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Langmuir 1996, 12, 4134-4144
Effect of Surfactant and Salt Concentrations on the Drainage and Collapse of Foams Involving Ionic Surfactants Eli Ruckenstein* and Ashok Bhakta Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received March 4, 1996. In Final Form: June 5, 1996X An improved drainage model for foams involving ionic surfactants is presented which accounts for the effect of surfactant and salt concentrations. The effect of surfactant and salt concentrations on collapse is due mainly to their influence on the maximum disjoining pressure in the liquid films. The disjoining pressure is computed as a sum of the van der Waals force, the electrical double-layer force, and a short range repulsive force. The dissociated surfactant molecules on the inner surface of a film give rise to the electrical charge, while the undissociated molecules behave as dipoles. Since the surface charge determines the electrical double-layer force and the dipoles give rise to the short range repulsive force, the degree of dissociation of adsorbed surfactant molecules plays an important role in determining the characteristics of the disjoining pressure isotherm. Conditions are identified under which various kinds of disjoining pressure isotherms arise. It is shown that, under certain conditions, a high salt concentration, instead of causing collapse as predicted by the conventional DLVO theory, increases the number of surface dipoles due to counterion binding and gives rise to a strong short range repulsive force. Thus, under certain conditions, high salt concentrations result in the formation of very stable Newton black films. Simulations for foams are carried out taking into account, in addition, the effect of surfactant and salt concentrations on the surface viscosity and surface tension. Results indicate that the surfactant and salt concentrations play a crucial role in determining whether a foam will collapse. They also significantly affect the drainage through the Plateau border channels via their effect on the surface viscosity.
Introduction Foams consist of polyhedral gas bubbles separated by thin liquid films or lamellae. The continuous phase liquid is distributed between the films and the Plateau border (PB) channels which are formed in the region where three contiguous films overlap. The curvature of the PB walls gives rise to a pressure difference in the liquid between the films and the PB’s, causing the liquid in the former to be sucked out. Thus, films become thinner with time and finally rupture. On the other hand, the liquid in the PB channels (which form a complex interconnected network) drains out of the foam under the influence of gravity. Drainage in the films and PB channels therefore plays a crucial role in the collapse of foams. Several theoretical models1-10 for foam drainage have appeared in the literature. Krotov1 was the first to recognize that under certain conditions a drainage equilibrium results when the PB suction gradient balances gravity. Narsimhan2 accounted for this effect in his drainage model for pneumatic foams. However, he used a quasi-steady state approximation to compute the profile of the liquid fraction in the freshly generated foam. Bhakta and Ruckenstein8 used a more appropriate unsteady state model to describe the drainage during foam generation. A feature that has largely been ignored by most drainage models is the collapse of foams. We have presented a * Author to whom correspondence must be addressed. X Abstract published in Advance ACS Abstracts, August 1, 1996. (1) Krotov, V. V. Colloid J. USSR 1981, 43, 33 (Engl Transl). (2) Narsimhan, G. J. Food Eng. 1991, 14, 139. (3) Bhakta, A.; Khilar, K. C. Langmuir 1991, 7, 1827. (4) Ramani, M. V.; Kumar, R.; Gandhi, K. S. Chem. Eng. Sci. 1993, 48, 455. (5) Haas, P. A.; Johnson, H. F. Ind. Eng. Chem. Fund. 1967, 6, 225. (6) Jacobi, W. H.; Woodcock, K. E.; Grove, C. S. Ind. Eng. Chem. 1956, 48, 9046. (7) Miles, G. D.; Sheklovsky, L.; Ross, J. J. Phys. Chem. 1945, 49, 93. (8) Bhakta, A. R.; Ruckenstein, E. Langmuir 1995, 11, 1486. (9) Bhakta, A. R.; Ruckenstein, E. Langmuir 1995, 11, 4642. (10) Bhakta, A. R.; Ruckenstein, E. Langmuir 1996, 12, 3089.
S0743-7463(96)00193-X CCC: $12.00
model in an earlier publication9 which neglects film drainage and accounts for collapse in initially homogeneous foams and concentrated emulsions. That model provided insight into the “phase behavior” of these systems at drainage equilibrium. In a later paper,10 we considered the collapse of pneumatic foams and presented a more complete treatment in which film drainage was taken into account and modeled using Reynold’s equation for the radial flow between parallel circular disks. Collapse of a film occurs due to the unbounded growth of mechanical and thermal perturbations on the film surfaces. Whether a surface wave is damped or undergoes catastrophic growth is determined mainly by the shape of the disjoining pressure isotherm. Thus, if the repulsive disjoining force which opposes film thinning increases in response to the local thinning due to the disturbance, the wave is damped and no rupture occurs. On the other hand, if the disjoining pressure decreases, the local thinning is accelerated and leads to rupture. Film rupture can therefore only occur if the disjoining pressure decreases when the film thickness decreases, i.e. when its derivative with respect to film thickness is positive. The characteristics of the disjoining pressure isotherm in individual films therefore play a crucial role in determining the collapse behavior of a foam. In our earlier papers, we have considered only van der Waals and double-layer forces. The latter was computed assuming an arbitrary fixed surface potential. However, the surface potential is expected to depend on the surfactant concentration. In foams involving ionic surfactants which we consider here, the surface charge is primarily the result of the dissociation of the adsorbed surfactant molecules, which is affected by the concentration of the counterions. In this paper, we compute the surface charge and potential by accounting for the dissociation of the adsorbed surfactant molecules. The double-layer force in this formulation is therefore strongly influenced by the adsorption behavior of the surfactant. © 1996 American Chemical Society
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a
c
b
d
Figure 1. Various types of disjoining pressure isotherms.
It has been observed11 that in many systems the resistance of a foam to collapse increases sharply at sufficiently high surfactant and salt concentrations. Experimental measurements of disjoining pressure isotherms of single films containing sodium dodecyl sulfate12 show that qualitatively different isotherms are obtained depending on the salt concentration. As the pressure on a film is increased, its thickness decreases and the repulsive disjoining pressure increases. When the thickness is small enough (about 10 nm), black spots appear which eventually cover the entire film, giving rise to the so-called common black film. A further increase in pressure however gives rise to different phenomena, depending on the salt concentration. At lower salt concentrations, the common black film ruptures, while at higher salt concentrations there is a sudden transition to a very stable Newton black film. Thus, in the latter case, the film rupture predicted by the DLVO theory is prevented by the emergence of a large short range repulsive force. In this paper, we use a simple model for the short range repulsive force proposed by Schiby and Ruckenstein13 which considers it to be a result of the (11) Scheludko, A. Adv. Colloid Interface Sci. 1967, 1, 391. (12) Exerowa, D.; Kolarov, T.; Khristov, K. Colloids Surf. 1987, 22, 171. (13) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435.
organization of water molecules due to an electric field generated by the surface. We consider this field to be generated by the undissociated surfactant molecules via their dipoles. Thus the increased counterion binding at high salt concentration actually makes the short range repulsive force stronger by raising the number of surface dipoles. We show that most effects of salt and surfactant concentration can be qualitatively accounted for by combining this theory (assuming additivity) with the DLVO theory. We first present a discussion of the disjoining pressure in single films and then show the effect of salt and surfactant concentration on bulk foams. Our improved model for bulk foam drainage also takes into account the effect of surfactant and salt concentrations on the surface viscosity and surface tension. In all our calculations, a model system involving sodium dodecyl sulfate and sodium chloride is considered. Single Films Disjoining Pressure in Single Films. As mentioned earlier, the shape of the disjoining pressure isotherm plays a crucial role in the collapse of a standing foam. Figure 1 shows some typical plots of disjoining pressure (Π) versus film thickness (xF). For the rupture of a thin film to occur
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via the growth of instabilities, dΠ/dxF must be positive. Thus a thin film with an isotherm as in Figure 1a will rupture only for xF < xFc. Since a film with an initial thickness xF > xFc can arrive to thicknesses less than xFc only when the capillary pressure in the Plateau borders exceeds the maximum disjoining pressure, it can rupture only if the capillary pressure exceeds the maximum disjoining pressure (Πmax). On the other hand, a film with an isotherm like that in Figure 1b will definitely rupture, since dΠ/dxF is always positive. Figure 1c shows an isotherm in which there are two maxima. The smaller one to the right is primarily due to the double-layer forces, while the other is due to a short range repulsive force. At high salt concentrations the double layer is compressed and the double-layer forces are overwhelmed by the van der Waals forces at all thicknesses. However, there is significant counterion binding, which increases the number of surface dipoles. Thus, at sufficiently high surfactant concentrations, when there are a large number of surface dipoles, the short range force will become significant and we will have an isotherm either like Figure 1a where the maximum is mainly due to the short range repulsive forces or like Figure 1d if the short range forces are large enough to overcome the van der Waals forces at short distances (a few angstroms). Our first goal is to explain the effect of surfactant and salt concentration using the DLVO theory in combination with the model of Schiby and Ruckenstein13 for the short range force. Since the disjoining pressure is influenced strongly by the adsorption of the surfactant molecules at the gas/liquid interface, we will first compute the surface density of surfactant (Γ). Calculation of the Surface Density of Surfactant. The surface density of surfactant (Γ) along with the degree of dissociation plays a crucial role in determining both the double-layer forces (via the surface charge) and the short range forces (via the dipole moment on the surface). It is therefore imperative to compute Γ as a function of the salt and surfactant concentrations. We compute Γ using the Frumkin adsorption isotherm:
(
)
Γ Γ exp -2a1 Γ∞ Γ∞ b1cR- ) Γ 1Γ∞
Thus, denoting Kd as the equilibrium constant, we have
Kd )
(RdΓ)cNa+ )
(1 - Rd)Γ
RdcNa+ 1 - Rd
(3)
In eq 3, the concentration cNa+ of the sodium ions near the surface can be expressed in terms of the surfactant concentration (cs) in the Plateau border channels and the salt concentration in the bulk (ce) as
( )
cNa+ ) (cs + ce) exp -
eψs kBT
(4)
and Γ can be computed from eq 1 using
( )
cR- ) cs exp
eψs kBT
(5)
where ψs is the surface potential. It may be noted that the use of eq 3 implies that the effect of the interactions between the adsorbed species has been ignored. Because these interactions provide a positive contribution to the free energy, they decrease the degree of dissociation. These effects have been considered for micelles by Ruckenstein and Beunen.15 Calculation of the Surface Charge. An approximate power series solution to the Poisson Boltzmann equation at moderate potentials has been obtained by Oshima and Kondo,16 in which the potential ψ at a distance x from the midplane is given by
tanh
[ ]
eψ ) γA1(κx) + γ3A2(κx) + γ5A3(κx) kBT
( )
(6)
eψs cosh(κx) ; A1(x) ) ; 4kBT κxF cosh 2 κxF κxF cosh(κx) tanh (κx) sinh(κx) 2 2 (7a) A2(x) ) κxF cosh3 2
γ ) tanh
(2)
where e is the protonic charge and the degree of dissociation Rd is provided by the equilibrium of the following (14) Fainerman, V. D. Colloids Surf. 1991, 57, 249.
R-Na|surface f R-|surface + Na+|aqueous phase
where
(1)
where cR- is the concentration of the surfactant anions near the interface, Γ∞ is the surface density of surfactant at saturation, and the constants a1 and b1 are empirical parameters which are available in the literature from experiments carried out for air/water interfaces in contact with a large amount of water. The experimental results, however, relate Γ to the bulk surfactant concentration cs and not cR-. At large ionic strengths, the double layer is completely compressed and cR- is equal to cs. The highest salt (NaCl) concentration for which data on sodium dodecyl sulfate are available14 is 1 M, for which Γ∞ ) 5 × 10-6 mol/m2, a1 ) -1.53, and b1 ) 881 m3/mol. These values were used in all our calculations. Calculation of the Electrical Double-Layer Force. The charge on the surface of a film is caused by the adsorbed surfactant molecules that are dissociated. Thus, if Rd is the degree of dissociation of the adsorbed surfactant molecules, the surface charge per unit area (σc) is given by
σc ) -ΓRde
reaction:
A3(x) )
( ) ( ) ( ) [ ( ) ( ) ( )] {( ) } ( )
A1(x) - (A1(x))3 3A2(x) + 1κxF κxF 2 2 4 cosh 4 cosh 2 2 κxF 2 - (κx)2 A1(x) κxF κxF 2 4 tanh 2 2 κxF 2 cosh4 2
(7b)
The midplane potential (Ψm) is therefore given by
tanh
[ ]
eψm ) γA1(0) + γ3A2(0) + γ5A3(0) kBT
(8)
The surface charge can be expressed in terms of the surface (15) Ruckenstein, E.; Beunen, J. A. Langmuir 1988, 4, 77. (16) Oshima, H.; Kondo, T. J. Colloid Interface Sci. 1988, 122, 591.
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and midplane potential as
[x { ( )
σc ) -
2 cosh
( )}]
eψs eψm - cosh kBT kBT
2 6 2δ2 - a2 + δ3 (a2 + δ2)3/2 δ2 + a2
(14)
A′ ) -1 + γp[Λ(δ) exp(Rδ) + Λ(0)]
(15)
B′ ) 1 - γp[Λ(δ) exp(-Rδ) + Λ(0)]
(16)
Λ((δ) ) κ0kBT (9) e
Equations 1-9 can be solved simultaneously for given bulk surfactant (cs) and salt (ce) concentrations, to yield the surface potential (ψs) for any given film thickness. Once ψs is known, the repulsive force can be calculated using the expression
ΠDL ) 16(cs + ce)RGT[γ2A12(0) + 2γ4A1(0)(A2(0) + A13(0)) + γ6(2A1(0) A3(0) + 8A13(0) A2(0) + 3A16(0) + A22(0))] (10) The force computed above was compared to that obtained by solving numerically the Poisson Boltzmann equation and was found to be appropriate for the conditions used here. Calculation of the Short Range Repulsive Force. Using an idealized model in which the water molecules are arranged in a hexagonal close packed configuration in parallel planes separated by a distance δ, Schiby and Ruckenstein13 have derived an expression for the short range repulsive force between two parallel plates. The model considers that the electric field generated by the surfaces orients the water molecules in the direction of the field and gives rise to a profile of mean dipole moment. They have obtained a simple expression for the polarization (dipole moment per unit volume) profile between the plates and used it to obtain the following expression for the force per unit area (Πh) between the plates:
Also,
For δ ≈ a ≈ 2.8 Å, A′ ) 0.268, B′ ) 2.53, and R ) 3.58 × 109 m-1. The electric field E° however depends on the characteristics of the surface and is computed as shown below. Calculation of E°. The inner surface of the film contains dissociated chains which behave as point charges and undissociated chains which behave as dipoles pointing into the liquid phase. If we consider both surfaces to be of infinite extent, the electric field at any point between the planes due to the surface charges will be 0. The reason for this is that the electric field due to a charged infinite plane is independent of distance and the fields from the charges on the two surfaces cancel each other. The field E° therefore depends solely on the dipoles present on the surface. The electric field at a distance ‘z’ from a semiinfinite plane with a dipole moment per unit area us is given by (see appendix)
E)
Thus, if µs is the dipole moment of a single undissociated surfactant molecule we have
us ) Γ(1 - Rd)NAµs
+ (A′ exp(-RxF) + B′)2 P1A′[exp(-RxF) - exp(-3RxF) - 2RxF exp(-2RxF)]
+ [A′ exp(-RxF) + B′] P2A′(exp(-RxF) - exp(-2RxF)) [A′ exp(-RxF) + B′]2
E) +
(11)
where A′, B′, and R are constants which can be computed (see eqs 13-16) and P1 and P2 are given by
P1 ) nγp[E°]2; P2 ) 2P1Rδ
(12)
In eq 12, n is the number of water molecules per unit volume, γp is the polarizability of water, and E° is the electric field generated by the surface which acts on the first layer of water molecules. Calculations indicate that the third term is dominant and that Πh is primarily an exponentially decaying force with a decay length of R-1 . The constants A′, B′, and R in eq 11 can be expressed in terms of the distance between adjacent water molecules (a) and the distance between the parallel planes of water molecules (δ) as follows:
R2 )
[1 - γp(2Λ(δ) + Λ(0))] 2
γpΛ(δ)δ where
(17)
(18)
where NA is Avogadro’s number. Combining eqs 17 and 18, we get
Πh ) P1[exp(-2RxF) - exp(-RxF) + RxF exp(-RxF)]
[A′ exp(-RxF) + B′]3 P2 exp(-RxF)
us 30z
(13)
Γ(1 - Rd)NAµs 30z
(19)
Since we are dealing with individual water molecules, the medium among them is vacuum and we have ) 1. The assumption in that theory is that the external electric field is important only in the first layer of molecules. Thus, if d is the distance between the plane containing the surfactant molecules and the first layer of water molecules, we need E at z ) d. Since µs/d is not known, we use it as a parameter. Thus we write
E° ) Γ(1 - Rd)Ed with Ed ) NAµs/(30d)
(20)
Since Ed is a parameter to be varied, it is helpful to have a reference value for Ed about which it can be varied. In the model of Schiby and Ruckenstein, the mean dipole moment of the water molecules close to the surface is given by
µ(0) )
γpE°(1 - exp(-RxF)) ) A′ exp(-RxF) + B′ γpΓ(1 - Rd)Ed(1 - exp(-RxF)) A′ exp(-RxF) + B′
(21)
Thus
Ed )
µ(0)(A′ exp(-RxF) + B′) γpΓ(1 - Rd)(1 - exp(-RxF))
(22)
Therefore, if µw is the dipole moment of water, Edmax )
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Figure 2. Three layers involved in the calculation of the van der Waals forces.
µw(A′ + B′)/Γ∞γp ) 3.65 × 1015 N‚m2/C serves as a useful reference value. It may be noted that we assume here that the short range repulsion force and the electric double-layer force are additive. An attempt to couple them nonlinearly has been made by Schiby and Ruckenstein.17 Calculation of the van der Waals Force. For the system shown in Figure 2, Donners et al.18 have expressed the results of the Lifshitz theory of van der Waals forces in terms of the simple expression
ΠVDW )
[
]
b + cxF 1 +q 3 xF 1 + dxF + exF2
(23)
where b, c, d, e, and q are constants which depend on the specifics of the particular system such as the media involved and the thickness ’p’ of the surfactant film. For air-water-air systems with dodecane monolayers (p ) 0.9 nm) at the air/water interfaces, they provide the following values of the constants: b ) -3.96 × 10-23, c ) -2.05 × 10-13, d ) 8.86 × 107, e ) 6.61 × 1015, q ) -1.8 × 10-22. All the quantities in eq 23 are in SI units. Since the structure of the hydrocarbon chain in sodium dodecyl sulfate is very similar to that of dodecane, we use these values in our calculation. It must be emphasized that xF refers to the thickness of the aqueous core of the film. Experimentally obtained disjoining pressure isotherms are usually presented in terms of the total film thickness, which includes the thickness of the surfactant layer. Thus, a correction (2.3 nm in the case of sodium dodecyl sulfate12) must be applied to xF to make a comparison with experimental results. Effect of Surfactant Concentration on the Disjoining Pressure. Being an ionic surfactant, the bulk concentration of sodium dodecyl sulfate affects the disjoining pressure in two ways: (a) It affects Γ and hence affects the surface charge(see eq 2). (b) It affects the ionic strength and the degree of dissociation and therefore the extent of the electrical double layer. Figures 3 and 4 show the effect of the surfactant concentration on the disjoining pressure isotherm for two different values of the dissociation constant Kd, viz Kd ) 158 mol/m3 and Kd ) 0.316 mol/m3. In both cases, Ed/Edmax was taken to be 0.04. It is clear that there is a qualitative difference in the effect of surfactant concentration in the two cases. In Figure 3, since the degree of dissociation is large, the number of dipoles which contribute to the short range repulsive force is small and it is too weak to overcome the van der Waals forces. Thus the increase in surfactant concentration primarily affects the disjoining pressure by changing the double-layer force. This is clear from the fact that the maximum disjoining pressure is very close to that (Πmax,dl) obtained when no short range repulsive force is included (see Table 1). In the system considered in Figure 4, the (17) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1983, 100, 277. (18) Donners, W. A. B.; Rijnbout, J. B.; Vrij, A. J. Colloid Interface Sci. 1977, 60, 540.
Figure 3. Effect of surfactant concentration on the disjoining pressure isotherms with Kd ) 158 mol/m3 and Ed ) 0.04Edmax.
Figure 4. Effect of surfactant concentration on the disjoining pressure isotherms with Kd ) 0.316 mol/m3 and Ed ) 0.04Edmax.
smaller degree of dissociation significantly increases the number of surface dipoles and hence the contribution of the short range polarization force. Thus the increase in surfactant concentration causes the isotherm to undergo a qualitative change from one containing a single maximum corresponding to the double-layer force to one containing two peaks. In Figure 4, the shorter peak at a larger film thickness is due to the electrical double layer while the steep peak to the left is due to the short range hydration force. Effect of Salt Concentration. The concentration of sodium chloride has the following effects: (a) It compresses the electrical double layer by raising the ionic strength. (b) It decreases the degree of dissociation of the adsorbed
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Langmuir, Vol. 12, No. 17, 1996 4139
Table 1. Effect of Surfactant Concentration on System Parameters for Ed ) 0.04Edmax and ce ) 0 cs (M)
Kd (mol/m3)
Πmax (N/m2)
xFm (nm)
Πmax,dl (N/m2)
xFm,dl (nm)
0.001 0.003 0.005 0.001 0.005
158 158 158 0.316 0.316
217 500 384 000 534 000 30 990 (A) 106 000a (B) 22 000a
2.624 1.855 1.402 3.97 0.828 3.56
217 200 378 000 506 000 30 970 21 900
2.64 1.94 1.7 3.97 3.62
a
Figure 6. Schematic of the experimental setup for foam drainage.
There are two maxima.
Table 2. Effect of Salt Concentration on System Parameters for cs ) 0.001 M, Kd ) 0.316 mol/m3, and Ed ) 0.05Edmax ce (M)
Πmax (N/m2)
xFm (nm)
Πmax,dl (N/m2)
xFm,dl (nm)
0 0.01
29 180 (A) 435 000a (B) 8 320a 614 800
4.58 0.671 4.03 0.612
29 140 82 950
4.58 4.05
0.1 a
There are two maxima.
Figure 5. Effect of sodium chloride concentration on the disjoining pressure isotherm with cs ) 0.001 M, Ed ) 0.05Edmax, and Kd ) 0.316 mol/m3.
surfactants (see eq 3) by causing increased binding of the counterions. (c) It affects the concentration of R- ions near the interface via the surface potential and hence Γ. In experiments with single thin liquid films, it has been observed by Exerowa et al.12 that at high salt concentrations, as the capillary pressure is increased, there is, instead of rupture, a sudden transition to very stable Newton black films. Figure 5 shows the effect of salt concentration on a system with cs ) 0.001 M. One can see that, with no salt (ce ) 0), there is just one peak corresponding to the electrical double layer. However, as the salt concentration is increased, the curve changes first to one in which there are two maxima and then to one having only a short range repulsion. This can be explained as follows: As the salt concentration increases, the short range repulsive force increases, since the binding of the sodium ions causes an increase in the number of dipoles on the surface. At the same time the double layer is compressed. At intermediate concentrations, the double layer is not too compressed and the increased short range
repulsive force gives rise to an additional strongly repulsive force at a smaller thickness. In this case, there is a jump transition when the capillary pressure is raised above the value at point A. However, at high concentrations, the double layer is so compressed that it now has a range comparable to that of the short range polarization force and the two combine to give a single peaked short range repulsion. These results are in qualitative agreement with the observations of Exerowa et al., in the sense that a jump transition is observed as the salt concentration is increased. We would like to point out however that Exerowa et al. observed these jump transitions only at salt concentrations above 0.165 M, much higher than those used in Figure 5. While the calculations here have been done for ionic surfactants, the model is applicable in principle to nonionic surfactants with short head groups as well. In these systems, the surface charge arises due to ion binding rather than the dissociation of the adsorbed surfactant molecules, as happens with ionic surfactants. The short range force in this case is likely to be independent of electrolyte concentration, since the surface dipole moment would be essentially unchanged by the ion binding. This could be the reason why Newton black films are formed at much lower ionic strengths with nonionic surfactants.19 It may be noted however that, in the case of nonionic surfactants with large head groups, steric forces are likely to play a significant role. Drainage and Collapse Behavior of Bulk Foams. In the previous sections, we discussed the effect of various parameters on the disjoining pressure isotherm in single films. We now consider bulk foams. As mentioned earlier, we have proposed models8-10 for the drainage and collapse of bulk foams. It was shown that, depending on the conditions, a draining foam arrives at a mechanical equilibrium when the opposing forces due to gravity and Plateau border suction gradient balance each other. Conditions were also identified under which foam collapse occurs. The improvement here over our previous papers is that we now discuss the effect of surfactant concentration. The salt and surfactant concentration affect the drainage behavior of bulk foam through three factors, viz. the maximum disjoining pressure (Πmax) in the films, which determines the critical thickness of film rupture, the surface viscosity (ηs), and the surface tension (σ). It may be noted that, unlike other properties, the surface viscosity does not affect the equilibrium behavior of the foam. It primarily affects the time required to achieve “equilibrium”. Before we proceed further with results for bulk foams, we present in brief the main features of our drainage model. More details can be found in our earlier papers.8-10 Theoretical Model for Drainage. Figure 6 shows a schematic diagram of a typical experimental setup used to produce foam by bubbling. An inert gas is bubbled at a fixed volumetric flow rate through a porous frit into the (19) Kolarov, T.; Khristov, K.; Exerowa, D. Colloids Surf. 1989, 42, 49.
4140 Langmuir, Vol. 12, No. 17, 1996
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surfactant solution. Foam is formed at the foam/liquid interface and moves up at a rate which depends on the superficial gas velocity. The space coordinate ‘z’ is selected to increase in the downward direction, and the plane z ) z1 represents the foam/gas interface at the top. We place the origin (z ) 0) on a plane (see Figure 6) which defines the upper boundary of the entire system (i.e. all the gas entering the system and all the liquid contained in the foam). The plane z ) z2 represents the foam/liquid interface. As before, coalescence is ignored and the foam is assumed to consist of identical pentagonal dodecahedra. The dimensions of these polyhedra are expressed in terms of the effective bubble radius (R), which is the radius of a sphere with a volume equal to that of a bubble. Since drainage in a foam takes place through capillarydriven flow in the films and gravity-driven flow in the PB channels, we need expressions for the flows in the films and PB channels. Film thinning is modeled using a Reynolds type equation for flow between two circular parallel disks, and the rate of film thinning is given by
-
dxF 2cf∆PxF3 ) Vf ) dt 3µR 2
(24)
F
In eq 24, t is the time, µ is the viscosity of the continuous phase, RF is the radius of the disk, and ∆P is the pressure difference causing the flow. The driving force ∆P ) (σ/rp) - Π is a net result of the suction pressure in the adjacent PB channels and the disjoining pressure (Π) in the films. The coefficient cf, which is a correction to the classical Reynolds expression accounting for the mobility of the film surfaces, is given by20
1 cf
(6µ + ηskn2RsxF)
∞
∑ n)1
) 32
(6µ + 6µRs + ηskn2RsxF)λn4
(25)
In eq 25, λn is the nth root of the equation J0(λn) ) 0, kn ) λn/RF,
( )
[ ] ( )
∂Γ ∂cs DxF
2Ds
-3Dµ 1+ Rs ) ∂σ Γ ∂cs
3Dµ ≈∂σ Γ ∂cs
( )
J0 is the 0th-order Bessel function of the first kind, D is the bulk diffusivity, and Ds is the surface diffusivity. The rupture of the film is deemed to occur9,10 when the film thickness corresponds to the maximum disjoining pressure Πmax. This film thickness is the critical film thickness, which is denoted by xFc. The average velocity of the continuous phase in a PB channel of radius rp and cross-sectional area ap is given by
u)
cvap
(
20µx3
( ))
∂ 1 Fg + σ ∂z rp
(26)
In eq 26 F is the continuous phase density, g is the accelaration due to gravity, and the factor cv is a coefficient which accounts for the effect of finite surface viscosity (ηs). Desai and Kumar21 have computed cv as a function of the inverse of the dimensionless surface viscosity (γs ) 0.4387µxap/ηs). (20) Ivanov, I. B.; Dimitrov, D. S. Colloid Polym. Sci. 1974, 252, 982. (21) Desai, D.; Kumar, R. Chem. Eng. Sci. 1982, 37, 1361.
Before we proceed with further details, some quantities need to be defined. Let np be the number of PB channels per bubble, nF be the number of films per bubble, l be the length of a PB channel, V be the volume of a bubble, R be the radius of a sphere of volume V, AF be the area of a film, and N be the number of bubbles per unit volume. The main conservation equation for the continuous phase is given by10
∂ 3 ∂NnpapuR (Nnpapl + NnFAFxF) ) ∂t 15 ∂z
(27)
Equation 27 simply expresses the fact that the accumulation of continuous phase in a given volume element is a result of the net inflow of liquid into the element from the PB channels above. The factor 3/15 results from the assumption that the PB channels are randomly oriented.8 Since eq 27 is a second-order partial differential equation in rp, we need two boundary conditions. The first boundary condition arises from the fact that at the foam/liquid interface, the PB radius is equal to the bubble radius, since the bubbles there are predominantly spherical. We therefore have
rp|z2 ) R
(28)
The second boundary condition applies to the PB channels at the top. Since the rate at which liquid is released due to collapse ((dz1/dt)|z1) equals the volumetric flow rate per unit area (3/15NnpapuR|z1) into the PB channels at the top, we have9,10
dz1 3 NnpapuR|z1 ) | 15 dt z1
(29)
where is the liquid fraction. The expression for the movement of the foam/gas interface is given by10
dz1 ) 0 {before collapse} dt Vf ) | {during collapse} ∂xF z1 ∂z
(30a)
These conditions arise from the fact that, before collapse starts, the film thickness is larger than the critical thickness (xFc) and the gas/liquid interface is fixed with respect to the origin (z1 ) constant ) 0), while, during collapse, the film thickness at the top is always equal to the critical thickness.10 A global conservation condition for the continuous phase gives the expression for the movement of the gas/liquid interface:8,10
dz2 (3/15)NnpuapR - G |z2 ) dt (1 - )
(30b)
where G is the superficial gas velocity, which becomes 0 when the gas supply is shut off. The above system of differential equations is solved numerically10 to obtain the profiles of rp, xF, and the positions of the system boundaries (z1 and z2) as a function of time. The space coordinate (z) is transformed so that the system boundaries are immobilized. The discretization in space is done using second-order finite differencing, and the resulting system of ordinary differential equations is solved using Gear’s backward difference formula. The initial profiles of rp and xF do not affect the results when the initial foam length is small, provided they satisfy the boundary conditions.
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Langmuir, Vol. 12, No. 17, 1996 4141
In our calculations, the initial foam length was taken to be 0.5 mm. It may be noted that it has been assumed here that the surfactant concentration in the entire system is constant and equal to that in the surfactant solution used to produce the foam. This is likely to be the case if the quantity of the surfactant solution is large enough that foam generation does not result in a perceptible change in its concentration. If depletion of the surfactant solution does occur, the concentration of the surfactant would be greater higher up in the foam. This would bring in several new features into the problem including a variation of Πmax and σ along the foam, which could result in internal collapse and coalescence. Another assumption made here is that adsorption equilibrium is established instantaneously. This assumption, while valid for small sized surfactants, is likely to break down with large molecules such as proteins which adsorb much more slowly. As mentioned earlier, the surface tension and the surface viscosity are affected by the surfactant concentration. Thus, we need ηs and σ as functions of Γ. Surface Tension. The Frumkin isotherm (eq 1), when combined with the Gibbs adsorption equation, provides the following relation for the surface tension:22,23
[ (
σ0 - σ ) -Γ∞RT log 1 -
) ( )] [ ( ) ]
Γ Γ + a1 Γ∞ Γ∞
2
+
eψs 4RGT(20RGT(ce + cs))1/2 cosh - 1 (31) NAe 2kBT where σ0 is the surface tension of pure water. Since the relation between Γ and the bulk concentrations depends on the dissociation constant of the adsorbed surfactant, we need an appropriate value of Kd to obtain reasonable values for σ. Since it was experimentally observed that in the absence of electrolyte the σ versus cs curve for sodium dodecyl sulfate flattens out at about σ ) 38 mN/m near the cmc (8 mM), we chose Kd ) 158 mol/m3 for our simulations, which provides the value σ ) 37 mN/m at cs ) 8 mM. Surface Viscosity. The surface viscosity is a function of Γ. To our knowledge there is no available theoretical relation between the surface viscosity and Γ. Experimentally measured values of ηs as a function of cs and of cs as a function of Γ are however available.24 We therefore combined the two sets to get a piecewise cubic polynomial fit of ηs versus Γ. Figure 7 presents the curve obtained. It may be noted that, in computing the surface tension and the surface viscosity, we have used equations which are valid for a single semi-infinite double layer. This assumption is justified, since the PB dimensions are large compared to the double layer thickness. Results In order to study the effect of surfactant concentration on bulk foam stability, simulations were first carried out for foams of solutions containing only sodium dodecyl sulfate (no salt). Three concentrations were considered, viz. cs ) 0.000 01 M, cs ) 0.0001 M, and cs ) 0.001 M. Table 3 presents the values of the relevant variables for these concentrations. In Table 3, the length Lmax is the height of an equilibrated foam in which, in addition to u being zero throughout the foam, the capillary pressure at the top equals the maximum disjoining pressure. At this (22) Ruckenstein, E.; Krishnan, R. J. Colloid Interface Sci. 1980, 33, 201. (23) Borwankar, R. P.; Wasan, D. T. Chem. Eng. Sci. 1988, 43, 1323. (24) Poskanzer, A. M.; Goodrich, F. C. J. Phys. Chem. 1975, 79, 2122.
Figure 7. Piecewise polynomial fit of surface viscosity as a function of Γ.
equilibrium, gravity is balanced by the PB suction gradient, and we have (see eq 26)
()
∂ σ ) -Fg ∂z rp
(32)
Since a collapsing foam which arrives at equilibrium must satisfy the conditions σ/rp ) Πmax at z ) z1 and σ/rp ) σ/R at z ) z2, integration of eq 32 from z1 to z2 gives
z2 - z1 ) Lmax )
σ + Πmax R Fg
(33)
In foams shorter than Lmax, no collapse will occur. Only drainage of the continuous phase will take place until the equilibrium distribution of continuous phase liquid is established. On the other hand, foams longer than Lmax will collapse until an equilibrated foam of length Lmax is established. The following features may be noted from Table 3: (1) The surface viscosity increases with the bulk surfactant concentration. (2) Πmax and hence Lmax increase with surfactant concentration. (3) The critical thickness (xFc) decreases slightly, since an increase in cs causes a compression of the double layer. (4) The surface tension (σ) decreases with an increase in cs. Figure 8 shows the distribution of the liquid fraction () in the foam column when the gas supply is just shut off. It is clear that the effect of surfactant concentration is quite significant. For higher cs, the average liquid hold up is much higher. It may also be noted that for lower cs the gradient of is much steeper. The primary reason for these differences is the surface viscosity. For higher surfactant concentrations, the drainage is much slower because of the higher surface viscosity. This results in a larger entrainment of liquid in the foam during bubbling. Figure 9 shows the variation in the length of the foam column (z2 - z1) with time, once the bubbling is stopped. One can see that for the foam with cs ) 0.000 01 M there is an abrupt change in slope at point A. This is because, at point A, the bubbles at the top of the foam begin to
4142 Langmuir, Vol. 12, No. 17, 1996
Ruckenstein and Bhakta
Table 3. Effect of Surfactant Concentration on System Parameters for Bulk Foam for ce ) 0, Kd ) 158 mol/m3, and Ed ) 0.05Edmax cs (M)
ηs (N‚s/m)
σ (mN/m)
ψs (mV)
Πmax (N/m2)
Lmax
xFc (nm)
0.00001 0.0001 0.001
7.055 × 10-10 4.02 × 10-9 3.3 × 10-8
71.69 70.68 61.91
-154.7 -166.6 -159.9
1423.4 14240.6 1.25 × 105
10.85 cm 1.42 m 12.73 m
22.3 9.46 3.45
Figure 8. Effect of surfactant concentration on the liquid fraction profiles in freshly generated foam with Ed ) 0.05Edmax, Kd ) 158 mol/m3, and ce ) 0.
collapse occurs because Lmax is 1.42 and 12.73 m for cs ) 0.0001 and 0.001 M, respectively. Effect of Salt Concentration. To see the effect of salt concentration on the drainage and collapse behavior of sodium dodecyl sulfate foams, we carried out simulations at three salt concentrations, ce ) 0.0001 M, ce ) 0.001 M, and ce ) 0.514 M, keeping the surfactant concentration fixed at 0.0001 M. Table 4 lists the values of the various variables for this set of calculations. We took Ed ) 0.01Edmax for this case. The following features may be noted: (1) The surface potential at the PB walls decreases with increasing salt concentrations. (2) Πmax and Lmax first increase and then decrease with increasing salt concentration. (3) The surface viscosity increases with increasing salt concentration. An increase in salt concentration causes a compression of the double layer and also results in increased binding of the sodium ions to the adsorbed surfactant ions. The accompanying decrease in surface potential means that the surfactant ion concentration near the surface increases and results in an increase in Γ which in turn raises the surface viscosity. Figure 10 shows the liquid fraction profiles for the above sets of conditions when the bubbling is just stopped. Again, one can see that the increase in surface viscosity with salt concentration significantly increases the liquid hold up in the freshly formed foam. Figure 11 shows the effect of salt concentration on the variation of foam height with time. The larger initial hold up causes a faster change in height for high salt concentrations. It may be noted that the collapse of the uppermost bubbles occurs for ce ) 0.514 M, since Lmax in this case is less than the initial height. To show the effect of the short range repulsive force, we carried out one simulation for ce ) 0.514 M with Ed ) 0.05Edmax. Figure 12 shows the comparison of the variation of foam height with time for this system along with that for Ed ) 0.01Edmax. One can see that while the drainage behavior of the two systems is practically identical, there is a qualitative difference with respect to collapse. There is no collapse for Ed ) 0.05Edmax. For this system, the short range repulsion force becomes significant at short range, leading to large values for Πmax and Lmax, thereby preventing collapse. Conclusion
Figure 9. Effect of surfactant concentration on the variation of foam length with Ed ) 0.05Edmax, Kd ) 158 mol/m3 and ce ) 0.
collapse and the foam height starts decreasing more rapidly. This feature is to be expected, since, for cs ) 0.000 01 M, Lmax ) 10.85 cm, which is less than the initial foam height of 20 cm. The abrupt change due to the collapse is not seen at the other two concentrations. No
The effect of surfactant and salt concentration on the drainage and collapse behavior of foams based on ionic surfactants is treated theoretically. The disjoining pressure in individual films is computed by considering, in addition to the van der Waals and the electrical doublelayer forces, a short range repulsive force due to the organization of water molecules near the surfaces. The electrical double layer is a result of the surface charge due to the dissociated adsorbed surfactant molecules, while the short range repulsive force is caused by the dipole moments of the undissociated adsorbed surfactant molecules. It is shown that, depending on the values of the parameters used, it is possible to have a disjoining pressure isotherm with two maxima at sufficiently high salt and surfactant concentrations. Under these conditions, instead of film rupture, there is a jump transition to very stable Newton black films. These results are in qualitative agreement with experimental observations. Simulations were carried out with bulk foams to study the effect of
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Langmuir, Vol. 12, No. 17, 1996 4143
Table 4. Effect of Salt Concentration on System Parameters for Bulk Foam for cs ) 0.0001 M, Kd ) 158 mol/m3, and Ed ) 0.01Edmax ce (M)
ηs (N‚s/m)
σ (mN/m)
ψs (mV)
Πmax (N/m2)
Lmax
xFc (nm)
0.0001 0.01 0.514
3.03 × 10-8 3.3 × 10-7 2.27 × 10-6
70.64 62.29 42.16
-158.3 -100.8 -23.7
26810 505723 1313.16
2.63 m 51.57 m 11.25 cm
6.95 1.45 1.2
Figure 10. Effect of sodium chloride concentration on the liquid fraction profiles in freshly generated foam with Ed ) 0.01Edmax, Kd ) 158 mol/m3, and cs ) 0.0001 M.
Figure 12. Comparison of the collapse behavior of foams with cs ) 0.001 M and ce ) 0.514 M for Ed ) 0.01Edmax and Ed ) 0.05Edmax.
concentrations, a larger amount of continuous phase is entrained with the foam during foam generation. It is also shown that, for a typical foam 20 cm high, there is significant collapse at low surfactant and salt concentrations and no collapse at high surfactant and salt concentrations. Acknowledgment. This work was supported by a grant from the National Science Foundation. Appendix
Figure 11. Effect of sodium chloride concentration on the variation of foam length with Ed ) 0.01Edmax and Kd ) 158 mol/m3 for a surfactant concentration maintained at 0.001 M.
The component of the electric field at point P in a direction perpendicular to the plane due to a dipole of moment ‘u’ at point A is given by
E) surfactant and salt concentrations. It is shown that salt and surfactant concentrations strongly influence the surface viscosity via Γ and hence affect the drainage behavior of the foam. At high surfactant and salt
2u cos2(R) 4π0y3
(1)
If us is the dipole moment per unit area, the electric field at point P due to a ring of radius r and thickness dr is
4144 Langmuir, Vol. 12, No. 17, 1996
Ruckenstein and Bhakta
given as
dE )
[
]
2us(2πrdr) cos2(R) 4π0y3
(2)
Since cos(R) ) z/y and y2 ) r2 + z2, we have
dE )
2us 2πrdr z2 2 4π0 (r + z2)3/2 y2
(3)
Thus the total electric field at P is given as
E)
us 0
2
u
∫0∞(r2 +rzz2)5/2 dr ) 3s0z
(4)
Nomenclature a ) distance between adjacent water molecules ap ) average cross-sectional area of a Plateau border channel a1 ) empirical parameter in the Frumkin adsorption equation A′ ) constant in the expression for the short range repulsive force A1, A2, A3 ) functions used in the calculation of the electrical double-layer force AF ) area of the surface of a film b1 ) empirical parameter in the Frumkin adsorption equation b ) empirical constant used in the calculation of the van der Waals force B′ ) constant in the expression for the short range repulsive force c ) empirical constant used in the calculation of the van der Waals force ce ) concentration of salt in the bulk cs ) concentration of surfactant in the bulk cR- ) concentration of surfactant anions near the surface cf ) coefficient accounting for the effect of the surface mobility on film drainage cv ) coefficient accounting for the mobility of the walls of a Plateau border channel d ) empirical constant used in the calculation of the van der Waals force; also distance between the plane containing the adsorbed surfactant molecules and the first plane of the water molecules D ) bulk diffusivity Ds ) surface diffusivity P1, P2 ) variables used in the calculation of the short range repulsive force q ) empirical constant used in the calculation of the van der Waals force e ) protonic charge; also empirical constant used in the calculation of the van der Waals force E ) electric field E° ) electric field due to the surface acting on the first layer of water molecules Ed ) parameter used in the calculation of the short range repulsive force Edmax ) upper limit for Ed G ) superficial gas velocity g ) gravity
Kd ) dissociation constant of the adsorbed surfactant molecules kB ) Boltzmann’s constant l ) length of a Plateau border channel Lmax ) equilibrium height attained by a collapsing foam N ) number of bubbles per unit volume np ) number of PB channels per bubble nF ) number of films per bubble n ) number of water molecules per unit volume NA ) Avogadro’s number q ) empirical constant used in the calculation of the van der Waals force R ) radius of a bubble RF ) radius of a film rp ) Plateau border radius RG ) molar gas constant T ) absolute temperature u ) average velocity in a PB channel us ) dipole moment per unit area V ) volume of a sphere of radius R Vf ) rate of film thinning x ) distance from the midplane xF ) film thickness xFc ) critical thickness of film rupture xFm ) film thickness corresponding to the maximum disjoining pressure xFm,dl ) film thickness corresponding to the maximum disjoining pressure when only electrical double-layer forces are present z ) space coordinate z1 ) coordinate of the foam/gas interface z2 ) coordinate of the foam/liquid interface Greek Letters R ) reciprocal decay length of the short range repulsive force Rd ) degree of dissociation of the adsorbed surfactant molecules δ ) distance between adjacent planes of water molecules ) dielectric constant; liquid fraction 0 ) permittivity of free space γP ) polarizability of water γs ) inverse of dimensionless surface viscosity ηs ) surface viscosity κ ) reciprocal Debye length λn ) nth root of the equation J0(λn) ) 0 µ ) bulk viscosity µs ) dipole moment of an adsorbed surfactant molecule µw ) dipole moment of a water molecule F ) density of the continuous phase σ ) surface tension σc ) surface charge per unit area σ0 ) surface tension of pure water ψs ) surface potential ψ ) potential ψm ) midplane potential Γ) surface density of surfactant Γ∞ ) surface density of surfactant at saturation Π ) disjoining pressure Πmax ) maximum disjoining pressure Πh ) short range repulsive force per unit area ΠDL ) electrical double-layer force per unit area ΠVDW ) van der Waals force per unit area Πmax,dl ) maximum disjoining pressure when no short range repulsive forces are present LA960193X
