Thickness and Shape of Films Driven by a Marangoni Flow

Transcription

1 Langmuir 1996, 12, Thickness and Shape of Films Driven by a Marangoni Flow X. Fanton, A. M. Cazabat,* and D. Quéré Laboratoire de Physique de la Matière Condensée, Collège de France, 11 place Marcelin-Berthelot, Paris Cedex 05, France Received May 16, In Final Form: August 6, 1996 X We study an example of spreading driven by surface tension gradients, i.e. the climbing of a film on a plane wall against gravity. We show which parameters control the thickness of the film. Moreover, we present a theoretical and experimental study of the crossover between the film and the macroscopic reservoir. 1. Introduction Coating surfaces with liquids is of practical interest in numerous applications (coating of fibers, wafers, or photographic films), and it is a major concern to predict the thickness of the deposited films. 1-3 In this paper, we study an example of spreading driven by a surface tension gradient induced by a temperature gradient (Marangoni effect 4 ): in capillary-rise geometry, a film climbs out of a reservoir on a vertical or tilted plane wall. 5 The thickness of the climbing film is the relevant parameter in applications, and it is necessary to have theoretical predictions for it. A model 6 has recently been proposed: the thickness is predicted to be independent of the viscosity of the fluid and to be imposed by the surface tension gradient and by the curvature of the meniscus out of which the film climbs. More precisely, the thickness is predicted to be proportional to the square of the gradient and to the third power of the curvature radius. These predictions were only partly checked: experimental results in ref 6 confirm the square dependence on the gradient. It is the aim of this paper to investigate the role of curvature and viscosity. As a matter of fact, the model predicts a very strong dependence on curvature, while the viscosity is expected not to play a significant role. Both predictions are of utmost importance for applications. After a theoretical presentation including a discussion of the meniscus shape and a critical description of the experimental setup, experimental results are presented and analyzed. Conclusions and perspectives follow. 2. Theoretical Approach 2.1. Static Meniscus. A fluid bounded on one side by a vertical wall forms a meniscus, whose shape is obtained by balancing Laplace and hydrostatic pressures: γc -Fgz ) 0 (1) where z is the height above the reservoir (Figure 1), γ is the surface tension of the fluid, F is its specific mass, and C is the curvature of the profile (positive if the curvature center is in the gas). It can be seen from eq 1 that the capillary length κ -1 is the characteristic length of the problem: X Abstract published in Advance ACS Abstracts, November 1, (1) Landau, L. D.; Levich, V. G. Acta Physicochim.URSS 1942, 17, 42. (2) Ludviksson, V.; Lightfoot, E. N. AIChE J.1971, 17, (3) de Ryck, A.; Thésé de l Université Paris 6, (4) Scriven, L. E.; Sterling, C. V. Nature 1960, 187, 186. (5) Cazabat, A. M.; Heslot, F.; Troian, S. M.; Carles, P. Nature 1990, 346, 824. (6) Carles, P.; Cazabat, A. M. J. Colloid Interface Sci. 1993, 157, 196. S (96)00488-X CCC: $12.00 Figure 1. Static meniscus formed by a liquid on a vertical plane wall in the free geometry configuration, tilt angle ) zero, without applied gradient. The z-axis and x-axis are respectively parallel and perpendicular to the plate. The profile of the film is defined by its thickness h(z). The curvature is maximum at the top of the meniscus and is equal to zero in the reservoir. κ -1 ) γ Fg For a silicone oil, we have κ mm. The curvature C is given by C ) where h(z) is the profile of the surface and h and h are respectively the first and second derivatives of the profile. In a free geometry configuration, i.e. when the reservoir is much wider than the capillary length, the shape of the meniscus can be calculated 7 by integrating eq 3, using eq 1. The curvature is calculated from this shape: it increases with decreasing thickness, i.e. approaching the wall, and is equal to zero in the reservoir. The maximum C 0 at the top of the meniscus is where it is assumed that the liquid completely wets the wall. If the wall is tilted by an angle R with respect to vertical, eq 4 must be corrected as If the width D of the reservoir is smaller than the capillary length (confined geometry), the curvature scales as D Driven Spreading of the Fluid. Dynamic Meniscus. If a downward temperature gradient is (7) Landau, L. D.; Lifshitz, E. M. Fluid Mechanics, 2nd ed.; Pergamon Press: New York, American Chemical Society (2) h (1 + h 2 ) 3/2 (3) C 0 ) κ 2 (4) C 0 ) κ 2(1 - sin R) (5)

2 5876 Langmuir, Vol. 12, No. 24, 1996 Fanton et al. Figure 3. Bottom part of the climbing film: the injection zone. The dynamic meniscus makes the transition between the flat part of the film (curvature ) zero) and the static meniscus (curvature of order C 0 at its top). Figure 2. Profile of the climbing film when a downward temperature gradient is applied along the plate, creating an upward surface tension gradient τ. From bottom to top: reservoir, flat part of the film (thickness h ), and bump before destabilization. The apparatus is tilted by an angle R with respect to vertical. applied to the previous system along the z-axis, the resulting upward surface tension gradient τ drives the spreading of liquid up on the wall (Marangoni flow). τ is a force per surface unit and equals τ ) dγ dz ) dγ dt dt dz Experimentally it is observed that a flat film indeed develops, with a bump at its top (Figure 2), whose origin was previously discussed. 8 The bump is unstable with respect to Rayleigh instability, 5,9,10 but before destabilization occurs, the profile of the bump, the thickness of the flat part of the film, and the climbing velocity remain constant. In particular, the bottom of the film (its flat part, the meniscus, and the reservoir) is in a steady state. In this paper, we restrict ourselves to this steady situation, and no time dependence interferes. In the following, we define h as the thickness of the flat part of the film. In the film and in the upper part of the meniscus, the slope dh/dz is much smaller than unity and the capillary numbers are very small (of the order 10-6 ). Thus the lubrication approximation can be used, and the flow rate Q (per unit width of the wall) in the steady state is given by 5 Q(h) ) h2 τ 2η - Fgh3 cos R + γh3 h 3η 3η where η is the dynamic viscosity of the fluid and h is the third derivative of the profile with respect to z (Figure 2). Finally the conservation of flux can be written, i.e. Q(h) ) Q(h ), from which a third-order equation for the profile is deduced: d 3 h dz ) 3τ h 2 - h 2 - Fg 3 2γ h 3 γ h 3 - h 3 (8) Carles, P.; Cazabat, A. M. Mater. Res. Soc. Symp. Proc. 1992, 248, 519. (9) Troian, S. M.; Herzbolzheimer, E.; Safran, S. A.; Joanny, J. F. Europhys. Lett. 1989, 10 (1), 25. (10) Brzoska, J. B.; Brochard-Wyart, F.; Rondelez, F. Europhys. Lett. 1992, 19 (2), 97. (6) (7) h 3 (8) It is worth discussing the different terms in eq 8 or equivalently in eq 7. The first term on the right hand side of eq 7 is the gradient (Marangoni) term. Gravity (second term) is opposed to the flow and can be neglected if the thickness h is smaller than τ/fg. The third term corresponds to curvature effects: let us detail its origin by analyzing the shape of the meniscus. As emphasized by Carles, 6 the problem is very similar to the coating problem first studied by Landau and Levich, dealing with a plate pulled out of a liquid bath. 1 Three zones can be defined around the region where the film forms (Figure 3): (1, upper part) flat part of the film, of constant thickness h ; (2, lower part) static meniscus not perturbed by the flow, with a curvature of order C 0 close to the top; (3, middle part) transition region, which makes the crossover between the flat part of the film and the static meniscus. In this region, called the dynamic meniscus, the curvature of the profile varies between 0 (flat part of the film) and C 0 (static meniscus). This curvature variation corresponds to a pressure variation and appears via the product γh in the flow (third term in eq 7) Thickness of the Flat Part of the Film. Scaling Relationship. Using the Landau-Levich results, we can predict how the film thickness h scales. In the coating problem, the film thickness is h C 0-1( ηu γ ) 2/3 (9) where U is the coating velocity. On the other hand, the velocity of the flat part of the film in our problem is given by eq 7, and if gravity can be neglected (for thicknesses smaller than τ/fg, i.e. smaller than 10 µm), one may write U ) Q h h τ η (10) Combining eqs 9 and 10 gives the scaling form for the film thickness: h τ2 γ 2 C 3 (11) The leading parameters appear to be the gradient, the surface tension, and the curvature of the meniscus. Conversely h is expected to be independent of the viscosity of the fluid Film Thickness: Quantitative Prediction. The scaling relationship (eq 11) can be turned into a

3 Films Driven by a Marangoni Flow Langmuir, Vol. 12, No. 24, quantitative relationship. The starting point is the equation of the profile (eq 8), which becomes, if gravity is neglected d 3 h dz )3τ h 2 -h 2 (12) 3 2γ h 3 Three boundary conditions can be added (for large z, h tends to h and h and h tend to zero), but the film thickness remains unknown, and the fourth condition for determining the thickness is obtained by matching the dynamic meniscus with the static one. The difficulty comes from the fact that the thickness where the matching occurs is also unknown, as in the Landau and Levich coating problem. To alleviate this difficulty, Landau and Levich proposed an asymptotic matching all the more justified since the deformation of the static meniscus is small (i.e. the velocity is small). The matching is done by writing the equality of the asymptotic second derivatives of the profile: ( d2 h 2) stat ) dz ( d2 h 2) dyn (13) hf0 dz hf This equality physically corresponds to pressure continuity, as the slopes dh/dz are much smaller than unity in this region. The first limit in eq 13 is C 0, namely the curvature at the top of the static meniscus given by eq 5, and the second will be calculated by integration of eq 12. For this calculation, putting adequate dimensionless X and Z variables in eq 12 gives with Numerically we take τ 0.2 Pa, γ 0.02 N/m, and h 1 µm; thus, the length h (γ/3τh ) 1/3, which appears to be the characteristic length of the dynamic meniscus along the z-axis, is about 30 µm. Equation 14 cannot be solved analytically, but the entire dimensionless profile can be calculated iteratively, starting from the flat part of the film. Near the flat part of the film (X ) 1), the dimensionless thickness X can be linearized as X ) 1 + ɛ, so that eq 14 becomes A solution of eq 16 is d 3 X dz ) X 2 (14) 3 2 X 3 h ) h X z ) h ( γ 3τh ) 1/3 Z (15) d 3 ɛ )-ɛ (16) 3 dz ɛ ) Ae -Z (17) Then the entire profile can be calculated step by step toward infinite X and the searched limit X is ( d2 X ) 1.15 (18) dz 2)Xf+ Compared to the Landau and Levich problem, where the asymptotic value of the dimensionless curvature is attained for X 20, X in our case reaches its asymptotic value very slowly (Figure 4), and we have X ) 1.10 for Figure 4. Dimensionless second derivative X versus dimensionless thickness (X ) h/h ). The asymptotic value of X is attained at X > In the Landau and Levich problem, the asymptotic value of X would be reached at X 20. Table 1. Physical Characteristics of PDMS PDMS 47V20 47V100 47V350 dynamic viscosity η at 25 C (Pa s) specific mass F at 25 C (kg/m 3 ) surface tension γ at 25 C (10-3 N/m) X 160, X ) 1.12 for X 400, and X ) 1.14 for X This low growth rate is specific of gradient-driven flows. Going back to dimensional variables, using eq 15 and combining eq 18 with eq 13 yields the film thickness h ) 13.7 τ2 τ 2 C 0 3 (19) which has the dimensional form predicted by eq 11, but the coefficient is now explicit. The whole analysis was performed under the assumption that gravity was negligible, which is valid if the film is thinner than τ/fg. Carles et al. 6 verified that h scales as the square of the gradient τ. The work was done with one silicone oil in the free meniscus configuration (tilt angle ) zero). In the confined geometry, 11 two reservoir widths were used: 1 and 2 mm, giving respective thicknesses in a 1/8 ratio, in agreement with a power of -3 for curvature dependence. But the curvature was not known precisely enough to give a quantitative verification of eq 19. Thus we performed new experiments to investigate the role of curvature and viscosity. The experimental setup was modified in order to be able to tilt the whole apparatus and to change the curvature accordingly. 3. Experimental Setup 3.1. Wetting Liquids. Three silicone oils, polydimethylsiloxane (PDMS), were used. Their properties are summarized in Table 1. These PDMS s are nonvolatile and wet completely silicon wafers. In addition, we have for the three oils dγ dt = N m -1 K -1 and df dt =-1kg m-3 K Experimental Procedure. Silicon wafers were cleaned using the following procedure: (a) rinsing with n-hexane, (b) rinsing with methanol, and (c) carefully wiping with lens-cleaning tissues, so that no residue of evaporation was left on the surface. Then the silicon wafer (1) was pinned by suction (Figure 5) to two brass blocks (2 and 3) maintained at a constant temperature by two circulating baths. We waited for thermal equilibrium, and then the meniscus was slowly put in the gradient zone by raising the external reservoir (4). Films were observed through (11) Carles, P.; Cazabat, A. M.; Kolb, E. Colloids Surf., A: Physicochem. Eng. Aspects 1993, 79, 65.

4 5878 Langmuir, Vol. 12, No. 24, 1996 Fanton et al. Figure 7. Dynamic meniscus. Experimental profile and numerical profiles, calculated via eq 8. A good agreement is reached if the gradient value is lowered. Experimental system: oil, 47V100; R)0 ; τ m ) 0.11 Pa; h ) 0.23 ( 0.06 µm. Calculated profiles: (a) τ ) 0.11 Pa (mean value); (b) τ ) 0.09 Pa (actual value). The values of the physical parameters γ and F are taken at the temperature of the lower brass block. Figure 5. Schematic outlook of the apparatus: (1) Silicon wafer pinned by suction to the brass blocks (2) and (3), which are connected to two circulating baths (not represented). The downward temperature gradient induces an upward surface tension gradient. The outer and inner reservoirs (4 and 5) are filled with silicone oil. (6) Microscope. The apparatus is tilted by an angle R with respect to vertical. Figure 6. Temperature variation along the wafer (z-axis). The gradient in the meniscus zone (close to the lower brass block) is smaller than the mean gradient τ m. a microscope (6) and recorded on video tape. Profiles were measured via equal thickness interference fringes with a He- Ne laser (λ ) 6328 Å) in normal incidence. The optical index of PDMS is n ) 1.4, and two successive black fringes correspond to a thickness difference λ/2n ) 226 nm. Image processing was performed with NIH software Mean Gradient and Actual Gradient. The mean surface tension gradient τ m is calculated as follows: τ m )- dγ T dt e (20) where e is the distance between brass blocks and T is the temperature gap. Numerically we have e ) 10 mm, T 30 C,and τ m 0.15 Pa. As the thermal conductivity of PDMS is three orders of magnitude lower than the silicon one, the temperature is imposed by the wafer and the local temperature is equal to that of the substrate (the films are less than 2 µm thick, and the wafer thickness is about 300 µm). In principle, the temperature along the wafer is expected to vary linearly between the brass blocks and to be constant elsewhere, but the real temperature variation does not have this ideal behavior (Figure 6). The gradient is not a constant, especially close to the lower brass block, i.e. in the region of the meniscus. In this last region, the actual gradient (given by the Figure 8. Experimental and calculated second derivative versus thickness in the dynamic meniscus. Same experimental system as Figure 7. Calculation parameters: Figure 7, curve b. Experimental data are in good agreement with the numerical curvature calculated with the actual gradient value. local first derivative of the temperature profile) is smaller than the mean gradient τ m. In the following, we detail the procedure to calculate the actual gradient which exists in the meniscus zone. 4. Experimental Results. Discussion and Analysis 4.1. Profile of the Meniscus. Determination of the Actual Gradient. Experimental profiles were measured up to about 10 µm in thickness and 180 µm in distance. They are reported in Figure 7, and they are compared with the numerical profile (curve a) obtained by integrating eq 8 via the iterative procedure explained above. For this calculation, the physical parameters F and γ are taken at the temperature of the reservoir and the gradient value is the mean value τ m (eq 20). The agreement is qualitative, but it can be improved if the parameters used for the calculation are shifted. This is not surprising, since the actual gradient in the meniscus is expected to be lower than the mean gradient τ m. For the same system, an excellent agreement between profiles (curve b) is reached if τ is shifted from 0.11 Pa (mean value) to 0.09 Pa. In the following, this new value is considered as the actual gradient value. Besides, the study of the profiles can be extended to the curvature variation in the dynamic meniscus. Close to the flat part of the film, the film is nearly parallel to the solid so that the curvature is just the second derivative of the profile (eq 3). Experimental profiles (Figure 7) were differentiated once, and local slopes on these derivatized profiles were measured to get second derivatives. In Figure 8 the experimental curvature is plotted versus thickness and is found to be in good agreement with numerical predictions calculated with the actual value of

5 Films Driven by a Marangoni Flow Langmuir, Vol. 12, No. 24, Figure 9. Experimental thickness h versus τ 2 /γ 2 C 0 3 for three silicone oils. Viscosity varies between and 0.34 Pa s. τ is the actual gradient value ( Pa), C 0 is the maximum curvature of the static meniscus (C 0 ) κ 2) and ranges from 980 to 690 m -1 when the tilt angle R is varied between 0 and 30, and γ is taken at the lower brass block temperature. the gradient. The curvature increases rapidly until h 2 µm; then the growth rate decreases, and for h 6 µm, the curvature reaches a maximum value C max 900 m -1, close to the static value C 0 ) 976 m -1 (eq 5). Curvature agreement is more significant than profile agreement, and confirms our model for the dynamic meniscus. As emphasized above, this description is satisfactory if we consider that the meniscus experiences a gradient smaller than the mean gradient τ m. Thus, the procedure allows us to remove the uncertainty in gradient and to know the actual gradient value in the meniscus. For all experiments, the actual gradient values were found to be about 20% lower than the mean ones. Since the gradient uncertainty is eliminated, and as the surface tension and the specific mass have a low variation with temperature (Table 1), all parameters in eq 19 are precisely known and we can now investigate the validity of this equation Thickness of the Flat Part of the Film. Three PDMS oils of different viscosities were used (Table 1). Using PDMS allows us to modify the viscosity of the wetting liquid in a 1/10 ratio with low variations of the other physical properties, namely the specific mass and the surface tension. Experiments were first performed with a constant gradient (τ m ) 0.16 Pa) and then with other gradient values (τ m ) Pa). The corresponding actual values were found to be between 0.09 and 0.15 Pa. The apparatus was tilted up to 30, yielding a curvature range from 980 to 690 m -1. In Figure 9, experimental thicknesses are plotted versus τ 2 /γ 2 C 3 0 on a linear scale. The gradient value is the actual one (see preceding paragraph), and the curvature C 0 is given by eq 5. Experimental data gather around the same straight line, with a slope equal to 11 ( 2. Results are qualitatively in good agreement with the theoretical model: h is independent of the viscosity and scales like τ 2 C Gravity could indeed be neglected in the flat part of the film, since the films remain thinner than τ/fg, which is on the order of 10 µm. The mean slope of the line in Figure 9 is slightly lower than that predicted by eq 19, 11 instead of Going back to dimensionless variables, this value corresponds to a dimensionless curvature X equal to 1.07, instead of the asymptotic value This point can be qualitatively well understood, in view of the low growth rate of the dimensionless second derivative. In Figure 4, it can be seen that 1.15 is reached only at very large X values (X > 2000). Thus using asymptotic matching, as in Landau and Levich theory, is probably excessive: in our case, the matching between menisci occurs before the asymptotic Figure 10. Estimation of the matching zone. Calculated curvatures are plotted versus thickness in the dynamic and in the static menisci. Calculation parameters: Figure 7, curve b C 0 ) κ 2 ) 976 m -1. The solid line represents the curvature along the interface. It is equal to the dynamic curvature close to the wafer and to the static curvature when thickness tends to infinity. The gray zone, where curvatures are close to each other, corresponds to the crossover between the dynamic and the static menisci. value is attained, for a dimensionless thickness X 70. De Ryck 12 recently studied another example of the Marangoni effect, where the surface tension gradient was created by a concentration gradient induced by evaporation. In the same way, he noticed that the matching between menisci occurs for a dimensionless curvature of about 1.08, in good agreement with our conclusions. These values result from a combination of the low growth rate of the dimensionless curvature, specific of gradient flows, and the role played by gravity in the dynamic meniscus, which was not taken into account for the calculation of the dimensionless curvature. Gravity logically leads to limiting the thickness of the climbing film. Thus, the model derived from Landau and Levich theory (eq 19) is in good agreement with the experimental results in first order, and the small discrepancy we reported is qualitatively well understood. A more quantitative explanation can be performed by estimating more precisely the transition zone between dynamic and static menisci Estimation of the Transition Zone. In Landau and Levich theory, matching between dynamic and static menisci is obtained by writing the equality of two asymptotic curvatures: knowing the exact matching thickness between menisci is not necessary. We introduce now another approach in order to estimate the matching zone between menisci. We compare on the same graph the curvature of the dynamic meniscus and the one of the static meniscus, and we still define the matching between menisci by writing the equality of curvatures. We shall see that this procedure provides an interpretation of the matching condition discussed above. In Figure 10, dynamic (same system as Figure 7, curve b) and static curvatures are plotted versus the film thickness. The curvatures were calculated with eq 3, i.e. without neglecting the first derivative. The solid line at low thicknesses is the dynamic curve, whose validity was shown until h 8 µm (Figure 8). In the same domain, the dotted line is the static curvature, assuming that its stat shape is not distorted and particularly that C hf0 ) κ 2, which is strictly correct only if the thickness of the climbing film tends to zero. Then for larger thicknesses the flow of liquid inside the meniscus becomes negligible and the shape of the interface is determined by gravity (static meniscus). Between those two domains, a region (12) De Ryck, A. Submitted to Phys. Fluids.

6 5880 Langmuir, Vol. 12, No. 24, 1996 Fanton et al. can be defined (h 20 µm) where curvatures are close to each other and of the same order as the maximum curvature of the static meniscus (C 900 m -1 for C 0 ) 976 m -1 ). In this system, h is equal to 0.23 µm, and a 20 µm thickness corresponds to a dimensionless thickness equal to about 90: matching between menisci indeed occurs for a dimensionless thickness of order 90, which confirms the previous results (matching for X 70). Comparing the dimensioned curvatures is the relevant procedure to estimate the transition zone between menisci. It allows us to predict more precisely where the matching actually takes place and to explain more quantitatively the discrepancy mentioned above. The matching actually occurs before the asymptotic value is reached, for a dimensionless thickness of about 80, in good agreement with experimental results. 5. Conclusion and Perspectives We have studied experimentally the climbing of a liquid film on a vertical or tilted wall, where the driving force is created by a temperature gradient which induces a surface tension gradient (Marangoni effect). New experiments showed that the thickness h of the flat part of the climbing film scales like C -3 0 (C 0 is the curvature of the meniscus out of which the film climbs), and is independent of the viscosity, as predicted by a theoretical model. 6 Moreover, we calculated in steady state the profile of the dynamic meniscus, which makes the transition between the flat part of the film and the static meniscus. By analyzing the shape of the film, we were able to introduce a new matching procedure and turn the former qualitative agreement into a quantitative one. We have developed a good understanding of the mechanisms that occur at the bottom of the film: how the fluid is injected and how the thickness h is chosen. In the upper part of the film, we observed at short times the formation of a bump between the contact line and the flat part of the film. The contact line is unstable at long times, this destabilization being analyzed as a Rayleigh-like instability of the bump. 5,9,10 A transition between stable and unstable regimes was predicted to occur for h ) τ/fg. 6 Thanks to the analysis presented in this paper, we have a quantitative knowledge of the experimental leading parameters and particularly of the gradient. It should allow us to study precisely the transition between stable and unstable regimes. Vuilleumier et al. 13 considered the well-known tears of wine experiment, where a surface tension gradient is induced by the evaporation of alcohol in a water-alcohol mixture. This experiment is very similar to ours, but they observed a destabilization of the meniscus at the crossover between the climbing film and the reservoir. The analysis of the meniscus shape performed in this paper is a first step toward a global understanding of these different behaviors. Acknowledgment. Support from CNRS (BDI Grant No ) is gratefully acknowledged. We thank also S. Villette and C. Gay for useful discussions. LA960488A (13) Vuilleumier, R.; Ego, V.; Neltner, L.; Cazabat, A. M. Langmuir 1995, 11 (10), 4117.