- Email: [email protected]
Equilibrium profile and rupture of wetting film on heterogeneous substrates
Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
Equilibrium profile and rupture of wetting film on heterogeneous substrates Radomir Slavchova , Boryan Radoeva,∗ , Klaus Werner St¨ockelhuberb,c b
a University of Sofia, Department of Physical Chemistry, 1164 Sofia, Bulgaria Max-Planck-Research Group for Colloids and Surfaces at the Institute of Ceramics, Glass and Construction Materials at the TU Bergakademie Freiberg, Chemnitzer Strasse 40, D-09599 Freiberg, Germany c Institute of Polymer Research Dresden, Hohe Strasse 6, D-01067 Dresden, Germany
Received 25 August 2004; accepted 20 December 2004 Available online 29 January 2005
Abstract There is abundant experimental evidence, indicating that the non-homogeneity of solid substrates plays substantial role in the behavior of wetting films. Drainage and rupture of wetting films are among the most comprehensively studied phenomena that are influenced by the characteristics of the solid substrate state. Experimental data on silica/water/air wetting films reported in previous studies show that film lifetime is strongly dependent on the substrate degree of hydrophobicity, while the rate of drainage was observed to remain practically unaffected. It has been found also, that both van der Waals and electrostatic forces in these wetting films are repulsive and according to the classical DLVO theory, such films should remain stable. The proposed explanation of the paradox (finite lifetimes in the presence of repulsive surface forces)—that the rupture of these films is caused by nanosized bubbles adhering to the hydrophobic silica surface, is a typical demonstration of the crucial effect of the surface non-homogeneities. The aim of this paper is to analyze the previously proposed model of wetting film with adhered to the solid substrate nanobubble and use the solution to discuss the consistency of the nanobubble rupture mechanism. © 2004 Elsevier B.V. All rights reserved. Keywords: Wetting films; Nanobubbles; DLVO forces; Rupture of thin films; Surfactant stabilization
1. Introduction The drainage and rupture of thin films between bubbles, droplets and solid particles are very important stages in the course of their collisions. For planar homogeneous films the mechanisms of these processes are, as a whole, well understood. It is established in particular that film drainage of such films obeys the so-called ‘Reynolds law’ [1–3] and the rupture occurs through the Scheludko–Vrij rupture mechanism [1,4,5]. According to this mechanism [6], besides the capillary forces, surface forces act also upon the surface fluctuation waves in thin films [1,7]. When repulsive forces prevail, fluctuation waves remain stationary and the films are stable. ∗
Corresponding author. Tel.: +359 2 8161 250. E-mail address: [email protected] (B. Radoev).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.12.031
On the contrary, if the resultant force is attractive, the wave’s amplitude grows up until touching the other surface, where the film ruptures [1,4]. In fact, the liquid films are neither plane-parallel nor homogeneous. It is clearly proved experimentally that only foam films of small radii (Rf 10−4 m) could drain while preserving their plane parallel shape. Films of larger radii are dimpled during their outflow, which drastically changes their hydrodynamic behavior and stability [8–10]. Wetting films are another example, which complex behavior, both with respect to thinning rate and rupture, cannot be correctly interpreted within the Scheludko–Vrij classical approach. Fig. 1 presents experimental data thickness of rupture versus time of silica/water/air wetting films discussed elsewhere [11,12]. The different points correspond to substrates of different hydrophobicity. As it can be seen, rupture thickness
136
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
Fig. 1. Rupture thickness vs. time of glas/water/air wetting film. The points are rupture coordinates (thickness and time) at different hydrophobicity; black solid line hw (t) is drainage under positive disjoining pressure Π ∞ (h∞ ) (Eq. (2.5)), the equilibrium thickness is 34 nm; gray lines are drainage under attractive hydrophobic pressure, models of Churaev [13] and Yoon [14].
and respectively lifetime show pronounced dependence on the degree of hydrophobicity of the silica substrate: the higher the hydrophobicity, the shorter the lifetime. However, the drainage kinetic curves hw (t) do not depend on the hydrophobization [12]. The solid line hw (t) in Fig. 1 is obtained as integral of the above mentioned Reynolds’ law: dhw 8h3 = − w2 p dt 3ηRf
(1.1)
with driving pressure p: p = Π − pm .
(1.2)
Here Π is the so-called disjoining pressure (see Eq. (2.1) below), and pm is an experimentally measurable quantity [1]. All other variables and parameters, such as film thickness hw , time t, viscosity η, film radius Rf are known or readily available from the experiment [12]. Inserting Π from Eq. (2.5) in Eqs. (1.1) and (1.2), after appropriate integration one obtains the solid line curve hw (t) in Fig. 1 Respectively, the experimental points hrupt versus time t in Fig. 1 are dispersed within the confidence interval around the theoretical drainage curve, which proves the correctness of the driving pressure model. The consistency of the theoretical curve hw (t) with the experimental points in Fig. 1 excludes any substantial attraction term, such as hydrophobic disjoining pressure [13,14] in the driving pressure p. According to the Scheludko–Vrij theory of fluid/fluid (foam and emulsion) films, the lack of attractive term in p is equivalent to stable film, which, as repeatedly pointed out, contradicts the wetting film behavior.
A reasonable step toward the solution of this paradox (rupture of films with finite equilibrium film thickness) was the model of wetting film with gas bubbles at the solid substrate, known in the literature also as nucleation mechanism [11,12]. Recently, a number of papers have been published that give clear evidence for the existence of such nanobubbles by means of IR spectroscopy [15], force measurements [16–18] and image-giving methods such as tapping-mode atomic force microscopy [19–21]. The formation of gas bubbles actually creates non-homogeneities in the wetting film. In the bubble region the film interaction parameters differ from those of wetting films (see Eqs. (2.3a) and (2.4)) and in the last stages before rupture, they are much closer to foam (air/water/air) films nature. In wetting (silica/water/air) film both van der Waals and electrostatic forces are repulsive (Eq. (2.5)), while in foam film the van der Waals interactions have attractive term (Eq. (2.4)). This attractive van der Waals term alone is responsible for film rupture. A general assumption in thin film models is that rupture occurs when the two film surfaces touch each other at least in a single point. Formally there are two ways of realizing this contact: (i) quasi-static, via equilibrium deformations of the film surface, and (ii) trough a spontaneous non-equilibrium, local corrugation. In the present paper only the equilibrium deformations will be analyzed, respectively—their dependence on the system parameters and especially their role in the film rupture. Therefore, this study is a necessary stage for the analysis of the problem, since equilibrium state is the starting point for any stability analysis. The spontaneous rupture mechanism concerns the problem of the stability of the system and will be the subject of a separate study. Section 2
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
is devoted to the pressure balance of the considered wetting film, and the problem of its equilibrium profile. Section 3 presents numerical results followed by their discussion. Special attention is paid to the possible effects leading to stabilization of wetting films by surfactants. The study finishes with Section 4 about the problems and perspectives in the future.
2. Pressure balance and equilibrium profile equation The equilibrium profile ζ is determined by the pressure balance at the upper fluid film surface (see Fig. 2) as follows [12]: pγ + Πel (r, ζ) + ΠvdW (r, ζ) = Π∞ .
(2.1)
Due to the circular shape of the bubble three phase contact (TPC) line, the film profile should be axially symmetrical, i.e. the profile ζ should only depend on the radial coordinate r. As it is well known, the complete expression for the capillary pressure pγ in Eq. (2.1) at this symmetry has the form: pγ = γ
1 d dζ/dr r , r dr 2 1 + (dζ/dr)
(2.2)
In all considered below cases the non-linear term (dζ/dr)2 was found to be much smaller than unity. The factor γ as usually is the surface tension. Because of the non-homogeneity in the wetting film caused by the bubbles, one has to define separately the disjoining pressure there as a function of the radial coordinate—Π el (r), Π vdW (r). Using the discussed elsewhere [12] expression of the electrostatic disjoining pressure valid for silica/water/air films, we have assumed the following explicit form of Π el (r) for the considered heterogeneous system: Bf e−κhf , r < a Πel (r) = (2.3a) Bw e−κhw , r > a
137
with constants B given from the double layer theory [22]: 4kT 2 zeψLG zeψLG tanh tanh , Bf = 2εε0 κ2 ze 4kT 4kT 2 zeψLG zeψSL 2 4kT tanh Bw = 2εε0 κ tanh . ze 4kT 4kT (2.3b) The subscript ‘f’ refers to the film region above the bubble at r < a (a—bubble TPC radius, see Fig. 2), called further ‘inner’ region, where a foam film is formed. The subscript ‘w’ refers to the outer region, i.e. at r > a. In the inner region hf (r) = h∞ − hb (r) − ζ(r) is the foam film thickness with hb (r) as the local bubble height. In the outer region hw (r) = h∞ − ζ(r) is the wetting film thickness. Here h∞ , which is constant with respect to r, is the wetting film thickness far from the bubble. The evaluations of Π el in the present study are carried out with a value for the solid–liquid potential Ψ SL = −30 mV [12] and for the liquid–gas potential Ψ LG = −35 mV [23]. As seen, in both foam and wetting regions the charge of the two film surfaces has the same sign and the electrostatic pressure is repulsive everywhere. A typical Debye length value has been used (1/κ = 9.38 nm) corresponding to monovalent electrolyte concentration 10−3 M. All other notations in Eq. (2.3b) have their conventional meaning and values: ε = 80 is the relative dielectric permittivity of water, ε0 = 8.85 × 10−12 F m is the dielectric permittivity of vacuum, z = 1 is the ion valence, e = 1.60 × 10−19 C is the elementary charge, k = 1.38 × 10−23 J/K is the Boltzmann constant, T = 298 K is the temperature. Formulae (2.3) present an approximation, which is correct at a sufficiently large distance from the TPC line. The problem here is with the behavior of the surface potentials Ψ SL , Ψ LG . They are assumed to be constant at the corresponding surface, which consequently leads to a jump of the potential at the TPC. Moreover, the TPC is a cross-line of three surfaces (solid–liquid; liquid–gas, solid–gas), which means that we have to account for the existence of a third (Ψ SG ) potential. As known from the electrostatics, electrical potentials are
Fig. 2. Geometric and dynamic parameters of equilibrium wetting film, cross-section of a detail with adhered nanobubble at the solid substrate: (a) geometric paramerters: ζ(r): film surface profile, hf = h∞ − ζ − hb : local foam film thickness, hb : bubble height, h∞ : wetting film thickness far from the bubble, a: three phase contact radius of the bubble, r: radial coordinate; (b) dynamic parameters: pγ : capillary pressure, Π el : electrostatic disjoining pressure, Π vdW : van der Waals disjoining pressure, Π ∞ = Π el + Π vdW : disjoining pressure in the wetting film far from TPC.
138
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
continuous functions everywhere including at the boundary, i.e. at TPC line. The boundary condition there defines a complicated electrostatic problem standing outside the scope of this study [23]. Nevertheless, we could use the model equations (2.3) as a rough approximation because the TPC region is relatively far from the upper film surface and its effect on the surface forces is supposed to be negligible. The most important for the interaction in the foam film, of course, is the region at the bubble apex, where relations (2.3) are practically correct. The problems with van der Waals disjoining pressure Π vdW (r) are connected with the fact that in the inner region (r < a, see Fig. 2a) one has to account for the van der Waals interactions in a multi-layer, solid/gas/liquid/gas system. Using the well-known relations for Π vdW in multilayer systems (see e.g. [7]) one obtains the following simple relation valid for the entire heterogeneous film, i.e. for both, the inner and the outer regions: 1 ASL ALL ΠvdW (r) = . (2.4) − 3 6π (h∞ − ζ(r))3 hf (r) In Eq. (2.4), ASL = 4,7 × 10−20 J, ALL = 3,7 × 10−20 are the Hamaker constants with values corresponding to quartz as solid phase and water as liquid phase [7,12]. Relation (2.4) is continuous when passing through the TPC region. Far from the TPC in the outer region, where hf (r → ∞) → h∞ , ζ(r → ∞) → 0, the disjoining pressure reaches its correct value for the silica/water/air wetting film: ΠvdW (r → ∞) →
ASL − ALL . 6πh3∞
(2.4’)
Eq. (2.4 ), combined with the corresponding expression for Π el Eq. (2.3a), has been used to calculate Π ∞ : Π∞ (h∞ ) = Bo e−κh∞ +
ASL − ALL . 6πh3∞
(2.5)
As far as ASL > ALL , the van der Waals term in Eq. (2.5) is repulsive. Note, that in the inner region Π vdW could change its sign and from repulsive pressure to turn in attractive one. In the cases of relative thick wetting films (h∞ hf ) the van der Waals term (Eq. (2.4)) tends to its foam film value (see also Fig. 4): ΠvdW (h∞ hf ) → −
ALL . 6πh3f
(2.4”)
Inserting the appropriate expressions of pressure terms from Eqs. (2.2) to (2.5) into the force balance (2.1) one obtains a differential equation for the film profile ζ(r) which will be numerically solved with boundary conditions ζ|r → ∞ → 0 and dζ/dr|r = 0 = 0.
3. Results and discussion Fig. 3 presents graphs of numerically calculated film surface profiles according to Eqs. (2.1)–(2.5) as a function of
Fig. 3. Film surface profiles ζ(r) at four different thicknesses h∞ of the wetting film; the bubble height at the apex is hb (r = 0) = 20 nm; the TPC angle is θ = 90◦ . At h∞ = 28 nm the profile is convex, at h∞ = 21.95 nm the film surface touches the bubble.
the radial co-ordinate r. Three geometric parameters determine the profile ζ: the wetting film thickness h∞ and two others, describing the bubble dimensions. As bubble dimensions here the height at bubble apex hb and the TPC angle, θ are chosen. The presented profiles of ζ(r, h∞ ) for four different h∞ clearly demonstrate the influence of the foam film thickness (hf = h∞ − hb − ζ, see Fig. 1). As seen, at relatively thick foam films (hf = 7–8 nm) the deformation ζ is convex, corresponding to repulsive forces. All other cases show concave profiles which means that the foam films are sufficiently thin and the attractive part of Π vdW (∼ALL , see Eq. (2.4)) is predominant over the other (repulsive) pressure terms. At h∞ = 21.95 nm the surface profile ζ(r = 0) touches the bubble apex which is equivalent to film rupture. As already mentioned, there are two different possible mechanisms through which the surfaces could come into contact. The results in Fig. 3 correspond to the steady state deformation mechanism, since they are solutions of the equilibrium balance (2.1). Generally, ‘equilibrium’ rupture thickness should be considered as the lower limit (smaller thickness), as compared to the actual one (any instability should lead to an earlier rupture). Fig. 4 presents disjoining pressure isotherms Π versus hf at the bubble apex, r = 0. The two isotherms clearly illustrate the concurrence of the repulsive and attractive terms of Π vdW . At larger bubble height the influence of van der Waals repulsion (which is the only one which depends on hb ) is weaker and the resultant disjoining pressure becomes more negative, but as a whole, the pressure depends weakly on the bubble height. It is clear that the largest deformation will be in the centre at r = 0, where the foam film is the thinnest, i.e. |ζ(0)| = max. Fig. 5 illustrates the deflection h∞ + ζ(0) versus h∞ . As seen from the graph, the surface deformations are negligible ( ζ < 1–3 nm) except shortly before the moment of rupture. The slope near the rupture point (hrupt ≈ 22 nm) is quasivertical ( ζ/ h∞ ≈ 102 ) which explains the experimental fact why no deformations are observed.
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
Fig. 4. Dependence of disjoining pressure Π on the foam film thickness hf (r = 0) for two different bubble heights hb (r = 0) = 10, 100 nm.
3.1. Influence of surfactants on the film stability A series of experiments have shown the strong stabilizing effect of surfactant on the wetting film [1,3,24]. As seen in Fig. 6, at constant degree of hydrophobization the film lifetime in the presence of sodium dodecylsulfate (SDS) is by two orders of magnitude longer. Note that the electrolyte concentration in the system with and without surfactant is selected in a way to keep the drainage along nearly the same h versus t curves (see solid and dotted lines of Fig. 6). Our nanobubble model describes qualitatively the wetting film rupture by locally formed unstable foam film. On the other hand, it is well known [1,5,12,24] that surfactants stabilize foam films and this effect could be a possible explanation for the longer lifetime of our wetting film. As known (e.g. [1,3,5]), the foam film stability depends on the surface forces, i.e. on the disjoining pressure Π. From the two DLVO terms of Π, van der Waals term, Π vdW actually do not depend on the surfactant concentration, while the electrostatic
139
Fig. 6. Influence of surfactant on the lifetime and rupture of wetting films on hydrophobized silica with a contact angle θ = 90◦ . The maximum lifetime without SDS is 3 s; addition of 10−5 M SDS increases the lifetime of the wetting film up to 300 s. The lines show the calculated drainage for the system without surfactant (solid line, 10−3 M KCl) and with surfactant (dotted line, 2 × 10−3 M KCl + 10−5 M SDS) [12].
term Π el is a function of surfactants via the dependence of surface potentials on the their concentration. Experiments [12] have shown for instance, that low SDS concentrations increases more than two fold the (absolute value of) surface potential Ψ LG (10−5 M SDS) = −83 mV [23]. Putting this potential value in the expression of Π el (Eq. (2.3)), together with Π vdW (Eq. (2.4)) one obtains for the pressure barrier with surfactant [Π el (r = 0) + Π vdW (r = 0)]max = 52,500 Pa at hf = 4.9 nm (calculated for bubble height hb = 30 nm and 2 × 10−3 KCl). Without surfactant Ψ LG = −35 mV, the pressure maximum is [Π el (r = 0) + Π vdW (r = 0)]max = 4325 Pa at hf = 9.6 nm (hb = 30 nm and 1 × 10−3 KCl, see Fig. 4). For these two systems, following the procedure described in Section 3, for hrupt one obtains negligible differences in hrupt for the two cases—with and without surfactant (see Table 1). As it can be seen, the difference of Ψ LG leads to insignificant difference in hrupt . Thus, in the framework of this rupture mechanism, the higher pressure barrier alone cannot explain the longer lifetime of the wetting film with surfactant. The quantitative discrepancy between the disjoining pressure barrier effect and the experimentally observed longer Table 1 Rupture thicknesses of wetting films with and without surfactant for different bubble height and the same TPC angle, θ = π/2 hb (nm)
Fig. 5. Dependence of the deflection h∞ − ζ(r = 0) on the wetting film thickness h∞ . The circle corresponds to the contact between the bubble surfaces (the dashed line hb = 20 nm) and the film.
30 35 40 45
hrupt (nm) Without surfactant (1 × 10−3 M KCl)
With 10−5 M SDS (2 × 10−3 M KCl)
32.22 37.33 42.42 47.50
31.29 36.16 41.00 45.81
In all cases the variations of hrupt are experimentally negligible.
140
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
lifetimes requires detailed analysis of the second possibility. How does the surfactant affect the size of the nanobubbles and their concentration at the substrate? From the theory it is known that in the case of the so-called homogeneous nucleation, i.e. when the new phase is formed in the bulk, the equilibrium radius of curvature of the new phase nucleus (the bubble radius rb in our case) is proportional to the liquid/gas surface tension: rb ∼ γ LG (see e.g. [25]). In our nanobubble model actually, the new phase nuclei are in contact both with bulk and surface phases and one speaks of heterogeneous nucleation. Consequently, here the equilibrium nucleus size (volume) depends on two parameters: bubble radius (of curvature) rb and (TPC) angle θ, or whichever combination of two geometric parameters defining the adhered at the solid surface bubble volume. As already many times discussed, important geometric parameter of our analysis is not rb but the bubble height, hb = rb (1 + cos θ). After substituting rb and using cos θ = (γ SG − γ SL )/γ LG (where γ SG and γ SL are the surface tensions of the corresponding solid/gas, solid/liquid surfaces), one derives hb ∼ γ LG [1 + (γ SG − γ SL )/γ LG ]. The surfactant lowers the surface pressure γ LG without changing γ SG and γ SL (electron spectroscopy for chemical analysis [12]). Thus should be expected hb to become smaller with addition of surfactant.
4. Concluding remarks The present study is only the first step in the analysis of the wetting film stability. After deriving the equilibrium deformations ζ(r), a future stability analysis should be carried out and Scheludko–Vrij wave mechanism must be generalized and applied to heterogeneous systems. Preliminary estimations were made, which have shown that, in contrast to homogeneous planar liquid films, in films with local attractive spots the length of the so-called unstable waves must be smaller than the size of the attractive spot, in order to be able to rupture the film. It has been mentioned above that neither the effect of the solid–gas surface charge, nor the boundary effects on the TPC line are considered. The model is developed only for TPC angles larger than 90◦ , and should be generalized (note that θ > 90◦ for more of the cited experiments). In the present model nanobubbles are considered as spherical, i.e. their deformation in the foam film region (where wetting film surface and bubble surface interact) is neglected but only the wetting film surface deformation ζ(r), is taken into account. The reason for that approximation follows from the well known nature of capillary surfaces: when bubbles (droplets) collide, the smaller particle deforms the larger one. From this viewpoint considering the bubble deformation one could expect only corrections to the results accounting deformation only of the wetting film surface, as in Section 3.
Nevertheless, in a more precise analysis the effects of bubble deformation should be accounted as well. The future work relies very much on the experiment. The hypothesis that we have made for dependence of the bubble size on the presence of surfactant, as well as the other assumptions, ought to be confirmed experimentally.
Acknowledgments This study is partly supported by the Bulgarian National Science Foundation Project Y-X-2/2003. K.W.S. is grateful for the financial support of the German Research Council (DFG), Special Research Area (SFB) 285.
References [1] A. Scheludko, Proc. Kon. Ned. Akad. Wet. B65 (1962) 87. [2] O. Reynolds, Phil. Trans. R. Soc. A177 (1886) 157. [3] B. Radoev, D.S. Dimitrov, I.B. Ivanov, Colloid Polym. Sci. 252 (1974) 50. [4] A. Vrij, Disc. Faraday Soc. 42 (1966) 44. [5] B. Radoev, A. Scheludko, E. Manev, J. Colloid Interface Sci. 95 (1) (1983) 254. [6] L. Mandelstam, Ann. Phys. 41 (1913) 609. [7] J. Israelachvili, Intermolecular Forces, Academic Press, London, NY, 1991. [8] E.D. Manev, Ann. Univ. Sofia, Fac. Chem. 75 (1981) 174. [9] K.J. Mysels, K. Shinoda, S. Frankel, Soap Films, Pergamon Press, N.Y., 1959; S. Frankel, K. Mysels, J. Phys. Chem. 66 (1962) 190. [10] E. Manev, R. Tsekov, B. Radoev, J. Disper. Sci. Technol. 18 (1997) 769. [11] H.J. Schulze, K.W. St¨ockelhuber, A. Wenger, Colloids Surf. A192 (2001) 61. [12] K.W. St¨ockelhuber, B. Radoev, A. Wenger, H.J. Schulze, Langmuir 20 (2004) 164. [13] N.V. Churaev, Adv. Colloid Interface Sci. 58 (1995) 87. [14] R.-H. Yoon, Int. J. Miner. Proc. 58 (2000) 129. [15] W. Gong, J. Stearnes, D. Fornasiero, R. Hayes, J. Ralston, J. Phys. Chem. Chem. Phys. 1 (1999) 2799. [16] J. Mahnke, J. Stearns, J. Ralston, Phys. Chem. Chem. Phys. 1 (1999) 2793. [17] N. Ishida, M. Sakamoto, M. Miyahara, K. Higashitani, Langmuir 16 (2000) 5681. [18] J.W.G. Tyrell, P. Attard, Langmuir 18 (2002) 160. [19] N. Ishida, M. Sakamoto, M. Miyahara, K. Higashitani, J. Colloid Interface Sci. 253 (2002) 112. [20] J.W.G. Tyrell, P. Attard, Phys. Rev. Lett. 87 (2001) 176104. [21] S. Lou, J. Gao, X. Xiao, X. Li, G. Li, Y. Zhang, M. Li, J. Sun, J. Hu, Chin. Phys. 10 (2001) 108. [22] G. Kar, S. Chander, T.S. Mika, J. Colloid Interface Sci. 44 (1973) 347. [23] Tz. Ivanov, B. Radoev, Colloids Surf. A 244 (2004) 19. [24] D. Exerowa, P.M. Kruglyakov, Foam and Foam Films, Elsevier, Amsterdam, 1998. [25] I.V. Markov, Crystal Growth for Beginners, World Scientific, Singapore, 1995.
Equilibrium profile and rupture of wetting film on heterogeneous substrates Radomir Slavchova , Boryan Radoeva,∗ , Klaus Werner St¨ockelhuberb,c b
a University of Sofia, Department of Physical Chemistry, 1164 Sofia, Bulgaria Max-Planck-Research Group for Colloids and Surfaces at the Institute of Ceramics, Glass and Construction Materials at the TU Bergakademie Freiberg, Chemnitzer Strasse 40, D-09599 Freiberg, Germany c Institute of Polymer Research Dresden, Hohe Strasse 6, D-01067 Dresden, Germany
Received 25 August 2004; accepted 20 December 2004 Available online 29 January 2005
Abstract There is abundant experimental evidence, indicating that the non-homogeneity of solid substrates plays substantial role in the behavior of wetting films. Drainage and rupture of wetting films are among the most comprehensively studied phenomena that are influenced by the characteristics of the solid substrate state. Experimental data on silica/water/air wetting films reported in previous studies show that film lifetime is strongly dependent on the substrate degree of hydrophobicity, while the rate of drainage was observed to remain practically unaffected. It has been found also, that both van der Waals and electrostatic forces in these wetting films are repulsive and according to the classical DLVO theory, such films should remain stable. The proposed explanation of the paradox (finite lifetimes in the presence of repulsive surface forces)—that the rupture of these films is caused by nanosized bubbles adhering to the hydrophobic silica surface, is a typical demonstration of the crucial effect of the surface non-homogeneities. The aim of this paper is to analyze the previously proposed model of wetting film with adhered to the solid substrate nanobubble and use the solution to discuss the consistency of the nanobubble rupture mechanism. © 2004 Elsevier B.V. All rights reserved. Keywords: Wetting films; Nanobubbles; DLVO forces; Rupture of thin films; Surfactant stabilization
1. Introduction The drainage and rupture of thin films between bubbles, droplets and solid particles are very important stages in the course of their collisions. For planar homogeneous films the mechanisms of these processes are, as a whole, well understood. It is established in particular that film drainage of such films obeys the so-called ‘Reynolds law’ [1–3] and the rupture occurs through the Scheludko–Vrij rupture mechanism [1,4,5]. According to this mechanism [6], besides the capillary forces, surface forces act also upon the surface fluctuation waves in thin films [1,7]. When repulsive forces prevail, fluctuation waves remain stationary and the films are stable. ∗
Corresponding author. Tel.: +359 2 8161 250. E-mail address: [email protected] (B. Radoev).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.12.031
On the contrary, if the resultant force is attractive, the wave’s amplitude grows up until touching the other surface, where the film ruptures [1,4]. In fact, the liquid films are neither plane-parallel nor homogeneous. It is clearly proved experimentally that only foam films of small radii (Rf 10−4 m) could drain while preserving their plane parallel shape. Films of larger radii are dimpled during their outflow, which drastically changes their hydrodynamic behavior and stability [8–10]. Wetting films are another example, which complex behavior, both with respect to thinning rate and rupture, cannot be correctly interpreted within the Scheludko–Vrij classical approach. Fig. 1 presents experimental data thickness of rupture versus time of silica/water/air wetting films discussed elsewhere [11,12]. The different points correspond to substrates of different hydrophobicity. As it can be seen, rupture thickness
136
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
Fig. 1. Rupture thickness vs. time of glas/water/air wetting film. The points are rupture coordinates (thickness and time) at different hydrophobicity; black solid line hw (t) is drainage under positive disjoining pressure Π ∞ (h∞ ) (Eq. (2.5)), the equilibrium thickness is 34 nm; gray lines are drainage under attractive hydrophobic pressure, models of Churaev [13] and Yoon [14].
and respectively lifetime show pronounced dependence on the degree of hydrophobicity of the silica substrate: the higher the hydrophobicity, the shorter the lifetime. However, the drainage kinetic curves hw (t) do not depend on the hydrophobization [12]. The solid line hw (t) in Fig. 1 is obtained as integral of the above mentioned Reynolds’ law: dhw 8h3 = − w2 p dt 3ηRf
(1.1)
with driving pressure p: p = Π − pm .
(1.2)
Here Π is the so-called disjoining pressure (see Eq. (2.1) below), and pm is an experimentally measurable quantity [1]. All other variables and parameters, such as film thickness hw , time t, viscosity η, film radius Rf are known or readily available from the experiment [12]. Inserting Π from Eq. (2.5) in Eqs. (1.1) and (1.2), after appropriate integration one obtains the solid line curve hw (t) in Fig. 1 Respectively, the experimental points hrupt versus time t in Fig. 1 are dispersed within the confidence interval around the theoretical drainage curve, which proves the correctness of the driving pressure model. The consistency of the theoretical curve hw (t) with the experimental points in Fig. 1 excludes any substantial attraction term, such as hydrophobic disjoining pressure [13,14] in the driving pressure p. According to the Scheludko–Vrij theory of fluid/fluid (foam and emulsion) films, the lack of attractive term in p is equivalent to stable film, which, as repeatedly pointed out, contradicts the wetting film behavior.
A reasonable step toward the solution of this paradox (rupture of films with finite equilibrium film thickness) was the model of wetting film with gas bubbles at the solid substrate, known in the literature also as nucleation mechanism [11,12]. Recently, a number of papers have been published that give clear evidence for the existence of such nanobubbles by means of IR spectroscopy [15], force measurements [16–18] and image-giving methods such as tapping-mode atomic force microscopy [19–21]. The formation of gas bubbles actually creates non-homogeneities in the wetting film. In the bubble region the film interaction parameters differ from those of wetting films (see Eqs. (2.3a) and (2.4)) and in the last stages before rupture, they are much closer to foam (air/water/air) films nature. In wetting (silica/water/air) film both van der Waals and electrostatic forces are repulsive (Eq. (2.5)), while in foam film the van der Waals interactions have attractive term (Eq. (2.4)). This attractive van der Waals term alone is responsible for film rupture. A general assumption in thin film models is that rupture occurs when the two film surfaces touch each other at least in a single point. Formally there are two ways of realizing this contact: (i) quasi-static, via equilibrium deformations of the film surface, and (ii) trough a spontaneous non-equilibrium, local corrugation. In the present paper only the equilibrium deformations will be analyzed, respectively—their dependence on the system parameters and especially their role in the film rupture. Therefore, this study is a necessary stage for the analysis of the problem, since equilibrium state is the starting point for any stability analysis. The spontaneous rupture mechanism concerns the problem of the stability of the system and will be the subject of a separate study. Section 2
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
is devoted to the pressure balance of the considered wetting film, and the problem of its equilibrium profile. Section 3 presents numerical results followed by their discussion. Special attention is paid to the possible effects leading to stabilization of wetting films by surfactants. The study finishes with Section 4 about the problems and perspectives in the future.
2. Pressure balance and equilibrium profile equation The equilibrium profile ζ is determined by the pressure balance at the upper fluid film surface (see Fig. 2) as follows [12]: pγ + Πel (r, ζ) + ΠvdW (r, ζ) = Π∞ .
(2.1)
Due to the circular shape of the bubble three phase contact (TPC) line, the film profile should be axially symmetrical, i.e. the profile ζ should only depend on the radial coordinate r. As it is well known, the complete expression for the capillary pressure pγ in Eq. (2.1) at this symmetry has the form: pγ = γ
1 d dζ/dr r , r dr 2 1 + (dζ/dr)
(2.2)
In all considered below cases the non-linear term (dζ/dr)2 was found to be much smaller than unity. The factor γ as usually is the surface tension. Because of the non-homogeneity in the wetting film caused by the bubbles, one has to define separately the disjoining pressure there as a function of the radial coordinate—Π el (r), Π vdW (r). Using the discussed elsewhere [12] expression of the electrostatic disjoining pressure valid for silica/water/air films, we have assumed the following explicit form of Π el (r) for the considered heterogeneous system: Bf e−κhf , r < a Πel (r) = (2.3a) Bw e−κhw , r > a
137
with constants B given from the double layer theory [22]: 4kT 2 zeψLG zeψLG tanh tanh , Bf = 2εε0 κ2 ze 4kT 4kT 2 zeψLG zeψSL 2 4kT tanh Bw = 2εε0 κ tanh . ze 4kT 4kT (2.3b) The subscript ‘f’ refers to the film region above the bubble at r < a (a—bubble TPC radius, see Fig. 2), called further ‘inner’ region, where a foam film is formed. The subscript ‘w’ refers to the outer region, i.e. at r > a. In the inner region hf (r) = h∞ − hb (r) − ζ(r) is the foam film thickness with hb (r) as the local bubble height. In the outer region hw (r) = h∞ − ζ(r) is the wetting film thickness. Here h∞ , which is constant with respect to r, is the wetting film thickness far from the bubble. The evaluations of Π el in the present study are carried out with a value for the solid–liquid potential Ψ SL = −30 mV [12] and for the liquid–gas potential Ψ LG = −35 mV [23]. As seen, in both foam and wetting regions the charge of the two film surfaces has the same sign and the electrostatic pressure is repulsive everywhere. A typical Debye length value has been used (1/κ = 9.38 nm) corresponding to monovalent electrolyte concentration 10−3 M. All other notations in Eq. (2.3b) have their conventional meaning and values: ε = 80 is the relative dielectric permittivity of water, ε0 = 8.85 × 10−12 F m is the dielectric permittivity of vacuum, z = 1 is the ion valence, e = 1.60 × 10−19 C is the elementary charge, k = 1.38 × 10−23 J/K is the Boltzmann constant, T = 298 K is the temperature. Formulae (2.3) present an approximation, which is correct at a sufficiently large distance from the TPC line. The problem here is with the behavior of the surface potentials Ψ SL , Ψ LG . They are assumed to be constant at the corresponding surface, which consequently leads to a jump of the potential at the TPC. Moreover, the TPC is a cross-line of three surfaces (solid–liquid; liquid–gas, solid–gas), which means that we have to account for the existence of a third (Ψ SG ) potential. As known from the electrostatics, electrical potentials are
Fig. 2. Geometric and dynamic parameters of equilibrium wetting film, cross-section of a detail with adhered nanobubble at the solid substrate: (a) geometric paramerters: ζ(r): film surface profile, hf = h∞ − ζ − hb : local foam film thickness, hb : bubble height, h∞ : wetting film thickness far from the bubble, a: three phase contact radius of the bubble, r: radial coordinate; (b) dynamic parameters: pγ : capillary pressure, Π el : electrostatic disjoining pressure, Π vdW : van der Waals disjoining pressure, Π ∞ = Π el + Π vdW : disjoining pressure in the wetting film far from TPC.
138
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
continuous functions everywhere including at the boundary, i.e. at TPC line. The boundary condition there defines a complicated electrostatic problem standing outside the scope of this study [23]. Nevertheless, we could use the model equations (2.3) as a rough approximation because the TPC region is relatively far from the upper film surface and its effect on the surface forces is supposed to be negligible. The most important for the interaction in the foam film, of course, is the region at the bubble apex, where relations (2.3) are practically correct. The problems with van der Waals disjoining pressure Π vdW (r) are connected with the fact that in the inner region (r < a, see Fig. 2a) one has to account for the van der Waals interactions in a multi-layer, solid/gas/liquid/gas system. Using the well-known relations for Π vdW in multilayer systems (see e.g. [7]) one obtains the following simple relation valid for the entire heterogeneous film, i.e. for both, the inner and the outer regions: 1 ASL ALL ΠvdW (r) = . (2.4) − 3 6π (h∞ − ζ(r))3 hf (r) In Eq. (2.4), ASL = 4,7 × 10−20 J, ALL = 3,7 × 10−20 are the Hamaker constants with values corresponding to quartz as solid phase and water as liquid phase [7,12]. Relation (2.4) is continuous when passing through the TPC region. Far from the TPC in the outer region, where hf (r → ∞) → h∞ , ζ(r → ∞) → 0, the disjoining pressure reaches its correct value for the silica/water/air wetting film: ΠvdW (r → ∞) →
ASL − ALL . 6πh3∞
(2.4’)
Eq. (2.4 ), combined with the corresponding expression for Π el Eq. (2.3a), has been used to calculate Π ∞ : Π∞ (h∞ ) = Bo e−κh∞ +
ASL − ALL . 6πh3∞
(2.5)
As far as ASL > ALL , the van der Waals term in Eq. (2.5) is repulsive. Note, that in the inner region Π vdW could change its sign and from repulsive pressure to turn in attractive one. In the cases of relative thick wetting films (h∞ hf ) the van der Waals term (Eq. (2.4)) tends to its foam film value (see also Fig. 4): ΠvdW (h∞ hf ) → −
ALL . 6πh3f
(2.4”)
Inserting the appropriate expressions of pressure terms from Eqs. (2.2) to (2.5) into the force balance (2.1) one obtains a differential equation for the film profile ζ(r) which will be numerically solved with boundary conditions ζ|r → ∞ → 0 and dζ/dr|r = 0 = 0.
3. Results and discussion Fig. 3 presents graphs of numerically calculated film surface profiles according to Eqs. (2.1)–(2.5) as a function of
Fig. 3. Film surface profiles ζ(r) at four different thicknesses h∞ of the wetting film; the bubble height at the apex is hb (r = 0) = 20 nm; the TPC angle is θ = 90◦ . At h∞ = 28 nm the profile is convex, at h∞ = 21.95 nm the film surface touches the bubble.
the radial co-ordinate r. Three geometric parameters determine the profile ζ: the wetting film thickness h∞ and two others, describing the bubble dimensions. As bubble dimensions here the height at bubble apex hb and the TPC angle, θ are chosen. The presented profiles of ζ(r, h∞ ) for four different h∞ clearly demonstrate the influence of the foam film thickness (hf = h∞ − hb − ζ, see Fig. 1). As seen, at relatively thick foam films (hf = 7–8 nm) the deformation ζ is convex, corresponding to repulsive forces. All other cases show concave profiles which means that the foam films are sufficiently thin and the attractive part of Π vdW (∼ALL , see Eq. (2.4)) is predominant over the other (repulsive) pressure terms. At h∞ = 21.95 nm the surface profile ζ(r = 0) touches the bubble apex which is equivalent to film rupture. As already mentioned, there are two different possible mechanisms through which the surfaces could come into contact. The results in Fig. 3 correspond to the steady state deformation mechanism, since they are solutions of the equilibrium balance (2.1). Generally, ‘equilibrium’ rupture thickness should be considered as the lower limit (smaller thickness), as compared to the actual one (any instability should lead to an earlier rupture). Fig. 4 presents disjoining pressure isotherms Π versus hf at the bubble apex, r = 0. The two isotherms clearly illustrate the concurrence of the repulsive and attractive terms of Π vdW . At larger bubble height the influence of van der Waals repulsion (which is the only one which depends on hb ) is weaker and the resultant disjoining pressure becomes more negative, but as a whole, the pressure depends weakly on the bubble height. It is clear that the largest deformation will be in the centre at r = 0, where the foam film is the thinnest, i.e. |ζ(0)| = max. Fig. 5 illustrates the deflection h∞ + ζ(0) versus h∞ . As seen from the graph, the surface deformations are negligible ( ζ < 1–3 nm) except shortly before the moment of rupture. The slope near the rupture point (hrupt ≈ 22 nm) is quasivertical ( ζ/ h∞ ≈ 102 ) which explains the experimental fact why no deformations are observed.
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
Fig. 4. Dependence of disjoining pressure Π on the foam film thickness hf (r = 0) for two different bubble heights hb (r = 0) = 10, 100 nm.
3.1. Influence of surfactants on the film stability A series of experiments have shown the strong stabilizing effect of surfactant on the wetting film [1,3,24]. As seen in Fig. 6, at constant degree of hydrophobization the film lifetime in the presence of sodium dodecylsulfate (SDS) is by two orders of magnitude longer. Note that the electrolyte concentration in the system with and without surfactant is selected in a way to keep the drainage along nearly the same h versus t curves (see solid and dotted lines of Fig. 6). Our nanobubble model describes qualitatively the wetting film rupture by locally formed unstable foam film. On the other hand, it is well known [1,5,12,24] that surfactants stabilize foam films and this effect could be a possible explanation for the longer lifetime of our wetting film. As known (e.g. [1,3,5]), the foam film stability depends on the surface forces, i.e. on the disjoining pressure Π. From the two DLVO terms of Π, van der Waals term, Π vdW actually do not depend on the surfactant concentration, while the electrostatic
139
Fig. 6. Influence of surfactant on the lifetime and rupture of wetting films on hydrophobized silica with a contact angle θ = 90◦ . The maximum lifetime without SDS is 3 s; addition of 10−5 M SDS increases the lifetime of the wetting film up to 300 s. The lines show the calculated drainage for the system without surfactant (solid line, 10−3 M KCl) and with surfactant (dotted line, 2 × 10−3 M KCl + 10−5 M SDS) [12].
term Π el is a function of surfactants via the dependence of surface potentials on the their concentration. Experiments [12] have shown for instance, that low SDS concentrations increases more than two fold the (absolute value of) surface potential Ψ LG (10−5 M SDS) = −83 mV [23]. Putting this potential value in the expression of Π el (Eq. (2.3)), together with Π vdW (Eq. (2.4)) one obtains for the pressure barrier with surfactant [Π el (r = 0) + Π vdW (r = 0)]max = 52,500 Pa at hf = 4.9 nm (calculated for bubble height hb = 30 nm and 2 × 10−3 KCl). Without surfactant Ψ LG = −35 mV, the pressure maximum is [Π el (r = 0) + Π vdW (r = 0)]max = 4325 Pa at hf = 9.6 nm (hb = 30 nm and 1 × 10−3 KCl, see Fig. 4). For these two systems, following the procedure described in Section 3, for hrupt one obtains negligible differences in hrupt for the two cases—with and without surfactant (see Table 1). As it can be seen, the difference of Ψ LG leads to insignificant difference in hrupt . Thus, in the framework of this rupture mechanism, the higher pressure barrier alone cannot explain the longer lifetime of the wetting film with surfactant. The quantitative discrepancy between the disjoining pressure barrier effect and the experimentally observed longer Table 1 Rupture thicknesses of wetting films with and without surfactant for different bubble height and the same TPC angle, θ = π/2 hb (nm)
Fig. 5. Dependence of the deflection h∞ − ζ(r = 0) on the wetting film thickness h∞ . The circle corresponds to the contact between the bubble surfaces (the dashed line hb = 20 nm) and the film.
30 35 40 45
hrupt (nm) Without surfactant (1 × 10−3 M KCl)
With 10−5 M SDS (2 × 10−3 M KCl)
32.22 37.33 42.42 47.50
31.29 36.16 41.00 45.81
In all cases the variations of hrupt are experimentally negligible.
140
R. Slavchov et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 261 (2005) 135–140
lifetimes requires detailed analysis of the second possibility. How does the surfactant affect the size of the nanobubbles and their concentration at the substrate? From the theory it is known that in the case of the so-called homogeneous nucleation, i.e. when the new phase is formed in the bulk, the equilibrium radius of curvature of the new phase nucleus (the bubble radius rb in our case) is proportional to the liquid/gas surface tension: rb ∼ γ LG (see e.g. [25]). In our nanobubble model actually, the new phase nuclei are in contact both with bulk and surface phases and one speaks of heterogeneous nucleation. Consequently, here the equilibrium nucleus size (volume) depends on two parameters: bubble radius (of curvature) rb and (TPC) angle θ, or whichever combination of two geometric parameters defining the adhered at the solid surface bubble volume. As already many times discussed, important geometric parameter of our analysis is not rb but the bubble height, hb = rb (1 + cos θ). After substituting rb and using cos θ = (γ SG − γ SL )/γ LG (where γ SG and γ SL are the surface tensions of the corresponding solid/gas, solid/liquid surfaces), one derives hb ∼ γ LG [1 + (γ SG − γ SL )/γ LG ]. The surfactant lowers the surface pressure γ LG without changing γ SG and γ SL (electron spectroscopy for chemical analysis [12]). Thus should be expected hb to become smaller with addition of surfactant.
4. Concluding remarks The present study is only the first step in the analysis of the wetting film stability. After deriving the equilibrium deformations ζ(r), a future stability analysis should be carried out and Scheludko–Vrij wave mechanism must be generalized and applied to heterogeneous systems. Preliminary estimations were made, which have shown that, in contrast to homogeneous planar liquid films, in films with local attractive spots the length of the so-called unstable waves must be smaller than the size of the attractive spot, in order to be able to rupture the film. It has been mentioned above that neither the effect of the solid–gas surface charge, nor the boundary effects on the TPC line are considered. The model is developed only for TPC angles larger than 90◦ , and should be generalized (note that θ > 90◦ for more of the cited experiments). In the present model nanobubbles are considered as spherical, i.e. their deformation in the foam film region (where wetting film surface and bubble surface interact) is neglected but only the wetting film surface deformation ζ(r), is taken into account. The reason for that approximation follows from the well known nature of capillary surfaces: when bubbles (droplets) collide, the smaller particle deforms the larger one. From this viewpoint considering the bubble deformation one could expect only corrections to the results accounting deformation only of the wetting film surface, as in Section 3.
Nevertheless, in a more precise analysis the effects of bubble deformation should be accounted as well. The future work relies very much on the experiment. The hypothesis that we have made for dependence of the bubble size on the presence of surfactant, as well as the other assumptions, ought to be confirmed experimentally.
Acknowledgments This study is partly supported by the Bulgarian National Science Foundation Project Y-X-2/2003. K.W.S. is grateful for the financial support of the German Research Council (DFG), Special Research Area (SFB) 285.
References [1] A. Scheludko, Proc. Kon. Ned. Akad. Wet. B65 (1962) 87. [2] O. Reynolds, Phil. Trans. R. Soc. A177 (1886) 157. [3] B. Radoev, D.S. Dimitrov, I.B. Ivanov, Colloid Polym. Sci. 252 (1974) 50. [4] A. Vrij, Disc. Faraday Soc. 42 (1966) 44. [5] B. Radoev, A. Scheludko, E. Manev, J. Colloid Interface Sci. 95 (1) (1983) 254. [6] L. Mandelstam, Ann. Phys. 41 (1913) 609. [7] J. Israelachvili, Intermolecular Forces, Academic Press, London, NY, 1991. [8] E.D. Manev, Ann. Univ. Sofia, Fac. Chem. 75 (1981) 174. [9] K.J. Mysels, K. Shinoda, S. Frankel, Soap Films, Pergamon Press, N.Y., 1959; S. Frankel, K. Mysels, J. Phys. Chem. 66 (1962) 190. [10] E. Manev, R. Tsekov, B. Radoev, J. Disper. Sci. Technol. 18 (1997) 769. [11] H.J. Schulze, K.W. St¨ockelhuber, A. Wenger, Colloids Surf. A192 (2001) 61. [12] K.W. St¨ockelhuber, B. Radoev, A. Wenger, H.J. Schulze, Langmuir 20 (2004) 164. [13] N.V. Churaev, Adv. Colloid Interface Sci. 58 (1995) 87. [14] R.-H. Yoon, Int. J. Miner. Proc. 58 (2000) 129. [15] W. Gong, J. Stearnes, D. Fornasiero, R. Hayes, J. Ralston, J. Phys. Chem. Chem. Phys. 1 (1999) 2799. [16] J. Mahnke, J. Stearns, J. Ralston, Phys. Chem. Chem. Phys. 1 (1999) 2793. [17] N. Ishida, M. Sakamoto, M. Miyahara, K. Higashitani, Langmuir 16 (2000) 5681. [18] J.W.G. Tyrell, P. Attard, Langmuir 18 (2002) 160. [19] N. Ishida, M. Sakamoto, M. Miyahara, K. Higashitani, J. Colloid Interface Sci. 253 (2002) 112. [20] J.W.G. Tyrell, P. Attard, Phys. Rev. Lett. 87 (2001) 176104. [21] S. Lou, J. Gao, X. Xiao, X. Li, G. Li, Y. Zhang, M. Li, J. Sun, J. Hu, Chin. Phys. 10 (2001) 108. [22] G. Kar, S. Chander, T.S. Mika, J. Colloid Interface Sci. 44 (1973) 347. [23] Tz. Ivanov, B. Radoev, Colloids Surf. A 244 (2004) 19. [24] D. Exerowa, P.M. Kruglyakov, Foam and Foam Films, Elsevier, Amsterdam, 1998. [25] I.V. Markov, Crystal Growth for Beginners, World Scientific, Singapore, 1995.