5674
Langmuir 1997, 13, 5674-5677
Hydrophobic Forces in Thin Liquid Films: Adsorption Contribution Roumen Tsekov* and Hans Joachim Schulze Research Group at the Freiberg University of Mining and Technology, Max Planck Institute for Colloids and Interfaces, 09599 Freiberg, Germany Received March 4, 1997. In Final Form: June 12, 1997X The contribution of adsorption to the hydrophobic forces in a thin liquid film is considered. It is shown that the corresponding disjoining pressure component depends exponentially on the film thickness. Expressions for both the magnitude and decay length of this force are proposed in terms of well-known thermodynamic characteristics. Comparison of the theory to existing experimental results confirms the importance of the adsorption component to the hydrophobic interaction.
Introduction Thin liquid films are basic structural elements of foams and emulsions, and for this reason, detailed knowledge of their dynamics is important for flotation, food industry, cosmetics, etc. The understanding of the forces between surfaces in liquids is also essential for the description of many related phenomena of intermolecular interactions1 such as wetting, swelling, and adhesion. Historically, the excess free energy in a thin liquid film as compared to the bulk liquid is described by the so-called disjoining pressure2
Π ) p(∞) - p(h) where p(h) is the pressure in a bulk liquid phase being equilibrated with a flat film with thickness h and p(∞) ) p(hf∞). The dependence of the disjoining pressure on the film thickness is a problem which has been the object of intensive investigations. The classical DLVO theory is based on the van der Waals and electrostatic forces only. During the last two decades, however, a number of new interesting effects of macroscopic interactions have been observed. Here, one can mention hydrophobic, hydration, protrusion, undulation and ion correlation forces.3 In foam films at surfactant concentrations above the cmc, the disjoining pressure isotherm exhibits possible periodic behavior, which is responsible for film stratification.4 The scope of the present paper is to present a simple explanation of the hydrophobic forces. The latter manifest themselves in a dramatic change in the stability of films when their surfaces are properly modified by surfactant layers.5 The first direct measurement of hydrophobic forces is dated more than a decade ago.6 However, despite several recent attempts to provide a theoretical model or explanation of the hydrophobic forces,7-11 their origin remains shrouded in mystery. The only well established X Abstract published in Advance ACS Abstracts, September 15, 1997.
(1) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1994. (2) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Plenum: New York, 1987. (3) Christenson, H. K. J. Dispersion Sci. Technol. 1988, 9, 171. (4) Bergeron, V.; Radke, C. J. Langmuir 1992, 8, 3020. (5) Churaev, N. V. J. Colloid Interface Sci. 1995, 172, 479. (6) Israelachvili, J. N.; Pashley, R. M. J. Colloid Interface Sci. 1984, 98, 500. (7) Claesson, P. M.; Blom, C. E.; Herder, P. C.; Ninham, B. W. J. Colloid Interface Sci. 1986, 114, 234. (8) Claesson, P. M.; Christenson, H. K. J. Phys. Chem. 1988, 92, 1650. (9) Podgornik, R. J. Chem. Phys. 1989, 91, 5840. (10) Ruckenstein, E.; Churaev, N. V. J. Colloid Interface Sci. 1991, 147, 535. (11) Tsao, Y.; Evans, D. F.; Wennerstro¨m, H. Langmuir 1993, 9, 779.
S0743-7463(97)00239-4 CCC: $14.00
Figure 1. Schema of the film.
experimental fact is that they vanish exponentially with increasing film thickness.6,12 More recent investigations11,13 show the existence of short- and long-range hydrophobic forces. The present work relates the hydrophobic interaction to the adsorption contribution and thus explains the existing experimental results. A new approach for treatment of the adsorption effects in thin films is developed which differs substantially from the existing ones.2 Theoretical Model Obviously, the hydrophobic forces have an interfacial origin. This is why they strongly depend on the surface properties and can be changed by surface modifications. Hence, they should be explored by the gain or loss of specific interfacial energy during the film thinning. For instance, if a hole is produced in a symmetric foam film, roughly speaking the system gains additional free energy equal to twice the hole area multiplied by the film surface tension. This should be a source of the acting attractive surface forces. By making the film surfaces more hydrophobic, one increases the surface tension and thus increases the attraction. This additional force is naturally called the hydrophobic force. For the sake of transparency we will consider first a twocomponent thin liquid film with a modified liquid/solid surface (see Figure 1). However, the generalization of our approach is obvious. We will restrict ourselves to the case of an isothermal system. Hence, the interfacial Gibbs-Duhem equation for the film reads14
d(σ1 + σ2) ) -Γ dµ - Π dh
(1)
where σ1 and σ2 are the surface tensions on the two film interfaces, Γ is the adsorption of the surfactant on the film/solid surface, and µ is the surfactant electrochemical potential. It is important not to mix up the surface tensions of the film σ1 and σ2 with the (12) Rabinovich, Ya. I.; Derjaguin, B. V. Colloids Surf. 1988, 30, 243. (13) Skvarla, J. J. Colloid Interface Sci. 1993, 155, 506. (14) Rusanov, A. I. Phasengleichgewichte und Grenzfla¨ chenerscheinungen; Akademie-Verlag: Berlin, 1978.
© 1997 American Chemical Society
Hydrophobic Forces in Thin Liquid Films
Langmuir, Vol. 13, No. 21, 1997 5675
film tension γ ) σ1 + σ2 + Πh. From eq 1 an alternative definition of the disjoining pressure follows
( ) ( )
Π)-
∂σ1 ∂h
∂σ2 ∂h
-
µ
(2)
µ
which is very important, since it relates the two specific interfacial parameters of the film, the disjoining pressure and the surface tensions. According to our simplified model, at constant temperature the gas/film surface tension depends only on the film thickness σ1(h) (for the sake of transparency we suppose that there is no adsorption on this interface, and a generalization will be given latter), while the h-dependence of the film/solid surface tension is carried by the adsorption as well, i.e. σ2(h,Γ). Thus, eq 2 can be split into two specific parts
[( ) ( ) ] ( ) ( )
Π)-
∂σ1 ∂h
+
µ
∂σ2 ∂h
∂σ2 ∂Γ
-
µ,Γ
h
∂Γ ∂h
µ
Figure 2. Schema of the three-phase contact.
(3)
One can recognize in the first part the van der Waals ΠVW and electrostatic ΠEL components of the disjoining pressure, while the last term represents a specific surfactant contribution. Since the adsorption Γ is located at the film surface, the interfacial origin of the last term is obvious and we attributed it to the hydrophobic interaction. Therefore, our definition of the hydrophobic disjoining pressure reads
( )( )
ΠHP ) -
∂σ2 ∂Γ
h
∂Γ ∂h
(4)
µ
The further employment of eq 4 requires an expression for the thickness dependence of the adsorption on the film/solid surface. It is evident from eq 1 that (∂Γ/∂h)µ ) (∂Π/∂µ)h. Thus, employing the thermodynamic definition (∂p/∂µ) ) c, one can obtain (∂Π/ ∂µ)h ) ∆c where ∆c ) c(∞) - c(h) is the difference between the surfactant concentrations in the bulk liquid equilibrated to films with infinite and finite thicknesses, respectively. Now, expanding ∆c in the power series of ∆Γ and keeping only the linear term, one could write a differential equation for the h-dependence of Γ
∂Γ Γ(∞) - Γ ) ∂h D
(5)
where Γ(∞) ) Γ(hf∞) is the adsorption on the solid surface from a bulk liquid phase and
D)
(∂Γ∂c )(hf∞)
The solution of eq 5 is
( Dh )
Γ ) Γ(∞) - [Γ(∞) - Γ(0)] exp -
∆E h exp D D
( )
(see Figure 2). The second relation in this equation is obtained by employing the Neuman-Young equation σsg - σsl ) σlg cos θ relating the surface tensions with the contact angle θ. Since the modulus of ∆E cannot be larger than σlg, it is reasonable to present ∆E0 as -σlg cos θ0. Hence,
∆E ) σlg(cos θ - cos θ0)
∆E ) ∆E0 - (σsl - σsg) ) ∆E0 + σlg cos θ where ∆E0 is a constant and σsg, σsl, and σlg are the surface tensions of the gas/solid, liquid/solid, and gas/liquid interfaces, respectively
∫
∞
0
Π dh
which is obtained by integration of eq 2, one can demonstrate that the value of θ0 is determined by the van der Waals and electrostatic components of Π
(6)
where E ) -Γ(∂σ2/∂Γ)h is the Gibbs elasticity of the adsorption layer and ∆E ) E(∞) - E(0). As seen, ΠHP from eq 6 decays exponentially with increasing film thickness, as observed in many experiments.2 The gradient D ) (∂Γ/∂c)(∞) of the adsorption with respect to the surfactant concentration in the bulk liquid has an effect as a decay length. In general, the difference ∆E ) -(∂σ2/∂Γ)∆Γ is expected to be proportional to ∆σ2. Keeping only the linear term in the power series of ∆E vs ∆σ2, one can write the following expression
(7)
As is seen, the sign of ΠHP depends on the value of the contact angle, and for this reason it deserves its name the hydrophobic disjoining pressure. If the solid surface is hydrophobic enough (θ > θ0), the resulting interaction is attraction (ΠHP < 0), while for hydrophilic surfaces (θ < θ0) our theory predicts a repulsive force (ΠHP > 0). Such a behavior is experimentally observed on hydrophobized mica surfaces separated by a water film,7 and the transition angle θ0 is estimated5 to be about 30°. Using eqs 6 and 7 and the general relationship between the contact angle and the disjoining pressure5
σlg(cos θ - 1) )
where Γ(0) is the adsorption on the gas/solid surface. Using this expression, one can rewrite eq 4 to obtain
ΠHP )
Figure 3. Principle adsorption isotherm from a bulk solution. The length D(c) ) ∂Γ/∂c is equal to the local slope of the curve.
σlg(cos θ0 - 1) )
∫
∞
0
(ΠVW + ΠEL) dh
The characteristic decay length D is a strong function of the surfactant concentration which is schematically shown in Figure 3. At low concentrations it could be extremely large (larger than microns). However, due to the inverse proportionality of the magnitude of ΠHP on D, these super-long-range interactions are weak and difficult to detect experimentally. At high concentrations D becomes of the order of nanometers. This correlates well with the existing measurements of short-range hydrophobic forces, which are usually observed at relatively close-packed surface layers (large Γ). Finally, at medium concentrations, the order of D could be dozens of nanometers and, perhaps, this is a reasonable explanation of the existence of the so-called longrange hydrophobic interactions.13
5676 Langmuir, Vol. 13, No. 21, 1997
Tsekov and Schulze
Table 1. Dependence of Specific Hydrophobic Disjoining Pressure Constants from Eq 9 on the Octanol Concentration in Films of a 5 µM Aqueous Solution of Dodecylamine Hydrochloride Separating Two Mica Surfacesa octanol (µM)
∆E1 (mN/m)
0.0 0.5 5.0
-8.0 -8.0 -8.0
aAll
∆E2 (mN/m)
D1 (nm)
D2 (nm)
-0.08 -0.08
1.4 1.2 1.2
6.8 4.0
data are taken from ref 16.
In the case of a multicomponent system we expect to have a contribution from each component to the hydrophobic disjoining pressure. Hence, accounting now for both film interfaces, one concludes that, in general, ΠHP is a superposition of exponential terms
Πik HP )
( )
∆Eik h exp Dik Dik
(8)
with Eik ) -Γik(∂σi/∂Γik)h and Dik ) (∂Γik/∂ck)(∞). The number of such terms is twice as larger as the number of components in the system. For symmetric films half of the terms are degenerate. An interesting point here is the case of a foam film of pure liquid. The borders of the film are placed on the so-called surfaces of tension, which are determined by the detailed force balance of the system. In general, they do not coincide with the equimolecular surfaces. Hence, solvent adsorption contributes also to the hydrophobic force. This force is an extra component to the usual van der Waals one which is calculated by the assumption that the liquid density F in the film is uniform and hence the solvent adsorption effect is not accounted for. Obviously, the pure solvent hydrophobic disjoining pressure is equal to
ΠHP ) -
ΠVW )
2σlg cos θ0 h exp D D
( )
where the two film surfaces are taken into account and, of course, the contact angle of a hole is 90°. As is seen, ΠHP could be either attractive or repulsive depending on the value of the angle θ0. In contrast to the case for the surfactants, however, the decay length here D ) (∂Γ/∂F) is difficult to handle. Hydrophobic forces in films of pure water have also been observed experimentally.15 Sometimes, hydrophobic forces in particular systems cannot be approximated only by exponential terms. The reason for such behavior could be found in a substantial contribution of the nonlinear terms in the power series of ∆c on ∆Γ. This will be reflected in a more complicated dependence of the adsorption on the film thickness and, hence, in nonexponential terms in ΠHP. Additional complications could arise from the h-dependence of the term (∂σ2/∂Γ)h in eq 4. In the present paper it is accepted to be constant because for small ∆Γ the main (linear) effect of the adsorption comes from (∂Γ/∂h)µ ) ∆c.
Interpretation of Some Existing Experimental Results The contributions from the terms of eq 8 to the hydrophobic disjoining pressure are not equivalent, due to different decay lengths Dik and Gibbs elasticities ∆Eik. Thus, one can describe well experiments with one or two exponents only corresponding to the leading terms in the sum
Π ) ΠVW + ΠEL +
( )
kar16 in their numerical fit of experimental results by eq 9. The considered system consists of two mica surfaces separated by a film of a 5 µM solution of dodecylamine hydrochloride (DAHCl) in water with various concentrations of octanol. As is seen, one can ascribe the first exponent in eq 9 to DAHCl, while the second exponential term is responsible for the octanol contribution. Table 1 confirms our expectations, since D2 increases with decreasing octanol concentration. Obviously, DAHCl is a better surfactant than octanol, which leads to the conclusion that, at the same concentrations, (∂Γ/∂c) of DAHCl is smaller than the octanol one. Indeed, Table 1 shows that D1 < D2. The only unexpected result for the present theory here is the decrease of D1 with increasing octanol concentration. A possible explanation could be found in a specific interaction of DAHCl and octanol molecules, leading to a change of the activity coefficient of DAHCl. Thus, the extended decay length Dk ) (∂Γk/∂ak) (ak is the activity of the k-th component) will depend on the other components in the system. This could be a reasonable explanation of the fact that D decreases with increasing electrolyte concentration.11 Let us consider now the applicability of the present theory to a second example of experimental results. Alexandrova and Tsekov17 have reported measurements for the lifetimes of thin liquid films formed from vesicle suspensions of dimyristoylphosphatidylcholine (DMPC) in 0.15 M NaCl solution in water on a quartz plate at 30 °C. Due to the high salt concentration, the electrostatic disjoining component can be neglected, while the van der Waals component is given by
( )
∆E1 ∆E2 h h exp + exp D1 D1 D2 D2
(9) Table 1 presents the values of the specific constants of hydrophobic interaction obtained by Yoon and Ravishan(15) Craig, V. S. J.; Nihnam, B. W.; Pashley, R. M. J. Phys. Chem. 1993, 97, 10192.
A 6πh3
Since the Hamaker constant A ) 7 x 10-21 J for the air/ water/quartz system is positive, ΠVW is repulsive and films without DMPC are stable in accordance with the experiments. In the present case adsorption occurs only on the gas/ liquid interface. According to the proposed model of film rupture,17 the hole formed between air and quartz is covered by a condensed monolayer of DMPD molecules. Thus, the adsorption there can be well approximated by the maximal adsorption on a single air/suspension interface, i.e. Γ(0) ) Γ∞(∞). Now, employing the fact18 that the surface tension of an air/suspension interface is nearly a linear function of Γ, one can replace the factor ∆E in eq 6 by the difference σ∞ - σlg, where σlg and σ∞ ) 24.2 mN/m are the surface tensions on an air/suspension interface at finite and at infinite DMPD concentrations, respectively. Therefore, the hydrophobic disjoining pressure in the present case is
ΠHP )
σ∞ - σlg h exp D D
( )
As is seen, ΠHP is negative, which corresponds to an attractive force. The total disjoining pressure is a sum of the van der Waals and hydrophobic components. Due to competition between these two forces, there is a region of film thickness where the film is unstable. This fact is schematically (16) Yoon, R.-H.; Ravishankar, S. A. J. Colloid Interface Sci. 1994, 166, 215. (17) Alexandrova, L.; Tsekov, R. Colloids Surf., A, in press. (18) McDonald, R. C.; Simon., S. A. Proc. Natl. Acad. Sci. (USA) 1987, 84, 4089.
Hydrophobic Forces in Thin Liquid Films
Langmuir, Vol. 13, No. 21, 1997 5677 Table 2. Calculated Film Thickness h and Hydrophobic Decay Length D According to Eqs 10 and 11 As a Function of the DMPC Concentration in Water Filmsa C (mg/L)
σlg (mN/m)
τ (s)
h (nm)
D (nm)
1.0 2.5 5.0 7.5 10.0 25.0
60.6 59.2 59.1 57.4 54.7 51.2
30.0 20.0 12.5 2.8 1.7 60.0
13.2 13.2 13.1 12.8 12.8 14.0
1.0 1.0 1.0 1.0 1.0 1.0
aThe data for the gas/liquid surface tension and the film lifetime are taken from ref 17.
second equation needed for calculation of h and D reads
x
48ησlg
Figure 4. Principle disjoining pressure isotherm for an air/ vesicle suspension/quartz film system; Pc is the capillary pressure.
shown in Figure 4. During the film drainage process the film can reach a thickness close to the maximum of Π. In this region the disjoining pressure becomes commensurable with the capillary pressure Pc in the meniscus. Since the film-thinning rate is proportional to their difference, one concludes that the film becomes practically static at this thickness. Hence, the first equation for determining the two unknown thicknesses h and D is
2σlg σ∞ - σlg h A + )Π) exp 3 R D D 6πh
( )
(10)
where R ) 1 mm is the radius of the curvature in the meniscus. As is seen from Figure 4, there are three points where eq 10 is fulfilled. Since the considered films rupture, we are interested only in points with a positive slope Π′ ) ∂Π/∂h > 0, corresponding to unstable surface waves. The lifetime of such a stationary film is given by19
τ)
48ησlg h3Π′2
where η ) 1 mPas is the bulk water viscosity. Hence, the
3
hτ
) Π′ ) -
σ∞ - σlg h A exp 4 D 2πh D2
( )
(11)
Using the experimentally measured surface tension and lifetime, a numerical solution of the nonlinear equations (eqs 10 and 11) is possible. The calculated results for h and D are presented in Table 2. The obtained film thicknesses are reasonable as compared to the equilibrium one for the film without vesicles, h ) [AR/12πσlg]1/3 ) 13.8 nm (σlg ) 71.15 mN/m), which is calculated from eq 10 without the hydrophobic part. Table 2 shows also that the decay length does not depend on the amount of DMPC in a wide region of concentrations. However, this is an expected result, since C is the amount of DMPC per unit volume dissolved in water. Due to vesicle formation, addition of DMPC molecules does not influence substantially the concentration of monomers c, and thus the value of D remains constant. Finally, the experiments demonstrate unlimited stability of the films at very low (below 0.1 mg/L) and very high (above 50 mg/L) concentrations of DMPC.17 The first result is obvious because the films are stable without DMPC. The second result could be explained by the fact that at high concentrations the surface tension tends to its minimal value σ∞ and thus the attractive hydrophobic component disappears. Acknowledgment. The financial support from the Alexander von Humboldt Foundation is gratefully acknowledged. LA9702392 (19) Ruckenstein, E.; Jain, R. K. J. Chem. Soc., Faraday Trans. 2 1974, 70, 132.
