Langmuir 2007, 23, 9213-9220
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Different Surface Corrugations Occurring During Drainage of Axisymmetric Thin Liquid Films Nicolas Anton†,‡ and Patrick Bouriat*,† Laboratoire des Fluides Complexes, UMR CNRS 5150, UniVersite´ de Pau, BP 1155, 64013 Pau Cedex, France, and Inserm U646, Inge´ nierie de la Vectorisation Particulaire, UniVersite´ d’Angers, Angers, F-49100 France ReceiVed March 9, 2007. In Final Form: April 30, 2007 This paper deals with the different surface corrugations observable during the thinning of axisymmetric thin and large aqueous films, stabilized by saponin. The films are observed using a thin film balance under a constant driving pressure. This device allows measurement of the thicknesses of the film surface shapes arising all along the drainage, as well as the following-up of their evolution before equilibrium is attained. Depending on the electrolyte (NaCl) concentration, three different sorts of corrugation were originally observed in such suspended thin liquid films. At the lowest NaCl concentrations, corresponding to repulsive potential between film walls, only the hydrodynamic corrugations deformed the film surfaces. Regarding the higher NaCl concentrations, when van der Waals forces become predominant, and following the thickness of the first-established thin film, the experiments disclose either that the thinner films are broken up by spinodal decomposition, or that the thicker ones are broken by nucleation and growth of black film. In addition, an original aspect of these works appears in the fact that these observations of the spontaneous decomposition of suspended thin films are relatively similar to those usually described for dewetting experiments on solid substrates, and are well fitted by the existing theoretical models.
Introduction Foam and emulsion drainage is of fundamental importance in a large number of applications, such as detergency, cosmetics, or even pharmaceutics. Investigations on drainage are often tackled by regarding the thinning of thin liquid films, separating two bubbles in a foam, or two flocculated emulsion droplets. These circular thin films are formed when bubbles or droplets approach each other and deform; they constitute the last barrier before coalescence. Film thinning can be globally described in three stages: (i) First, when the film is still relatively thick (approximately thicker than 100 nm), its drainage is governed by the driving pressure ∆p, which is equal, in this case, to the hydrodynamic pressure ∆p′ (i.e., the pressure difference between the film center and periphery, due to the viscous flow). (ii) Second, as the draining film becomes thinner (below 100 nm), the disjoining pressure Π, defined as the repulsive force between the two interfaces per unit area, becomes significant and can accelerate (Π < 0) or slow down (Π > 0) film drainage as ∆p ) ∆p′ + Π. (iii) Finally, drainage stops when ∆p′ ) 0, that is, when ∆p ) Π, and thus the thin film is stabilized at its equilibrium thickness, eeq. Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory1,2 provides a disjoining pressure calculation as a function of the film thickness ef, only considering the long-range components of the intersurface forces, that is, electrical double-layer repulsion and van der Waals attraction. Therefore, on one hand, if electrostatic forces are strong enough to outweigh the van der Waals ones, the resulting equilibrium film remains relatively thick (generally between 10 and 50 nm), and is called a common * To whom correspondence should be addressed. E-mail address:
[email protected]. † Universite ´ de Pau. ‡ Universite ´ d’Angers. (1) Derjaguin, B. V.; Landau, L. D. Acta Physicochim. USSR 1941, 14, 633662. (2) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of the Lyophobic Colloids; Elsevier: Amsterdam, 1948.
black film.3 On the other hand, in the case of weak electrostatic repulsion, the film thins until it is stabilized by short-range repulsive forces. A very thin film of a few nanometers, called Newton black film or Perrin film4 is then formed. More recently, non-DLVO forces from diverse phenomena were reported.5 For instance, the oscillatory structural forces6,7 due to the presence of micelles within the films, induce stepwise thinning in which each stabilized thickness corresponds to the stratification of micelle layers. In addition to disjoining pressure, it is also necessary to consider capillary and hydrodynamic phenomena in order to describe drainage dynamic. Owing to the complexity of these involved phenomena, a thinning film rarely exhibits only a uniform thickness, but generally presents many surface corrugations.8 Moreover, since the film hydrodynamic is largely influenced by the film surface rheology, two cases have to be distinguished: (i) The first one concerns fluid surfaces, generally obtained with low molecular weight surfactants. Surface fluidity favors planeparallel film drainage in comparison with rigid interfaces, by decreasing the velocity gradient within the film and therefore the viscous resistance of fluid flow. Then, during the film thinning, unstable dimples immediately and randomly move and are rapidly sucked in the Plateau border creating a plane-parallel film. (ii) The second case concerns the films that exhibit rigid surfaces, generally observed with surface-active macromolecules. Descriptions of such film drainage are based on the Reynolds model9 which provides the driving pressure needed to ensure the drainage, at a given velocity, for plane-parallel liquid film held between two disks. This model was widely used and extended, including (3) Jones, M. N.; Mysels, K. J.; Sholten, P. C. Trans. Faraday Soc. 1966, 62, 1336-1348. (4) Perrin, J. Ann. Phys. 1918, 10, 160-184. (5) Petsev, D. N.; Denkov, N. D.; Kralchevsky, P. A. J. Dispersion Sci. Technol. 1997, 18, 647-659. (6) Nikolov, A. D.; Wasan, D. T. J. Colloid Interface Sci. 1989, 133, 1-12. (7) Bergeron, V. Curr. Opin. Colloid Interface Sci. 1999, 4, 249-255. (8) Ivanov, I. B.; Dimitrov, D. S. Thin Liquid Films; Marcel Decker: New York, 1988. (9) Reynolds, O. Philos. Trans. R. Soc. London 1886, 177, 157-234.
10.1021/la700698d CCC: $37.00 © 2007 American Chemical Society Published on Web 06/13/2007
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the influence of the potential interfacial properties of the film surface as well.10-13 It was thus established that, in the case of highly viscoelastic surface, thin liquid films exhibit an axisymmetric profile during the drainage. The influence of DLVO conditions on dimple formation was studied by Joye et al. 14 These authors were able to predict successfully, as a function of time, the axisymmetric shape of the dimple generally formed at the early stage of thin film formation. Then, it was shown, on one hand, that when van der Waals forces are negligible compared to the repulsive electrostatic ones, the dimple initially formed drains toward the Plateau border and rapidly disappears, leading to the stabilization of a planeparallel film. On the other hand, when van der Waals forces are predominant, the film near the Plateau border is intensively pinched, creating a very thin barrier impeding the dimple to drain. It forms a bispherical lens in a metastable equilibrium with a very thin parallel film. The study of ref 14 concerns small radius films, that is, where corrugation length is smaller or on the order of the film radius. For practical use, it would also be important to study the evolution of larger axisymmetric films, where several dimples can be formed. The current paper precisely studies the different regimes governing the drainage of such large axisymmetric thin films made with saponin. Observations of films thinning are undertaken with a thin film balance. Playing on the repulsive component of the disjoining pressure, by a gradual screening with electrolyte, we disclose here three different regimes of surface corrugations that can occur in draining axisymmetric films. The great analogy between the phenomenon variety encountered in such systems (spinodal decomposition, nucleation and growth, and rim formation) and those encountered with the problem of dewetting solid substrates is also discussed. Experimental Section Materials. Saponin was purchased from Sigma and used as received. It is a technical grade product, extracted from Quillaja saponaria bark, a tree also known as the “soap tree”. Deep NMR and chromatographic studies on Quillaja saponin structure were performed by Nyberg et al.,15 disclosing the presence of a triterpenoid nucleus as a hydrophobic component, and the side chains of carbohydrates (i.e., sugar) as hydrophilic components. As a consequence, this molecule acts as a surfactant; it appears as an excellent foaming agent and forms very stable foams. This saponin powder is dissolved under a weak magnetic stirring, in Ultrapure water obtained from a MilliQ filtration system (Millipore, SaintQuentin-en-Yvelines, France). Afterward, the saponin aqueous solution was filtered twice on a 0.2 µm Millipore filters, and a selected amount of electrolyte, sodium chloride from Prolabo (Fontenay-sous-Bois, France), was finally added. Methods. Thin Film Balance. The experimental apparatus worked out for studying the thinning of a thin liquid film under a controlled driving pressure is schematically presented in Figure 1. It is largely inspired by the thin film balance, originally created by Mysels, Scheludko, and Exerowa.16-18 The liquid film made with the saponin aqueous solution is established within a hole (of 2 mm diameter) (10) Ivanov, I. B.; Dimitrov, D. S. Colloid Polym. Sci. 1974, 252, 982-990. (11) Singh, G.; Hirasaki, G. J.; Miller, C. J. Colloid Interface Sci. 1996, 184, 92-105. (12) Danov, K. D.; Valkovska, D. S.; Ivanov, I. B. J. Colloid Interface Sci. 1999, 211, 291-303. (13) Zapryanov, Z.; Malhotra, A. K.; Aderangi, N.; Wasan, D. T. Int. J. Multiphase Flow 1983, 9, 105-129. (14) Joye, J. L.; Miller, C. A.; Hirasaki, G. J. Langmuir 1992, 8, 3083-3092. (15) Nyberg, N. T.; Kenne, L.; Ro¨nnberg, B. S. B. Carbohydr. Res. 2000, 323, 87-97. (16) Mysels, K. J.; Jones, M. N. Discuss. Faraday Soc. 1966, 42, 42-50. (17) Exerowa, D.; Scheludko, A. Chim. Phys. 1971, 24, 47-50. (18) Exerowa, D.; Kolarov, T.; Khristov, K. H. R. Colloids Surf. 1987, 22, 171-185.
Anton and Bouriat
Figure 1. Schematic representation of the “thin film balance” for observing the thinning of suspended liquid films under a controlled driving pressure. drilled in a porous glass disk (the pore diameters are from 10 to 20 µm). The porous glass was previously submerged into the aqueous solution to ensure its saturation. The pressure of the Plateau border (PP.b.) is directly controlled with a capillary tube welded to the porous disk. Then, the disk is shut up in a transparent cell, hermetically sealed, which also contains some saponin solution in order to prevent film evaporation. The inner gas pressure (Pgas) is controlled by another capillary tube. These two capillary tubes have the same diameter of 4 mm. Therefore, the driving pressure ∆p of the thin film is given by ∆p ) Pgas - PP.b., which immediately relates the driving pressure and the height difference between the liquid level in the capillary tubes ∆H, by ∆p ) ∆Fg(∆H + Hfilm)
(1)
where ∆F is the density variation between aqueous solution and gas, g is the gravitational constant, and Hfilm is the vertical relative position of the film compared to the surface of the solution. A thick film is initially formed in the hole, and the thinning suddenly starts when a driving pressure is applied to the film through the increase in ∆H according to eq 1. In that way, for each experiment, the value of ∆H was adjusted in a similar manner (by adding liquid to the external capillary tube), up to the wanted driving pressure value, that is, ∆p is fixed around 1200 Pa in this study. This relatively low value of the driving pressure was chosen to ensure the slowness of the hydrodynamic phenomena and to approach driving pressures existing in foam columns. Film thickness is determined by an interferometric method, initially developed by Scheludko and Platikanov,19,20 where thin film acts as a Pe´rot-Fabry reflecting blade. An incident sodium light considered to be monochromatic (λ0 ) 589 nm) arrives on the film at normal angle. The reflective intensity I is collected normally and provides film thickness ef according to ef )
(
)
λ0 ∆ arcsin 2πn 1 + 4R(1 - ∆)/(1 - R)2
1/2
(2)
where n is the refractive index of the solution, ∆ is the normalized reflected intensity given by ∆ ) (I - Imin)/(I - Imax) (with Imin and Imax respectively being the minimum and maximum possible values for the reflected intensity), and, finally, R is the reflection factor given by R ) (n - n0)2/(n + n0)2 (with n0 being the refractive index of the gaseous atmosphere surrounding the film). So, in order to appreciate the different surface shapes, such as dimples or surface corrugations that occur during film drainage, theoretical variation of the normalized reflected intensity ∆ as a function of film thickness ef, according to eq 2, is reported in Figure 2. Therefore, a succession of minima and maxima reflects a variation in the film thickness, and, for instance, a lens-like shape on the film will provide a succession of concentric bright and dark rings. On the other hand, a succession of weakly contrasted, clearer and darker (19) Scheludko, A. Kolloid Z. 1957, 155, 39-44. (20) Scheludko, A.; Platinakov, D. Kolloid Z. 1961, 175, 150-158.
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specific area of saponin when adsorbed at the air/water interface at the cmc; assuming a single solute in solution, Γ can be determined from the Gibb’s adsorption equation:
Γ)-
d(γ) 1 NakBT d(ln C)
(3)
Results and Discussion
where C is the saponin concentration in the bulk aqueous phase, Na is the Avogadro number, kB is the Boltzmann constant, T is the temperature, and γ is the interfacial tension. One finds Γsapo. ∼ 3.25 mg‚m-2, a value fully consistent with similar studies in the literature for Quillaja saponin,21,22 assuming a single solute in solution. This value is compared with the surface excess concentration for typically fluid and rigid interfaces, that is, low molecular weight amphiphile interfaces (e.g., sodium dodecyl sulfate, SDS) and protein interfaces (e.g., human serum albumin, HSA), respectively. Indeed, for SDS, a value of ΓSDS ∼ 0.92 mg‚m-2 was measured (data not shown), and, for HSA, the literature provides23 a value of ΓHSA ∼ 11.6 mg‚m-2. These macromolecules form a relatively compact interface called a close-packed interface; it is due to both their higher molecular weight and the cooperative nature of their adsorption.24 In addition, the interfacial behavior of HSA involves global structure and conformation changes, which globally favors the increase in the surface concentration (in comparison with saponin). In that way, the surface concentration of saponin at saturation appears as being intermediate between those for low molecular weight surfactants and macromolecules; it is thus also coherent with a close-packed molecular interfacial organization. In the following sections, for all experiments, a saponin concentration in water slightly above the cmc, C ) 2.0 g‚L-1, was chosen to ensure the presence of such a saturated closepacked, interfacial layer and thus an axisymmetric film drainage. Hydrodynamic Corrugations. Figure 4 presents a 2 g‚L-1 saponin film thinning, containing 10-3 M NaCl, under a driving pressure of about 1200 Pa. The four pictures were taken at 51, 103, 131, and 179 s after having established the constant driving pressure. The first remark concerns the darker regions located at film periphery. These regions must not be interpreted as having thinner thicknesses but as being non-normal to the incident light. The observed films are distorted near their periphery owing to the high applied driving pressure; the resulting Plateau border is so small that the film periphery comes into contact with the porous glass granular wall. Thus, a part of the film peripheral region appears warped, like a “warped bicycle wheel”. Figure 4a presents the first established thin film. It exhibits a succession of luminous and dark rings. Since the reflected intensity difference between these rings is relatively weaker than Imax - Imin, this intensity fluctuation indicates an axisymmetric undulation of the surfaces, as illustrated in Figure 4e. These surface corrugations are still existing throughout all the drainage (see Figure 4b,c), and definitively disappears when the drainage stops (see Figure 4d). It follows therefrom that these surface corrugations result from the hydrodynamic flux within the film. Although the exact origins of these surface undulations differ among the authors,25,26 Tsekov27 quantified the influence of such
Saponin Behavior at the Air/Water Interface. A study of the equilibrium surface tension γ of saponin aqueous solutions at the air/water interface was performed as a function of saponin concentration, and reported in Figure 3. Typically, a logarithmic decrease in γ is observed until it reaches a steady state above the critical micelle concentration (cmc) between 0.5 and 0.8 g‚L-1. Such logarithmic decrease allows one to evaluate the
(21) Mitra, S.; Dungan, S. R. J. Agric. Food Chem. 1997, 45, 1587-1595. (22) Mitra, S.; Dungan, S. R. Colloids Surf., B 2000, 17, 117-133. (23) Chen, P.; Prokop, R. M.; Susnar, S. S.; Neumann, A. W. In Proteins at Liquid Interfaces; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1998; Chapter 8, pp 303-339. (24) Murray, B. M. In Proteins at Liquid Interfaces; Mo¨bius, D., Miller, R., Eds.; Elsevier: Amsterdam, 1998; Chapter 5, pp 179-220. (25) Malhotra, A. H.; Wasan, D. T. AIChE J. 1987, 33, 1533-1541. (26) Ruckenstein, E.; Sharma, A. J. Colloid Interface Sci. 1987, 119, 1-13.
Figure 2. Theoretical variation of the normalized reflected intensity ∆ as a function of film thickness ef, according to eq 2.
Figure 3. Equilibrium surface tension at the air/water interface for various bulk concentrations of saponin. concentric rings will be interpreted as a radially undulated film. Film thinning is recorded by means of a CCD camera, and the video file is processed to give the values of I, Imax, and Imin. Thicknesses of the different zones of the film are thus precisely evaluated as a function of time. As schematically presented in Figure 1, both incident and reflected lights have to exhibit a normal angle with regard to the film. Practically, it was done by using a binocular microscope. One ocular was removed in order to set up the sodium light instead, and a CCD video camera was adapted to the other one. Finally, the transparent cell was put down on a plate, for which the horizontality can be adjustable by means of three micrometric screws, thus incident and reflected angles can be obtained very close to the normal one. Furthermore, the raw images provided by the camera are processed pixel-by-pixel in order to collect the mean intensity of the selected region of the film. The maximum and minimum reflected intensity (Imax and Imin) are beforehand measured by analyzing the video sequence when the thin film is forming and thus when I passes through pronounced maxima and minima, as it is shown in Figure 2. Surface Tension Measurements. Surface tension measurements were performed with a drop tensiometer device (Tracker, Teclis, Longessaigne, France) by analyzing, at equilibrium, a pendant drop of the aqueous saponin solution in air. Temperature was maintained at 25 °C, and the surface tension was allowed to stabilize before measurement, about 1 h for the lowest concentrations. The measurements were performed three times.
9216 Langmuir, Vol. 23, No. 18, 2007
Anton and Bouriat
Figure 5. Disjoining pressure as a function of the film thickness described by the DLVO theory and calculated using eq 7 with ψ ) -30 mV. The three electrolyte concentrations presented here are those used to attain the three different modes of surface corrugation. The curve labeled van der Waals was calculated with ψ ) -30 mV.
Figure 4. (a-d) Formation and thinning of a saponin liquid film, saponin concentration of 2.0 g‚L-1, [NaCl] ) 10-3 M, and ∆p fixed at 1200 Pa. (e) Schematic of the transversal cross-section of the corrugated thin film.
corrugations on film drainage velocity V and found V ) n3/2VRe, where VRe is the drainage velocity of a plane Reynold’s film and n is the number of bright and dark concentric domains observed in the film. This last expression clearly shows that these corrugations favor film drainage. The observations in the present study (Figure 4) give the experimental number of domains nexp ) 11. Remarking that the thin liquid film corrugation versus the radial coordinate was related to the zeroth order Bessel function, Tsekov found that n and the wavenumber q of surface corrugation in the film were approximately linked by
qR n) 4
(4)
where R is the film radius (the hole radius in our case). An estimation of the surface wavenumber was proposed by this author as
q)
( ) 24ηV γef4
1/4
(5)
where η is the viscosity and V is the thinning rate of the film. Then it follows, accordingly, the relationship linking the number of domains and the experimental conditions of film formation:
n)
R 24ηV 1/4 4ef γ
(
)
(6)
Using the experimental average film thickness ef ) 72 nm and the experimental thinning rate value Vexp ) 0.4 nm‚s-1 (both determined from Figure 4), the use of eq 6 provides n ∼ 14 for a 1 mm film radius of surface tension γ ) 35 mN‚m-1. The coherence between these two results confirms that the film thinning presented in Figure 4 is governed by hydrodynamic corrugations. Furthermore, it is noteworthy that the experimental (27) Tsekov, R. Colloids Surf., A 1998, 141, 161-164.
thinning rate Vexp corresponds to the one given by the Reynolds expression, that is, VRe ) (2ef3∆p)/(3ηR2), which gives 0.37 nm‚s-1 (with η ) 0.8 cp). On one hand, it proves that the air/ water interface is effectively rigid in the presence of adsorbed saponin. On the other hand, this result involves the fact that the Tsekov’s formula (V ) n3/2VRe) is not applicable in our case, but, nevertheless, it has no repercussions on the validity of eqs 5 and 6, since they were established independently from the former equation. One explanation could be that, this formula cannot be applied to this system because the surfaces may not be sufficiently undulated, and then the film-thinning behavior would be more approached by the Reynolds model. The thickness of the flat equilibrated film obtained when drainage stops (such as in Figure 4d) is ef ) 21 nm. The equilibrium thickness is reached precisely when ∆p ) Π. This provides the determination of the surface potential, assuming that Π is correctly described by the DLVO theory. For a 1:1 symmetrical electrolyte such as NaCl, the DLVO theory gives
Π ) 64nvkBT tanh2
( ) ( )
ef A eψ exp - -1 4kBT κ 6πef3
(7)
where nv is the electrolyte concentration, e is the electron charge, ψ is the electrostatic surface potential, ef is the aqueous film average thickness, A is the Hamaker constant (equal to 3.7 × 10-20 Nm for the air/water/air system), and κ-1 is the Debye length given by κ-1 ) x0rkBT/nve2, with 0 and r being the permittivity of free space and the dielectric constant of water, respectively. In our experimental conditions and results (ef ) 21 nm, Π ) 1200 Pa, and [NaCl] ) 10-3 M), the electrostatic surface potential value is given by eq 7 as ψ ) -30 mV. Accordingly, disjoining pressure can be deduced as a function of film thickness and electrolyte concentration from eq 7. Figure 5 presents such theoretical curves relative to the NaCl concentrations used to typically obtain the three different regimes of surface corrugation presented in this paper. The electrostatic repulsive forces are predominant for [NaCl] ) 10-3 M, and they are outweighed by the van der Waals contribution at 10-2 and 2 × 10-2 M for thicknesses above 20 nm. In this latter case, the disjoining pressure reduces to van der Waals contribution, that is, Π ) -A/6πef3. Regime of Spinodal Decomposition. Figure 6 presents the thinning of a film formed for ∆p ) 1200 Pa, with an aqueous solution containing 10-2 M NaCl and a saponin concentration of 2 g‚L-1.
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Langmuir, Vol. 23, No. 18, 2007 9217
sufficiently narrow, they are also broken-up by the PlateauRayleigh instabilities (see Figure 6d,e). Finally, the bright spots then formed are forced, due to black film progression, to gather the trapped liquid into small lenses in a metastable equilibrium with the continuous black film. The profile of these lenses is given to be circular by the Laplace law, and the equilibrium contact angle θE is provided by the FrumkinDerjaguin theory:28
cos(θfE) ) 1 +
∫0Π(e ) efdΠ(ef)
1 2γ
eq
(8)
where ef is the thickness of the black film surrounding the spherical lens. Since the surrounding black film prevents central liquid drainage, and since the instabilities are spontaneously amplified with time, these surface corrugations cannot be related to the hydrodynamic drainage of the central thick film. Theoretical developments regarding the spontaneous amplification of surface corrugations were first introduced by Frenkel29 and Vrij,30 and more recently reported by Valkovska et al.31 According to Frenkel,29 the corrugation amplification is the consequence of the film instability in the case of (dΠ/def) > 0 (precisely our case for thick films at 10-2 M NaCl; see Figure 5). The presence of surface corrugation gives rise to two antagonist pressures: the Laplace pressure and the disjoining pressure. Then, for a small radial deformation of the film u(r) defined as ef ) e0 + u(r) (e0 being the average film thickness), the Laplace pressure is given by
PL ) -
γ 2 ∇ u(r) 2
(9)
and the disjoining pressure variation is given by
δΠ ) Figure 6. The formation and thinning of a saponin liquid film containing 2.0 g‚L-1 of saponin and 10-2 M of NaCl, under a driving pressure of 1200 Pa.
Figure 6a presents the formation of a film thicker than those previously observed in Figure 4. This film, whose thickness is about 100 nm, is separated from the Plateau border by a thinner black film of about 30 nm thickness. This black film thickness diminishes and attains a stable value corresponding to the equilibrium thickness relative to Π ) 1200 Pa for 10-2 M NaCl concentration (see Figure 5). Such peripheral black film formation has already been observed and analyzed in detail by Joye et al.,14 who showed that this phenomenon is a direct consequence of the predominance of the attractive van der Waals forces. Due to the direct Plateau border proximity, film periphery rapidly drains, forming the annular black film. This black film acts as a barrier that impedes the drainage of the central thick film (also called a pancake). In spite of this barrier, the trapped thin film still thins to attain a thickness average of about 60 nm at 186 s in Figure 6b. At this time, the film barrier has attained its equilibrium thickness, and the central thick film is then prevented from further drainage. Afterward, the evolution of the pancake trapped in the film by this concentric black film appears fundamentally different from the one previously observed in Figure 4 once the drainage is stopped. Film corrugations are now amplified with time instead of vanishing (see Figure 6b,c). In Figure 6d, they become so amplified that the thinner rings approach their equilibrium thickness, and they break-up to form stable black films. The brighter (thicker) zones gradually collect the liquid from the black film expansions, and then, when the bright rings become
dΠ u(r) def
(10)
Equalization of these two pressures is only possible when (dΠ/ def) > 0 and provides a differential equation whose solution is a zeroth-order Bessel function of the form
u(r) ∝ J0(r/ξ)
(11)
where ξ is the characteristic length given by
ξ)
x
γ dΠ 2 def
(12)
Actually, during the formation of film corrugations, the increase of disjoining pressure is not exactly balanced by the Laplace pressure, but it is given by PL ) δΠ + ∆p′, (∆p′ being the hydrodynamic pressure). According to the local mass conservation of draining thin liquid films, the local thickness variation is provided, in that case, by
∂ef 1 ) ∇(ef3∇(∆p′)) ∂t 12η
(13)
and, then, the temporal evolution of the film deformation is given by (28) Morrow, N. Interfacial Phenomena in Petroleum RecoVery; Dekker: New York, 1991; Vol. 36, pp 23-76. (29) Frenkel, J. Kinetic Theory of Liquids; Dover: New York, 1955. (30) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23-33. (31) Valkovska, D. S.; Danov, K. D.; Ivanov, I. B. AdV. Colloid Interface Sci. 2002, 96, 101-129.
9218 Langmuir, Vol. 23, No. 18, 2007
Anton and Bouriat
[
]
γef3 2 2 ∂u(r, t) u(r, t) )∇ ∇ u(r, t) + 2 ∂t 24η ξ
(14)
Equation 14 is easily solved, assuming that the local deformation relaxes exponentially as u(r, t) ) uqJ0 exp(t/τ), where uq is a constant and τ is the characteristic time of growth associated with a surface corrugation that exhibits the wavenumber q. It results
(
)
γef3 2 2 1 1 )q q - 2 τ 24η ξ
(15)
It follows therefrom that the corrugations are dramatically amplified for wavenumbers such that q < (1/ξ) (i.e., τ > 0), whereas they vanish when q > (1/ξ). In order to choose a representative wavenumber, Vrij30 considered that the growing mode observed corresponds to the fastest mode, which implies that the initial deformations are due to thermal effects. This fastest growing mode has been calculated by maximizing eq 15. Consequently, the wavenumber for the fastest growing mode is
qM )
1 ξx2
(16)
The corresponding characteristic time of corrugation amplification is then calculated with eq 15 and is given by
τM )
96η 4 ξ γef3
Figure 7. Thinning of a saponin liquid film with a saponin concentration of 2.0 g‚L-1, [NaCl] ) 2 × 10-2 M, and ∆p fixed at 1200 Pa.
Accordingly, the characteristic time for this regime of instability can be determined by combining eq 15 and the wavenumber of the hydrodynamic corrugations qexp given by eq 4. Remarking that, typically, q , 1/ξ and calculating ξ with eq 12 and 7, one finds
(17)
It follows that ξ ) 4.3 µm from eq 12, and that τM ) 7.4 s for a 60 nm film thickness from eqs 7 and 17. Thus, such a characteristic time is not coherent with our experiments, since corrugations are growing for several minutes, as shown in Figure 6. Moreover, a significant difference appears when the experimental qexp is compared to the theoretical qM (corresponding to the fastest growing mode) wavenumber values. In fact, on the central thick film of radius R ) 0.6 mm, the simple observation shows that n ) 5 domains and, using eq 4, provides qexp ) 33 mm-1. On the other hand, the fastest growing mode according to eq 16 and ξ ) 4.3 µm provides qM ) 164 mm-1. Finally, this ensures that the instability observed in Figure 6c is not a corrugation of the fastest mode (i.e., initiated by thermal fluctuations), but the amplification of a preexisting mode. Such a preexisting mode may be of hydrodynamic origin, since, at the first stage of thin film formation, the films are thick enough to neglect the DLVO forces, and only hydrodynamic phenomena govern the thin film formation (corresponding to those illustrated in Figure 4). This assumption is reinforced by the fact that, for the same hydrodynamic pressure, the wavenumbers observed in Figures 4 and 6 appear to be comparable, being 44 and 33 mm-1, respectively. As a last remark, whereas qexp appears slightly lower for the case shown in Figure 6, it is only due to the diminished thinning rate resulting from the quick formation of the thinner film barrier near the Plateau border, following eq 5. Then, the formation and growth of the concentric corrugations observed in Figure 6a-d should be summarized as follows: First, during thin film formation, the film corrugates to favor drainage according to Tsekov.27 Second, when the film is thin enough so that van der Waals forces become influent, simultaneously (i) the drainage stops with the pinching of the film periphery and the formation of the surrounding black film, and (ii) the amplification of the initial hydrodynamic corrugation starts according to eqs 13-15.
τ)
24πηef Aq2
(18)
In fact, for a film of ef ) 60 nm average thickness that exhibits initial corrugations characterized by a wavenumber qexp ) 33 mm-1 (from Figure 6b), eq 18 provides τ ) 112 s, which is finally in good agreement with the film destabilization reported in Figures 6b-d. Additionally, these hydrodynamic phenomena shown in Figure 6 appear to be globally similar to those described by Reiter,32 but for the dewetting process on a solid substrate of unstable thin liquid (polystyrene solutions) films. This author reported and analyzed such spontaneous amplification of a surface undulation originated from thermal fluctuations and generated by van der Waals forces, leading to the film break-up and to dewetted hole nucleation. The equilibrium configuration of that system after such a dewetting process appears to be very similar to the one we observed in Figure 6f: a solid substrate spotted with liquid lenses. This phenomenon was called spinodal decomposition by analogy with spontaneous decompositions observed in some metastable mixtures. Although thin liquid film decomposition differs from the dewetting of solid substrate, these phenomena are all the same, closely related to whether black films are regarded as dewetted holes. Likewise, the comparison is achieved when we consider the parallel drawn between the description of the thin film decomposition, here given by eq 14, and the similar description for the classical spinodal decomposition provided by the works of Cahn and Hilliard.33 Eventually, we propose to also call spinodal decomposition the phenomenon of film decomposition by corrugation amplification observed in Figure 6. Regime of Nucleation and Growth. Figure 7 shows the representative behavior of the film drainage for ∆p ) 1200 Pa, with an aqueous solution containing a higher electrolyte concentration, 2 × 10-2 M NaCl, and a saponin concentration (32) Reiter, G. Langmuir 1993, 9, 1344-1351. (33) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1959, 31, 688-699.
Surface Corrugations of Axisymmetric Thin Films
Figure 8. Experimental value of the rim progress on the film surface, in the case illustrated in Figure 7.
Langmuir, Vol. 23, No. 18, 2007 9219
the uniform black film and the central thick film of uniform thickness (illustrated in Figure 7a). Gradually, the rim progresses toward the center until the collected liquid forms a spherical lens in equilibrium with the surrounding black film (see Figure 7b). Interferometric measurements confirm that the radial profile of this lens is bispheric, consistent with Laplace’s law, and provides the equilibrium contact angle value as θfE ∼ 7.5 × 10-3 rad. Figure 8 reports the distance between the rim periphery and the Plateau border as a function of time. The rim progression on the film surface appears constant until the lens formation is -1 achieved, and the rim velocity is given as V exp rim ) 0.210 µm‚s . Actually, a parallel can also be drawn between the hydrodynamic phenomena observed in Figure 7 for thin liquid films and the dewetting of a solid substrate by a liquid film. In this latter case, it was shown that the growth of a hole initiated in a non-wetting or partially wetting liquid film deposited on a solid substrate by spin-coating also leads to rim formation.34-36 Such rim formation and kinetics have been extensively studied and modeled, and37-39 the main result40 is the fact that the rim velocity Vrim is constant and given by
γ f 3 (θ ) 6η D
(19)
θfE ln A 1/2 1+ ln B
(20)
Vrim ) and
θfD )
Figure 9. Profiles of liquid velocity within the rims, in the case of (a) dewetting on solid substrate with a mobile air/liquid interface, and (b) a thin film with rigid air/liquid interfaces.
of 2 g‚L-1. The electrostatic screening is now so important that the attractive van der Waals forces largely dominate the film formation. The surrounding black film is created faster than that for the above-described regime, and then the film is immediately strongly pinched at its periphery. As a result, the quantity of liquid trapped in the central pancake is greater than that in the one presented in the last section, and therefore the pancake thickness is more important (about 90 nm in Figure 7a) when its thickness is about 60 nm, as in the pancake shown in Figure 6b. This difference between the initial pancake thicknesses is precisely the origin of the change in the film-thinning behaviors between the cases shown in Figures 6 and 7. In the latter case, the central film is so thick that attractive van der Waals forces are too weak to induce a destabilization by spinodal decomposition. The thick film, which is nevertheless unstable, is now destabilized by a process of nucleation and growth of black film. Actually, the initiation of the black film nucleation corresponds to the black film formation at the film periphery (previously presented), and its growth can be described, in this case, as follows; First, the black film progresses gradually by squeezing the inner trapped liquid, and the solution expelled by the black film expansion does not uniformly fill the central pancake, but progressively accumulates at its periphery. Then, this liquid excess at the pancake periphery forms a so-called rim, which separates
[
]
where ln A and ln B are logarithmic factors related to the viscous dissipation at each side of the rim. At the dry solid/rim frontier, ln A ) ln L/a, where L is a length typically on the order of the rim thickness, and a is a length of molecular scale. Generally, ln A ∼ 10-20 in the case of solid substrate dewetting problems.39 At the rim/wetting film frontier, ln B ) ln L/e ∼ 1, as film thickness e is only slightly higher than rim thickness L. Although eq 19 was developed to describe the kinetics of a rim progression in the case of a liquid film dewetting a solid substrate, it can also be applied to describe rim progression for the axisymmetric liquid film shown in Figure 7, because these two problems are driven by the same capillary forces and present an analogous geometry. Since it is common to consider the lubrication approximation in the case of dewetting on solid substrate, the rigid air/liquid interfaces of our suspended axisymmetric liquid films are assumed to play a role similar to that of the liquid/solid interfaces. Thus, the mobile air liquid interfaces are transposed in our case as the radial plane of symmetry of the film (illustrations are provided in Figure 9 with the velocity profile). In both experiments, the logarithmic factors ln B regarding the connection of the rim to the thick film can be taken as ln B ∼ 1 since, in both cases, rims only remain slightly larger than film thicknesses. On the other hand, logarithmic factors ln A concerning the connection between the rim and the black film must be reevaluated because black film (34) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Phys. ReV. Lett. 1991, 66, 715-718. (35) Brzoska, J. B.; Shahidzadzeh, N.; Rondelez, F. Nature 1992, 360, 719721. (36) Andrieu, C.; Sykes, C.; Brochard-Wyart, F. J. Adhesion 1996, 58, 15-24. (37) de Gennes, P.-G. C. R. Acad. Sci. Paris 1986, 303, 1275-1277. (38) Brochard-Wyart, F.; di Meglio, J. M.; Que´re´, D. C. R. Acad. Sci. Paris 1987, 304, 553-558. (39) Brochard-Wyart, F.; de Gennes, P.-G. AdV. Colloid Interface Sci. 1992, 39, 1-11. (40) de Gennes, P.-G.; Brochard-Wyart, F.; Que´re´, D. Gouttes, Bulles, Perles et Ondes; Belin: Paris, 2002.
9220 Langmuir, Vol. 23, No. 18, 2007
thickness is much larger than molecular scale. Typically, in our experiments, black film thickness is about a ∼ 10 nm, and rim thickness is about L ∼ 400 nm, thus ln A ) ln L/a ∼ 3.3. Now using these adapted logarithmic factors, eq 19 predicts, for the experimental system relative to Figure 7 (where γ ) 35 mN‚m-1 and θfE ∼ 7.5 × 10-3 rad), that the rim should progress on the film surface at Vrim ) 0.24 µm‚s-1. This calculated value of rim velocity is very consistent with the experimental one V exp rim ) 0.21 µm‚s-1. This result finally confirms that eq 19, initially written to describe the kinetic behavior of a rim formed when a thin film dewets a solid substrate, is also transposable to describe the rim progression on a black film.
Conclusion This study brought out original behaviors occurring in the thinning of large axisymmetric saponin thin films. Depending on electrolyte concentration, it was possible to observe different drainage modes. At lowest concentrations, hydrodynamic corrugations were observed. For higher concentrations, playing on
Anton and Bouriat
the strength of the repulsive contribution between the film interfaces, it was possible to control the whole amount of liquid trapped in the central dimple and thus the initial thickness of the central pancake. Then, two fundamentally different instability regimes causing pancake decomposition were observed. Each of these two regimes (spinodal decomposition and nucleation and growth) is, to a certain extent, well fitted by the models existing to describe the classical behaviors on solid substrates, and adapted to axisymmetric thin films. Then, the characteristic time of spinodal decomposition was calculated assuming that the instability observed is the result of the amplification of preexisting surface corrugations induced by hydrodynamic phenomena at the time of film formation. In regards to nucleation and growth regime, rim velocity was evaluated accounting for the fact that rim progression creates a black film of appreciable thickness instead of a dewetted hole. Certainly, these phenomena should be of some importance for the study of the lifetime of foams. LA700698D
