Langmuir 1995,11, 4117-4121
4117
Tears of Wine: The Stationary State R. Vuilleumier, V. Ego, L. Neltner, and A. M. Cazabat* Colkge de France, Physique de la MatiBre Condenshe, 11, place Marcelin Berthelot, 75231 Paris Cedex 05, France Received March 14, 1995. In Final Form: May 24, 1995@ Water-ethanol mixtures spontaneously produce strong convective effects, driving the liquid upward in thin films and downward in regular rows of drops known as “tears of wine”. We present here a study of this stationary state, with a specific investigation of the interfacial instability which develops at the crossover between the reservoir and the thin film. We interpret it as a Rayleigh-like instability, the wavelength of which is mainly controlled by the extension of this crossover zone.
Introduction The dynamics and stability of spreading films driven by surface-tensiongradients have recently been the subject of a number of studies. Extensive analyses are available in situations where the gradient z is externally controlled,172for example, if z is induced by a temperature gradient imposed along the film ~ u b s t r a t e . l - Less ~ has been done with “self-adjusting”gradients, like those that develop in mixtures of liquids differing by both surface tension and ~ o l a t i l i t y . ~ - ~ A well-known example of this latter case is the celebrated “tearsof wine”.8 As water has a higher surface tension than alcohol, a concentration gradient of alcohol creates a gradient of surface tension. The phenomena are conveniently studied with a water-ethanol mixture in watch glasses5y7or along tilted plates. The concentration gradient builds up in the thin part of the liquid, i.e., a t the meniscus, which is depleted in alcohol, more volatile than water. The resulting surface driving force raises a thin film along the tilted glass surface. At short times, the climbingfilm presents a self-similar profile of parabolic shape and obeys a diffisive-likewetting law. The fluid velocity and the surface tension gradient along the film have been studied. In addition, the existence of a star instability a t the meniscus and a fingering instability at the front of the film has been rep~rted.~ At longer times, a stationary situation is obtained. The film stops at a certain height, which for an extended surface and a completely wetting liquid would be determined by the balance between surface tension gradient and gravity. In practice, the edge of the substrate and its wettability are the relevant parameters. The upwardly directed flow creates drops which when large enough roll downward under gravity. The thickness of the thin film seems roughly constant along its length, and the wavelength of the star instability does not change any more (Figure 1). The present paper deals with this stationary situation. As in the previous case,7we do not provide an extensive, quantitative analysis of the problem, because too many ca Abstract published inAdvance ACSAbstracts, August 15,1995. (1) Ludviksson, V.; Lightfoot, E. N. MChE J. 1971,17, 1166. (2) Carles, P.;Cazabat, A. M. J.Colloid Interface Sci. 1993,157,196.
(3) Cazabat, A. M.; Heslot, F.; Troian, S. M.; Carles, P. Nature, 1990, 346, 024. (4) Brzoska, J. B.; Brochard-Wyart F.; Rondelez, F. Europhys. Lett. 1992, 19, 97. (5)Thomson, J . Philos. Mag. 1855,lO (ser. 41,330;Loewenthal, M. Philos. Mag. 1931, 12 (ser. 71, 462. (6) Neogi, P. J. Colloid Interface Sci. 1985, 105, 94. (7) Fournier, J. B.; Cazabat, A. M. Europhys. Lett. 1992,20, 517. (8)Proverbs, XXIII:31.
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Figure 1. Projected overview of the watch glass at long time for a 20% ethanol-rich mixture. Note the darker zone at the
meniscus between reservoir and thin film, indicating a dip in the profile, and the forked dendrites of the star instability. parameters interplay. Our aim is to extract the more significant ones and to propose a coherent draft of the problem.
Experimental Observations The main riddle left after the previous study was the star in~tability,~ which we try to understand now. A preliminary investigation of flows in the meniscus (talcum powder is dispersed in the liquid and the particle motions are monitored with a binocular) did not show any circulation rolls, excluding the case of a Bhard-Ma-
0743-7463/95/2411-4117$09.00/0 1995 American Chemical Society
Vuilleumier et al.
4118 Langmuir, Vol. 11, No. 10, 1995
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“Tear” drop
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Figure 2. Schematic picture of the system. The tilted plate is the flat bottom of a borosilicated Petri dish, diameter 10 cm. The insert shows the shape of the dendrites in this case, which we measure as the “wavelength”.
rangoni instability. Therefore, we assumed that the star instability is a Rayleigh-likeinstability, like those observed in driven films when the free surface profile is nonm o n ~ t o n o u s . ~ - ~As, ~a- matter ~~ of fact, projected overviews of the watch glass show a darker zone a t the meniscus, Le., a t the transition between reservoir and flat film, where the concentration gradient builds up. This means that the liquid surface is depressed there (Figure 1). To support this interpretation, we performed further experiments in a planar geometry where the film climbs from a reservoir on a tilted plate (the bottom surface of a tilted Petri dish). The slope of the plate determines the length (L) of the transition zone. In the case of a Rayleighlike instability, the wavelength (A) is proportional to Lm3J1 Various alcohol concentrations (C) were investigated. A schematic view of the system is given in Figure 2, where the relevant parameters are introduced: a is the slope of the plate, e the thickness of the flat part of the film, x the position along the film with the origin a t the meniscus, and L the length of the transition zone. L is inversely proportional to the curvature (r)ofthe meniscus. Here, r = K sin a x Ka a t low a values, where K is the inverse capillary length K = @
(Here, g is the gravitational acceleration, 8 the density, and y the surface tension.) With this planar geometry, the instability is no longer star-like,but formed of parallel, forked dendrites, as shown in the insert. The “wavelength” I is measured on the reservoir side. The stationary state can be seen as the long time behavior of either a film climbing on a initially dry plate or a film draining from a n initially wetted plate. We have verified that both procedures led to the same final state. The second procedure has been preferred because it is faster and less sensitive to the surface cleaning.’ Note that the draining film displays specific instabilities a t short times in the meniscus. They fade out with time and will not be considered here. The setup used in a previous study is convenient for the measurement of the instability wavelength the liquid surface is illuminated with a n optical fiber and we observe the projection image on a white screen 3 cm under the glass. Typical 1vs l / a plots are given in Figure 3a,b for ~~~~
~~
(9) Huppert, H. E . Nature 1982,300, 427.
(101Silvi, N.; Dussan, V., E. B. Phys. Fluids 1985,28, 5. (11)Troian, S. M.; Herbolzheimer, E.; Safran, S. A,; Joanny, J. F. Europhys. Lett. 1989,10, 25. (12)Goodwin, R.; Homsy, G. M.Phys. FZuids,A 1991,3,515.
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Figure 3. Measured wavelength versus the inverse of the tilt angle (a)for two alcohol concentrations: (a) C = 0.5, (b) C = 0.9. Table 1. Wavelength Dependence 1 = kda C 0.3 0.5
kc (cm)
C
kc (cm)
0.088
0.7 0.9
0.074 0.067
0.071
alcohol concentrations C = 0.5 and 0.9 of the mixture. A linear fit A = kJa accounts satisfactory for the data, with k, weakly dependent on C, but in a nontrivial way (see Table 1). As L increases as l / a a t low a, we conclude that A is proportional to L in this range. In systems with an imposed gradient, the film thickness increases with the curvature radius of the meniscus (i.e., with L), at least for relatively thin films.2 The same is expected here. However, in the case of “self-adjusting” gradients, film thickness and gradient are linked because the concentration in the film in the stationary state depends on the thickness. Therefore, data on local concentration in the film are useful. To this end, minute samples of liquid were taken from the film with a syringe, and the optical index was measured using an Abbe refractometer. For a given concentration (C)of the reservoir, a first series of samples was taken a t a given distance from the meniscus, x 3 cm, for different slopes a. A second series was taken a t given a but for different x values. Typical results are given in Figures 4a,b for the first series and Figure 5a-c for the second one. The trends in Figure 4a,b are relatively clear: the larger the slope, the thinner the film, and the lower the stationary alcohol concentration.
Langmuir, Vol. 11, No. 10, 1995 4119
Tears of Wine: The Stationary State
lo 80 o]
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Figure 4. Film concentration measured 3 cm from the meniscus,versus the inverse of the tilt angle (a)for two alcohol concentrations: (a)C = 0.5, (b)C = 0.7. At low tilt angle (right part), the concentration is close to the one of the reservoir because the film is relatively thick and the alcohol evaporation does not significantly change the concentration. At large tilt angle, the film gets depleted in alcohol and ultimately dewets for C < 0.15.
The second series of measurements is more delicate: as a matter of fact, the flow in the film is significantly disturbed by the liquid uptake. In the middle of the film, Le., far from the meniscus and from the forming tears, relevant data can be obtained. But both a t small and large x values, difficulties may arise, the trend being to overestimate C(x)a t low x and underestimate it at large x . The uptake has to be done inside a hollow cylinder isolating the liquid sample. To get independent complementary information, we studied the flow pattern in the system. Fine talcum powder is added to the mixture, and the flow pattern is observed with a binocular. In the reservoir, no flow is detectable. Close to the meniscus, a flow pattern develops, mainly directed toward the film, but with a weak backflow at the bottom surface. In the meniscus and inside the film, the velocity is toward increasing x without backflow, the downward flow being entirely due to the "tears". In the following, we shall have to analyze the results in terms of Couette or mixed Poiseuille-Couette flow. An obvious question is the meaning of the measured velocity, U: is it the surface or the average velocity? In reality, the velocity is rather inhomogeneous in the film a t given x (and it is probably the same with C). Our tendency is probably to follow the motion of the fastest particles. Without further information, we have to assume
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Figure 5. Film concentration measured at given tilt angle versus the distance x to the meniscus: (a)C = 0.7, a = 10";(b) C = 0.7, a = 6";(c) C = 0.5, a = 7". The continuous line corresponds to the value of z calculated from the measured value of the velocity in the film. The dotted line is an extrapolation toward the value of the concentration in the reservoir and shows that the gradient increases at the crossover.
that the measured value U is between the average value ( u ) and the maximum value (umax) of the velocity. Because of its large field depth, the binocular does not allow measurement of film thicknesses. With a microscope, we checked that the film thickness was fairly constant along the film (say, for x between 1.5 and 4.5 cm), with typical values ranging between 20 pm (large a, large C) and 100 pm or more (low a, low C). Possibly due
Vuilleumier et al.
4120 Langmuir, Vol. 11, No. 10, 1995
to the heterogeneity of the film, we did not obtain precise thickness values with this procedure. As our system is in a stationary, but not in an equilibrium, state (open air without draft), we also measured the surface tension of the reservoir itself, because the surface is slightly depleted in alcohol. A weak negative shift of 0.02-0.03 on the C value a t the surface with respect to the equilibrium value (closed box with saturated atmosphere) was detected.
Interpretation of the Data: Thin Film The average velocity and the thickness have been found to be fairly constant along the film, which means that the gradient t is also constant. This allows us to use the relations between thickness ( e ) , gradient (t),meniscus curvature (r),and gravitational acceleration &)previously obtained in this case.1,2 The average velocity ( u ) of the film can be calculated from the Navier Stokes equation. In a stationary situation, and using the lubrication approximation, it can be written as ?g sin a h2 I yh2h”’
u = - -t h
317
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Here, h is the local thickness of the film ( h = e in the flat part), h”’ the third derivative, d3hldx3, y the surface tension, g the density, and 17 the viscosity. The problem of the selection of the thickness ( e ) in our system is very similar to the well known Landau-Levich prob1em,l3J4where a film is entrained on a plate driven out of a liquid bath at a given velocity, UO.Actually, the governing equations are the same, except that the surface tension related term, i.e. thl217,in the velocity is replaced by UO.For thin films, the thickness ( e )is controlled by the curvature (I7of the meniscus a t the bottom of the film, i.e., by the competition between the surface tension gradient and the curvature of the source zone. For thick films, gravity is the controlling parameter. A detailed analysis can be found in refs 1 and 2. The results are as follows: (1) For films controlled by curvature,2 the flow in the film is a Couette flow, with
e x 13x 1 3 y2r3 Y K a
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(2) For films controlled by gravity,l the flow in the film is a complex Poiseuille-Couette flow, with
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(3) The threshold between curvature-controlled and gravity-controlled films is found a t the value Xth of the gradient t:
Zth
1 g3:
sin a
-,y
C = 0.7, a = lo”, zth = 0.04 Pa, U = 0.6 mm/s curvature-controlled film: e = 5 2 p , t = 0 . 0 5 5 P a if U = v e = 3 3 p m , t = 0 . 0 4 4 P a if U = u m a x In both cases, the calculated gradient is higher than the threshold value, which makes the calculation inconsistent. gravity-controlled film: e = 7 4 p m , t = O . l l P a if U = v e = 4 4 p m , t = 0.065 Pa if U = u, In these cases, the calculations are self-consistent: the film is controlled by gravity. As a complementary information, the flux calculated from these values has been compared with the one measured from the volume of the drops formed. For 3 cm of contact line, the calculated flux is J x 4 p L s-l if U = u , J = 1pL s-l if U = umax. From the drops, a measured value of J = 0.8 pL s-l is obtained, in better agreement with the latter hypothesis. We conclude that the measured value is probably umm, as expected. The continuous line on the figure represents the concentration values calculated with t = 0.065 Pa and is in fairly good agreement with the measured ones. We note that the gradient increases a t the crossover to the meniscus. For Figure 5b:
C = 0.7, a = 6”,
tth
= 0.015 Pa, U = 0.75 “/s
curvature-controlled film: e = 9 8 p m , t = 0.033 Pa if U = u e = 6 2 p m , z = 0 . 0 2 6 P a if U=umax
gravity-controlled film: e = 1 1 2 p m , t=O.lOPa if U = u e = 6 5 p m , t = 0 . 0 6 P a if U=u,,, Both values are acceptable. The continuous line on the figure represents the concentration values calculated with t = 0.06 Pa, in good agreement with the measured ones. An increase in gradient is observed a t the crossover to the meniscus.
--ega
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For Figure 5a:
As in the previous case, these results are inconsistent with the hypothesis.
P
t2
The film thickness is controlled by curvature if t < Tth and by gravity if z t t h . In our case, the gradient is not known a priori: we have to consider the two possibilities and check the coherence of the results. The analyses correspond to the conditions of the Figure 5a-c. Let us present them successively.
2 3 2 K
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(13)Landau, L. D.; Levich, V. G.Acta Physicochim. URSS 1942,17, 42. (14)White, D. A,; Tallmadge, J. A. Chem. Eng. Sci. 1966,20, 33.
For Figure 5c:
C = 0.5, a = 7”,
tth= 0.02 Pa,
U = 0.7 “Is
curvature-controlled film: e = 9 3 pm, t = 0.044 Pa if U = u e = 6 0 p m , t = 0.036 Pa if U = u, As in the previous cases, these results are inconsistent with the hypothesis.
Tears of Wine: The Stationary State
Langmuir, Vol. 11, No. 10, 1995 4121
gravity-controlled film:
t
e = 9 7 , u m , z = O . l l P a if U = v e = 5 6 p m , z = 0 . 0 6 3 P a if U=v,,
Thickness
Both values are acceptable. The continuous line on the figure represents the concentration values calculated with t = 0.063 Pa, in very good agreement with the measured ones. An increase in gradient is observed a t the crossover to the meniscus. In conclusion, we obtain a coherent picture of velocity and concentration in the central part of the stationary films. As expected, the alcohol concentration in the films is lower a t low film thicknesses. The gradient in the central part of the films does not change strongly with slope or alcohol concentration. The main change occurs a t the crossover to the meniscus, where a larger gradient develops in the case of thin films, the concentration of which is significantly lower than the one of the reservoir (the weak difference between the surface and bulk concentration in the reservoir has been neglected).
Discussion of the Data: Interfacial Instability The analysis of the meniscus instability in terms of a Rayleigh-like instability is not straightforward and deserves further comment. The references given before deal with the instability of a n advancing contact line, induced by the presence of a bump in the film profile. The wavelength A increases like the transverse size L of the bump (Figure 6). In fact, although the analyses (and the instability pattern) depend on the driving force (gravity, surface tension g ~ a d i e n t , ~ , ~the , ~ WL - l ~ratio ) is always close to 4,15like in the true static Rayleigh instability of a liquid cylinder. This suggests that the wavelength of the instability depends mainly on the interface profile and not on dynamical parameters. Let us now compare the profiles in Figures 2 and 6. What influences the surface-tension-driven instabilities is the interface profile, whatever the side ofthe liquid and gas phases. We conclude that a hollow in the profile is just as unstable as a bump. Let us discuss further the origirrof this dip. What created the hollow a t the meniscus is the presence of a gradient in the surface tension gradient t a t the crossovers between the reservoir and meniscus and between meniscus and film: t i s 0 in the reservoir (no flow),has a maximum in the meniscus, and is more o r less constant in the film. A positive gradient in t is present between the reservoir and meniscus. This results in a downwardly directed component for the film velocity close to the surface, which ultimately depresses the profile. Similarly, a negative gradient in t is present between meniscus and film: a hollow in the profile results. This is specificto our problem of “self-adjusting” gradients and would not be observed with constant imposed or in the LandauLevich pr0b1em.l~ We observed the same instability in binary mixtures, where the more volatile compound has the lower surface tension. At given slope, the instability in the acetone(15)Cazabat, A. M.; Fournier, J. B.; Carles, P. In Lecture Notes in
Fluid Physics; Velarde, M. G., Christov, C. I., Eds.; World Scientific: London, 1994.
Figure 6. (bottom) Schematic profile of an instable “bump”, as can be found in refs 3 and 11;(top) thickness profile of the
crossover between meniscus and thin film, as seen in Figure 2.
water mixture (C = 0.75) has the same wavelength as that in the ethanol-water mixture at the same C. Similar values are obtained with polydimethylsiloxane-hexane mixtures. In this latter case, the difference in surface tension values is only 3 mN/m, while it is around 49 mN/m for the ethanol-water mixture. This again suggests that the slope of the plate, which controls the length L, is the leading parameter for the wavelength of the instability.
Conclusion The “tears of wine” problem has been studied in the stationary situation, where the evaporation of alcohol continuously drives a film along a tilted plate, the back flow being due to drops falling down back to the liquid reservoir. The interfacial instability of the meniscus is proposed to be a Rayleigh-like instability, the wavelength ofwhich is controlled by the slope ofthe plate. This picture is supported by the presence of a dip in the film profile, shown by optical observation, which has to be related with a specific behavior of the surface tension gradient. As a matter of fact, measurements of concentrations and velocities in the films lead to a coherent picture with a maximum of the surface tension gradient at the meniscus. This maximum explains the observed dip. We conclude that our assumption of a Rayleigh-like instability is well supported by experiments. The next step will be to investigate the transient situation, where the meniscus shape and therefore the wavelength of the instability change with time and are more sensitive to alcohol concentration. Acknowledgment. Enlightening discussions with M. Benamar, Y. Couder, J. B. Fournier, C. Gay, V. Goutal, and J. P. Hulin are gratefully acknowledged. LA950201U
