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Langmuir 1998, 14, 2554-2561
Spreading and Instabilities Induced by a Solutal Marangoni Effect X. Fanton* and A. M. Cazabat† Laboratoire des Fluides Organise´ s, CNRS-URA 792, Colle` ge de France, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France Received November 25, 1997. In Final Form: February 13, 1998 We study the spreading of liquid films driven by surface tension gradients induced by evaporation from a two-component mixture. The films climb from a macroscopic reservoir on a plane tilted surface and their length L is found to depend linearly on the square root of time t: L(t) ) (Dt)0.5. We develop a semiquantitative analysis that shows which parameters control the value of D for ideal mixtures and for nonideal ones. We report also experimental results about the time evolution and the spatiotemporal behavior of the interfacial instability that develops at the meniscus between the reservoir and the film. Results agree well with previous experimental and theoretical studies.
Introduction A surface tension gradient along a fluid-fluid interface is well-known to give rise to strong convective effects. Such gradients may occur with pure liquids when a temperature inhomogeneity exists in the system, this is the so-called thermal Marangoni effect.1,2 A solutal Marangoni effect is produced by concentration gradients of surface-active materials at the interface, due, for example, to chemical reactions,3,4 to evaporation,5 or to dilatation/compression of the surface.6 In both cases, instabilities can develop at the free interface, those instabilities have been the subject of many theoretical7-9 and experimental10-12 studies. We present an example of a solutal Marangoni effect induced by evaporation from a two-component mixture, where the two miscible liquids differ by both their volatility and their surface tension (the more volatile compound has the lower surface tension). As the mixture is put in a watch glass or in a tilted Petri dish, the liquid thickness is not uniform and preferential evaporation of one species creates a concentration gradient that drives the spreading of a film on the glass surface (Figure 1). Besides the spreading, we observe a star instability at the meniscus, at the crossover between the reservoir and the film. The white lines, denoted here as dendrites (Figure 1), are thicker than the dark parts, and optical contrast is due to the successively converging and diverging lenses created by this nonmonotonic thickness profile. One sees also an instability of the contact line, on which we will not comment in this paper. * Corresponding author.
[email protected]. †
[email protected]. (1) Schwabe, D.; Moller, U.; Schneider, J.; Scharmann, A. Phys. Fluids A 1992, 4 (11), 2368. (2) De Saedeleer, C.; Garcimartin, A.; Chavepeyer, G.; Platten, J. K. Phys. Fluids 1996, 8 (3), 670. (3) Vedove, W. D.; Sanfeld, A. J. Colloid Interface Sci. 1981, 84 (2), 318. (4) Kozhoukharova, Z. D.; Slavchev, S. G. J. Colloid Interface Sci. 1992, 148 (1), 42. (5) Wilson, S. K. Surface Coatings Int. 1997, 4, 162. (6) Sorensen, T. S. In Lecture Notes in Physics; Sorensen, T. S., Ed.; Springer-Verlag: Berlin, 1979; Vol. 105, p 1. (7) Scriven, L. E.; Sternling, C. V. J. Fluid Mech. 1964, 19, 321. (8) Smith, M. K.; Davis, S. H. J. Fluid Mech. 1983, 132, 119. (9) Mercier, J. F.; Normand, C. Phys. Fluids 1996, 8 (6), 1433. (10) Villers, D.; Platten, J. K. J. Fluid Mech. 1992, 234, 487. (11) Daviaud, F.; Vince, J. M. Phys. Rev. E 1993, 48 (6), 4432. (12) Favre, E.; Blumenfeld, L.; Daviaud, F. Phys. Fluids 1997, 9 (5), 1473.
Figure 1. Projected overview of a heptane/dodecane mixture, heptane volume fraction φHp ) 70%, in a watch glass (a), after t ) 12 s. The length L(t) of the spreading film (c) is measured between the contact line (b) and the meniscus (d), which makes the crossover between the spreading film and the reservoir (e). Two instabilities develop at the contact line (b) and at the meniscus (d). In the meniscus, the white lines (called dendrites) correspond to a thickness larger than the dark parts. This instability is characterized by the wavelength λ, the mean distance between consecutive dendrites.
This study is the continuation of previous works by Fournier et al.13 and Vuilleumier et al.14 on nonideal ethanol/water mixtures, and in order to characterize quantitatively these complex spreading phenomena, we performed new experiments with ideal15,16 alkane mixtures, namely hexane/dodecane, heptane/dodecane, and octane/dodecane. Complementary experiments were done with nonideal ethanol/water and 2-propanol/water mixtures. In the first part of this paper, we will write the relations between evaporation and spreading through concentration gradient, and we will calculate a velocity characteristic of this solutal Marangoni effect. The second part will be devoted to the experimental observations for alkane mixtures, i.e., the kinetics of spreading and the (13) Fournier, J. B.; Cazabat, A. M. Europhys. Lett. 1992, 20 (6), 517. (14) Vuilleumier, R.; Ego, V.; Neltner, L.; Cazabat, A. M. Langmuir 1995, 11 (10), 4117. (15) Caldin, E. F. An introduction to chemical thermodynamics; Oxford University Press: Oxford, U.K., 1958; Chapter 9. (16) Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The properties of gases and liquids, 3rd ed.; McGraw-Hill: New York, 1977; Chapter 8.
S0743-7463(97)01292-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/04/1998
Spreading of Liquid Films by the Marangoni Effect
Langmuir, Vol. 14, No. 9, 1998 2555
a
Figure 2. Liquid film of thickness h(z,t) flowing on a flat surface at a mean velocity v(z,t). The liquid is a mixture of I + II, with species I much more volatile than II.
time evolution of the meniscus instability. In the third part we show that despite the interplay of many parameters, we can propose a coherent description of the problem, which accounts very well for the experimental results obtained for the spreading of ideal and nonideal mixtures. In the last part, we will report some observations about the meniscus instability, including the motion of dendrites along the meniscus. Theoretical Part Let us discuss the relation between the concentration gradient and the velocity in an evaporating flowing film. Consider the one-dimensional case of a homogeneous liquid film of thickness h(z,t) on a flat surface, flowing at a mean velocity v(z,t) in the z-direction at time t (Figure 2). The film is an ideal mixture of liquids I + II, where I is a volatile component, and II is considered as nonvolatile. For ideal solutions, the volume fraction is the relevant parameter for writing matter conservation,17 and we suppose that the film composition depends only on z and t. Let φ be the volume fraction of I (eq A4 in the Appendix). Following the analysis proposed by de Ryck,18 the matter conservation in the flowing film is for I:
∂(φh) ∂(φhv) + )-φA ∂t ∂z
(1a)
∂((1 - φ)h) ∂((1 - φ)hv) + )0 ∂t ∂z
(1b)
and for II:
where the evaporation velocity A (>0) is given by eq A7. In stationary state the equations become
d(φhv) )-φA dz
(2a)
d((1 - φ)hv) )0 dz
(2b)
Combining eqs 2a and 2b gives the concentration gradient along the z-direction induced by evaporation in the flowing film
Aφ(1 - φ) dφ )dz hv
(3)
In usual liquids, volatile species have a lower surface tension than nonvolatile ones, and the concentration (17) Cussler, E. L. Diffusion - Mass transfer in fluid systems; Cambridge University Press: Cambridge, U.K., 1984; Part 1. (18) A. de Ryck, Seminar at College de France, Paris 1996; preprint 1996, to be published.
b
Figure 3. (a) Borosilicated watch glass of diameter 12.5 cm. The liquid film spreads from the reservoir in a 2-D radial geometry, so that there is no side-wall effect. The initial volume of the reservoir is equal to 1 mL, and the local angle with respect to horizontal in the meniscus region is about 9°. (b) Tilted borosilicated Petri dish, diameter 11 cm. The liquid film climbs between parallel lines of nonwettable pen drawn on the surface, so that the films spreads in a 1-D geometry. The tilt angle with respect to horizontal is denoted β.
gradient induced by evaporation creates a surface tension gradient in the opposite direction. Because of this solutal Marangoni effect, the velocity v is not externally imposed but depends on the concentration gradient. Neglecting gravity in the film gives
v)
h dσ dφ h dσ ) 2η dz 2η dφ dz
(4)
where η(φ) is the dynamic viscosity of the film and σ(φ) is the surface tension. Combining (3) and (4) gives the characteristic selfadjusted Marangoni velocity vc of the spreading in stationary state
vc2(φ) ) -
Aφ(1 - φ) dσ 2η dφ
(5)
with dσ/dφ < 0. It is worth commenting on the different origins of the spreading that appear in eq 5. The solutal Marangoni effect occurs because two components are present in the solution, because one of the component evaporates, and because pure components have different surface tension values. The shearing velocity depends on the thickness h (eq 4) while vc2 does not, because of the form of the concentration gradient dφ/dz (eq 3). In this approach, only concentration gradients are taken into account, meaning that any thermal Marangoni effect due to evaporation is ignored. Experimental Observations We studied the growth of macroscopic liquid films on tilted surfaces, either watch glass (Figure 3a) or tilted Petri dish (Figure 3b). Surfaces were put in sulfochromic acid for 24 h, rinsed with distilled water, dried in an oven at 500 °C for 1 h, and stored before use in a closed box at high humidity. Covering high-energy sites with water
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a
Figure 4. Schematic view of the film in the meniscus zone. The surface tension of I is lower than that of II, and the downward concentration gradient built by evaporation in the thinning zone creates an upward surface tension gradient. Observations14 show an upward flow near the surface and a return flow at the bottom.
molecules prevents surface contamination,19 and complete wetting with pure compounds is then obtained. We deposit 1 mL of an initially homogeneous mixture of two liquids I + II in the glass (Figure 3a or b). As the liquid thickness is inhomogeneous in the meniscus (Figure 4), the concentration of (I) is lower in the thinner regions, and this downward concentration gradient creates an upward surface tension gradient, which drives the spreading of a film up on the glass. The films are 1-10 µm thick, their thickness profiles have a parabolic shape,13 and they spread on the glass at typical velocities of 1 mm/s. All experiments were performed on a watch glass, except where otherwise stated. The patterns are visualized by shadowgraphic imaging. Surfaces (put in a box to avoid air currents) are illuminated from above by an optical fiber, and the images projected on a ground glass screen are recorded on a videotape via a CCD camera. A typical record is given in Figure 1. From this, the film growth and the instability of the meniscus are analyzed. To study the spatiotemporal evolution of the dendrites, a mathematical circle of radius R is drawn in the meniscus (Figure 5a) via imageprocessing software.20 This circle is decomposed in 512 pixels, and the gray level (0-255) of each pixel is recorded during 512 successive time steps at a chosen frequency. The recorded pixels are then used to form an image (512 × 512), on which a line represents space and a column represents time (Figure 5b). For alkane mixtures, the length L(t) of the climbing film is closely fitted by a square root of time law (Figure 6); thus the kinetics of spreading is characterized by a coefficient Dexp, which has the dimension of a diffusion coefficient
L(t) ) xDexpt
(6)
Dexp depends on the volatility and on the composition of the mixture (Figure 7). At a given concentration, Dexp increases with increasing volatility, and for a given pair, it has a nonmonotonic variation versus concentration, with a maximum value at about φ ) 60%. No spreading occurs for pure liquids (Dexp vanishes for φ ) 0 or φ ) 100%), meaning that only the solutal Marangoni effect takes place. We also studied the spreading of heptane/dodecane mixtures in a tilted Petri dish, with a tilt angle of 9°, i.e., close to the one on the watch glass (Figure 3a). The comparison between experiments on the watch glass and (19) Villette, S.; Valignat, M. P.; Cazabat, A. M.; Jullien, L.; Tiberg, F. Langmuir 1996, 12, 825. (20) The image-processing software was kindly lent by L. Limat and J. M. Flesselles.
b
Figure 5. (a) Sketch of the dendrites in the meniscus. A mathematical circle of radius R is drawn across the dendrites with a image-processing software. The gray values along this line are recorded at successive times to form an image (Figure 5b). On that image, a horizontal line corresponds to the perimeter of the circle and a vertical one represents time. (b) Spatiotemporal evolution of the meniscus instability for an octane/dodecane mixture, φOc ) 70%. The dendrites appear in white on dark field. We call R the radius of the circle drawn in the meniscus (Figure 5a). At short times (t < 50 s), the number of the dendrites (2πR/λ) decreases; then it reaches a mean constant value (2πR/λs). At long times, dendrites move either clockwise or anticlockwise, and the local slopes of the trajectories correspond to azimuthal velocities, in deg‚s-1 (or in mm‚s-1). Coalescence and creation of dendrites occur simultaneously during motion, so that the number of dendrites remains roughly constant versus time.
the Petri dish shows that spreading dynamics is independent of the geometry (open and full triangles in Figure 7). A typical evolution of the wavelength λ of the meniscus instability versus time is plotted in Figure 8 and two regimes appear, with a crossover at time tinst. The instability builds up at t < tinst, and then the wavelength keeps a constant value λs for t > tinst. In fact, a stationary state is reached between the meniscus and the surrounding atmosphere for t larger than tinst. For alkane mixtures, tinst is constant to first approximation for a given pair, i.e., independent of φ, but it depends strongly on the volatility of the lighter compound (Table 2). We check that the free diffusion lengths corresponding to these times (8-19 mm, given by xDgastinst) are of the order of the characteristic dimensions of the watch glasses (see Figure 3a). At the stationary state (t > 17 s), we measured the wavelength λs as a function of the composition for heptane/
Spreading of Liquid Films by the Marangoni Effect
Figure 6. Film length measured from the meniscus as a function of xt of three alkane mixtures. Hexane/dodecane films exhibit the fastest spreading velocities, and octane/dodecane the lowest. Such a linear square root of time dependence was also noticed by Fournier13 for ethanol/water mixtures.
Figure 7. Experimental Dexp (markers) and calculated Dc (full lines) values for three light alkane/dodecane mixtures versus volume fraction φ of the lighter alkane. Spreading was in a watch glass, except where otherwise stated. Experimental temperatures: hexane, 24.0 ( 0.5 °C; heptane, 26.0 ( 0.5 °C; octane, 22.0 ( 0.5 °C. Solid curves were calculated with eq 8, where φ is the volume fraction of the reservoir. The t0 value for each pair is given in Table 2.
Langmuir, Vol. 14, No. 9, 1998 2557
Figure 9. Stationary wavelength λs for (heptane/dodecane) mixtures versus the heptane volume fraction φ of the reservoir. The dotted line is obtained from ref 18 (eq 14).
Figure 10. Drift velocity of heptane/dodecane mixtures versus the heptane volume fraction φ of the reservoir. Experimental data (markers) are closely fitted by the calculated velocity vHp(φ) ) 0.72vHp c (φ) (solid line). Table 1. Vapor Pressures (at 25 °C) from ref 21 species
Pvap (103 Pa)
species
Pvap (103 Pa)
water ethanol 2-propanol hexane
3.17 7.95 6.01 20.1
heptane octane dodecane
6.08 1.86 0.016
Table 2. Characteristic Time tinst and Crossover Time t0 for Alkane Mixtures light species hexane heptane octane
Figure 8. Wavelength λ of the meniscus instability as a function of time for a heptane/dodecane mixture, φHp ) 70%. The crossover time tinst between the short-time and stationary state is equal to 17 s and the stationary wavelength λs is equal to 2.5 mm. Solid lines are guides for the eyes.
dodecane mixtures (Figure 9). λs decreases monotonically with φ, from 3.5 mm at φ ) 30% down to 0.9 mm at φ ) 98%, with a more rapid variation close to φ ) 100%. Finally, this instability presents a complex dynamical regime, as the ones already reported for other interfacial instabilities in the case of the thermal Marangoni effect.1,11-12 The spatiotemporal pattern presented in Figure 5b for an octane/dodecane mixture shows a rather
tinst (s)
t0 (s)
9 17 50
18 44 116
intricate evolution, but one from which many results can be extracted. At short times (t < 50 s), the number of dendrites decreases by coalescence between neighbors (the wavelength λ increases); then at long times, their number remains roughly constant (constant wavelength λs) while some of the dendrites merge and others appear. These results correspond to the ones previously reported, but one also obtains new information. The lifetime of the dendrites varies between 10 and 250 s, and the azimuthal motion of dendrites is evidenced by the continuous white lines, which show both clockwise and anticlockwise drift of more than 80°. A drift velocity vd is extracted from the slope of the trajectories, and its variation versus concentration for heptane/dodecane mixtures is presented in Figure 10. Finally, direct observations show that the moving dendrites are slightly tilted and they exhibit an angle ψ ) 17° with radii.
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Table 3. Evaporated Loss of Mass of Pure Species species
dm/dt (10-4 kg‚m-2‚s-1)
species
dm/dt (10-4 kg‚m-2‚s-1)
water ethanol 2-propanol
0.6 0.9 0.8
hexane heptane octane
4.0 1.3 0.35
Interpretation of Data We try now to understand the spreading dynamics within the framework of the evaporation approach previously described. In the experimental part, we reported the growth of alkane films, and we showed that the shorttime behavior is characterized by a coefficient Dexp (eq 6), i.e., that the spreading velocity is not constant with time. On the other hand we calculated in stationary state a constant velocity vc, which contains ingredients relevant to solutal Marangoni effects (eq 5). Those two approaches can be made compatible by connecting the short-time behavior (measured and not calculated) with the stationary-state behavior (calculated and not measured) for a crossover time t0. We shall then introduce a calculated coefficient Dc(φ) obtained from stationary-state equations
Dc(φ) ) vc2(φ)t0
(7)
and compare it with the experimental coefficient Dexp(φ). For alkane mixtures, the surface tension varies linearly with the composition, then dσ/dφ is equal to the opposite of the surface tension difference ∆σ ) σII - σI between pure species (∆σ > 0), and we finally write Dc as
Dc(φ) )
Aφ(1 - φ)∆σ t0 2η(φ)
(8)
where t0 is a priori unknown but it is supposed to be independent of φ, i.e., the same for a given pair. Evaporation velocities of pure species are calculated (see eq A7) from weight measurements (Table 3). For the calculation of Dc(φ), viscosity and surface tension are taken from ref 21 and φ is taken as the volume fraction of the mixture in the reservoir. With the values of the time t0 reported in Table 2, Dc(φ) values for each alkane mixture agree very well with experimental data (Figure 7), meaning that the solutal Marangoni effect is well described by our model. Gravity was neglected in the calculation (eq 4), and this may explain that Dexp values are lower than Dc’s at low concentrations (films are thicker). At this stage, t0 is still a fitting parameter for each curve, but it is worth discussing its value. t0 increases with decreasing volatility, from 18 s for hexane to 116 s for octane, and it is of the same order as tinst, to which it is roughly proportional:
to ≈ 2.4tinst
(9)
Our analysis in terms of short-time behavior and stationary state was guided by the observation of the time evolution of the meniscus instability (Figure 8), and the experimental relation between t0 and tinst (eq 9) gives credit to our approach. t0 is no more an arbitrary fitting parameter, but it has a physical meaning since it indeed corresponds to a crossover time for the film, just as tinst is the crossover time for the meniscus. All of this gives a coherent picture of this self-adjusted gradient experiment, and we can comment more on the short-time behavior by looking carefully at the way (21) Riddick, J. A.; Bunger, W. B. Organic solvents; 3rd ed.; WileyInterscience: 1970.
evaporation at the liquid surface occurs in the first period of experiments. The atmosphere is initially free of volatile species, and evaporation from the moving film may look similar to the case of the evaporation limited by convection.22 But here an approach in terms of velocity boundary layer and concentration boundary layer cannot be used, because of the relatively slow spreading kinetics of the films. Experimental coefficients Dexp range between 1 and 18 mm2/s, of the same order as the kinematic viscosity of air (about 15 mm2/s at room temperature23). Moreover, the binary gas diffusion coefficients in air Dgas (typically around 8 mm2/s for alkanes) are likewise very close to Dexp; then mass transfer in the air is a mixed diffusion/ convection process. Since all physical parameters describing the system (kinematic viscosity of air, mass diffusion coefficient in air, coefficient Dexp) are of the same order of magnitude, no process prevails and a complete description of these spreading phenomena is complex. Considering this complexity, our semiquantitative analysis of the self-adjusted gradient problem is very satisfactory, because it explains the influence of the volatility and of the composition on the spreading kinetics of alkane mixtures. The major characteristics of the spreading are well analyzed for ideal mixtures; we will now extend this methodology for the behavior of nonideal alcoholic ones. We studied the spreading and the meniscus instability of (ethanol/water) and (2-propanol/water) mixtures. The analysis of these solutions is more complex and less quantitative. First, because of the nonideality, writing matter conservation equations with the mole fraction is more adequate than with the volume fraction. For the same reason, the definition of the evaporation velocity A is slightly different from alkanes (eq A8). Second, as alcohol and water have relatively close vapor pressures (Table 1), we shall write evaporation of both species, but with different boundary conditions. The alcohol concentration far away from the liquid interface is equal to zero, whereas the water concentration far away is given by the relative humidity (RH) of ambient air. We introduce the activity coefficient γi for each species (i), with ai ) γixi, and we use a formalism similar to the one for alkanes; then calling x the mole fraction of alcohol gives for the Marangoni velocity
vc2(x) )
1 dσ [-Aaγax(1 - x) + Aw(γw(1 - x) 2η(x) dx RH)x] (10)
where the subscripts a and w correspond respectively to alcohol and water. The first part in eq 10 describes evaporation of alcohol, and the second one, evaporation of water, with the corresponding evaporation velocities. Coefficients of activity were calculated with the UNIQUAC method developed by Reid and Prausnitz.16 Let us first comment on the influence of the composition for curves (2) (ethanol/water) and (4) (2-propanol/water) in Figure 11. Velocities are equal to zero for x ) 0 (pure water); they reach a high maximum value (out of scale) at very low x, and then they decrease for intermediate concentrations, because of the rapid decrease of |dσ/dx| and of the activity coefficients (Tables 4 and 5). The velocity of 2-propanol/water shows a more pronounced variation. Then curves pass through a minimum, respectively at xEth ) 75% and at x2-Prop ) 67%, because the (22) Incropera, F. P.; de Witt, D. P. Fundamentals of heat and mass transfer, 3rd ed.; Wiley: New York, 1990; Chapters 6 and 7. (23) Handbook of Chemistry and Physics; Chemical Rubber Publishing Co.: Cleveland, OH, 1957.
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Langmuir, Vol. 14, No. 9, 1998 2559
Figure 11. Square of the calculated Marangoni velocities vc for alcoholic mixtures versus the alcohol mole fraction x (eq 10) at various relative humidities. The humidity and temperature of curves (2) and (4) correspond to the experimental conditions (see Figure 12). At very low concentration the curves coincide. Table 4. Activity Coefficients of Ethanol and Water in Liquid Mixture as a Function of Ethanol Mole Fractiona xEth (%)
γEth
γwater
xEth (%)
γEth
γwater
0 1 2 3 5 10 20 30
5.5 4.8 4.3 3.8 3.2 2.3 1.6 1.3
1 1.0 1.0 1.0 1.0 1.0 1.1 1.2
40 50 60 70 80 90 100
1.2 1.1 1.1 1.0 1.0 1.0 1
1.3 1.3 1.4 1.5 1.5 1.6 1.7
a
The activity coefficients were calculated from ref 16.
Table 5. Activity Coefficients of 2-Propanol and Water in Liquid Mixture as a Function of 2-Propanol Mole Fractiona x2-Prop (%)
γ2-Prop
γwater
x2-Prop (%)
γ2-Prop
γwater
0 1 2 3 5 10 20 30
107 69.2 47.8 34.8 20.7 8.5 3.2 2
1 1.0 1.0 1.0 1.0 1.1 1.3 1.5
40 50 60 70 80 90 100
1.5 1.2 1.1 1.1 1.0 1.0 1
1.8 2.1 2.3 2.6 2.9 3.1 3.4
a
The activity coefficients were calculated from ref 16.
sign of net water evaporation from the mixture becomes negative, or equivalently because water absorbs from the atmosphere. The minimum value for 2-propanol/water is negative, meaning that the surface tension gradient is directed downward instead of upward, i.e., that dσ/dz is directed toward the reservoir. The finite values of vc2 of pure alcohol solutions (x ) 100%) are due to water absorption into the liquid. Comparing curves (1) and (3) in Figure 11 shows the influence of the humidity for (ethanol/water) mixtures. At a given x, the velocity is highest for the highest RH. When RH decreases, water evaporation increases, and the local concentration gradients are lowered, as well as the Marangoni velocity. When RH ) 0%, vc2 decreases monotonically with increasing x and it vanishes at xEth ) 100%, as expected from previous remarks. We will now compare these predictions with experimental observations obtained with ethanol/water and 2-propanol/water mixtures, for intermediate alcohol concentrations (Figure 12). After the heating in the cleaning procedure, the glass is not wetted by the mixtures with the lowest alcohol concentrations13 and the films do not spread properly. At x > 4%, Dexp decreases with increasing
Figure 12. Experimental Dexp (markers) and calculated Dc (solid lines) coefficients (eq 12) for alcoholic mixtures versus the alcohol mole fraction x of the reservoir. Experimental conditions: ethanol, T ) 21.0 ( 0.5 °C, RH ) 51%; 2-propanol, T ) 24.0 ( 0.5 °C, RH ) 28%. Table 6. Characteristic Instability Times Ethanol/Water Mixtures xEth (%)
tinst (s)
xEth (%)
tinst (s)
11.7 17.0
19 28
23.6 41.8
35 45
2-Propanol/Water Mixtures x2-Prop (%)
tinst (s)
x2-Prop (%)
tinst (s)
4.0 5.5 9.2
18 21 31
19.0 35.4 67.9
72 100 no spreading
alcohol concentration, with a more pronounced variation for 2-propanol/water. This latter mixture exhibits a very noticeable behavior at x2-Prop ) 68%: no film spread at this concentration (and consequently no instability developed), and we observed a retraction of the liquid on the surface. In parallel, we studied the time evolution of the meniscus instability. tinst depends greatly on the alcohol concentration (Table 6a,b), while for alkane mixtures tinst is constant for a given pair (Table 2). The concentration dependence seems to be specific of the nonideal mixtures and it makes the comparison between calculations and experimental results more difficult, but we shall use the same approach as for alkane mixtures. We keep these experimental variations, and we calculate coefficients Dc introducing a crossover time t0(x):
Dc(x) ) vc2(x) t0(x)
(11)
and since alkane results suggests that t0 is proportional to tinst, Dc is here written as
Dc(x) ) vc2(x)µtinst(x)
(12)
where µ is a constant of proportionality between t0 and tinst. With µ2-Prop ) 0.7 and µEth ) 1.0, calculated Dc(x) values for both alcoholic mixtures (where x is the concentration of the reservoir) have a very satisfactory shape and a magnitude that agrees well with experimental values (graph 12). We also account for the liquid retraction that we observed at x2-Prop ) 67%. For these nonideal mixtures, the coefficients of proportionality µ are lower than for the ideal alkane mixtures (µ ) 2.4), but a quantitative comparison between those two families may not be very significant regarding the slightly different treatments we used. The present study is also in agreement with experimental results obtained by Fournier et al.,13 who
2560 Langmuir, Vol. 14, No. 9, 1998
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studied the spreading of (ethanol/water) mixtures at various humidities. Therefore the spreading behaviors of alcoholic mixtures are well described by our model, which accounts very satisfactorily for the role of the mixture composition and of the ambient conditions. In conclusion, we did not derive the square root of time law for the spreading film, but by introducing a crossover time consistent with experimental observations, we could develop a model that describes well the spreading kinetics of ideal and nonideal mixtures. Our analysis explains the influence of the internal parameters such as the composition and the volatility, and the role of external factors such as the temperature and the humidity. Note that we obtained a good agreement despite our (1) neglecting gravity and (2) taking φ (or x) as the composition of the reservoir. Discussion Characteristics of the Meniscus Instability for Alkane Mixtures. The study of the time evolution of the meniscus instability allowed us to describe satisfactorily the kinetics of spreading, and we will now focus on a few characteristics of this instability for ideal alkane mixtures. We will first discuss the values of the characteristic time tinst, and then we will report a few hypotheses about instability mechanisms. Finally, we will comment on the azimuthal motion of the dendrites. We checked that the values of tinst (Table 2) give diffusion lengths compatible with the dimensions of watch glasses. To modify the kinetics of evaporation, we measured the time evolution of the meniscus instability of octane/ dodecane mixtures (φ ) 60%) at reduced total static pressures between 1.33 × 104 and 1.01 × 105 Pa. The vapor pressure of octane is very low in comparison with the total pressure (Table 1); thus we keep a low concentration of volatile species in the gas phase, and the relevant coefficient is still Dgas of octane/air. As expected, tinst decreases with the decreasing pressure, but in an unanticipated way. A least-squares fit gives the power law
tinst ∝ 2.5P0.36
(13)
(R2 ) 0.939). Equation 13 means that tinst does not vary as the inverse of the gas diffusion coefficient, as would be expected from free diffusion equations. This is probably due to the complexity of the transient state on which we commented in the previous part. Another question concerns the origin of the meniscus instability. Vuilleumier et al.14 proposed to consider it as a Rayleigh-like instability of the meniscus. Its interpretation was supported by the linear relation he found between the wavelength instability and the inverse of the tilt angle. Experiments with heptane/dodecane mixtures give identical results about the tilt angle dependence ((λs ∝ 1/β), but this analysis does not consider the influence of the composition (Figure 9). More recently, de Ryck18 performed similar experiments with isooctane/decane mixtures in a tilted Petri dish. Introducing the curvature Γ of the meniscus, he proposed the following relation for the wavelength of the meniscus instability:
λs ∝
(
1 Aη∆σ φ(1 - φ) Γ σ2
)
1/6
(14)
At low β values, Γ ∝ β, and eq 14 accounts for the experimental tilt angle dependence of λs. Equation 14 is plotted in Figure 9. This model does not take into account the spatiotemporal evolution of the instability, but it
agrees very satisfactorily with our experimental results in shape and in order of magnitude. The agreement with data is poor for φ < 40% and φ > 90%, but it is difficult to conclude about the significance of this discrepancy. At very low concentrations (φ < 30%), the contact line does not separate from the meniscus, and we could not get reliable data in this concentration range. In conclusion, this model gives interesting results, but a more precise approach including the dynamical behavior of the instability is needed. Finally, we can comment on the very puzzling dynamical behavior of the meniscus instability. We reported in the experimental part the intricate spatiotemporal behavior of the dendrites (Figure 5b), and we showed that drift velocities vd could be extracted from these graphs. Mean values of vd are plotted in Figure 10 as a function of the composition for heptane/dodecane mixtures. The data are slightly scattered at intermediate concentrations, where they do not show any clear composition dependence, but a more rapid evolution appears for φ > 90%. Since the meniscus instability develops always in parallel with the spreading, we compare in Figure 10 vd(φ) and the characteristic Marangoni velocity vc(φ), where φ is the volume fraction of the reservoir. Experimental data are closely fitted by the curve vHp(φ) ) 0.72vHp c (φ) (solid line) and the same agreement is found for octane/dodecane and hexane/dodecane, with respectively vOc(φ) ) 0.70vOc c (φ) and vHx(φ) ) 0.60vHx c (φ). The motion of the dendrites along the meniscus is controlled by the solutal Marangoni effect, and this result is another illustration of the intrinsic connection between the meniscus instability and the film growth. In this part, we reported new experimental results about the meniscus instability. They are not fully explained yet, but their consistency with our description of the solutal Marangoni effect clearly asks for further analysis. Complementary experiments are under way to clarify the influence of the pressure on the dynamics and the relation between the drift velocity of dendrites and the film growth. Conclusion The spreading of a two-component liquid film on a tilted plane surface induced by evaporation is satisfactorily understood for ideal mixtures and for nonideal ones. We show that the spreading kinetics is controlled by internal parameters (composition, volatility) and by external factors (temperature, humidity). The interfacial instability of the meniscus that develops in parallel with the spreading shows an intricate time evolution and further work is needed to conclude about its precise origin. Complementary experiments are under way. Appendix Kinetics of Evaporation. Consider the evaporation of a volatile liquid (I) in air, at constant pressure P and temperature T. We call yI the mole fraction of I in the gas phase, and Dgas the diffusion coefficient of I in air at pressure P. If the concentration yI,S at the gas/liquid interface is higher than the concentration yI,∞ far away from the liquid, transfer will occur in the gas phase. In stationary state, and for yI , 1 the molar flux of evaporation NI is written
NI ) k(yI,S - yI,∞)
(A1)
The coefficient k (m/s) is called the mass transfer coefficient, and it depends on the way matter exchange takes place.22
Spreading of Liquid Films by the Marangoni Effect
Langmuir, Vol. 14, No. 9, 1998 2561
At thermodynamic equilibrium yI, S is calculated from Pvap I , the vapor pressure of I. If the liquid is pure, it is written
yI,S )
Pvap I (T) P
(A3)
If the mixture is ideal, the molar activity a is equal to the mole fraction x. Thus, if yI,∞ ) 0, the loss of matter by evaporation from a liquid mixture is the loss of matter from a pure solution multiplied by the activity of evaporating species in the liquid. Previous equations show the influence of the different physical parameters on the evaporation process, such as the boundary conditions or the temperature. Matter Conservation Equations in a Flowing Film. Definition of the Evaporation Velocity. Consider the liquid film in Figure 2. If the mixture I + II is ideal, partial molar volumes V h i are constant, and the volume fraction of species is the relevant parameter for writing matter conservation during transport phenomena.17 We define φ the volume fraction of I as
φ)
VI VI + VII
1 φ 1-φ ) + V hφ V hI V h II
(A2)
More generally, if evaporation occurs from a mixture, yI, S depends15 on the amount of species I in the liquid mixture. Let x be the mole fraction of I in the liquid phase, and a its activity, then yI, S is
Pvap I (T) (yI,S)mixture )a P
of I from the film is equal to the loss of matter from a pure solution multiplied by its mole fraction x in the mixture (eq A3). We define the average molar volume V h φ as
(A4)
where Vi is the volume of species i in the mixture. The volatile species evaporates from the moving film and diffuses in the gas phase, where the concentration yI,∞ is equal to zero. As the solution is ideal, the activity is equal to the mole fraction; then the loss of matter by evaporation
(A5)
h I by and x is related to φ via V h φ and V
hφ xV h I ) φV
(A6)
Then the loss of matter by evaporation is equal to -φA, where A is called the evaporation velocity (m/s) of pure volatile species. A is given by
A)
V h φ dm M dt
(A7)
where M is the molecular weight of I and dm/dt is the mass evaporated from a pure solution of I per time and surface unit deduced from weight measurements. A is the mass transfer coefficient of this problem. If the mixture is not ideal, V h i is not constant and the definition of an average molar volume has no more meaning. Then we define the evaporation velocity as
A)
1 dm F dt
(A8)
Here F is the specific gravity of the solution, and as previously, dm/dt is the mass evaporated from a pure solution per time and surface unit. Acknowledgment. We gratefully acknowledge A. de Ryck for communicating his preprint (1996) prior to publication. We thank also L. Limat and J. M. Flesselles for the image processing software and for enlightening discussions. Research supported by CNRS under BDI grant 1000881. LA971292T
