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Shock Separation in Wetting Films Driven by 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 July 13, 2000. In Final Form: September 5, 2000 Previous investigation has shown that in certain cases thermally driven wetting films may spread over surfaces without the usual fingering at the contact line. Numerical simulations suggest that the contact line region may be composed of two shock waves, a trailing Lax shock and a leading undercompressive shock, which advance at different speeds and separate. Results are presented here that show that the separation speed is indeed similar to that predicted by numerical simulations. The role of the thickness of the prewetted film on the solid surface is also investigated.
Introduction The behavior of wetting films driven by gravity or surface tension gradients has been the subject of numerous investigations.1-9 In our previous paper10 we investigated the behavior of wetting films driven to climb inclined solid surfaces by thermal gradients. Typically, these films had a flat region adjacent to the bulk liquid at the bottom and a bump adjacent to the contact line at the top. The important parameter was the thickness, hf, of the flat region of the film. For thin films, the bump could be considered as stationary for a period of time, although ultimately it was subject to a fingering instability.11-12 As the thickness, hf, of the film was increased (by increasing the thermal gradient or by increasing the tilt angle of the solid from the vertical) a threshold thickness was reached. Above this threshold fingering of the contact line was no longer observed but the bump continued to grow in width with time.14 As observed by Bertozzi, Shearer, and coworkers,14-16 this was in fact due to the presence of two shock fronts, a Lax shock and an undercompressive shock, advancing at differing speeds, causing them to separate over time. A schematic diagram of a typical predicted profile14 of the bump above the threshold is shown in Figure 1. A prewetted film of thickness b is present on the substrate. The leading shock is the undercompressive shock wave that connects the flat film of thickness b to another flat film of thickness hUC. The trailing shock is a Lax shock that connects the flat film of thickness hUC to the flat film of thickness hf and has the greatest height of the profile. Although the undercompressive shock is predicted to be (1) Ludviksson, V.; Lightfoot, E. N.; AIChE J. 1971, 17, 1166. (2) Huppert, H. E. Nature 1982, 300, 427. (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) Hocking, L. M. J. Fluid Mech. 1990, 211, 373. (6) de Bruyn, J. R. Phys. Rev. A 1992, 46, R4500. (7) Brenner, M. P. Phys. Rev. E 1993, 47, 4597. (8) Bertozzi, A. L.; Brenner, M. P. Phys. Fluids 1997, 9, 530. (9) de Bruyn, J. R. Phys. Fluids 1997, 9, 1599. (10) Schneemilch, M.; Cazabat, A. M. Langmuir, in press. (11) Cazabat, A. M.; Heslot, F.; Troian, S. M.; Carles, P. Nature 1990, 346, 824. (12) Cazabat, A. M.; Heslot, F.; Carles, P.; Troian, S. M. Adv. Colloid Int. Sci. 1992, 39, 61. (13) Carles, P.; Cazabat, A. M. J. Coll. Int. Sci. 1993, 157, 196. (14) Bertozzi, A. L.; Mu¨nch, A.; Fanton, X.; Cazabat, A. M. Phys. Rev. Lett. 1998, 81, 5169.
Figure 1. A typical profile of a double shock structure. The leading shock wave is the undercompressive shock. The Lax shock has the greatest height of the profile. The flat film thickness is hf and the thickness of the prewetted film is b. The bulk liquid reservoir is at infinity on the left.
stable, the Lax shock is actually expected to develop a fingering instability albeit with a larger wavelength than observed for thin films. Behind the Lax shock is a small depression, which is joined to the flat film. The flat film of thickness hf extends to the bulk liquid reservoir at the bottom of the substrate. As in our earlier publication, our aim was to determine the influence of finite size effects on the behavior of wetting films as compared to the theoretical predictions of the model, which were based on an ideal system (infinite substrate, constant gradient, etc.). Previously,10 the presence of the threshold and the predicted changes in stability of the contact line were confirmed by systematically eliminating possible size effects in the nonideal experimental system. In the current paper the focus is on the behavior above the threshold, in particular on the separation of the two shock waves, and on the role of the prewetted film thickness, b. That shock wave separation was not observed previously was due to the relatively short time available for the experiment. To increase the duration and allow the waves to separate it was necessary to increase the separation between the respective thermal reservoirs. This introduced some complications, that will be discussed prior to the presentation of the results proper. Experimental Apparatus The experimental apparatus, illustrated in Figure 2, was identical to that used in the previous investigation.10 The liquid
10.1021/la0009893 CCC: $19.00 © 2000 American Chemical Society Published on Web 11/16/2000
Shock Separation in Thermal Gradient Film Wetting
Langmuir, Vol. 16, No. 25, 2000 9851 τ Fg cos R
(2)
τ2 e1 ) 11 2 3 γC
(3)
e0 ) and
Figure 2. The experimental apparatus. The silicon wafer (1) is attached by suction to the upper holder (2) that is maintained at a temperature slightly lower than ambient. The lower holder (3) is adjacent to the bulk liquid reservoir (4) 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 outer reservoir (5) 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. used was a nonvolatile silicone oil poly(dimethylsiloxane) (PDMS) with viscosity η ) 0.1 Po, density F ) 966 kg m-3, and surface tension γ ) 20.9 mN m-1 at 20 °C, which completely wetted the oxidized silicon wafer substrates. Silicon wafers that had been previously immersed in PDMS were cleaned with lint free tissue paper and rinsed with hexane. This procedure produced a coating of PDMS of molecular thickness. Thicker prewetted films were prepared by spin-coating as described below. The films were imaged with a CCD camera and a microscope arrangement. The speed of the film was gauged by taking measurements of the length at regular time intervals. A laser illuminated the film in order to measure the thickness from the interference fringes. The vertical distance between adjacent fringes was determined by the equation
Here, τ is the thermally induced surface tension gradient, F the liquid density, g the acceleration due to gravity, γ the liquid surface tension, C the curvature of the meniscus where the flat film meets the liquid reservoir, and R the inclination of the substrate with respect to the vertical. e1 is determined by the balance between the surface tension gradient and gravity, as in the case of thick films. e0 corresponds to the thin film case where Laplace pressure is dominant and the balance is between the surface tension gradient and the curvature of the meniscus. The motion of the film was modeled by a partial differential equation which describes the height of the film, h(x,t), as a function of the position, x, with respect to the direction of flow and the time, t. The depth averaged fluid velocity of the film, U(h), was obtained from the Navier-Stokes equation in the lubrication approximation
U(h) )
hτ Fgh2 cos R γh2∂3h + 2η 3η 3η ∂x3
and by taking into account the conservation of mass.
∂h ∂ + [hU] ) 0 ∂t ∂x
(5)
To rescale, values of thickness are divided by a height H, values of time by a time T, and length values by a length L where
H)
3τ 2Fg cos R
T)
(
)
2η 4 τγFg cos R τ2 9
1/3
L)
(
)
3γτ 2F g cos2 R 2 2
1/3
where η is the fluid viscosity. The velocity in the flat part of the film is described by the following equation
Uf )
λ ∆h ) 2n
(4)
hfτ Fghf2 cos R 2η 3η
(6)
Equation 6 can be written in dimensionless form where λ is the wavelength of the laser beam and n is the optical index of PDMS. For the current system λ ) 632.8 nm and n ) 1.4. In principle it is possible to determine the film thickness at any position by counting the number of fringes between that position and the contact line. In practice, however, the fringes tended to occur too close together in the region adjacent to the contact line where the profile was steep and could not be resolved by the magnification system. The interference fringes formed concentric rings around the point of the maximum height of the profile. This provided a convenient method for locating the position of the trailing shock, as shown in Figure 3. The speed was then calculated by measuring the change in position of the wave with time. The speed of the leading shock could be approximated by the speed of the contact line. The separation speed was then the difference between the two. Theoretical Description. According to well established theory,1-13 the flat film thickness, hf, is determined at the connection between the bulk liquid reservoir and the flat film according to the following equation
(
e0 hf ) e0 1 + 2e1 where
x
e20
)
e0 + 2 e 4e1 1
(1)
Uf* ) hf* - (hf*)2
(7)
where the * superscript denotes a reduced variable. Traveling wave solutions connect (1) the flat film height, hf, to the undercompressive shock height, hUC, and (2) the undercompressive shock height, hUC, to the prewetted film ahead of the contact line of thickness b. It is necessary to assume the existence of this precursor film to avoid the well-known singularity that arises for a moving contact line. The evolution of the profile with time is described by the following equation, in reduced form (omitting the *)
ht + (h2 - h3)x ) -(h3hxxx)x
(8)
where ht is the time derivative and hx is the space derivative. Equation 8 has a nonconvex flux function
f(h) ) h2 - h3
(9)
The flux diagram is shown in Figure 4. Point A (here at the origin) corresponds to the preexisting film thickness, b. Point B corresponds to the thickness of the flat part of the film hf. When the flat film thickness, hf, is below a certain threshold value, ht, there exists a single Lax shock that is subject to a fingering instability. When hf is greater than ht, double shock structures
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Figure 3. An image of the double shock structure, at an angle of 65° after a period of 9 min. The flat film is connected to the reservoir by the meniscus. It can be seen that although the film is not truly flat the variation in thickness, evident from the pattern of interference fringes, is small compared to the average thickness. There is a small depression between the flat film and the bump. The position of the Lax shock can be identified by the rings that form at the thickest part of the bump. Since the structure has not had a lot of time to develop, the Lax shock is close to the middle of the bump. described above.) The leading shock is undercompressive and connects the precursor film b to hUC. The trailing shock, connecting hUC to the flat film thickness, is a classical Lax shock. The height of the undercompressive shock, hUC, is shown on the flux diagram as point C. The slope of the chord BC corresponds to the speed of the Lax shock while the slope of the line AC corresponds to the speed of the undercompressive shock. The speed at which they separate is thus the difference in the two slopes. Note that the threshold, ht and the height hUC are completely determined by the thickness of the prewetted film, b. In particular, the height hUC is predicted to be independent of the thickness hf. The transition from Lax shocks to double shock structures happens because of the interaction of surface tension with the nonconvex flux in eq 8. Changing either the flux (h2 - h3) to a convex flux (as in h2 or h3 alone) or removing the surface tension term on the right-hand side would remove this transition. Hence this kind of dynamics was not observed in previous simpler experiments (such as flow down a plane).
Figure 4. A typical flux diagram. Point A corresponds to the prewetted film thickness, b, which is close to zero. Point B corresponds to the flat film thickness, hf, and point C corresponds to the thickness of the undercompressive shock, hUC. The slope of the line AC is the speed of the undercompressive shock and the slope of the line BC is the Lax shock speed. comprising an undercompressive and a Lax shock are observed. (The simple threshold described here applies only to dynamic simulations of eq 8 starting from jump or monotone initial conditions. Other more complicated initial profiles might converge to unusually shaped traveling wave solutions that presumably are not seen in typical experiments. In fact, in ref 16 a cascade of bifurcation with unusual stable traveling waves are described, but were only accessible for very special initial conditions. In the current experiment there should be a simple threshold as
Results As already stated, in order for the shock separation to become readily apparent the duration of the experiment had to increase. To accomplish this it was necessary to increase the distance between the thermal reservoirs. The next issue to be addressed, then, was to determine the effective limits of the maximum and minimum separation at which a constant temperature gradient can be maintained. Presumably, when the separation is too small there is coupling between the two reservoirs that results in an actual gradient lower than the calculated value, as illustrated in Figure 5a. On the other hand, when the separation is too large, the coupling with the surroundings plays a role, which would result in a significantly variable effective gradient, locally larger than that calculated, as shown in Figure 5b. To estimate the effective limits of the distance between the reservoirs, a series of experiments on thin, stationary
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Figure 6. The dimensionless contact line velocity as a function of the dimensionless flat film thickness at an inclination of 0° from the vertical. The velocity was measured at the tips of the fingers formed at the contact line. The flat film thickness was calculated using eq 1. The separation was varied from 5 mm to 15 mm and the temperatures of the reservoirs were varied so as to maintain a constant theoretical thermal gradient of 4400 K/m. The mean temperature varied from 29° to 51°. The solid line is the theoretical dependence that has been established previously. The agreement is acceptable only at separations of 10 and 12.5 mm which indicates the range of validity of this parameter. The value at a separation of 15 mm falls above the solid line, while the points corresponding to 7.5 and 5 mm separations fall below the line.
Figure 5. An illustration of the temperature profile between the edges of the thermal reservoirs that are represented by the vertical lines. (a) The situation when the separation, d, is large (>12.5 mm). Here, the actual thermal gradient is locally greater than calculated. (b) The coupling between the holders when the separation, d, is too small (
