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Assessment of Hydrodynamic and Molecular-Kinetic Models Applied to the Motion of the Dewetting Contact Line between a Small Bubble and a Solid Surface Chi M. Phan, Anh V. Nguyen,* and Geoffrey M. Evans Discipline of Chemical Engineering, The University of Newcastle, University Drive, Callaghan, New South Wales 2308, Australia Received January 9, 2003. In Final Form: May 12, 2003 A high-speed camera was used to observe the motion of the three-phase contact (TPC) line for a small rising bubble ruptured by a submerged horizontal glass plate. The experimental data for the radial position of the TPC line as a function of time were used to examine both hydrodynamic and molecular-kinetic models previously developed for wetting/dewetting processes. It was found that both models were not able to describe the experimental data using the physically consistent values of parameters. A better fit could be obtained if the equilibrium contact angle, obtained from the TPC versus time measurements, was used in place of the thermodynamic contact angle determined by the Wilhelmy plate technique; however, the parameters obtained by this fitting could not be physically justified. Importantly, the equilibrium contact angle was found to be a function of the bubble radius and was significantly different from the Young contact angle. This radius dependence of contact angles and other curvature-dependent effects, which are not considered in the hydrodynamic and molecular-kinetic models, may be the cause of the deviations from the experimental results.
Introduction Wetting is the process whereby a liquid displaces a gas phase on a solid surface. On the other hand, dewetting is an opposite process in which a gaseous phase displaces a liquid phase. These processes have been investigated by a number of researchers, for example, refs 1-5. Both wetting and dewetting are central to a number of industrial processes, ranging from liquid coating6 to tertiary oil recovery and froth flotation separation of ink particles, oil droplets, and minerals.7 The vast majority of investigations of wetting and dewetting processes have focused on the motion of the three-phase contact lines with a large to infinite (2D cases) radius of curvature. A much smaller fraction of studies have addressed the wetting and dewetting processes with a small radius of the three-phase contact perimeter. This class of wetting and dewetting processes characterized by a small radius of the wetting perimeter includes wetting transition, phase formation and nucleation or condensation,2 and bubble-particle interaction in the mineral flotation separation,8 where the formation or the disappearance of a multiphase contact occurs. These wetting and dewetting processes with a small radius are still far from being completely understood. In this paper, the radial position of the three-phase contact (TPC) line is investigated for a small bubble * Corresponding author. Fax +61 2 4921 6920. E-mail
[email protected]. (1) Blake, T. D.; Shikhmurzaev, Y. D. J. Colloid Interface Sci. 2002, 253, 196. (2) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (3) Cazabat, A. M. Adv. Colloid Interface Sci. 1992, 42, 65. (4) Petrov, J.; Radoev, B. Colloid Polym. Sci. 1981, 259, 753. (5) Schneemilch, M.; Hayes, R. A.; Petrov, J. G.; Ralston, J. Langmuir 1998, 14, 7047. (6) Blake, T. D.; Ruschak, K. J. In Liquid Film Coating; Kistler, S. F., Schweizer, P. M., Eds; Chapman & Hall: London, 1997; p 63. (7) Schulze, H. J. Physico-Chemical Elementary Processes in Flotation; Elsevier: Amsterdam, 1984. (8) Nguyen, A. V.; Schulze, H. J.; Ralston, J. Int. J. Miner. Process. 1997, 51, 183.
contacting the underside of a horizontal solid surface. In the study, an experimental system is used to monitor the motion of the dewetting area, bounded by a TPC line, following rupture of the liquid film between the solid surface and the bubble. The TPC radius is recorded as a function of time until a stable equilibrium position is reached. The experimental results are then compared to model predictions, based on the hydrodynamic and molecular-kinetic theories, which relate the dynamic contact angle to the TPC line velocity for wetting conditions. Modeling of the Dewetting Process In our experiments, the motion and equilibrium position of the TPC line between a small air bubble and a solid surface are measured as a function of time. Consequently, it is of interest to develop model(s) to describe the TPC movement (or the TPC line radial position). In general, the radial position of the TPC line, r, as a function of time, t, is determined by
r(t) )
∫0tU dτ
(1)
where U is the instantaneous velocity of the TPC line and τ is the time integration variable. From eq 1, to predict the radial position of the TPC line as a function of time, U needs to be determined. In this paper, we use the velocity of the TPC line predicted from either hydrodynamic theory or molecular-kinetic theory, as described below. This will allow the comparison between the theories and the experimental data. Hydrodynamic Model. The hydrodynamic theory solves the continuity and Stokes equations governing the fluid dissipation and relates the TPC velocity to the dynamic contact angle θ. The TPC velocity is usually scaled to give the capillary number, Ca, by the expression
Ca t
Uµ σ
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truncating the solution at molecular dimensions, resulting in the inner region with the characteristic length scale L, which is of the order of 1 nm. Therefore, the order of ln(R/L) is equal to about ln 106 = 14, which is recommended in the literature.3 The radial position of the TPC line is obtained by substituting eq 6 into eq 1:
r(t) )
Figure 1. TPC line for a gas bubble submerged beneath a planar solid surface (the inset shows the contact angles of the outer and inner regions).
where σ is the gas-liquid surface tension and µ is the liquid viscosity. The hydrodynamic theory9-11 gives the following equation for the capillary number and the dynamic contact angle: 2
g(θm) - g(θ) ) Ca[ln(R/L) + Q] + O(Ca )
(3)
where R/L is the ratio of the characteristic macroscopic to slip length; Q is a term that accounts for dissipation in the inner zone10 and is smaller than unity; θ is the dynamic contact angle (of the outer region); and θm is the (microscopic) contact angle of the inner region, as shown in Figure 1. For air-liquid systems, the function g(x) in eq 3 is defined by
g(x) t (1/2)
∫0x [(z/sin z) - cos z] dz
(4)
This equation cannot be analytically integrated, but an approximation can be obtained employing a Maclaurin series expansion, which gives
g(x) ) x3/9 + O(x5)
(5)
This approximation is very accurate. For x e 0.8π, the relative error is smaller than 4%. Inserting eq 5 for θ and θm into eq 3 and solving eqs 2 and 3 for U gives
U)
(θm3 - θ3)σ 9 ln(R/L)µ
+ ‚‚‚
(6)
In eq 6, the dissipation term Q was omitted as it is insignificant in comparison with ln(R/L), which is an adjustable parameter whose magnitude can be estimated from the expected range of R and L, as given below. The hydrodynamic theory accounts for the change in the dynamic contact angle through the hydrodynamic bending of the gas-liquid meniscus at a distance close to the three-phase contact line as not to be macroscopically visible, resulting in the macroscopic region with the characteristic length scale R. In our case, R is of the order of the bubble size, which is about 1 mm. It is known that the hydrodynamic equations lead to singularity at the contact line.12 The wetting line singularity is avoided by (9) Huh, C.; Scriven, L. E. J. Colloid Interface Sci. 1971, 35, 85. (10) Cox, R. G. J. Fluid Mech. 1986, 168, 169. (11) Shikhmurzaev, Y. D. J. Fluid Mech. 1997, 334, 211.
∫0t [θm3 - θ3(τ)] dτ
σ 9 ln(R/L)µ
(7)
Molecular-Kinetic Model. The molecular model for wetting/dewetting processes is based on the principle of individual molecular displacements at the contact line.6,13,14 The velocity dependence of the dynamic contact angle is due to the disturbance of adsorption equilibrium at the three-phase contact line, driven by the imbalance of the tension forces. The TPC line velocity is expressed by6
U ) 2Kλ sinh
{
σλ2 (cos θ0 - cos θ) 2kBT
}
(8)
where K is the frequency of the molecular displacement; λ is the average molecular jumping distance; kB is the Boltzmann constant; T is absolute temperature in Kelvin; and θ0 is the Young (thermodynamic) contact angle, defined by
σsg - σsl σ
θ0 ≡ acos
(9)
where σsl and σsg describe the specific interfacial energy of the solid-liquid and solid-gas interfaces. In eq 8, both K and λ are usually not known precisely and are treated as adjustable parameters. If n is the number of active adsorption sites per unit area of the solid surface, then we have the estimation λ ∼ n-1/2, which shows that λ is of the order of (1 nm) molecular dimensions. In addition, the available experimental data14,15 indicate that within the three-phase zone, molecular displacements occur at lower frequencies than within the bulk liquid. Taking the usual value of 10-9 s for the relaxation time of water molecules, K has to be smaller than 109 s-1. A typical value of K is from 106 to 107 s-1.15 Similar to hydrodynamic theory, the radial position of the TPC line based on molecular theory is given by
r(t) ) 2Kλ
{
∫0t sinh
}
σλ2 (cos θ0 - cos θ) dτ 2kBT
(10)
Contact Angles. In the previous sections, three different contact angles (dynamic, microscopic, and thermodynamic) have been used to obtain the TPC velocity. First, the macroscopic dynamic contact angle can be determined from the radial position of the TPC line as a function of time or vice versa. Providing the bubble remains spherical, which is the case of small bubbles used in our experiments, and the TPC contact line has a circular shape with the center remaining in the same position, the (12) Dusan, W. E. B. Annu. Rev. Fluid Mech. 1979, 11, 371. (13) Cherry, B. W.; Holmes, C. M. J. Colloid Interface Sci. 1969, 29, 174. (14) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (15) Blake, T. D.; Clarke, A.; Ruschak, K. J. AIChE J. 1994, 40, 229.
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following expression can be used:
r)
{
(3V/π) sin3 θ
}
1/3
2 + 3 cos θ - cos3 θ
(11)
where V is the constant volume of the bubble. Experimentally, V is determined from the equilibrium TPC radius, re, and the equilibrium contact angle, θe, by
πre3 2 + 3 cos θe - cos3 θe V) 3 sin3 θ
(12)
e
In eq 12, θe is calculated using
() re Rb
θe ) arcsin
(13)
where Rb is the bubble radius once the TPC line reaches equilibrium (becomes stationary). While the macroscopic dynamic contact angle can be determined directly from the TPC position, it is not so straightforward to estimate the microscopic dynamic contact angle, θm. This angle is determined by the molecular interactions in the inner (molecular) region.2,16,17 It can be estimated from the disjoining pressure, Π(h), of intermolecular interactions between the gas-liquid and solid-liquid interfaces. The Frumkin-Derjaguin equation gives18
∫H∞Π(h) dh
cos θm ) 1 + (1/σ)
(14)
The disjoining pressure in this equation is a function of the film thickness, h, of the inner region and has a number of components due to the van der Waals, double-layer, and hydrophobic/hydrophilic interactions.18 The film thickness H in the lower integration limit in eq 14 is the smallest magnitude of the root of the equation Π(H) ) 0 that involves the algebraic sum of the above-described components of the disjoining pressure. Details of the calculation are given in ref 18. For a glass surface and air bubbles in pure water, our estimation gives θm ) 24°, which is reasonably close to the values obtained for similar systems.18 In the estimation, we used the values of -60 and -55 mV for the surface potentials of the water-glass and water-air interfaces and 10-20 J for the Hamaker constant of the system. The value of the microscopic contact angle is also close to the Young contact angle of 26° on the glass surface, which was determined by using the Wilhelmy plate technique described in the next section. This is in line with the practice of replacing the microscopic contact angle in the hydrodynamic model by the Young contact angle measured with the Wilhelmy plate technique. In this paper, θm in eq 7 is also replaced by θ0 and gives
r(t) )
∫0t[θ03 - θ3(τ)] dτ
σ 9 ln(R/L)µ
Figure 2. Schematic of experimental apparatus.
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Experimental Section Apparatus. The experimental setup is shown in Figure 2. It comprised a high-speed digital camera system, water container, microscope glass slide, and needle connected to a syringe. The (16) Voinov, O. V. J. Colloid Interface Sci. 2000, 226, 5. (17) Teletzke, G. F.; Davis, H. T.; Scriven, L. E. Chem. Eng. Commun. 1987, 55, 41. (18) Churaev, N. V.; Sobolev, V. D. Adv. Colloid Interface Sci. 1995, 61, 1.
Figure 3. An image showing the determination of the Young contact angle at the glass surface by the Wilhelmy plate technique after the receding TPC line becomes stationary. slide was horizontally submerged just below the water free surface. The camera was located above the glass slide. The water used was cleaned with a (Millipore) Milli-Q Plus system. The microscope slide was cleaned with 10% hydrochloric solution and then rinsed copiously with Milli-Q water before each experiment. The needle used was a PrecissionGlide 30 G × 1/2 needle. Procedure. An air bubble was produced and released from the needle by increasing pressure through the syringe. The size of the produced air bubble is about 1.5 mm in diameter. Each bubble hit the underside of the glass slide and bounced 3-5 times, depending on the bubble size, before resting and rupturing. Following rupture, the TPC line movement was recorded at 1000 frames per second until it reached an equilibrium position approximately 30 ms later. All measurements were carried out at room temperature (∼22 °C) maintained by an air-conditioning system. The air used to produce the bubbles was compressed through the needle manually and was free from any oil contaminants that could affect this delicate type of measurement. The Young Contact Angle. The Young contact angle, θ0, was obtained experimentally by the Wilhelmy plate technique by immersing the slide vertically into water before partly withdrawing it. The slide was then held stationary so that the meniscus could be recorded (see Figure 3) which was later digitized to obtain the slope of the interface profile at the TPC line. This contact angle is independent of the bubble size and corresponds to the equilibrium contact angle of the systems where the radius of curvature of the TPC line has no effect on contact angles.
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Figure 4. Images of the TPC line at 2, 4, 7, 10, 20, and 30 ms after the film rupture. Data Analysis. The data analysis involved first calculating the equilibrium contact angle, θe, in accordance with eq 13 and using the experimental values for the equilibrium TPC and bubble radius. The volume of air bubbles was then calculated using eq 12. For each experiment, the initial TPC was recorded. The moment when the TPC line was first observed was taken to be time t ) 0. The TPC radial position for time after t ) 0 was determined from the images using an image analysis software developed by the supplier of the high-speed camera system. The modeled TPC radius was calculated from eq 11 and either eq 7 or eq 10 employing a numerical integration procedure. The dynamic contact angle, θ, at the moment t ) 0 was calculated from eq 11 using the TPC radius at t ) 0 and the bubble volume. The procedure was iterated until the time t ) teq, when the TPC equilibrium radius was reached. The modeled profile can be compared with the experimental TPC radius profile. The calculation was first carried out with the typical values of the system parameters estimated from the physics involved for both hydrodynamic and molecular-kinetic models. These parameters were also allowed to change to best fit the experimental data. In this adjustment, Solver (Microsoft Excel) was used to minimize the total squares of deviations. The adjusted parameters were then analyzed to check whether they were physically consistent.
Results and Discussion General Observation of the TPC Line. A series of images of the motion of the TPC line immediately following rupture of the bubble at the solid surface is shown in Figure 4. It was found that the smallest recorded rupture radius was about 0.05 mm, with the TPC line remaining relatively circular throughout the expansion period. The buoyancy affects the bubble rise and the bubble-plate interaction before the dewetting process taking place as discussed in Procedure. For the small bubbles used in our experiments, the buoyancy effects on the deformation of the gas-liquid interface and the contact angle can be neglected in a manner similar to that of gravitational effects in the more common static drop technique. Indeed, if the buoyancy effects were significant, they would decrease the equilibrium contact angle relative to the Wilhelmy or static drop methods, rather than increase it as is illustrated below.
Figure 5. Equilibrium contact angle versus inverse bubble radius. The solid line describes eq 16, with θ0 equal to 26°.
The Young and Equilibrium Contact Angles. The Young contact angle determined by the Wilhelmy plate technique was about 26°. It is important to note that this Young contact angle is significantly smaller than the equilibrium macroscopic contact angle, θe, between the bubble and the surface and that θe is strongly dependent on the bubble size. The experimental data measured with bubbles with diameters up to 2 mm are shown in Figure 5. Also included in the figure is the Young contact angle, θ0, at 1/Rb ) 0. A simple regression analysis yielded the following relationship:
θe - θ0 ) 4.283
( ) 1 Rb
3
(16)
where θ0 is taken as the measured value of 26° and the bubble radius is in millimeters. Equation 16 is plotted in Figure 5, where it can be seen that a reasonably good fit is obtained over the range of the experimental data. However, the relationship obviously does not apply in the limit as Rb f 0 because it predicts 1/Rb f ∞ and, hence, θe - θ0 f ∞, which is physically inconsistent.
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Figure 6. Experimental data (points) and hydrodynamic model prediction (lines) for TPC radius versus time.
Figure 7. Experimental data (points) and molecular-kinetic model prediction (lines) for TPC radius versus time.
This paper uses different terms for the static contact angles, including the Young (thermodynamic) contact angle and the equilibrium contact angle. Ideally, these two contact angles are identical. The Young contact angle is thermodynamically defined by the Young eq 9, indicating that the contact angle would be a unique property of the solid-liquid-air system under consideration. However, in practice a range of static contact angles are usually possible, and this effect is caused by a number of factors such as the roughness and heterogeneity of the surface and the size of bubbles and droplets. Since the size effect is not included in the Young eq 9, the Young contact angle is the angle between a planar liquid-solid interface and an initially planar liquid-gas interface as measured in this paper. The equilibrium contact angle is a general term in this regard and may be influenced by the size effect as shown by eq 16 and in Figure 5. Moreover, since the solid surface is not perfectly flat and homogeneous, the contact angle hysteresis is often significant. Therefore, there may exist metastable contact angles and stable ones. In this situation, the Young contact angle may not ideally be the contact angle measured by the Wilhelmy plate technique (or any other technique). The stable contact angle is the angle given by the Wenzel and/or Cassie equations, depending on the source of hysteresis. Because no hydrophobizing reagents and other chemicals are used in our experiments, the patchy surface structure and heterogeneity effect should be minimal. However, the microscopic roughness of the microscope slide cannot be eliminated. Thus, both the (“Young”) contact angle measured with the Wilhelmy technique and equilibrium contact angle measured from the stationary bubbles represent the receding contact angles, which might correspond to the dewetting process. The strong dependence of the measured equilibrium contact angle on the bubble size, which is empirically described by eq 16, may be due to the “slip” and “stick” behavior of the contact lines amplified by the condition of the constant bubble volume during the contact line motion as discussed by Marmur.19 Motion of the Three-Phase Contact Line. The threephase contact line radial position as a function of time was determined experimentally by direct observation. The data were compared with predictions based on both (a) hydrodynamic and (b) molecular models and are presented in Figure 6 and Figure 7, respectively. In Figure 6, it can be seen that the TPC radius increased rapidly immediately following the bubble rupture, and after approximately 0.01
s the rate of increase in the TPC radius decreased significantly. After approximately 0.02 s, the TPC radius reached a constant (equilibrium value). The solid line shown in the graph describes eq 15 of the hydrodynamic model prediction with ln(R/L) ) 14. Clearly from the graph, the model fits only the first five experimental data. The difference between the model and the experimental data at long time of the TPC motion is caused by the significant difference between the Young contact angle and the equilibrium macroscopic contact angle between the bubble and the surface, as shown in Figure 5 and by eq 16. Least-squares regression analysis has been used to obtain a best-fit value of the hydrodynamic model by changing the model parameters. The regression analysis was carried out using Solver in Microsoft Excel. The best fit gives θ0 ) 36°, which is close to the experimental data of 35.93° for the equilibrium contact angle between the bubble and the particle, and ln(R/L) ) 40.33. The result of the regression analysis is shown by the dotted line in Figure 6. Clearly from the graph, there is significant improvement in the model agreement when the best-fit parameters are used. However, the best-fit value of ln(R/L) ) 40.33 gives R ) Le34.33 ) 1 × 1011.52 mm, where the typical value L ) 1 nm is considered. This magnitude of the macroscopic length scale R, which has to be the bubble size, is absolutely inconsistent with the physics involved in the hydrodynamic theory. In the dewetting experiments, in particular in the presence of hydrophibizing reagents with patchy adsorption, dendritic shapes of the dewetting area are often observed.12 The hydrodynamic description of this possible caterpillar-like motion at the contact line is deficient at present. However, for the results reported in this paper, the dendritic shapes of transient dewetting contact lines were not observed. Indeed, circular shapes were obtained (see Figure 4). Therefore, it can be concluded at this point that the disagreement between the experimental data and the hydrodynamic model is not due to the deviation from the circular shape of transient dewetting contact lines. In Figure 7, the experimental TPC radius versus time data (same as Figure 6) are compared with the molecularkinetic model predictions, described by eq 10, using the experimental value of θ0 and the typical values λ ) 1 nm and K ) 3 × 106 s-1. A similar outcome to that for the hydrodynamic model predictions was obtained, in that the model fits only the first few experimental data. The same least-squares regression analysis was used to best fit the model. The regression analysis gives the best-fitted values θ0 ) 36, λ ) 0.5 nm, and K ) 1.376 × 108 s-1. The best-fitted value for K gives a relaxation time of water
(19) Marmur, A. Colloids Surf., A 1998, 136, 209.
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molecules at the dewetting TPC line on the hydrophobic surface close to that of water molecules in the bulk water, which is difficult to accept since at the hydrophobic interface the bounding of water molecules is weaker than the bounding in the liquid phase and the relaxation time should be higher. General Discussion. There is a significant difference between the experimental data and both the hydrodynamic and molecular-kinetic models, which cannot be improved by the best fitting. The cause of this difference between the models and the experimental data is not known at present. In the previous papers on this topic,20,21 the line tension effect was considered. The obtained magnitude of line tension was about 1 µJ/m, which agrees with the experimental data determined with the sessile drop technique.22,23 This magnitude of line tension has been a source of debate over a number of years because it is higher than the expected magnitude by 2-3 orders.24-27 Clearly from eq 16 and Figure 5, the equilibrium contact angle strongly depends on the bubble radius and is significantly larger than the equilibrium (Young) contact angle between the solid surface and the unbounded gasliquid interface. Both the current hydrodynamic and (20) Stechemesser, H.; Nguyen, A. V. Colloids Surf., A 1998, 142, 257. (21) Nguyen, A. V.; Stechemesser, H.; Zobel, G.; Schulze, H. J. J. Colloid Interface Sci. 1997, 187, 547. (22) Amirfazli, A.; Kwok, D. Y.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1998, 205, 1. (23) Duncan, D.; Li, D.; Gaydos, J.; Neumann, A. W. J. Colloid Interface Sci. 1995, 169, 256. (24) Mingins, J.; Scheludko, A. J. Chem. Soc., Faraday Trans. 1 1979, 75, 1. (25) Stockelhuber, K. W.; Radoev, B.; Schulze, H. J. Colloids Surf., A 1999, 156, 323. (26) Yakubov, G. E.; Vinogradova, O. I.; Butt, H.-J. J. Adhes. Sci. Technol. 2000, 14, 1783. (27) Marmur, A. Colloids Surf., A 1998, 136, 81.
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molecular-kinetic models do not consider the influence of local curvature of the TPC line on the overall TPC motion in full. Usually in the hydrodynamic model, one radius of the local curvature is considered and the other is omitted to simplify the analysis. This is true for two-dimensional cases only but is not the case for the dewetting process between a small bubble and a flat surface considered in this paper. Clearly, further investigation is needed. Conclusions The experimental apparatus allowed the direct observation of the motion of the TPC line for a rising bubble being contacted by a submerged horizontal glass plate. Following rupture of the bubble, the TPC line expanded rapidly in a circular profile before decreasing in velocity and eventually reaching an equilibrium position. Both hydrodynamic and molecular models describe well the motion of the TPC line only in the short time period at the beginning. A significant difference between the models and the experimental data was observed at longer times of the TPC motion. The equilibrium contact angle was found to be a function of the bubble radius and was significantly different from the Young contact angle determined by the Wilhelmy plate method. This radius dependence of contact angles and other curvature-dependent effects, which are not considered in the hydrodynamic and molecular-kinetic models, may cause the deviations of the model predictions from the experimental data. Acknowledgment. The authors acknowledge the Australia Research Council for financial support and Professors Graeme J. Jameson, Emil Manev, and John Ralston for useful comments and advice. LA034038B
