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A Qualitative Theory of Wimples in Wetting Films Roumen Tsekov*,† and Olga I. Vinogradova*,‡,§ Institute of Physical Chemistry, University of Karlsruhe, 76131 Karlsruhe, Germany, Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany, and A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119991 Moscow, Russia Received August 10, 2005. In Final Form: October 31, 2005 It has long been known that hydrodynamic pressures in a thin draining liquid film can cause inversion of the curvature of a drop surface as it approaches another surface, creating a so-called dimple. However, it was recently found that a different shape, dubbed a wimple, can be formed if a fluid drop, which is already in the field of repulsive surface forces, is abruptly pushed toward the wall. The drop shape might include a central region in which the film remains thin, surrounded by a ring of greater film thickness bounded at the outer edge by a barrier rim. Here we present a qualitative theory of the wimple formation. It is shown that this is mainly driven by the film hydrodynamics, and a qualitative criterion for the wimple/dimple transition is derived.
1. Introduction It has long been known that during the approach of a fluid drop to a solid, or of two fluid drops toward each other, hydrodynamic pressures in an intervening fluid medium can cause the drop’s curvature to be inverted, creating a so-called dimple in the surface of the drop(s).1-5 The dimpling is well understood and has been predicted both numerically and theoretically.6-11 However, a completely new observation has been made recently. The new result is that if a drop starts close to contact with a solid surface, being separated by a wetting film, and is then abruptly pushed close to a solid, a more complex shape, a wimple, can be observed at the initial stage of approach.12 In a wimple, the drop surface buckles into a depressed ring with the central peak (see Figure 1). To the best of our knowledge, this phenomenon, which may have important consequences for drop coalescence, micro- and nanofluidic devices, and perhaps may even change our notions about membrane fusion, has not previously been noted experimentally or predicted theoretically. In this paper, we make an attempt to investigate the origin of wimpling. Our goal is to obtain a simple criterion for wimple formation in the first stage of the wetting film evolution. A precise discussion of this phenomenon requires a numerical solution of nonlinear equations of * Corresponding authors. E-mail:
[email protected] (R.T.);
[email protected]. (O.V.). † University of Karlsruhe. ‡ Max Planck Institute for Polymer Research. § Russian Academy of Sciences. (1) Derjaguin, B. V.; Kussakov, M. M. Acta Phys. Chim. URSS 1939, 10, 25-30. (2) Platikanov, D. J. Phys. Chem. 1964, 68, 3619-3624. (3) Hartland, S. Chem. Eng. Sci. 1969, 65, 82-89. (4) Joye, J. L.; Miller, C. A.; Hirasaki, G. J. Langmuir 1992, 8, 30833092. (5) Connor, J. N.; Horn, R. G. Faraday Discuss. 2003, 123, 193-206. (6) Frankel, S. P.; Mysels, K. J. J. Phys. Chem. 1962, 66, 190-191. (7) Dimitrov, D. S.; Ivanov, I. B. J. Colloid Interface Sci. 1978, 64, 97-106. (8) Yiantsios, R. G.; Davis, R. J. Fluid Mech. 1990, 217, 547-573. (9) Abid, S.; Chesters, A. K. Int. J. Multiphase Flow 1994, 20, 547573. (10) Tsekov, R.; Ruckenstein, E. Colloids Surf., A 1994, 82, 255261. (11) Tsekov, R.; Letocart, P.; Evstatieva, E.; Schultze, H. J. Langmuir 2002, 18, 5799-5803. (12) Clasohm, L. Y.; Connor, J. N.; Vinogradova, O. I.; Horn, R. G. Langmuir 2005, 21, 8243-8249.
capillary hydrodynamics. Here, we try to avoid this complicated approach by using simple (but tentative) arguments to clarify the major role of hydrodynamic pressure modulation. Our main result is an analytical expression for the film profile, which enables us to parametrically study the wimple and dimple selforganizations and the transition between these two viscous-dissipative structures. 2. Analysis We consider the situation with a drop that is initially close to the solid wall, separated by a thin wetting film of background fluid. Then the drop and the solid are driven toward each other. The initial flattened film changes the form, so that the local thickness H is becoming nonuniform. In general, it could be presented as a sum, H ) h - ζ, of the average film thickness h and the local deviation ζ. If ζ is relatively small (ζ , h) and flat (|∂rζ| , 1) the shape of the film can be described by the following normal force balance:13
σ∂r(r∂rζ)/r + (Π′ - ∆Fg)ζ + ∆P ) 0
(1)
in which σ is the surface tension on the film/drop interface, Π′ is the derivative of the disjoining pressure Π on the average film thickness, and ∆F is the difference in the mass densities of the drop and the background fluid. The driving pressure ∆P being equal to the difference in the normal components of the pressure tensor in the drop and the film, respectively, depends substantially on the bulk and interfacial hydrodynamics. In the frames of the limit ζ , h, one can expand the driving pressure in a series of ζ:
∆P ) ∆P0 + aζ + ‚ ‚ ‚
(2)
The pressure difference ∆P0 refers to a flattened film with ζ ) 0. Since the main friction appears in the film, one could express ∆P0 from the Reynolds theory of hydrodynamic lubrication as
∆P0 ) (pσ - Π)(2r2/R2 - 1)
(3)
in which R is the film radius, and pσ is the capillary (13) Horn, R. G.; Bachmann, D. J.; Connor, J. N.; Miklavcic, S. J. J. Phys.: Condens. Matter 1996, 8, 9483-9490.
10.1021/la052178a CCC: $30.25 © 2005 American Chemical Society Published on Web 11/11/2005
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Figure 1. The interference fringes reflecting the wimple shape evolution. (original data of L. Y. Clasohm, which led to results published in ref 12).
Figure 2. Plot of the dimensionless film thickness perturbations y vs x from eq 5 for b ) 5 (dimple, dashed curve) and b ) 45 (wimple, solid curve).
pressure in the meniscus adjacent to the film. Note that pσ is not the equilibrium pressure, and it could evolve in time as well as the average film thickness h. Substituting expressions 2 and 3 in eq 1 leads to
σ∂r(r∂rζ)/r + (a + Π′ - ∆Fg)ζ ) (pσ - Π)(1 - 2r2/R2) (4) The analytical solution of this equation under the 2 obvious conditions ∫R 0 ζdr ) 0 and (∂rζ)r)0 ) 0 is
y ) 8/b2 + (1 - 2x2)/b - 4J0(xbx)/bxbJ1(xb) (5) in which y ) ζσ/(pσ - Π)R2, x ) r/R, and b ) (a + Π′ ∆Fg)R2/σ. We remark that, in the majority of cases, the disjoining pressure gradient is much larger than the gravity term, so that the latter can safely be ignored. 3. Discussion In Figure 2, y from eq 5 is plotted as a function of x for b ) 45 (wimple) and b ) 5 (dimple), and these plots are quite similar to the experimentally observed plots presented in Figure 1 in the second and forth panels, respectively. Obviously, at the beginning of the process, a is relatively large, and for this reason a wimple forms. When the process advances, the system relaxes, and the value of a decreases. Hence, a wimple transforms to a dimple. Of course, at the end, the equilibrium takes place with pσ ) Π(he). In this case, a also tends to zero, and the equilibrium film is flattened due to the highly negative value of Π′(he). In general, a is a function of the film thickness and the rate of drainage. The exact dependence follows from the rigorous nonlinear hydrodynamics of the film and the drop, coupled by relevant boundary conditions.
Figure 3. Dimensionless curvature at the film center as a function of b.
To discriminate better between the formation of a dimple and a wimple, the dimensionless curvature at the center of the film is calculated using eq 5:
[∂x(x∂xy)/x]x)0 ) (4/b)[xb/J1(xb) - 2]
(6)
As seen here, it depends only on the parameter b, and this dependence is plotted in Figure 3. If b is smaller than λ22, where λ2 ) 3.83 is the second root of the Bessel function J1, the curvature from eq 6 is positive. Hence, in this case, a dimple is formed. If λ22 < b < λ32 ≈ 49, the curvature becomes negative, thus pointing out the appearance of a wimple. Obviously, the wimple formation requires a large value of b, which can originate from strong hydrodynamics with large a. The discontinuities in Figure 3 are due to the zero values of J1. However, the limit J1(xb) f 0 also corresponds to the discontinuity of the thickness perturbation ζ, which violates our linearized analysis that is valid for ζ , h. Hence, we are not able to exactly say what happens with the films in the dimple/wimple transition region. An interesting region is b > 49. Here, the curvature in the film center becomes positive again. However, this new kind of dimple possesses several minima and maxima of the local film thickness. Since such multidimple viscousdissipative structures appear at very high values of b, special arrangements should be carried out to generate the necessary hydrodynamic flow with an extremely large value of a. For this reason, these structures are not reported yet in the literature. However, they have a very strong impact on the rate of drainage of the films.14 Note that, since the formation of wimples requires large b, our qualitative analysis, which is accurate only at small a, is valid only for relatively large drops/large films14 with a radius larger than x12σh/(pσ-Π). The wimpling for (14) Tsekov, R.; Evstatieva, E. Prog. Colloid Polym. Sci. 2004, 126, 93-96.
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smaller films would be possible only at very large a, which is beyond the scope of the asymptotic theory and this paper. Finally, we would like to stress that, although the shape evolution reflects mostly the important pressure changes caused by hydrodynamics, another way to increase b is to have a low σ and/or large R. It is well known that large films during their drainage exhibit various dissipative structures, such as rings, channels, and even broken radial symmetry. We cannot exclude that these structures have a physical origin similar to that described here. In summary, we have provided simple theoretical arguments showing that the commonly accepted notions about the shape evolution of a drop approaching a solid surface are incomplete. Besides the well-known phenomenon of dimpling, other structures, such as the recently discovered wimple or even more complex rippled shapes,
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can be formed. Although driven mainly by hydrodynamics, these dissipative structures reflect a complex interplay between the surface tension of the drop and the disjoining pressure in the wetting film. The existence of these structures and a possibility of transitions between them can have dramatic implications for coalescence phenomena, flows in micro- and nanofluidic devices, and more. Acknowledgment. This work was supported by a DFG priority program “Micro and Nanofluidics” (Vi 243/ 1-1). We thank L. Y. Clasohm for providing the interference fringes reflecting a film evolution, and R. G. Horn for helpful remarks on this manuscript. LA052178A
