ARTICLE pubs.acs.org/Langmuir
Electro-Marangoni Effect in Thin Liquid Films Stoyan I. Karakashev* and Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1 James Bourchier Blvd, Sofia 1164, Bulgaria ABSTRACT: The present paper reports a new drainage model accounting for the electro-Marangoni effect in thin liquid films stabilized by ionic surfactants. It was shown that the liquid outflow during the film drainage drifts charges from the diffuse part of the electrical double layer toward the film rim and thus generates a streaming potential along the film plane. This creates reverse fluxes near the film surfaces due to the requirement for zero electric current in the film. In a previous paper on this model (Tsekov et al. Langmuir, 2010, 26, 4703), the immobile surfaces were assumed. Here, the film surfaces were considered mobile, and surface velocity is controlled by an electro-Marangoni number. It was also shown that the motion of the charges makes the film surfaces more mobile, and they flow in reverse direction to the overall liquid outflow. The theory was validated by experimental data on drainage of planar foam films stabilized by cationic (tetrapentyl ammonium bromide) and anionic (sodium dodecyl sulfate) surfactants. A good agreement between the theoretical prediction and experimental data was found.
1. INTRODUCTION This work aims at studying the dynamic behavior of the electrical double layer (EDL) in thin liquid films (TLFs) stabilized by ionic surfactants. The Marangoni number is a well-known parameter in the TLF drainage theory, controlling the mobility of the film surfaces. However, the motion of charges in the EDL diffuse and surface parts during the film drainage has been disregarded for many years due to the complexity of the problem. Scheludko1 first introduced the Reynolds model for drainage of planar films with immobile surfaces, Radoev et al.2 assumed mobile surfaces of planar films accounting for the Marangoni effect, and Ivanov and Dimitrov3 and Barber and Hartland4 added to the theory the effect of the film surface viscosity. These findings were summarized in ref 5. The evolution of thin films with dimples was analytically described by Tsekov and Ruckenstein,6 while Sharma and Ruckenstein,7 Manev et al.,8 and Tsekov and Evstatieva9 developed analytical models accounting for the film surface inhomogeneity during the drainage. The evolution of inhomogeneous thin films was numerically simulated by Joye et al.,10 Li,11 Manica et al.,12,13 Chan et al.,14 and Dagastine et al.15 All these works were based on the conventional theory of considering the charges in the EDL immobile. Hence, it is valid only for TLFs stabilized by nonionic surfactants. According to the conventional theory, the Marangoni effect, estimated by the Marangoni number, decreases the mobility of film surfaces during drainage. At a large enough Marangoni number, the film surfaces become immobile and hence such planar films, stabilized by nonionic surfactants, drain according to the well-known Reynolds equation. The Reynolds drainage is the slowest possible drainage in the conventional drainage theory. r 2011 American Chemical Society
However, recent experiments16,17 showed that planar TLF, stabilized by ionic surfactants, drain even slower than the prediction of the Reynolds equation. These experimental findings were in line with the numerical simulations performed earlier in ref 18. It was found in our former work19 that the reason for this slower drainage is the emergence of a streaming potential across the film radius, originating from the outflow of charged liquid from the EDL diffuse part. This causes the occurrence of reversed fluxes close to the film surfaces, thus decreasing the overall velocity of the film drainage. This model presumed immobile film surfaces similar to the Reynolds equation. The present work reports an advancement of the model from ref 19 by assuming mobility of the film surfaces. The electrostatic and van der Waals contributions were accounted for in the Navier-Stokes equation, along with the assumption for nonzero surface velocity. Thus a new Marangoni number was derived accounting for the electrical effects originating from the motion of charges. The effect of the electrical charges on surface mobility was introduced here by an electro-Marangoni number. This new model was validated with experiments on drainage of planar TLFs stabilized by the ionic surfactants tetrapentyl ammonium bromide (TPeAB) and sodium dodecyl sulfate (SDS). A good juxtaposition between experimental data and theory was found. A comparison between the present and the former theory19 was performed. Since the following theory advances our previous analysis, more details can be found in ref 19. Received: November 9, 2010 Revised: January 10, 2011 Published: February 10, 2011 2265
dx.doi.org/10.1021/la1044656 | Langmuir 2011, 27, 2265–2270
Langmuir
ARTICLE
2. THEORY Since the liquid films are very anisotropic structures, the overall electric potential in the film φ(r,z) can be presented as a superposition of tangential streaming ψ(r) and normal double layer potentials: ZeΓðrÞ coshðkzÞ φðr, zÞ ¼ ψðrÞ þ ε0 εk sinhðkh = 2Þ
Fvr dz ¼ - ε0 ε -h = 2
ZeΓk ¼ sinhðkh = 2Þ
ðZeΓÞ2 ε0 εk2 ðφs - ψÞ2 ¼ 2 2ε0 ε sinh ðkh = 2Þ 2 cosh2 ðkh = 2Þ
ð4Þ
cothðkh = 2ÞΠEL g
1 Dr f2½ðkzÞ2 - ðkh = 2Þ2 4ηk2
vr ¼
þ ½4 sinhðkh = 2Þ coshðkzÞ - sinhðkhÞ - kh cothðkh = 2ÞΠEL g
1 Dr f½ðkzÞ2 - ðkh = 2Þ2 2ηk2
Dr ðrvr Þ = r þ Dz vz ¼ 0
ð11Þ
and integrating on z yields the normal component of the hydrodynamic velocity: n z Dr rDr 2½ðkzÞ2 = 3 - ðkh = 2Þ2 vz ¼ 2 4ηk r þ kh cothðkh = 2Þ - 2ðp - ΠVW - ΠEL Þ sinhðkzÞ - sinhðkhÞ - kh þ 4 sinhðkh = 2Þ kz o cothðkh = 2ÞΠEL ð12Þ Finally, substituting this expression in the kinematic relation ∂th/2 = νz (z = h/2) yields Dt h ¼
h Dr ðrDr f½ðkhÞ2 - 6kh cothðkh = 2Þ þ 12 12ηk2 r ðp - ΠVW - ΠEL Þ þ 3½sinhðkhÞ þ kh - 4 sinh2 ðkh = 2Þ = ðkh = 2Þ cothðkh = 2ÞΠEL gÞ ð13Þ
Usually κh > 1, and in this case eq (13) simplifies to ( " #) 1 sinhðkhÞ 3 Dt h ¼ Dr rh Dr p - ΠVW þ 3 ΠEL ð14Þ 12ηr ðkhÞ2 To determine the gradient of the electrostatic disjoining pressure, one can use the interfacial stress balance
ðp - ΠVW - ΠEL Þ þ 2½coshðkzÞ - coshðkh = 2Þ coshðkh = 2ÞΠEL g þ vs
ð10Þ
Introducing eq (10) into the continuity equation
ð6Þ
As seen, the electro-kinetic effect can be conveniently expressed via the electrostatic disjoining pressure and its effect is stronger than ΠEL at large film thicknesses. Since the local dynamic and disjoining pressures do not depend on z, one can integrate eq (6) twice to obtain vr ¼
ð9Þ
Hence, the radial component of the hydrodynamic velocity acquires the form
ð5Þ
Note that ΠEL depends on r via ψ(r), due to nonhomogeneous adsorption along the film surfaces. Using eq (2) the radial momentum balance can be further developed to obtain Dr ½p - ΠVW - ΠEL þ coshðkzÞ coshðkh = 2ÞΠEL ¼ ηDz 2 vr
-h = 2
ðp - ΠVW - ΠEL Þ þ ½sinhðkhÞ - kh
where the electrostatic repulsive disjoining pressure is introduced via the relation ΠEL ¼
ð8Þ
1 Dr f4½ðkh = 2Þ cothðkh = 2Þ - 1 4ηk2
vs ¼
ð3Þ
coshðkzÞ Dr ψ sinhðkh = 2Þ
Dz ð p - ΠVW Þ ¼ 0
coshðkzÞvr dz ¼ 0
þ kh cothðkh = 2Þ - 2ðp - ΠVW - ΠEL Þ
where p and ΠVW are the local dynamic and van der Waals disjoining pressures, respectively, and F is the local charge density. Using the Poisson equation in the lubrication approximation ε0ε∂z2φ = -F and the electric potential from eq (1) leads to simplifications:
¼ ηDz 2 vr
hZ= 2
Substituting eq (7) here results in
Dr ½p - ΠVW þ ε0 εðDz φÞ2 = 2 þ FDr φ ¼ ηDz 2 vr
Dr ðp - ΠVW - ΠEL Þ - ZeΓk
ðDz 2 φÞvr dz -h = 2
ð2Þ
In the case of a constant surface potential φs, its dependence along the film radial position follows the streaming potential ψ(r). In the frames of the lubrication approximation, the film electro-hydrodynamics obeys the Navier-Stokes-Reynolds equations:19 Dz ½ p - ΠVW þ ε0 εðDz φÞ2 = 2 þ FDz φ ¼ 0
hZ= 2
hZ= 2
ð1Þ
The last expression describes the electric double layer at relatively low surface potential φs,19 where κ is the Debye parameter and Γ is the local adsorption of the ionic surfactant charging the film surface. The surface charge density can be expressed from eq (1) as ZeΓ ¼ ε0 εk tanhðkh = 2Þðφs - ψÞ
where νs is the surface velocity, which can be determined from the condition of zero electric current in the film:
~ s ¼ - kB TΓvs = Ds ηðDz vr Þs ¼ - Dr σ ¼ ΓDr μ
ð7Þ 2266
ð15Þ
dx.doi.org/10.1021/la1044656 |Langmuir 2011, 27, 2265–2270
Langmuir
ARTICLE
Figure 1. The dimensionless radial velocity Vr of liquid outflow along the dimensionless normal coordinate x of the film obtained from eq (19) at κh = 10.
where Ds is the surface diffusion coefficient of the surfactant. To simplify the consideration, the surfactant adsorption Γ, being a function of r, is represented in eq (15) by its average value Γ h , since the surface velocity νs is usually small. The radial velocity νr from eq (10) and surface velocity νs from eq (9) are introduced into eq (15) yielding for the case of κh > 1 the following relation between the gradients of dynamic and disjoining pressures: 2kh 1 þ Ma Dr ðp - ΠVW Þ ð16Þ Dr ΠEL ¼ sinhðkhÞ 2 þ Ma For the sake of compactness, the new electro-Marangoni number is introduced in eq (16) via the relation Ma* kBTΓ h/ ηDsκ. The difference between Ma* and the conventional Marangoni number is that the film thickness h is replaced by the Debye length 1/κ. Substituting eq (16) in eq (14) yields 1 1 þ Ma 6 Dr rh3 1 Dt h ¼ ðp Π Þ ð17Þ D r VW 12ηr 2 þ Ma kh Standard integration of eq (17) results in the following expression for the film rate of thinning V = -∂th: V 1 þ Ma 6 ¼ 1VRe 2 þ Ma kh
ð18Þ
2 where VRe = 2h3(p - Π VW - Π EL)/3ηR is the Reynolds velocity, h h 2 2 2 Π h EL = ε0εκ φs /2 cosh (κh/2) is the mean disjoining pressure, and R is the film radius. In the case of large electro-Marangoni number, the film surface becomes immobile and eq (18) reduces to V/VRe = 1 - 6/κh. Note that the last correction, derived for a constant surface potential, is twice smaller than that obtained at constant surface charge density.19 In the opposite case of low electro-Marangoni number, eq (18) reduces to V/VRe = 1 - 3/κh. Hence, the electro-kinetic effect is twice weaker due to rearrangement of the surfactant ions on the film surface. Thus, in the realistic case, when neither the surface potential nor the surface charge density are constants and the electro-Marangoni number is unknown, one can propose the following expression for the film thinning rate V/VRe = 1 - n/κh, where n is a dimensionless fitting parameter.
Figure 2. The dimensionless radial velocity Vr of liquid outflow along the dimensionless normal coordinate x of the film obtained from eq (19) at Ma* = 2.
The theory described above is illustrated by calculating the dimensionless radial velocity of the liquid outflow from eq (10) Vr -
2ηk2 vr ¼ ðkh = 2Þ2 ð1 - 4x2 Þ - kh Dr ðp - ΠVW Þ 1 þ Ma coshðkhxÞ kh 2 1 ð19Þ þ 2 þ Ma sinhðkh = 2Þ
where x = z/h is a dimensionless normal coordinate. An example of the profile of Vr across the film during the film drainage at different values of the electro-Marangoni numbers Ma* is presented in Figure 1. It is apparent in Figure 1 that the liquid outflow at x = 0 has maximal velocity and is directed from the center toward the periphery of the film, while the liquid outflow close to the film surfaces has reverse direction. Moreover, when the film surfaces become more mobile, the flow is directed from the periphery toward the center of the film at small and intermediate values of the electro-Marangoni number Ma*. The larger the electro-Marangoni number, the less mobile the film surfaces are. The change of the dimensionless velocity profile at different values of κh is presented in Figure 2. It is evident that ratio film thickness/inverse Debye length is a significant factor in controlling the velocity profile. The thicker the diffuse part of the EDL, the slower the rate of film drainage. These effects originate from the streaming potential. Using the well-known Reynolds parabolic profile of the pressure, one can derive from eq (16) the following expression giving the local deviation of the electrostatic disjoining pressure from the average electrostatic one across the film radius: ! ΔΠEL 1 þ Ma 2kh r2 2 2 -1 ¼ ð20Þ 2 þ Ma sinhðkhÞ Δp R where ΠEL is local electrostatic disjoining pressure depending on the local adsorption of charged surfactant and the local surface 2 2 2 potential, ΔΠEL = ΠEL- Π EL, Π EL = ε0εκ φs /2 cosh (κh/2) is h h average electrostatic disjoining pressure, where φs is the surface potential constant over the film area, and Δp is the driving pressure due to the meniscus. An example of the deviation of the local electrostatic disjoining pressure from the average one at different electro-Marangoni numbers is presented in Figure 3. It is apparent that the local electrostatic disjoining pressure in the center of the film is smaller 2267
dx.doi.org/10.1021/la1044656 |Langmuir 2011, 27, 2265–2270
Langmuir
ARTICLE
Figure 5. The ratio Ma*/Ma as a function of the film thickness.
Figure 3. Deviation of the local electrostatic disjoining pressure from the average one across the film radius at different electro-Marangoni numbers and κh = 5 (see eq (20)).
Figure 6. Scheme of the interferometric setup for experiments on TLFs and its model presentation (not to scale).
Figure 4. Deviation of the local electrostatic disjoining pressure from the average one across the film radius at different κh and Ma* = 2 (see eq (20)).
than the average one corresponding to reduced adsorption of charged surfactant as compared to the equilibrium. The latter means that the local surfactant adsorption on the film surface is controlled mainly by two factors: (i) electric effect driving the surface charges toward the film periphery due to the generation of streaming potential across the film radius, and (ii) the film surface motion (see Figures 1 and 2) bringing the surface charges back to the center of the film. It is evident in Figure 3 that the electric effect dominates. Thus the local adsorption of the charged surfactant increases toward the film periphery giving rise to the local electrostatic disjoining pressure as well. At the very film rim, the local electrostatic disjoining pressure is larger than the average one due to the accumulation of surface charges. The overall dependence weakens upon the decrease of the electro-Marangoni number (see Figure 3) as a result of the enhanced mobility of the film surface driving the surface charges back toward the center of the film. The same dependence of the local deviation on the electrostatic disjoining pressure, but at different κh, is presented in Figure 4. One can see the dependence ΔΠEL/Δp versus r/R becomes stronger upon the film drainage. The latter can be explained with the decrease of the film surface velocity (see Figure 2) during the film drainage. Thus the inverse flux of charged surfactant, which compensates the surface charges, which are already shifted toward the film rim by the
streaming potential, decreases, making the overall dependence ΔΠEL/Δp versus r/R stronger. The functional dependence of the ratio between the electroMarangoni number to the classical Marangoni number Ma*/Ma on the film thickness h is presented in Figure 5. It is apparent that Ma*/Ma < 1 at high κh, and it increases upon the film drainage exceeding unity at κh < 10. Evidently the motion of electrical charges due to film drainage decreases the Marangoni number as compared to the conventional case of an immobile EDL.5
3. EXPERIMENT Experiments on planar TLF drainage with two ionic surfactants were conducted: • SDS; • SDS þ 0.02 M NaCl; • TPeAB. The surface tensions of the solutions were measured via the Wilhelmy plate method at a temperature of 25 °C. The micro-interferometric method was used to determine the transient behavior of foam films. The full description of the experimental apparatus was previously reported20,21 and is not fully repeated here. Briefly, the apparatus consists of a glass cell, for producing horizontal foam films, normal to gravity. First, a droplet of surfactant solution was formed inside the film holder. Then the amount of liquid was regulated by means of a gastight microsyringe connected to the film holder through a glass capillary. Finally, a microscopic film was formed between the apexes of the doubleconcave meniscus by pumping out the liquid from the drop. The foam films were kept in contact with saturated water vapors (see Figure 6). The film radius was kept constant within a range of (1 μm. A metallurgical inverted microscope was used for illuminating and observing the film and the interference fringes (the Newton rings) in reflected light by means of wavelength λ = 546 nm and a digital camera 2268
dx.doi.org/10.1021/la1044656 |Langmuir 2011, 27, 2265–2270
Langmuir
ARTICLE
Figure 8. Theoretical predictions and experimental data on the kinetics of the thinning of foam films stabilized by 10-6 M SDS þ 0.02 M NaCl and 5 10-4 M SDS.
Figure 7. Basic stages from the drainage of small (Rf < 0.05 mm) foam films (not to scale). system connected with computer for storage of the data. The interferograms were processed offline using the “Image J” software for image processing delivering the pixel signal from a given small area of the film as a function of the time, thus producing temporal interferograms, which were used for further calculation of the film thickness versus time. At least 10 evolutions of foam films at each surfactant concentration were recorded. The film thickness was calculated with precision of (2 nm. The film radii were purposely made below 50 μm for achieving good planarity. Hence, the recorded foam films were kept planar during the drainage of the foam films (see Figure 6). The basic stages from the drainage of such films are depicted in Figure 7. First, the films become dimpled at sufficiently high thickness. The dimple relaxes fast, allowing the film to acquire planar shape (see Figures 6 and 7).
4. RESULTS AND DISCUSSION Equation (18) can be presented in the following form: dh 2h3 1 þ Ma 6 ¼ 1 V¼ðpσ - ΠÞ ð21Þ dt 2 þ Ma kh 3ηR 2 where pσ = 2σ/Rc is the capillary pressure, and Rc is the radius of the film holder. Equation (21) was numerically integrated to obtain the transient film thickness using the fourth-order RungeKutta algorithm. A program was written using the Visual Basic for Application programming language available in Microsoft Excel. Examples about the coincidence between experiment and theory are shown in Figures 8 and 9. The predictions of the present theory (Ma* > 0), of the theory in ref 19 without the electro-Marangoni
Figure 9. Theoretical predictions and experimental data on the kinetics of the thinning of foam films stabilized by 5 10-4 M TPeAB and 5 10-3 M TPeAB.
effect, and of the Reynolds equation were compared to the experimental data. It is apparent that in all cases the film drains slower than the prediction of the Reynolds equation. The new theory predicts faster drainage as compared to that from ref 19. 2269
dx.doi.org/10.1021/la1044656 |Langmuir 2011, 27, 2265–2270
Langmuir The importance of electrostatic effect increases with increase of the ionic strength. Unfortunately, the increase of the ionic surfactant concentration also results in tangential immobilization of the film surfaces due to Marangoni effect, and that is why the results differ slightly from the case reported in our previous paper.19 To magnify the effect, we added some results at low surfactant concentrations and sufficient ionic strength (0.02M) due to added electrolyte. One can see that in this case the difference between the predictions of our former theory19 and the present one becomes essential (see Figure 8). It should be noted that the new theory is valid for κh > 1 (see Figure 2). For this reason, the minimal surfactant concentration (for the cases of no added salt) chosen is 5 10-4 M. The electrostatic repulsion between the film surfaces was considered under the regime of constant surface potential. However, at a minimal electrolyte concentration of 5 10-4 M, the solution of eq (18) is weakly sensitive to the value of the surface potential. Hence, the theoretical curves were calculated avoiding any fitting procedure. It is evident as well that the prediction of the new theory is closer to the experimental data than the theory reported in ref 19, which shows that there is sufficient surface mobility of the film interfaces. The present theory reveals an electro-dynamic effect, caused by the emergence of streaming potential across the film radius, expressed in decreasing the speed of film drainage. This effect was ignored for many years due to the complexity of the problem. On the macroscale, it can be exploited for predicting the durability of dispersed systems (gas or oil emulsion, suspension, etc.) under dynamic conditions.
5. CONCLUSIONS The present work reports a new effect in the drainage theory of TLFs, quantified by the electro-Marangoni number. It was reported in the literature (e.g., refs 16 and 18) that TLFs, stabilized by ionic surfactants, drain slower than the prediction of the Reynolds equation. This effect cannot be explained by the conventional drainage theory, which assumes no dynamic effects of the EDL. Numerical simulations18 reported inverted fluxes close to the film surfaces, slowing down the drainage of TLFs. However, this result appears to be a numerical solution of the full set of hydrodynamic equations accounting for the electrical dynamic effects. The real physical reason for this effect was still obscure. A recent work19 explained this effect by the emergence of streaming potential along the film radius, which causes inverted fluxes to keep the requirement for zero electric current in the film. It presumed immobile film surfaces during the drainage. The present work assumes mobile film surfaces, thus investigating the effect of the motion of charges on surface mobility. It was established that the motion of charges makes the film surfaces more mobile; however, the film surfaces flow in the reverse direction to the overall liquid outflow (see Figures 1 and 2). In addition, the stationary levels of zero tangential velocity of the liquid outflow shift toward the film center upon the film drainage (see Figure 2). Our calculations showed that the local surfactant adsorption on the film surface is controlled mainly by two opposite effects: (i) the electric effect driving the surface charges toward the film’s periphery due to the generation of streaming potential across the film radius, and (ii) the film surface motion (see Figures 1 and 2) bringing the surface charges back to the center of the film. The electric effect dominates. The reductions of the electroMarangoni number and the film thickness (during the drainage)
ARTICLE
increase the surface velocity from the film’s periphery toward the film center, thus increasing the local adsorption of charged surfactant in the central area of the film. The electro-Marangoni number has smaller values as compared to the classical Marangoni number (see Figure 5). The present theory is valid for κh > 1. It was validated by experimental data on drainage of planar foam films with one cationic (TPeAB) and one anionic (SDS and SDS þ 0.02 M NaCl) surfactant. The validation established a better coincidence between theory and experiment as compared to ref 19, although the difference in both coincidences is small in the case of small ionic strength and essential in the case of enough high ionic strength (e.g., 0.02 M).
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]fia.bg.
’ ACKNOWLEDGMENT S.I.K. gratefully acknowledges the EC “Marie Curie Actions” for financial support through DETLIF - Project No. 200688/ 2009. ’ REFERENCES (1) Scheludko, A. Adv. Colloid Interface Sci. 1967, 1 (4), 391–464. (2) Radoev, B. P.; Dimitrov, D. S.; Ivanov, I. B. Colloid Polym. Sci. 1974, 252 (1), 50–55. (3) Ivanov, I. B.; Dimitrov, D. S. Colloid Polym. Sci. 1974, 252 (11), 982–990. (4) Barber, A. D.; Hartland, S. Can. J. Chem. Eng. 1976, 54 (4), 279– 284. (5) Ivanov, I. B., Ed. Thin Liquid Films; Marcel Dekker: New York, 1988. (6) Tsekov, R.; Ruckenstein, E. Colloids Surf., A 1994, 82 (3), 255–261. (7) Sharma, A.; Ruckenstein, E. Colloid Polym. Sci. 1988, 266 (1), 60–69. (8) Manev, E.; Tsekov, R.; Radoev, B. J. Dispersion Sci. Technol. 1997, 18 (6 & 7), 769–788. (9) Tsekov, R.; Evstatieva, E. Prog. Colloid Polym. Sci. 2004, 126, 93–96. (10) Joye, J.-L.; Hirasaki, G. J.; Miller, C. A. J. Colloid Interface Sci. 1996, 177 (2), 542–552. (11) Li, D. Chem. Eng. Sci. 1996, 51 (14), 3623–3630. (12) Manica, R.; Connor, J. N.; Carnie, S. L.; Horn, R. G.; Chan, D. Y. C. Langmuir 2007, 23 (2), 626–637. (13) Manica, R.; Klaseboer, E.; Chan, D. Y. C. Soft Matter 2008, 4 (8), 1613–1616. (14) Chan, D. Y. C.; Klaseboer, E.; Manica, R. Soft Matter 2010, 6 (8), 1809–1815. (15) Dagastine, R. R.; Webber, G. B.; Manica, R.; Stevens, G. W.; Grieser, F.; Chan, D. Y. C. Langmuir 2010, 26 (14), 11921–11927. (16) Karakashev, S. I.; Ivanova, D. S. J. Colloid Interface Sci. 2010, 343 (2), 584–593. (17) Karakashev, S. I.; Tsekov, R.; Ivanova, D. S. Colloids Surf. A 2010, 356 (1-3), 40–45. (18) Valkovska, D. S.; Danov, K. D. J. Colloid Interface Sci. 2001, 241 (2), 400–412. (19) Tsekov, R.; Ivanova, D. S.; Slavchov, R.; Radoev, B.; Manev, E. D.; Nguyen, A. V.; Karakashev, S. I. Langmuir 2010, 26 (7), 4703– 4708. (20) Karakashev, S. I.; Nguyen, A. V. Colloids Surf. A 2007, 293, 229–240. (21) Exerowa, D.; Kruglyakov, P. M. Foam and Foam Films: Theory, Experiment, Application; Marcel Dekker: New York, 1998; p 796. 2270
dx.doi.org/10.1021/la1044656 |Langmuir 2011, 27, 2265–2270
