with Poiseuille’s formula:
1 To whom correspondence should be addressed at present address: Centre de Recherche en Mod´elisation Mol´eculaire Bat. Materia Nova, Rue N. Copernic 7600 Mons, Belgium. Fax: +32 65373881. E-mail: ahmed.hamraoui@galileo. umh.ac.be.
199
2πr γ cos(θ ) = πr 2 ρgh(t) + 8π ηh(t)
∂h(t) . ∂t
[1a]
However, for capillaries with a large inner diameter, that is, when the rate of the capillary rise is large enough, dynamic inertia effects can no longer be neglected (6). An inertia term 0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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must then be added to Eq. [1a], which now becomes ∂h(t) 2πr γ cos(θ) = πr 2 ρgh(t) + 8πηh(t) ∂t µ ¶ ∂h(t) ∂ h . + πr 2 ρ ∂t ∂t
[1b]
Given the small inner diameter of the capillary in the present study we expect the effect of the inertia term to be minor. However, this will be justified at the end of the discussion section. At static conditions the equilibrium height, h e , of a liquid within a capillary follows by balancing the Laplace pressure, 1p = 2γR = (2γ cos(θ 0 )/r ), (R is the radius of curvature of the meniscus) and the hydrostatic pressure, 1p = 1ρgh e ≈ ρgh e : he =
2γ cos(θ 0 ) . rρg
[2]
We have solved Eq. [1a] numerically to fit our experimental curves, and in many cases we found only a semiquantitative agreement. While the correct equilibrium height h e is obtained with θ 0 = 0, the initial rate of the rising liquid suggests a lower acting force. Since the surface tension is constant for a pure liquid, a correction can be introduced as a dynamic contact angle: W (t) = γ cos[θ (t)] = γ − β
∂h(t) . ∂t
Equation [1a] then becomes µ ¶ ∂h(t) ∂h(t) 2πr γ − β = πr 2 ρgh(t) + 8πηh(t) . ∂t ∂t
[3]
[4]
Here θ(t) is the dynamic contact angle, and β is a constant. The constant β will be discussed below. An expression for β can be based on the molecular kinetics theory that accounts of the rate, U (t), of a liquid flowing over a substrate (3). In the present notation, · ¸ κ0s λ h– γ (cos θ 0 − cos θ(t)) ∂h(t) = 4π sinh , [5] U (t) = ∂t ηv 2nkT where θ 0 and θ (t) are the equilibrium and dynamic contact angles, respectively; in our case θ 0 = 0. κ0s is the rate constant of the surface wetting adsorption process, n and λ are the number of adsorption sites per unit area and the average distance between the sites, respectively (λ ≈ n −(1/2) ≈ 1 nm). The ratio of kT , the thermal energy, to the Planck’s constant, h–, is the infrared (thermal) frequency, and v is molecular volume of the fluid. If the capillary driving force is small compared to the thermal energy, the rate can be written as U (t) = 2π
We then obtain the expression of the constant β = (kT v/ 2π λ3 κ0s h–)η. From this expression its clear that β is a molecular parameter, and we define it as the friction coefficient for the liquid that flows over the solid surface in a dry capillary or over a thin liquid film on the surface in a prewetted capillary. EXPERIMENTAL
Materials and sample preparation. Ethanol of spectroscopic quality, obtained from Kemetyl, Sweden (ethanol content 99.5 wt%), was used as received. Glass capillaries (Capillary tubing, DURAN) of radius r = 0.295 mm were used. The water used was deionized, passed through a Millipore water purification unit, and then equilibrated with charcoal to remove residual organic contaminants. The charcoal used was washed with large volumes of boiling water. All glassware was washed in hot (70◦ C) solution of sodium dichromate in sulfuric acid and rinsed with charcoal-treated Millipore water. Finally the glassware was boiled for 8 h in the purified water. The preparation of the capillaries, with different wetting properties, will be discussed below under Results. Equipment. A schematic drawing of the experimental set-up is given in Fig. 1. The stroboscope lamp was used to obtain multiple exposures on each frame record of the video. The camera EHD KAM02, EHD Physikalische Technik, Damme, Germany was connected to a video recorder (JVC HR-S9400 E/H, JVC, Yokohama, Japan). In this way we were able to capture the initial fast rise of the liquid, with a time resolution of the order of 1/100 s. For the highest observed rate (ca. 200 mm/s), this will correspond to a 2-mm displacement of the meniscus. Each video frame was analyzed by NIH Image analyzing software (National Institutes of Health, MD) on a Macintosh PowerPC connected to the video. The capillary was installed in a Teflon holder, which was mounted on a translation stage. This allowed the capillary to be moved down to the interface with minimal vibrations. The use of a hydrophobic Teflon beaker, filled to the edge, ensured that the meniscus of the air/liquid interface was convex, which in turn facilitates the imaging of the interface as the tip of
κ0s λ3 h– γ (1 − cos θ(t)), kT ηv
or, solved for ¸ kT v η U (t). W (t) = γ − 2πλ3 κ0s h– ·
FIG. 1. The experimental set-up.
DYNAMIC CONTACT ANGLES AND FRICTION COEFFICIENT
201
the capillary approaches. All experiments were carried out at 25◦ C. RESULTS
The experimental data for the capillary rise of ethanol as a function of time are shown in Fig. 2, together with the best fits to the Washburn-Lucas equation (Eq. [1a]) and the corrected equation (Eq. [4]). The experimental points during the initial stage, that is, when the rate of the capillary rise is high, cannot be adequately described by the Washburn-Lucas equation. However, when the correction term (Eq. [3]) is introduced, good agreement between theory and experimental data can be achieved. The inset in Fig. 2 presents the correction in terms of the dynamic contact angle (θ(t)). The correction term becomes smaller and smaller as the liquid rises higher up in the capillary and at the equilibrium position the contact angle reduces virtually to zero. We note that this change of the contact angle is also observed directly from the shape of the meniscus in the images of the capillary rise. With the aid of Eq. [3], the dynamic contact angle can be expressed in terms of the constant β. For ethanol a value of β = 0.04 was obtained. Qualitatively, a similar observation was made for water; here the disagreement between experimental data and Eq. [1a] is even much more pronounced (Fig. 3a). Consequently, the correction term in the form of a dynamic contact angle initially takes on higher values (inset in Fig. 3). By applying Eq. [3], θ(t) converts to β = 0.2. Similar behavior was observed for mixtures of water and ethanol, with β varying depending on the ethanol content (and hence the surface tension), Fig. 5. All the results in Figs. 2–4 were obtained with capillaries that have been carefully washed in a hot (70◦ ) solution of sodium dichromate in sulfuric acid and later rinsed with charcoal-treated Millipore water. Finally they were boiled for 8 h in the purified
FIG. 2. The rise of ethanol within a capillary (internal radius 0.295 mm). The dashed line shows the prediction by the Washburn-Lucas equation (Eq. [1]), while the full line describe the best fit of Eq. [4] to the data g = 9.81 m s−2 , η = 1.17 10−3 Pa s, γ = 22 mN m−1 , and β = 0.04. Inset: The dynamic contact angle vs time.
FIG. 3. The same as Fig. 2, but for a water η = 10−3 Pa s, γ = 72 mN m−1 . (a) β = 0.2 (prewetted tube). Inset: The dynamic contact angle vs time. (b) β = 0.45 (dry tube). Inset: The dynamic contact angle vs time.
FIG. 4. The same as Fig. 2, but for water–ethanol mixture, 7%wt of ethanol, η ≈ 10−3 Pa s, γ = 50.37 mN m−1 , and β = 0.14.
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FIG. 5. liquid.
The constant β (Eq. [3]) as a function of the surface tension of the
water. The remaining water was removed by strongly shaking off the capillary (prewetted tube). For some other experiments, capillaries were treated in a different way. The cleaning and rinsing steps were the same as before, while remaining water was removed by flame heating (dry tube). With these capillaries the correction factor took on a higher value, β = 0.45, for pure water, Fig. 3b. However, for pure ethanol the values of β were independent of the pretreatment of the capillary. In Table I, are summarized the β values and deduced values of the adsorption rate constant κ0s ; these values are in same the range of magnitude as in Ref. (3). (For estimations of the parameters in the expression for β for other liquid/solid interfaces see Ref. (3, p. 287–290). Finally it should be noted that from the h e value we could directly calculate the equilibrium surface tension of the liquid by use of Eq. [2]. In our experiments we obtained excellent agreement between the experimentally determined γ values and tabulated (literature) data. This verifies adequate purity of the chemicals, glassware, and capillaries used. DISCUSSION
The rate of the capillary rise according to the data in Fig. 3 is initially about 0.2 m/s and decreases to zero speed at the equilibrium height, h e . This value one can compare with kinetic studies of the spreading of a drop over a glass plate, where it has been observed that the water front can reach rates of up to 0.1 m/s (3). TABLE I β Values and Deduced Values of the Adsorption Rate Constant κs0
Liquid
Solid
Surface tension (mN m−1 )
Water Water Ethanol Water–ethanol mixture
Prewetted tube Dry tube Dry tube Dry tube
72 72 22 50.37
β
κ0s (s−1 )
0.2 0.45 0.04 0.14
1.58 × 1010 7.04 × 1009 7.92 × 1010 6.79 × 1010
Close to the equilibrium state, the dynamic contact angle is small, and the movement of the contact line is slow. It is known that the movement of a macroscopic contact line can be preceded by a precursor film (7). The first experimental evidence for the existence of such films was based on a reduced static friction observed near droplets spreading on solid surfaces (8). Interference techniques and many other experimental methods have since been used to confirm the existence of precursor films (9, 10). Ellipsometry measurements have revealed that precursor films are usually much thinner than 1 µm (10). In fact it has been shown that, in extreme cases, the precursor film advances molecular layer by molecular layer (10). This is important, as it has been suggested that, in the case of spontaneous spreading of droplets, the dynamics depends on the friction coefficient of the first molecular layer of the liquid on the solid (11). Given this, during the initial stage of the capillary rise, when the liquid moves fast, it can catch up with the precursor film. We then might expect to observe a finite contact angle. The process is limited by the speed at which the precursor film forms. In other words, it depends on the friction between the moving liquid and the solid surface. Ngan et al. (12) measured the effect of the channel width on the rising of a meniscus between two parallel glass plates. They found that the formation of a thin film ahead of the wetting line depends on the width of the channel, that is, on the flow rate. Starov et al. (13) studied the dynamic contact angle of a meniscus advancing in a prewetted capillary. They found that the dynamic contact angle depends on the diameter of the capillary, as well as on the “microstructural forces.” These forces were included in the disjoining pressure term. The thickness of the preexisting film was obtained by assuming equilibrium between this film and a static meniscus in the capillary tube (13). We have shown that the Washburn-Lucas equation can be used to describe our experimental data only if a correction factor (Eq. [3]) is added. Furthermore, as illustrated in Fig. 5, it is evident that the factor β is a function of the surface tension of the liquid. The correction factor indeed takes on the form of a frictional force. The mere fact that a significant variation of θ was observed, implies a frictional force. We can express this force as µ ¶ dh(t) n 0 . F = γ [cos(θ ) − cos(θ )] = β dt Here n occurs as an adjustable parameter. Zhou et al. (14), Stokes et al. (15), and Mumley (16) used a similar equation to describe the displacement of two immiscible fluids in a capillary tube. They arrived at values of the exponent in the range of 0.33 ≤ n ≤ 0.5. However, for our system, when we displace air with a liquid, we observe a linear relation between the force and the speed (see Fig. 7b), hence n = 1. The cosine of the dynamic contact angle can be expressed as a linear function of the capillary number, Ca = (η/γ∞ )U (t), as (14–16) cos θ = 1 −
β Ca. η
[6]
DYNAMIC CONTACT ANGLES AND FRICTION COEFFICIENT
203
a surface. They found that the friction coefficient of the first layer of a liquid on a solid is two times larger than between additional liquid layers. In our capillary wetting study we found β = 0.2 for the prewetted capillaries and β = 0.45 for the dry one (Fig. 3). These results demonstrate that the wetting front advances directly on a “dry” solid surface. Friction properties of the liquid/substrate interface become important. It was also found that the rate of spreading of a liquid drop increases with the relative humidity (20). We will now discuss the influence of the inertia term. This effect can act in the same direction as the dynamic contact angle, by reducing the rate compared to predictions of the classical Washburn-Lucas equation. To illustrate how large this effect is within the time frame we have captured, the different contributions to W (t), based on Eq. [1b] and experimental data from Fig. 3b, are plotted in Fig. 7 versus time (a) and rate of capillary rise, U (t) (b). On purpose we have taken height versus time data from the case where water rises in a flame–dried capillary as
FIG. 6. (a) Capillary number vs θ 2 . (b) Square root of Capillary number vs θ.
Near the equilibrium value of θ (case of complete wetting and small contact angles) this equation simplifies to β θ 2 ≈ 2 Ca. η
[7]
This relation, which follows from the Blake theory, can also be derived from the hydrodynamic theory when a liquid displaces a gas. Our data are in good agreement with Eq. [7] (Fig. 6). For systems that obey the Tanner (18) law, Ca = kθ 3 , the contact angle changes relatively less with the rate, U (t). When the front of the precursor film moves fast, the viscosity, η, is expected to dominate wetting kinetics (11). However, when no precursor film is present, or when it moves too slowly, the frictional force controls the spreading; here β is the important parameter (11). The constant β, interpreted as a friction coefficient, depends only on the nature of the liquid–solid interaction, which in turn depends on the surface preparation. In fact we observed that the value of β is higher for water in a dry capillary compared to the prewetted one. Vou´e et al. (11), used the stratified droplet model of de Gennes (19) to interpret data from measurements of the friction coefficients between a droplet and
FIG. 7. The contribution of gravity (– – –), viscosity (— — —), and inertia (---) to W (t) = γ cos[θ(t)] (—). The experimental data from Fig. 3b has been used to calculate W (t) and the different contributions according to Eq. [1b]. W (t) is shown versus time, t, (a) and rate of capillary rise, U (t), (b).
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we here observed the largest difference between the WashburnLucas equation and the experimental data. Only in the initial stage of the capillary rise, the inertia term can give a significant contribution to W (t). It is therefore clear that the inertia term cannot fully explain the lower rate of capillary rise compared to the classical Washburn-Lucas equation. Therefore the introduction of a dynamic contact angle is justified for the narrow capillaries used in this study. However, we note that during the very initial stage of capillary rise, the inertia term will influence W (t). This effect is expected to become progressively more important for capillaries of larger radii. Whether static contact angles in the systems studied are zero or small but finite may be important to investigate in further detail. Given this we could speculate more on the nature of the apparent friction coefficient. Further attention to the vapor pressure control may be important for understanding the effect. CONCLUSION
In conclusion, our experiments are in good agreement with the modified Washburn-Lucas equation. We have introduced a dynamic contact angle term that is linearly dependent on the velocity of the capillary rise as shown in Fig. 7b. In this linear approximation the correction takes on the form of a three-phase line friction coefficient. It is clear that at the initial stage the friction between the rising liquid and the capillary surface controls the rate of capillary rise. Here the surface preparation has a major influence. ACKNOWLEDGMENTS It is our privilege to acknowledge J. De Coninck, M. De Ruijter, T. D. Blake, F. Tiberg, and A. Fogden for fruitful and stimulating discussions. This research is supported by the European Commission, Marie-Curie Grant FMBICT972513.
REFERENCES 1. Dussan, V. E. B., Annu. Rev. Fluid Mech. 11, 371 (1979); Marsh, J. A., Garoff, S., and Dussan, V. E. B., Phys. Rev. Lett. 70, 2778 (1993).
2. Cox, R. G., J. Fluid Mech. 168, 169 (1986); Voinov, O. V., Fluid Dyn. 11, 714 (1976). 3. Blake, T. D., in “Wettability” (J. C. Berg, Eds.), p. 251. Marcel Dekker, New York, 1993; Blake, T. D., AIChE Spring Meeting, New Orleans, LA, 1998, paper 1a. 4. Joos, P., van Remoortere, P., and Bracke, M., J. Colloid Interface Sci. 136, 189 (1990); van Remoortere, P., and Joos, P., J. Colloid Interface Sci. 141, 348 (1991). 5. Washburn, D. W., Am. Phys. Soc. 17, 374 (1921); Lucas, R., Kolloidn. Zh. 23, 15 (1918). 6. Duarte, A. A., Strier, D. E., and Zanette, D. H., Am. J. Phys. 64, 413 (1996); Qu´er´e, D., Europhys. Lett. 39, 533 (1997); Qu´er´e, D., Rapha¨el, E., and Ollitrault, J. Y., Langmuir 15, 3679 (1999). 7. De Gennes, P. G., Rev. Mod. Phys. 57, 827 (1985). 8. Hardy, W. P., Philos. Mag. 38, 49 (1919); Hardy, W. P., Proc. Roy. Soc. Ser. A 100, 550 (1922). 9. Williams, R., Nature 266, 153 (1977). 10. Heslot, F., Cazabat, A. M., and Levinson, P., Phys. Rev. Lett. 62, 1286 (1989); Cazabat, A. M., Fraysse, N., and Heslot, F., Coll. Surf. 52, 1 (1991); Heslot, F., Fraysse, N., and Cazabat, A. M., Nature 338, 640 (1989); Heslot, F., Cazabat, A. M., and Levinson, P., Phys. Rev. Lett. 65, 599 (1990). 11. De Ruijter, M., De Coninck, J., and Blake, T., private communications; Cazabat, A. M., Valignat, M. P., Villette, S., De Coninck, J., and Louche, F., Langmuir 13, 4754 (1997); Vou´e, M., Valignat, M. P., Oshanin, G., Cazabat, A. M., and De Coninck, J., Langmuir 14, 5951 (1998); Vou´e, M., Valignat, M. P., Oshanin, G., and Cazabat, A. M., Langmuir 15, 1522 (1999). 12. Ngan, C. G., and Dussan, V. E. B., J. Fluid Mech. 118, 27 (1982). 13. Starov, V. M., Churaev, N. V., and Khvorostyanov, A. G., Coll. J. USSR 39, 176 (1979); Derjaguin, B. V., Acta Physicochim. USSR 12, 181 (1940); Derjaguin, B. V., and Churaev, N. V., Coll. J. USSR 38, 402 (1976) [English translation]. 14. Zhou, M. Y., and Sheng, P., Phys. Rev. Lett. 64, 882 (1990); Sheng, P., and Zhou, M. Y., Phys. Rev. A 45, 5694 (1992). 15. Stokes, J. P., Higgins, M. J., Kushnick, A. P., Bhattacharya, S., and Robbins, M. O., Phys. Rev. Lett. 65, 1885 (1990). 16. Mumley, T. E., Radke, C. J., and Williams, M. C., J. Colloid Interface Sci. 36, 189 (1990). 17. Kistler, S. F., in “Wettability” (J. C. Berg, Eds.), p. 311. Marcel Dekker, New York, 1993. 18. Tanner, L. H., J. Phys. D 12, 1473 (1979). 19. de Gennes, P. G., and Cazabat, A. M., C. R. Seances Acad. Sci. S´er. 2 310, 1601 (1990). 20. Tiberg, F., and Cazabat, A. M., Langmuir 10, 2301 (1994); Villette, S., Valignat, M. P., Cazabat, A. M., Jullien, L., and Tiberg, F., Langmuir 12, 825 (1996).
Can a Dynamic Contact Angle Be Understood in Terms of a Friction Coefficient? Ahmed Hamraoui,1 Krister Thuresson, Tommy Nylander, and Vassili Yaminsky∗ Physical Chemistry 1, Center for Chemistry and Chemical Engineering, Lund University, P.O. Box 124, SE-221 00 Lund, Sweden; and ∗ Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Institute of Advanced Studies, Australian National University, Canberra, ACT 0200, Australia Received May 10, 1999; accepted March 13, 2000
The kinetics of capillary rise of pure water and ethanol as well as mixtures thereof that under static conditions wet glass capillaries has been studied by a high-speed imaging technique. To adequately describe the experimental data, a rate-dependent dynamic contact angle must be added to the Washburn-Lucas equation. This result is discussed in terms of a molecular friction coefficient at the front of the liquid flowing over the substrate. Dynamic inertia effects are significant at the initial stages of spreading and may have further importance for capillaries of larger radii. In addition, it was found that the preparation and cleaning of glass capillaries, which are essential for the liquid substrate interaction, have a significant effect on the initial rate of rise of the liquid. °C 2000 Academic Press Key Words: capillary rise; wetting; friction; dynamic contact angle.
INTRODUCTION
The wetting of a surface by a liquid can be described by classic thermodynamics using Young’s equation. Here the contact angle is related to the interfacial energy of the solid-liquid, liquid-vapor, and solid-vapor interfaces. However, in many real situations this equilibrium state is not reached, at least not within time frames of interest. Instead, the wetting process is controlled by rates of spreading of a liquid over the substrate. Since the contact angle depends on the speed and direction of movement of the contact line, an adequate description of the wetting process becomes quite complex. The influence of the hydrodynamic forces (1) on the dynamic contact angle in the microscopic region near the moving line of the three-phase contact is not yet understood. Two different models have been presented: the hydrodynamic theory of Cox and Voinov (2), and the molecular kinetic approach proposed by Blake (3). The main difference between the two models is how the frictional force contribution to the velocity dependence of the dynamic contact angle is calculated. The molecular kinetics model takes into account
the adsorption/desorption rates at the moving three-phase line. In the hydrodynamic approach the moving liquid is regarded as a macroscopic continuum that at the wedge forms a film on which classic hydrodynamic theory, like Navier-Stokes equation, is applied. The wetting front of a pure liquid may change the properties of the substrate. If this process takes place on the same time scale as the wetting front advances, the contact angle is no longer constant, but will vary with the velocity of the front. In this paper we have focused on the factors controlling such behavior. For this purpose the wetting of well-defined substrates under different conditions has been studied with the capillary rise method using high-speed imaging. To interpret the data, we have introduced a modification to the well-known Washburn-Lucas equation. Corrections for the Washburn-Lucas equation have been considered earlier (4), but these do not apply over the whole experimental time scale. The observed contact angle dynamics is discussed in terms of the friction factor of the liquid advancing over the substrate. THEORETICAL
The main forces that act on a liquid rising up a capillary tube are due to the surface tension, 2πr γ cos(θ ), the gravity, mg = πr 2 ρgh(t), and the viscosity of the liquid, 8π ηh(t)(∂h(t)/∂t). Here r is the radius of the capillary, γ is the surface tension of the liquid, θ is the contact angle, g is the acceleration due to gravity, h(t) is the height of the liquid within the capillary at time t, and, finally, ρ and η are the density and the viscosity of the liquid, respectively. The Washburn-Lucas (5) equation is obtained by balancing these three forces, in analogy with Poiseuille’s formula:
1 To whom correspondence should be addressed at present address: Centre de Recherche en Mod´elisation Mol´eculaire Bat. Materia Nova, Rue N. Copernic 7600 Mons, Belgium. Fax: +32 65373881. E-mail: ahmed.hamraoui@galileo. umh.ac.be.
199
2πr γ cos(θ ) = πr 2 ρgh(t) + 8π ηh(t)
∂h(t) . ∂t
[1a]
However, for capillaries with a large inner diameter, that is, when the rate of the capillary rise is large enough, dynamic inertia effects can no longer be neglected (6). An inertia term 0021-9797/00 $35.00
C 2000 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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must then be added to Eq. [1a], which now becomes ∂h(t) 2πr γ cos(θ) = πr 2 ρgh(t) + 8πηh(t) ∂t µ ¶ ∂h(t) ∂ h . + πr 2 ρ ∂t ∂t
[1b]
Given the small inner diameter of the capillary in the present study we expect the effect of the inertia term to be minor. However, this will be justified at the end of the discussion section. At static conditions the equilibrium height, h e , of a liquid within a capillary follows by balancing the Laplace pressure, 1p = 2γR = (2γ cos(θ 0 )/r ), (R is the radius of curvature of the meniscus) and the hydrostatic pressure, 1p = 1ρgh e ≈ ρgh e : he =
2γ cos(θ 0 ) . rρg
[2]
We have solved Eq. [1a] numerically to fit our experimental curves, and in many cases we found only a semiquantitative agreement. While the correct equilibrium height h e is obtained with θ 0 = 0, the initial rate of the rising liquid suggests a lower acting force. Since the surface tension is constant for a pure liquid, a correction can be introduced as a dynamic contact angle: W (t) = γ cos[θ (t)] = γ − β
∂h(t) . ∂t
Equation [1a] then becomes µ ¶ ∂h(t) ∂h(t) 2πr γ − β = πr 2 ρgh(t) + 8πηh(t) . ∂t ∂t
[3]
[4]
Here θ(t) is the dynamic contact angle, and β is a constant. The constant β will be discussed below. An expression for β can be based on the molecular kinetics theory that accounts of the rate, U (t), of a liquid flowing over a substrate (3). In the present notation, · ¸ κ0s λ h– γ (cos θ 0 − cos θ(t)) ∂h(t) = 4π sinh , [5] U (t) = ∂t ηv 2nkT where θ 0 and θ (t) are the equilibrium and dynamic contact angles, respectively; in our case θ 0 = 0. κ0s is the rate constant of the surface wetting adsorption process, n and λ are the number of adsorption sites per unit area and the average distance between the sites, respectively (λ ≈ n −(1/2) ≈ 1 nm). The ratio of kT , the thermal energy, to the Planck’s constant, h–, is the infrared (thermal) frequency, and v is molecular volume of the fluid. If the capillary driving force is small compared to the thermal energy, the rate can be written as U (t) = 2π
We then obtain the expression of the constant β = (kT v/ 2π λ3 κ0s h–)η. From this expression its clear that β is a molecular parameter, and we define it as the friction coefficient for the liquid that flows over the solid surface in a dry capillary or over a thin liquid film on the surface in a prewetted capillary. EXPERIMENTAL
Materials and sample preparation. Ethanol of spectroscopic quality, obtained from Kemetyl, Sweden (ethanol content 99.5 wt%), was used as received. Glass capillaries (Capillary tubing, DURAN) of radius r = 0.295 mm were used. The water used was deionized, passed through a Millipore water purification unit, and then equilibrated with charcoal to remove residual organic contaminants. The charcoal used was washed with large volumes of boiling water. All glassware was washed in hot (70◦ C) solution of sodium dichromate in sulfuric acid and rinsed with charcoal-treated Millipore water. Finally the glassware was boiled for 8 h in the purified water. The preparation of the capillaries, with different wetting properties, will be discussed below under Results. Equipment. A schematic drawing of the experimental set-up is given in Fig. 1. The stroboscope lamp was used to obtain multiple exposures on each frame record of the video. The camera EHD KAM02, EHD Physikalische Technik, Damme, Germany was connected to a video recorder (JVC HR-S9400 E/H, JVC, Yokohama, Japan). In this way we were able to capture the initial fast rise of the liquid, with a time resolution of the order of 1/100 s. For the highest observed rate (ca. 200 mm/s), this will correspond to a 2-mm displacement of the meniscus. Each video frame was analyzed by NIH Image analyzing software (National Institutes of Health, MD) on a Macintosh PowerPC connected to the video. The capillary was installed in a Teflon holder, which was mounted on a translation stage. This allowed the capillary to be moved down to the interface with minimal vibrations. The use of a hydrophobic Teflon beaker, filled to the edge, ensured that the meniscus of the air/liquid interface was convex, which in turn facilitates the imaging of the interface as the tip of
κ0s λ3 h– γ (1 − cos θ(t)), kT ηv
or, solved for ¸ kT v η U (t). W (t) = γ − 2πλ3 κ0s h– ·
FIG. 1. The experimental set-up.
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the capillary approaches. All experiments were carried out at 25◦ C. RESULTS
The experimental data for the capillary rise of ethanol as a function of time are shown in Fig. 2, together with the best fits to the Washburn-Lucas equation (Eq. [1a]) and the corrected equation (Eq. [4]). The experimental points during the initial stage, that is, when the rate of the capillary rise is high, cannot be adequately described by the Washburn-Lucas equation. However, when the correction term (Eq. [3]) is introduced, good agreement between theory and experimental data can be achieved. The inset in Fig. 2 presents the correction in terms of the dynamic contact angle (θ(t)). The correction term becomes smaller and smaller as the liquid rises higher up in the capillary and at the equilibrium position the contact angle reduces virtually to zero. We note that this change of the contact angle is also observed directly from the shape of the meniscus in the images of the capillary rise. With the aid of Eq. [3], the dynamic contact angle can be expressed in terms of the constant β. For ethanol a value of β = 0.04 was obtained. Qualitatively, a similar observation was made for water; here the disagreement between experimental data and Eq. [1a] is even much more pronounced (Fig. 3a). Consequently, the correction term in the form of a dynamic contact angle initially takes on higher values (inset in Fig. 3). By applying Eq. [3], θ(t) converts to β = 0.2. Similar behavior was observed for mixtures of water and ethanol, with β varying depending on the ethanol content (and hence the surface tension), Fig. 5. All the results in Figs. 2–4 were obtained with capillaries that have been carefully washed in a hot (70◦ ) solution of sodium dichromate in sulfuric acid and later rinsed with charcoal-treated Millipore water. Finally they were boiled for 8 h in the purified
FIG. 2. The rise of ethanol within a capillary (internal radius 0.295 mm). The dashed line shows the prediction by the Washburn-Lucas equation (Eq. [1]), while the full line describe the best fit of Eq. [4] to the data g = 9.81 m s−2 , η = 1.17 10−3 Pa s, γ = 22 mN m−1 , and β = 0.04. Inset: The dynamic contact angle vs time.
FIG. 3. The same as Fig. 2, but for a water η = 10−3 Pa s, γ = 72 mN m−1 . (a) β = 0.2 (prewetted tube). Inset: The dynamic contact angle vs time. (b) β = 0.45 (dry tube). Inset: The dynamic contact angle vs time.
FIG. 4. The same as Fig. 2, but for water–ethanol mixture, 7%wt of ethanol, η ≈ 10−3 Pa s, γ = 50.37 mN m−1 , and β = 0.14.
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FIG. 5. liquid.
The constant β (Eq. [3]) as a function of the surface tension of the
water. The remaining water was removed by strongly shaking off the capillary (prewetted tube). For some other experiments, capillaries were treated in a different way. The cleaning and rinsing steps were the same as before, while remaining water was removed by flame heating (dry tube). With these capillaries the correction factor took on a higher value, β = 0.45, for pure water, Fig. 3b. However, for pure ethanol the values of β were independent of the pretreatment of the capillary. In Table I, are summarized the β values and deduced values of the adsorption rate constant κ0s ; these values are in same the range of magnitude as in Ref. (3). (For estimations of the parameters in the expression for β for other liquid/solid interfaces see Ref. (3, p. 287–290). Finally it should be noted that from the h e value we could directly calculate the equilibrium surface tension of the liquid by use of Eq. [2]. In our experiments we obtained excellent agreement between the experimentally determined γ values and tabulated (literature) data. This verifies adequate purity of the chemicals, glassware, and capillaries used. DISCUSSION
The rate of the capillary rise according to the data in Fig. 3 is initially about 0.2 m/s and decreases to zero speed at the equilibrium height, h e . This value one can compare with kinetic studies of the spreading of a drop over a glass plate, where it has been observed that the water front can reach rates of up to 0.1 m/s (3). TABLE I β Values and Deduced Values of the Adsorption Rate Constant κs0
Liquid
Solid
Surface tension (mN m−1 )
Water Water Ethanol Water–ethanol mixture
Prewetted tube Dry tube Dry tube Dry tube
72 72 22 50.37
β
κ0s (s−1 )
0.2 0.45 0.04 0.14
1.58 × 1010 7.04 × 1009 7.92 × 1010 6.79 × 1010
Close to the equilibrium state, the dynamic contact angle is small, and the movement of the contact line is slow. It is known that the movement of a macroscopic contact line can be preceded by a precursor film (7). The first experimental evidence for the existence of such films was based on a reduced static friction observed near droplets spreading on solid surfaces (8). Interference techniques and many other experimental methods have since been used to confirm the existence of precursor films (9, 10). Ellipsometry measurements have revealed that precursor films are usually much thinner than 1 µm (10). In fact it has been shown that, in extreme cases, the precursor film advances molecular layer by molecular layer (10). This is important, as it has been suggested that, in the case of spontaneous spreading of droplets, the dynamics depends on the friction coefficient of the first molecular layer of the liquid on the solid (11). Given this, during the initial stage of the capillary rise, when the liquid moves fast, it can catch up with the precursor film. We then might expect to observe a finite contact angle. The process is limited by the speed at which the precursor film forms. In other words, it depends on the friction between the moving liquid and the solid surface. Ngan et al. (12) measured the effect of the channel width on the rising of a meniscus between two parallel glass plates. They found that the formation of a thin film ahead of the wetting line depends on the width of the channel, that is, on the flow rate. Starov et al. (13) studied the dynamic contact angle of a meniscus advancing in a prewetted capillary. They found that the dynamic contact angle depends on the diameter of the capillary, as well as on the “microstructural forces.” These forces were included in the disjoining pressure term. The thickness of the preexisting film was obtained by assuming equilibrium between this film and a static meniscus in the capillary tube (13). We have shown that the Washburn-Lucas equation can be used to describe our experimental data only if a correction factor (Eq. [3]) is added. Furthermore, as illustrated in Fig. 5, it is evident that the factor β is a function of the surface tension of the liquid. The correction factor indeed takes on the form of a frictional force. The mere fact that a significant variation of θ was observed, implies a frictional force. We can express this force as µ ¶ dh(t) n 0 . F = γ [cos(θ ) − cos(θ )] = β dt Here n occurs as an adjustable parameter. Zhou et al. (14), Stokes et al. (15), and Mumley (16) used a similar equation to describe the displacement of two immiscible fluids in a capillary tube. They arrived at values of the exponent in the range of 0.33 ≤ n ≤ 0.5. However, for our system, when we displace air with a liquid, we observe a linear relation between the force and the speed (see Fig. 7b), hence n = 1. The cosine of the dynamic contact angle can be expressed as a linear function of the capillary number, Ca = (η/γ∞ )U (t), as (14–16) cos θ = 1 −
β Ca. η
[6]
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a surface. They found that the friction coefficient of the first layer of a liquid on a solid is two times larger than between additional liquid layers. In our capillary wetting study we found β = 0.2 for the prewetted capillaries and β = 0.45 for the dry one (Fig. 3). These results demonstrate that the wetting front advances directly on a “dry” solid surface. Friction properties of the liquid/substrate interface become important. It was also found that the rate of spreading of a liquid drop increases with the relative humidity (20). We will now discuss the influence of the inertia term. This effect can act in the same direction as the dynamic contact angle, by reducing the rate compared to predictions of the classical Washburn-Lucas equation. To illustrate how large this effect is within the time frame we have captured, the different contributions to W (t), based on Eq. [1b] and experimental data from Fig. 3b, are plotted in Fig. 7 versus time (a) and rate of capillary rise, U (t) (b). On purpose we have taken height versus time data from the case where water rises in a flame–dried capillary as
FIG. 6. (a) Capillary number vs θ 2 . (b) Square root of Capillary number vs θ.
Near the equilibrium value of θ (case of complete wetting and small contact angles) this equation simplifies to β θ 2 ≈ 2 Ca. η
[7]
This relation, which follows from the Blake theory, can also be derived from the hydrodynamic theory when a liquid displaces a gas. Our data are in good agreement with Eq. [7] (Fig. 6). For systems that obey the Tanner (18) law, Ca = kθ 3 , the contact angle changes relatively less with the rate, U (t). When the front of the precursor film moves fast, the viscosity, η, is expected to dominate wetting kinetics (11). However, when no precursor film is present, or when it moves too slowly, the frictional force controls the spreading; here β is the important parameter (11). The constant β, interpreted as a friction coefficient, depends only on the nature of the liquid–solid interaction, which in turn depends on the surface preparation. In fact we observed that the value of β is higher for water in a dry capillary compared to the prewetted one. Vou´e et al. (11), used the stratified droplet model of de Gennes (19) to interpret data from measurements of the friction coefficients between a droplet and
FIG. 7. The contribution of gravity (– – –), viscosity (— — —), and inertia (---) to W (t) = γ cos[θ(t)] (—). The experimental data from Fig. 3b has been used to calculate W (t) and the different contributions according to Eq. [1b]. W (t) is shown versus time, t, (a) and rate of capillary rise, U (t), (b).
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we here observed the largest difference between the WashburnLucas equation and the experimental data. Only in the initial stage of the capillary rise, the inertia term can give a significant contribution to W (t). It is therefore clear that the inertia term cannot fully explain the lower rate of capillary rise compared to the classical Washburn-Lucas equation. Therefore the introduction of a dynamic contact angle is justified for the narrow capillaries used in this study. However, we note that during the very initial stage of capillary rise, the inertia term will influence W (t). This effect is expected to become progressively more important for capillaries of larger radii. Whether static contact angles in the systems studied are zero or small but finite may be important to investigate in further detail. Given this we could speculate more on the nature of the apparent friction coefficient. Further attention to the vapor pressure control may be important for understanding the effect. CONCLUSION
In conclusion, our experiments are in good agreement with the modified Washburn-Lucas equation. We have introduced a dynamic contact angle term that is linearly dependent on the velocity of the capillary rise as shown in Fig. 7b. In this linear approximation the correction takes on the form of a three-phase line friction coefficient. It is clear that at the initial stage the friction between the rising liquid and the capillary surface controls the rate of capillary rise. Here the surface preparation has a major influence. ACKNOWLEDGMENTS It is our privilege to acknowledge J. De Coninck, M. De Ruijter, T. D. Blake, F. Tiberg, and A. Fogden for fruitful and stimulating discussions. This research is supported by the European Commission, Marie-Curie Grant FMBICT972513.
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