Langmuir 2004, 20, 8423-8425
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How Does a Thin Volatile Film Move? I. Leizerson* and S. G. Lipson Department of Physics, Technion - Israel Institute of Technology, 32000 Haifa, Israel Received January 11, 2004. In Final Form: July 4, 2004 When a water film evaporates from a mica substrate, an interface similar to a solidification front develops, separating two films of different thicknesses. We show experimentally that the evolution dynamics is controlled mainly by material diffusion through the vapor phase rather than by hydrodynamic flow through the film. Our results illustrate the role of different contributions to pattern formation of volatile liquid films.
Introduction Pattern formation in a rupturing liquid film on a rigid substrate demonstrates a rich variety of phenomena.1 Although most work has been done on nonvolatile systems where mass is conserved, the study of the evolution of volatile films has introduced a new dimension, in which the type of behavior is controlled by the vapor pressure. In particular, two-phase behavior has been observed mimicking at different vapor pressures dendritic growth, viscous fingering, and spinodal decomposition. The behavior of nonvolatile films2,3 has been explained in terms of the substrate-fluid interaction and agrees well with both theoretical4 and numerical calculations,5 for which hydrodynamic flow leads directly to a diffusive type of behavior.4,6 For volatile films, the relative importance of evaporative and hydrodynamic effects has been studied theoretically7 for thin films and experimental results for sessile drops were reported.8 Here we report the first simple direct observation proving that material transport between different regions of the film through the vapor phase is an order of magnitude faster than hydrodynamic film flow and therefore the mechanism of film evolution should be different. This suggests a new method of controlling the spreading of thin volatile films by using the ambient pressure of a neutral gas. Theory A cleaved mica surface is known to be wetted by water; under saturated vapor pressure, a uniform macroscopically thick film of water covers the surface completely. As this water evaporates, it breaks up into interesting patterns which were predicted by Cazabat et al.9 and Brochard-Wyart et al.10 as resulting from antagonistic * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Kataoka, D. E.; Troian, S. M. Pattering liquid flow on the microscopic scale. Nature 1999, 42, 794-797. (2) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Phys. Rev. Lett. 1991, 66, 715-718. (3) Sharma, A.; Reiter, G. J. Colloid Interface Sci. 1996, 178, 383399. (4) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 325-359. (5) Oron, A.; Davis, S. H.; Bankoff, S. G. Rev. Mod. Phys. 1997, 69, 931-980. (6) Brochard-Wyart, F.; Daillant, J. Can. J. Phys. 1991, 68, 10841088. (7) Sharma, A. Langmuir 1998, 14, 4915-4928. (8) Zheng, L.; Wang, Y.-X.; Plawsky, J. L.; Wayner, P. C. 2002, 18, 5170-5177. (9) Cazabat, A.-M. Contemp. Phys. 1987, 28, 347-364. (10) Brochard-Wyart, F. Soft Matter Physics; Springer: New York, 1995; pp 7-44.
van der Waals and polar surface forces between the water and the substrate.11 The patterns were shown both theoretically12 and experimentally13 to result from the coexistence of two films of very different thicknesses (h1 , h2), between which water is exchanged during the evolution. The model explains the observed similarity to diffusion-limited solidification or viscous fingering.14,15 The effective equation of motion proposed for the volatile case12 has two contributions: hydrodynamic and evaporative. Neglecting fourth (and higher) order derivatives and introducing small deviation δh from film thickness h2, the following approximate expression, assuming uniform temperature, was derived:
(
)
(
)
h23 d2g RγLV 2 ∂δh dg ) - µvap + ∇ δh + R ∂t 3η dh2 F F dh
(1)
where g is the interaction energy per unit area between the film and the substrate:12
g(h) ) SLWd02/h2 + SP exp[(d0 - h)/l]
(2)
where SLW and SP are spreading pressures due to van der Waals and polar interactions, l is a screening length, d0 is the molecular diameter, and F is the density of molecules in water. Figure 1 shows µfilm ) F-1 dg/dh as a function of h for typical values of the interaction parameters.11 η is viscosity, µvap ) kBT ln(pvap/psat) is the chemical potential of water vapor at pressure pvap, and R is a kinetic sticking parameter. The terms containing R thus depend on the vapor transport, and the remaining term describes hydrodynamic flow. A physical picture is that just below the saturated vapor pressure water evaporates from the convex regions of the film (the rim) and condenses in concave regions where the increase in film thickness is penalized minimally by the substrate interaction. As a result, most of the condensation is where d2g/dh2 is smaller, that is, at the thicker film. This causes expansion of the thin film regions because the water then moves outward. We can clearly distinguish diffusive behavior of film thickness with an equivalent 2-D diffusion coefficient which is the coefficient of ∇2δh in eq 1:
D ) h3/3η d2g/dh2 + RγLV/F
(3)
The coefficient R is related to the probability of water molecules to transfer between the thick flat film and a (11) Sharma, A.; Jameel, A. J. Colloid Interface Sci. 1993, 161, 190. (12) Samid-Merzel, N.; Lipson, S. G.; Tanhauser, D. S. Phys. Rev. E 1998, 57, 2906.
10.1021/la0499113 CCC: $27.50 © 2004 American Chemical Society Published on Web 09/03/2004
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Langmuir, Vol. 20, No. 20, 2004
Letters
Figure 1. The chemical potential of film F-1 dg/dh as a function of film thickness h.
Figure 2. An interferometric image showing the rims around thin film patches growing out of a thick film.
bulk source during evaporation and condensation. For a given partial water vapor pressure, R depends on the total pressure ptot in the cell, which controls the mean free path of a molecule in the gas, and is given in general form18 as R ∝ exp(-ptotLd02/kBT), where L is a constant given by the cell size and geometry and d0 is the molecular radius. On setting the 2-D diffusion eq 1, it follows that the normal velocity of the interface between the two regions is proportional to D and therefore has the general form
V ) V0 + V1 exp(-βptot)
(4)
where V0 is the hydrodynamic contribution and V1 exp(-βptot) is the vapor term. We have used the dependence of the effective diffusion constant on the total pressure in the cell to investigate the relative contribution of the two terms in eq 3. Experiment The experiments were carried out in an evacuated system at 18 °C, into which water vapor could be introduced by evaporation from a separate chamber (13) Leizerson, I.; Lipson, S. G.; Lyushnin, A. V. Nature 2003, 422, 395-396. (14) Ben-Jacob, E. Nature 1990, 343, 523-530. (15) Ihle, T.; Muller Krumbhaar, H. Phys. Rev. Lett. 1993, 70, 30833086.
Figure 3. Velocity of front propagation in dewetting phenomena as a function of total pressure in the cell.
Figure 4. Thickness profile between regions of two film heights (ref 19) measured across a rim (the dashed line in Figure 2). Mass transfers through the vapor by evaporation in region A and condensation in region B.
containing distilled water.16 In the present experiment, after the system had been evacuated to below 10-4 Torr, water vapor and a controlled background pressure of N2 gas were introduced. The mica substrate is prepared by cleavage in situ and then exposed to water vapor only. The temperature, T, of the experimental cell is controlled by an arrangement of thermoelectric coolers. The mica sample under investigation is attached by a thin layer of high-vacuum black wax to a pedestal in the center of the cell. The sample temperature, Ts, is controlled independently by an additional thermoelectric cooler. If Ts < T, water vapor condenses from cell walls to sample surface and vice versa. The interference pattern created during evaporation was observed by a 10-magnification microscope, and the picture in Figure 2 shows a typical pattern in a spinodally decomposed water layer. We found that the velocity of propagation is equal on all the fronts and constant during their growth, as was observed by Elbaum and Lipson.17 (16) Leizerson, I.; Lipson, S. G.; Lyushnin, A. V. Phys. Rev. E 2003, 68, 051601-6. (17) Elbaum, M.; Lipson, S. G. Phys. Rev. Lett. 1994, 72, 3562. (18) Born, M.; Bormann, E. Phys. Z. 1920, 21, 578-581. (19) Leizerson, I.; Lipson, S. G. Appl. Phys. Lett. 2003, 83, 260-262.
Letters
Figure 3 shows measurements of the velocity of front propagation as a function of the total pressure in the cell, each measurement being averaged over about five different fronts of patches of different sizes at each pressure. The partial pressure of water vapor was constant (18 Torr). The best fit to the experimental results is given by the form of eq 4 and shown in Figure 3 with V0 ) 0.68 µm/s, V1 ) 11.52 µm/s, and β ) 0.0132 (Torr)-1. From these, the relative importance of the evaporative and hydrodynamic terms in eq 3 can be determined: V1 exp(-βptot)/V0 which has value of >13 at the vapor pressure at 18 °C (18 Torr). This situation can be compared to slow crystal growth: the excess of solvent surrounding a growing crystal tip diffuses into the whole solution instead of being localized ahead of the tip as occurs in fast crystallization. These experiments demonstrate that diffusive behavior results mainly from mass transfer through the gas
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phase, from a region of positive curvature to a negative one (Figure 4), which occurs much faster than hydrodynamic flow. Since film spreading has usually been associated with the wetting properties alone, our observation suggests a new approach to controlling spreading properties of a volatile liquid film. Instead of changing the contact angle or other liquid properties such as viscosity, this can be done by varying the total vapor pressure above the film. Acknowledgment. We acknowledge the technical assistance of S. Hoida. This project was supported by the Minerva Foundation for Nonlinear Science and the Technion Fund for the Promotion of Research. LA0499113
