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Wetting Films in Thermal Gradients† M. Schneemilch* and A. M.Cazabat Colle` ge de France, Physique de la Matie` re Condense´ e, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France Received February 28, 2000. In Final Form: May 2, 2000 Previous experiments revealed some unusual behavior of wetting films driven across inclined surfaces by thermal gradients. Below a certain threshold film thickness, the film, predictably, formed a bump that was stationary but subject to a fingering instability. Above the threshold the bump was, unexpectably, stable but nonstationary. This new structure was identified by Bertozzi, Mu¨nch, and Shearer as that of an undercompressive shock. In this study we investigated the extent to which their proposed model, dealing with infinite films spreading across infinite, ideal substrates, can be applied to real systems of finite dimensions. It was found that while the velocity and width of the bump were not subject to finite size effects, the profile of the bump was not accurately predicted by the model.
Introduction The behavior of wetting films driven by gravity or surface tension gradients has long been investigated.1-14 In many cases, the film had a well-extended flat part. A bump in the thickness profile developed at the liquidsolid contact line, ultimately giving birth to a fingering instability. In some cases, however, the scenario was different, as when films were driven by thermally induced gradients in a capillary rise geometry, i.e., climbing on an inclined flat solid surface. While thin films behaved the usual way,10-12 thick films did not show any bump, and the contact line was stable.9 The intermediate situation was expected to be a progressive fading out of the bump and of the associated instability.11 However, experiments revealed a more complex behavior.14 One way to thicken the films was to increase the tilt angle of the solid substrate from the vertical position. The unexpected result of that procedure was an increase of the bumpswhich can be understood in hindsight because tilting the substrate reduces the gravity component along the filmsand the fading out of the instability, which was more surprising. Moreover, the overall behavior of the film was qualitatively different: after a short transient state, thin films climbed at a constant rate and the bump profile did not change significantly with time before destabilization occurred. Therefore, there was a time interval where they could be treated as stationary. Thicker † Part of the Special Issue “Colloid Science Matured, Four Colloid Scientists Turn 60 at the Millennium”.
(1) Huppert, H. E. Nature 1982, 300, 427. (2) Hocking, L. M. J. Fluid Mech. 1990, 211, 373. (3) Silvi, N.; Dussan, V. E. Phys. Fluids 1985, 28, 5. (4) Troian, S. M.; Herbolzheimer, E.; Safran, S. A.; Joanny, J. F. Europhys. Lett. 1989, 10, 25. (5) de Bruyn, J. R. Phys. Rev. A 1992, 46, R4500. (6) Brenner, M. P. Phys. Rev. E 1993, 47, 4597. (7) Bertozzi, A. L.; Brenner, M. P. Phys. Fluids 1997, 9, 530. (8) de Bruyn, J. R. Phys. Fluids 1997, 9, 1599. (9) Ludviksson, V.; Lightfoot, E. N. AIChE J. 1971, 17, 1166. (10) Cazabat, A. M.; Heslot, F.; Troian, S. M.; Carles, P. Nature 1990, 346, 824. (11) Cazabat, A. M.; Heslot, F.; Carles, P.; Troian, S. M. Adv. Colloid Interface Sci. 1992, 39, 61. (12) Carles, P.; Cazabat, A. M. J. Colloid Interface Sci. 1993, 157, 196. (13) Fanton, X.; Cazabat, A. M.; Que´re´, D. Langmuir 1996, 12, 5875. (14) Bertozzi, A. L.; Mu¨nch, A.; Fanton, X.; Cazabat, A. M. Phys. Rev. Lett. 1998, 81, 5169.
films also climbed at a constant rate after the transient state, but the bump grew continuously with time. There was no trend to build up a well-defined profile, even for a limited time interval, and the process was clearly nonstationary. This new hydrodynamical regime was nicely interpreted by Bertozzi, Shearer, and co-workers.14-16 In the range of parameters involved, the solution of the Navier Stokes equation was no longer stationary, but became an undercompressive shock wave. Therefore, our experimental observations, which could have been considered as a mere academic curiousity with no obvious practical consequences, were in fact connected to a fundamental problem in the mathematical community. See ref 17 for a general review article about liquid films for a mathematical audience. However, the mathematical analysis dealt with a model system: infinite, perfect substrate, fluid reservoir at infinite distance from the contact line, gradient constant everywhere ... conditions which were not met in the experiment. Although the interpretation of data in terms of undercompressive shock14 was clearly unquestionable, it was useful to check the incidence of finite size effects in the experiment on the quantitative comparison between the predictions of the model system and the measured parameters. This was the aim of the present study. The Experimental Setup The structure of the experimental setup was described elsewhere10-14 and was kept unchanged; see Figure 1a. However, a new setup was constructed, with increased mechanical and thermal stability and a larger liquid reservoir, allowing the use of 5 cm wide substrates, whereas previously the maximum width was 2.5 cm. The distance between holders, which was the upper limit of the length of the film, was between 0.8 and 1 cm. The temperature of the upper holder, the surface of which was exposed to the atmosphere, was chosen to be slightly lower than ambient. The lower holder, i.e., the liquid reservoir, was almost closed; therefore the thermal coupling with the surrounding was weak. The temperature here was 55 °C. Reproducibility studies were done systematically in order to optimize the choice of parameters. The values above resulted from these studies. The liquids were nonvolatile silicone oils, poly(dimethylsiloxane)s (PDMS). The less viscous oil had a viscosity η ) 0.02 (15) Mu¨nch, A.; Bertozzi, A. L. Phys. Fluids 1999, 11, 2812. (16) Bertozzi, A. L.; Mu¨nch, A.; Shearer, M. Physica D 1999, 134, 431. (17) Bertozzi, A. L. Notices of the American Math. Soc. 1998, 45, 689.
10.1021/la0002785 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/27/2000
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a
Langmuir, Vol. 16, No. 23, 2000 8797 with the vertical varied between 0 and 70°. In that range, the component of gravity normal to the substrate could be neglected.14-16
Theoretical Analyses of Thermally Driven Climbing Films
b
Let us now recall the main theoretical predictions and experimental observations for films driven by the thermal Marangoni effect, where a constant gradient τ in the surface tension γ was induced by a constant thermal gradient along the film. The analysis was done separately for the lower part of the film, i.e., the crossover between reservoir and flat film, and the upper part of the film, i.e., the crossover between flat film and bump, terminated by the contact line at the solid. The situation is illustrated schematically in Figure 1b. For the lower part, the problem was to connect a meniscus characterized by a macroscopic curvature C to an infinite flat film in the presence of a constant surface tension gradient τ. The procedure, reminiscent of the Landau Levich treatment of a film entrained on a plate pulled out a liquid bath,18 involved the calculation of the film thickness hf far from the meniscus. It was assumed that a stationary state was reached, with a translational invariance in the horizontal direction. The driving term was the surface tension gradient τ. Antagonist effects were due to the gravitational acceleration g and to the negative Laplace pressure in the meniscus. For thick films, where gravity was dominant, the corresponding thickness was9
c e0 )
τ Fg cos R
(1)
Laplace pressure was dominant for thin films and the corresponding thickness was12,13
e ≈ 11
τ2 γ2C3
(2)
The thickness hf was obtained in the general case as12,19 Figure 1. (a) The experimental apparatus. The silicon wafer (1) is attached by suction to the upper holder (2) which is maintained at a temperature slightly lower than ambient. The lower holder (3) is adjacent to the bulk liquid and is at a higher temperature. The temperature of each holder is maintained by a constant flow of water at the appropriate temperature. Once the silicon wafer has equilibrated, the bulk liquid level is raised by adjusting the vertical position of the reservoir (4) until the static meniscus extends into the region of thermal gradient. The resultant surface tension gradient, τ, drives the film toward the upper holder. The film is imaged with a microscope and camera. The entire apparatus can be tilted through an angle R to increase the film thickness. (b) A schematic illustration of the relevant mathematical parameters. The curvature of the interface at the junction of the bulk liquid reservoir and the flat film is denoted by C. hf is the thickness of the flat part of the spreading film and b is the thickness of the film on the prewetted substrate. (c) A schematic illustration of the double shock structure predicted to occur in thick spreading films. The leading shock wave is the undercompressive shock and the trailing shock wave is a Lax shock. P, a density F ) 950 kg‚m-3, and a surface tension γ ) 20.6 10-3 N‚m-1. For the more viscous oil, η ) 0.1 P, F ) 966 kg‚m-3, and γ ) 20.9 × 10-3 N‚m-1. They wetted completely the substrates, which were oxidized silicon wafers. The surface tension gradients τ were in the range 0.15-0.20 Pa. The tilt angle R of the substrate
[
h f ) e0 1 +
(
)]
e0 e0 e02 + 2 2e1 e1 4e1
1/2
(3)
This equation was successfully checked for films in a wide range of thicknesses,12,13 even if the upper part of the film was nonstationary.20 This meant that the lower and upper parts of the film did not interfere too much, as long as the film really had a flat part. For the present study, we shall see that the flat film exists only for R less than typically 70°. For the upper part, the problem was to describe the vicinity of the contact line at the tip of a semi-infinite film of thickness hf driven on the substrate by a constant surface tension gradient. The film thickness far from the contact line was given, and the unknown was the mean local velocity U. To remove the divergence of viscous dissipation at the contact line, without introducing explicitly molecular effects in the hydrodynamic treatment, a flat preexisting film of thickness b was assumed to be present ahead of the contact line. The problem, then, was to connect two infinite films of thicknesses hf and b by a front (18) Landau, L. D.; Levich, V. G. Acta Physicochim. USSR 1942, 17, 42. (19) White, D. A.; Tallmage, J. A. Chem. Eng. Sci. 1965, 20, 33. (20) Fanton, Xavier. Ph.D. Thesis, University Paris VI, 1998.
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advancing at velocity U. Initially, the contact line was assumed to be straight and the velocity U was calculated. Second, the stability of that solution was checked. The evolution of the thickness profiles obeys the continuity equation
∂ ∂h + [hU] ) 0 ∂t ∂x
(4)
where x is the axis along the flow, assumed to be onedimensional, and h(x,t) is the local film thickness. The average velocity U(h) was obtained from the Navier Stokes equation
U(h) )
γ 2 ∂3h hτ Fgh2 cos R + h 2η 3η 3η ∂x3
(5)
The behavior of the front was assumed to be controlled by the relative strength of capillarity and gravity. Associated dimensionless variables were introduced for a more general discussion.14-16 Thicknesses were divided by a characteristic thickness H, times by time T, and x values by a length L, with
H) T)
3 3τ ) e 2Fg cos R 2 0
2η 4 γτFg cos R τ2 9
L)
[
[
]
1/3
In the following, hf* and b* refer to the reduced flat film thicknesses and U* the reduced velocity. Note that in the flat part of the film
Uf )
hfτ Fghf2 cos R 2η 3η
Uf* ) hf* - (hf*)2
(8)
Ca ) ηUf/γ
(9)
The constant K was numerically calculated by Troian and co-workers.21,22 The accepted value, based on calculations in which gravity was ignored, is now around 18. Films above the threshold were predicted to feature two shock waves, an undercompressive shock and a Lax shock, as depicted in Figure 1c. The former, leading shock wave was predicted to be stable.14-16 The velocity of the contact line was predicted to be less than the velocity of the flat film, because the liquid accumulated in the bump. Therefore, the process was nonstationary. The bump was predicted to grow in width but not in thickness. Intuitively, one would have expected some Rayleigh-like instability, because a bump was present. In hindsight, it can be seen that since increasing the thickness decreases the velocity of the film, contact line and bump instabilities are antagonist, and the bump is stabilized. We shall report now about the behavior of these films, mainly thick ones but also thin ones, in order to complete previous experiments10-13 and see how the real systems, where the width and length of the film are finite, compare with the predictions of the model. Experimental Results and Discussion
1/3
]
3γτ 2(Fg cos R)2
λ ) Khf[3Ca]-1/3
(6) (7)
For small hf*, i.e., thin films, a stationary solution was found. For hf* larger than a threshold value, this was no longer the case, as first pointed out by Bertozzi.14-16 The threshold value depends somewhat on b*, varying from 0.2 to 0.15 for b* in the range 10-3 to 10-4. For the moment, the problem has been treated as onedimensional, with x as the space variable, assuming translational invariance in the y direction, parallel to the straight contact line. The next step was a linear stability analysis for perturbations in the y direction. Films below the threshold were predicted to be unstable.14-16,21,22 A thickness modulation in the y direction developed in the bump, reminiscent of the Rayleigh instability of a cylinder, and ultimately gave birth to a fingering instability at the contact line. The linear analysis could predict the wavelength of that instability but could give no information on the further growth of the fingers or their shape. The wavelength of the fingering instability was found to scale with the thickness hf and to depend on the capillary number Ca of the flow (21) Kataoka, D. E.; Troian, S. M. J. Colloid Interface Sci. 1997, 192, 350. (22) Kataoka, D. E.; Troian, S. M. J. Colloid Interface Sci. 1998, 203, 335.
Let us first summarize briefly the expected behavior: For thin films, i.e., hf* < 0.15, a stationary state was predicted. In other words, the film profile did not change with time; the height and the width of the bump should remain constant and the velocity of the contact line, Uf, could be calculated from the known value of hf*. This stationary state should not be stable: a thickness modulation of the bump parallel to the contact line should develop and give birth to a fingering instability, the wavelength of which could be predicted. Above the threshold, the velocity of the contact line was expected to be less than Uf. The bump was predicted to grow in width but not in height14-16 as it accumulates liquid. The leading edge of the bump was predicted to be stable. Let us now present the experimental results: The measured values of the dimensionless contact line velocity U* are plotted in Figure 2 versus the dimensionless flat film thickness, hf*, for both liquids. The continuous line was the calculated Uf* value. Small thicknesses (less than the threshold of 0.15) were measured from interference fringes in the case of the 20 cP liquid. Above the threshold and for all 100 cP data the thickness was calculated from the measured temperature gradient. The overall behavior was as expected. Below the threshold, we observed a constant growth rate for the film, and the contact line velocity U* (the dimensionless velocity of the tips of the fingers) agreed well with the stationary value U*. Therefore, there was no accumulation of the liquid in the bump and the obvious assumption for calculation was that of a stationary profile. A look at the published profiles11 showed that it was not strictly true. It was, however, an acceptable approximation, but for a limited time only, because the instability quickly developed. Above the threshold, the contact line velocity was constant, but less than the calculated flat film velocity. This agreed with the predicted behavior. No instability developed; i.e., the bump was stable but nonstationary. The scattering of the U* data was fairly large above the threshold, and especially for the low-viscosity oil. It may
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Figure 2. The dimensionless velocity of the contact line versus the dimensionless flat film thickness. The line corresponds to the predictions of the stationary model. The threshold between stationary and nonstationary behavior is at a dimensionless thickness of around 0.15. For the 20 cP liquid, the film thickness below the threshold was measured from interference fringes. In all other cases the thickness was calculated from the measured temperature gradient.
be expected that a nonstationary state was less reproducible than a stationary one in experiments. Moreover, the length where the gradient was constant was only 10 mm, and the actual velocities increased with the film thickness (although not the dimensionless velocities). As a significant portion of this length was consumed during the formation of the film and the bump from the meniscus, the time interval where a comparison with the model was meaningful was relatively short. Therefore, some scattering of data above the threshold was understandable. However, the scattering may also be linked to finite size effects. At greater film thicknesses the width of the substrate was more critical. The films were fully developed in the central region of the substrate but the speed and volume of the bump decreased toward the edges. With a decrease of the width of the sample (by drawing nonwetting ink lines on the substrate) the bump could be made to extend around the edges, joining with the bulk liquid. When this occurred, the speed of the film was observed to increase. Below the threshold, where the wavelength was small compared to the width of the substrate, the dependence of the contact line speed on the substrate width was not manifest. The question now was how the instability faded out when we approached the threshold. Did the relative amplitude of the modulation decrease (contact line position as well as bump thickness) or did the wavelength diverge? The latter case would be critical on substrates of finite width. The wavelength is plotted versus X ) hf[3Ca]-1/3 in Figure 3a and the ratio between wavelength and X versus X in Figure 3b. With increasing X, the wavelength clearly tended to be larger than the theoretical value, calculated using the accepted value of 18 for the constant term, K. Since the calculations from which this value was derived ignored the component of gravity, the better agreement for thin films was not surprising. As could be seen in Figure 3b, a divergence of the wavelength very close to the threshold was not probable but could not be completely discarded from the experimental data. However, we did
Figure 3. (a) The measured wavelength of the fingering instability versus the nonconstant term, X, in eq 8, X ) hf[3Ca]-1/3. The solid line has a slope equal to 18, the accepted value of the constant term, K. Note that the wavelength was only manifest for unstable cases below or on the threshold and was highly variable in the latter cases where average values were recorded. (b) The ratio of the measured wavelength of the fingering instability and X versus X. The horizontal line has an abscissa value equal to 18, the accepted value of the constant term, K.
not believe that the wavelength changed significantly, but rather that it was no longer defined. When we approached the threshold hf* ∼ 0.15, the amplification leading to the wavelength selection failed, and measurements became questionable. The largest wavelengths measured were around 4 mm, while the width of the substrate was 50 mm. Therefore, we did not believe that finite size effects played a role in the fading out of the instability. Let us consider the relative modulation of the position of the contact line. Such a study was done previously for very thin films, the thickness of which was around 1000
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Figure 4. The ratio of the growth rate of the fingers and the velocity of the contact line versus the dimensionless flat film thickness. Error bars correspond to the standard deviation of measurements made at the same film thickness.
Å or less.23 These very thin films were stable. We observed that the transition from stable to unstable films was quite sharp and that contact line and bump behaved the same. We used here the same parameter to characterize the instability below the threshold. It was the ratio between the growth rate of the fingers and the velocity of the contact line and is plotted versus hf* in Figure 4. It vanished at very thin film thickness, as already stated, and also at the threshold between stationary and nonstationary flow. Again the transition was rather sharp, with a large scattering of the data, as expected. A few more words about the very thin films, around hf* ∼ 0.01. Although it was clear that the hydrodynamical analysis failed, the qualitative predictions of the model were interesting.16 In that case, the preexisting film of thickness b had a physical meaning: this was the precursor film spreading ahead of the main film in complete wetting.24,25 The model16 predicted that the bump disappeared when the thickness b became close to h∞. This was exactly what was observed. Finally, let us investigate the behavior of the bump above the threshold h∞ > 0.15. We first measured the width of the bump and the length of the film as a function of time. The ratios of the growth rates of bump width and film length are plotted versus hf* on Figure 5. The result was quite satisfactory, except that it pointed out the limit of the experiment: for hf* ∼ 0.3, the bump and the film grow at the same rate, the flat film was very short, and the meniscus and contact line were no longer decoupled. Therefore, investigations of larger hf* will require a different experimental procedure. This was the case in the study of Ludviksson and Lightfoot,9 where the film was growing at the bottom of a draining film. Let us now investigate the height of the bump. This was in fact the weak point of the comparison between the theoretical model, where the substrate was infinite, with experiment, where edge effects were unavoidable. (23) Carles, P.; Kolb, E.; Cazabat, A. M. Colloid Surf. A 1993, 79, 65. (24) Hardy, H. W. Philos. Mag. 1919, 38, 49. (25) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.
Schneemilch et al.
Figure 5. The ratio of the growth rate of the bump width (in the direction of flow) and the velocity of the contact line versus the dimensionless flat film thickness. The values shown are for the 20 cP liquid at inclination angles ranging from 50° to 65°.
Figure 6. The ratio between the height of the bump and the flat film thickness versus time. The height of the bump was measured by counting the formation of interference fringes at the top of the bump after the flat film has formed and so does not include the flat film thickness. The time t ) 0 corresponds to the point when deformation of the static equilibrium meniscus first becomes noticeable. Only values for the 20 cP liquid are shown.
The ratio between bump thickness and flat film thickness for the more viscous oil is plotted versus time in Figure 6 (it would look the same versus reduced time). The first growth, until 400 s, was the formation of the bump from the flat film. After that transient state the bump was expected to widen but not to thicken. However, the thickness continued to increase, for all values of hf*, until the film reached the end of the region of the thermal gradient. Since the time taken for the film to traverse this region was relatively short (between 800 and 1200 s), we cannot discount the possibility that the behavior of the
Wetting Films in Thermal Gradients
film thickness in the latter part of the experiment was disturbed by the proximity to the end of the thermal gradient. To clarify the issue, the experiment must be conducted over a greater length. This then requires an increase in the temperature difference in order to maintain the same thermal gradient. Many problems then arise, both for the experimental setup and for the theoretical model: in the formulas, the surface tension gradient was introduced only as a boundary condition at the free surface. With increasing temperature difference, surface tension and viscosity become explicitly local variables. These problems are currently being addressed. We conclude that for the bump height, the experimental observations did not agree with the predicted behavior, but we cannot state whether the discrepancy was due to a shortfall of the model or to the limitations of a finite substrate. Subtle changes in bump thickness were evident below the threshold11 but were obscured by the onset of the fingering instability. Above the threshold they were obvious. However, we do not really know how the bump thickens. One might assume that lateral flow contributed to the growth of the bump thickness and might account for some of the variability in velocity. Anyway, the important point was that the qualitative behavior did not change. In particular, a nonstationary flow and a stable bump were always observed above the threshold. The only significant consequence of the experimental difficulties was that the measured value of the velocity was not fully reproducible, which prevented us from quantifying the influence of the thickness b* of the preexisting film. b* was small, as expected, but that was all we could conclude. We were at least able to demonstrate that the changes in bump thickness were controlled by the flat film thickness. This is shown in Figure 7, where the growth rate of bump volume was plotted versus the lack of flow with respect to the stationary state, i.e., hf[Uf - U]. The precise shape of the bump was not known; therefore the volume was estimated by assuming that the three points corresponding to the front, the back, and the top of the bump lay on the perimeter of a circle. The conclusion was obviously that the thickness of the flat film governed the flow and our use of the stationary model to calculate film thickness in the nonstationary case was justified. Even if a one-dimensional analysis was an approximation, it actually picked up the real physics of the problem. Conclusion The present study defined the limits of applicability of the model proposed by Bertozzi et al.,14-16 which dealt with infinite films and substrates, to a real experiment, with obvious size limitations. The agreement was excellent for the overall behavior of the climbing film: stationary
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Figure 7. The rate of change in the volume of the bump per unit length of contact line versus the product of the flat film thickness and the difference between the calculated and measured contact line velocities. The velocity is calculated according to the stationary model. Only values for the 20 cP liquid are shown.
behavior below the threshold with an unstable bump, and nonstationary behavior above the threshold, with a stable bump. It was fairly good for the quantitative predictions of velocity and bump length. However, the bump height was clearly more sensitive to finite size effects and we were unable to verify quantitatively the predicted evolution of the bump profile. In this investigation, as in our preliminary data (ref 14, Figure 3), the extent of the region of thermal gradient, i.e., the distance between the respective holders, was apparently insufficient for the separation of the two waves in the double shock structure to become readily apparent. Numerical simulations (ref 14, Figure 5) were used to extrapolate beyond the accessible range of the experiment and formed the basis for the prediction of undercompresssive shock formation. In investigations already underway, we have endeavored to increase the range of the experiment and so observe directly the formation of the double shock structure. Acknowledgment. The authors wish to express their gratitude for the invaluable assistance provided by Andrea Bertozzi in the interpretation of these data. LA0002785
