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Finger Instability in Wetting-Dewetting Phenomena I. Leizerson,*,† S. G. Lipson,† and A. V. Lyushnin‡ Department of Physics, TechnionsIsrael Institute of Technology, 32000 Haifa, Israel, and Department of Theoretical Physics, Perm State Pedagogical University, Perm 614600, Russia Received June 2, 2003. In Final Form: October 11, 2003 We present a study of finger instability observed in a propagating front resulting from dewetting of a thin water film on a mica substrate. The phenomenon results from a longitudinal instability in a thick cylindrical rim along the front, whose unusual properties are determined by domination of long-range van der Waals interaction between the film and the substrate over the effect of curvature. The interpretation is borne out by quantitative measurements and supported by computer simulations.
Introduction
Theory
A wetting liquid film spreads over a solid surface under the influence of surface interaction forces, a surface-tension gradient, or gravitation. The spreading process has attracted considerable attention in recent years because it is an interesting example of the development of nonlinear instability and due to the importance of this mechanism for the control of liquid flow in microfabricated devices.1,2 Ever since the hydrodynamic instability of a liquid cylinder to longitudinal periodic fluctuations was analyzed by Rayleigh, it has served as a prototype for a variety of phenomena. Similar behavior was observed in the moving front of a liquid film covering a solid substrate. The origin of this instability can be traced to a thick cylindrical rim created along the moving front by the hydrodynamics of the fluid. This is responsible for the front corrugation, which results in the formation of finger instability.3,4 Such instability is also observed in gravity-driven films,5,6 in thermally driven films (Marangoni effect),1,2 and in films with spreading surfactants.7 In the present paper we focus on the investigation of the effect of van der Waals forces on the dynamics of the fingering instability of a “contact line” created through dewetting of an evaporating water film on a mica substrate. Surface tension effects alone cannot give a satisfactory explanation for a simultaneous reduction of the curvature radius of the rim and an increasing mean distance between fingers observed in the experiment. We shall demonstrate that the long-range van der Waals forces between the liquid and the substrate are responsible for this behavior and have a strong effect on the fingering instability. The interpretation is supported by measurements and computer simulations and leads to good agreement between experiment and 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 theoretically by Cazabat et al.8 and Brochard-Wyart et al.9 and have been demonstrated experimentally to be a result of antagonistic van der Waals and polar surface forces between water and a substrate,10-12 represented by an excess surface free energy per unit area, for thickness h
* To whom correspondence may be addressed. E-mail: philya@ techunix.technion.ac.il. † TechnionsIsrael Institute of Technology. ‡ Department of Theoretical Physics, Perm State Pedagogical University. (1) Kataoka, D. E.; Troian, S. M. Pattering liquid flow on the microscopic scale. Nature 1999, 42, 794-797. (2) Cazabat, A. M.; Heslot, F.; Carles, P.; Troian, S. M. Adv. Colloid Interface Sci. 1992, 39, 61. (3) Kataoka, D. E.; Troian, S. M. A theoretical Study of instabilities at the Advancing Front of thermally Driven Coating Films. J. Colloid Interface Sci. 1997, 192, 350-362. (4) Oron, A.; Davis, S. H.; Bankoff, S. G. Long-scale evolution of thin liquid films. Rev. Mod. Phys. 1997, 69, 931-980. (5) Bertozzi, A. L.; Brenner, M. P. Linear stability and transient growth in driven contact lines. Phys. Fluids 1997, 9 (3), 530-539. (6) Ye, Y.; Chang, H.-C. A spectral theory for fingering on a prewetted plane. Phys. Fluids 1999, 11, 2494-2515. (7) Matar O. K.; Troian, S. M. Spreading of surfactant monolayer on a thin liquid film: Onset and evolution of digitad structures. Chaos 1999, 9, 141-153.
g(h) ) SLW/h2 + SP exp[(d0 - h)/l]
(1)
where SLW and SP are spreading pressures due to van der Waals and polar interactions, l is a screening length, and d0 is the molecular diameter. It was shown that a thermodynamic potential analogous to the Gibbs free energy, which is a function of the film thickness, can be defined for such a system.11 For a range of surface force parameters, this potential has minima at two different film thicknesses. Their relative depth depends on the evaporation rate and determines which of the thicknesses is in equilibrium and which is metastable. One minimum corresponds to a molecularly thin film of thickness h1 and the other to a macroscopically thick film, h2. These thicknesses were recently measured for the first time.13 Phase equilibrium between the two films is similar to that of liquid and solid phases in solidification, and a first-order phase transition between them occurs. Since two films together cover the surface continuously, there is no defined contact angle between them.13 At different rates of evaporation, observed patterns of film-thickness evolution11 for such a system are analogous to viscous fingers, spinodal decomposition, or doublon-like patterns similar to those calculated for solidification in an isotropic two-dimensional system.15,16,17 The heterogeneous nucleation of a thin film on the edge of a step on the mica surface, or at a linear defect, is straightforward. At a step, for example, the water thickness is increased in the region of concave curvature of the step (capillary condensation) and decreased on its (8) Cazabat, A.-M. How does a droplet spread? Contemp. Phys. 1987, 28, 347-364. (9) Brochard-Wyart, F. Soft Matter Physics; Springer: Berlin, 1995; pp 7-44. (10) Sharma, A.; Jameel, A. Nonlinear Stability, Rupture and Morphological Phase Separation of thin Fluid Films on Apolar and Polar Substrates. J. Colloid Interface Sci. 1993, 161, 190. (11) Samid-Merzel, N.; Lipson, S. G.; Tannhauser, D. S. Pattern formation in drying water films. Phys. Rev. E 1998, 57, 2906. (12) Elbaum, M.; Lipson, S. G. How Does a Thin Wetted Film Dry up? Phys. Rev. Lett. 1994, 72, 3562. (13) Leizerson, I.; Lipson, S. G.; Lyushnin, A. V. Wetting properties: When larger drops evaporate faster. Nature 2002, 422, 395-396.
10.1021/la034955h CCC: $27.50 © 2004 American Chemical Society Published on Web 12/13/2003
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Figure 1. Geometry of a cylindrical water slice on a mica surface, with a periodic instability of wavelength ζ.
exposed edge, which will therefore be a line of nucleation of the thin film. Growth of the thin film region continues because the thick film evaporates continuously and water diffuses into it in order to bring the thick film to equilibrium thickness h2. However, due to the effect of viscosity, the water partially gathers into a semicylindrical rim that has a thickness higher than h2. This propagating cylindrical “bump” is sensitive to longitudinal periodic fluctuations leading to finger creation. To investigate the stability of the rim, we assume it to have the geometrical form of a longitudinal slice cut from the rim of a cylinder with its plane surface in contact with the substrate14 (Figure 1). In terms of the maximum height z of the rim above the substrate plane, the cross section at any x is the segment of the circle of curvature K. The segment has width 2(2z/K)1/2 and cross-section area 4/3(2z3/ K)1/2. We now use the lubrication approximation for laminar flow in a planar film on a substrate, with no slip at the substrate and a free upper surface at z, as an approximation for viscous flow in the segment. In this case, the flux in the x direction per unit width in y is related to the pressure gradient along x by
J)-
z3 dp 3η dx
(2)
( ( ) ) (() )
∂ 2 z3 ∂t 3 K
1/2
)
∂ z J ∂x K
1/2
)-
( ())
1 ∂ dp z7 3η ∂x dx K
1/2
(3)
leading after some algebra to the following expression
(
) (
)
2z d2p z dK z2 dp dz 7 3 z dK ∂z (4) + ) - 2 2η dx dx K K dz K dx2 K K dz ∂t
(
)
interaction to g, and we retain only the van der Waals component g ) Slwd02/z2.10 A linear stability analysis for small fluctuations z ) h + u exp(iqx + st) gives the dispersion relation
s)
((
We can write the pressure in term of z(x,y), taking g(h) into account
d2z dg p ) - γ 2 + γK dz dx
(5)
in which the first term on the right-hand side is the curvature in the xz plane, the second is in the yz plane, and the third is pressure due to the interaction between substrate and film. The rim thickness is high enough (>100 Å) to neglect the polar contribution of the water-mica (14) Lipson, S. G. Pattern formation in drying water films. Phys. Scr. 1996, T67, 63. (15) Ben-Jacob, E. From snowflake formation to the growth of bacterial colonies. part I: Diffusive patterning in non-living systems. Contemp. Phys. 1993, 34, 247-273. (16) Ben-Jacob, E. The formation of patterns in non-equilibrium growth. Nature 1990, 343, 523-530. (17) Ihle, T.; Muller Krumbhaar, H. Diffusion-limited fractal growth morphology in thermodynamical two-phase systems. Phys. Rev. Lett. 1993, 70, 3083-3086.
lw
2
)
6S d0 2 h3 dK 2 q + γ -q4 q (6) dz Kη(3/K + h) h4
)
The maximum value of s is obtained for finger wavelength given by
(
ζ ) 2π/q ) 2π/ -
Ignoring the evaporation of the water, which is slow compared to the rate of development of the instability, allows us to assume conservation of water mass, giving
-
Figure 2. Interferograms of a propagating finger instability (right to left) with ∆t )1.7 s between frames.
)
lw 2 dK 6S d0 + dz γh4
1/2
(7)
The relative importance of the two terms in (7) must be judged according to the values of the parameters. We approximate |dK/dz| as K/h. For typical values of Slw ) 0.015 J/m2, d0 ) 0.2 nm,18 a rim height (h ∼ 300 Å), and curvature K ∼ 200 m-1, the first term in (7) is negligible, and the fingering wavelength is determined mainly by the rim thickness, as was anticipated theoretically by Kargupta and Sharma.19 Experiments The experiments were carried out in a high vacuum system, into which water vapor could be introduced by evaporation from a separate chamber containing distilled water. The temperature, T, of the copper 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 the cell walls to the sample’s surface and vice versa if Ts > T. The experiments were carried out around T ) 0 °C on mica substrates prepared by cleavage in situ and then exposed to water vapor. The pattern created during evaporation was observed by a 10-magnification reflection microscope. The water (18) Israelachvili, J. Intermolecular and surface forces, 2nd ed.; Academic Press: New York, 1992. (19) Kargupta, K.; Sharma A. Dewetting of Thin Films on Periodic Physically and Chemically Patterned Surfaces. Langmuir 2002, 18 (5), 1893-1903.
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Figure 3. Dependence of mean distance between fingers (ζ) on the mean rim height in two independent experiments.
film thickness was measured using a two-beam interferometric technique, involving reflections at the vaporfilm and film-mica interfaces creating an imaging interferogram. The approach is suitable for measurement of film thicknesses in range of 100 Å < h < 400 Å (a wavelength of λ ) 546 nm was used), where the total reflected light intensity varies linearly with h and thickness measurement is most sensitive. The intensity variations observed are due to interference alone and not to vignetting of relected light, since typical surface slopes are less than 10-2 rad, and the NA (numerical aperture) of the optical system used was 0.1. Figure 2 shows an example of a developing instability. It originates from a step on the mica, out of the field of view, and propagates with an increasing mean distance during this period, between fingers (ζ). The bump’s mean radius of curvature decreases from 7 to 3 mm, which in regular Rayleigh instability should cause a corresponding decrease in the longitudinal instability wavelength. However, for known values of Slw ) 0.015 J/m2 and d0 ) 0.2 nm,18 the low typical height of the rim (
