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Extremes of Some Foam Properties and Elasticity of Thin Foam Films near the Critical Micelle Concentration Anatoly I. Rusanov,* Valery V. Krotov, and Aleksandr G. Nekrasov Mendeleev Center, St Petersburg State University, 199034 St. Petersburg, Russian Federation Received October 6, 2003. In Final Form: December 8, 2003 The elasticity of open and closed thin foam films is analyzed. The elasticity modulus of a closed film is shown to be additive with respect to contributions from Gibbs elasticity and disjoining pressure. A detailed expression for the film elasticity modulus explains the pronounced maxima of foaminess and foam stability near the critical micelle concentration observed earlier in many experiments. A theory of transversal elasticity of thin foam films is formulated under conditions excluding the action of Gibbs elasticity. Near the critical micelle concentration, the theory predicts maxima of the transversal elasticity modulus and of the films thickness as functions of concentration at a given disjoining pressure. The prediction has been verified experimentally by measuring the film thickness in equilibrium foam as a function of height.
Introduction The motivation of this paper is rooted both in experimental and theoretical reasons related to foams and foam films. Experimentally, there are some facts (extremes of a number of the foam and foam film properties) whose origin remains unclear up to the present. This can be caused by the insufficient state of the theory of foam films, which needs further development. Among the foam properties, the foam stability looks the most important. Numerous methods of quantitative characterization of the foam stability are well described and have been reviewed in the literature.1-6 The authors contributed to this area by investigating the evolution time for the foam cells at a given horizontal level. The evolution time is introduced as follows. The internal stability of foam depends on the lifetime of internal films separating foam cells. Every breakup of an internal film means the coalescence of neighboring cells with the corresponding enlargement of a foam cell. The resulting state of polydispersity is characterized by the cell volume ratios close to the ratios of integer numbers. Reasonably assuming the number of bursting films per unit time to be proportional to the total film number, the process of cell growing should proceed exponentially. So we may write
Rt ) R0 exp (t/Τ) where Rt and R0 are the average cell radius and its initial value, respectively, t is time, and Τ is a coefficient that we called the evolution time and used as a characteristic of the foam stability. The evolution time is a time needed for the enlargement of a foam bubble by e times. Since the probability of film bursting practically does not depend on the film area, Τ is very close to being a constant, even * Corresponding author. E-mail:
[email protected]. (1) Bikerman, J. J. Foams; Springer: Heidelberg, Germany, 1973; Chapter 4. (2) Foams; Akers, R. J.; Ed.; Academic Press: New York, 1976. (3) Foams: Fundamentals and Applications in the Petroleum Industry; Schramm, L. L., Ed.; American Chemical Society: Washington, DC, 1994. (4) Khan, R. K. Foam: Theory, Measurements and Applications; Surfactant Science Series; Dekker: New York, 1996. (5) Exerova, D.; Kruglyakov, P. Foam and foam films; Elsevier: Amsterdam, 1998. (6) Krotov, V. V.; Rusanov, A. I. Physicochemical Hydrodynamics of Capillary Systems; Imperial College Press: London, 1999; p 372.
Figure 1. Concentration dependence of the evolution time for the foam of the SDS solution.
in polydisperse foam. Using an optical technique for measuring the transmission factor and the theory of multiple light scattering in foams, the foam cell radius is determined as a function of time. In this way, the evolution time can be found from experiment. The experimental setup and results have been described elsewhere.7,8 The investigation discovered a maximum of the evolution time near the critical micelle concentration (cmc) for the foam prepared from the sodium dodecyl sulfate (SDS) solution (Figure 1). Similar observations of peaks of the foam stability were still earlier reported in the literature1,9 as obtained by traditional measuring the foam-column height decrease in time. However, the explanation has not been given so far and remains to be a challenge to theory. Additional data were obtained8,10 from studying the dense bubble monolayer on the surface of the SDS solution. The bubble monolayer can be regarded as a kind of twodimensional foam, but, in any case, is an object of a specific interest in colloid science. In this case, the average bubble lifetime, τ (that we called foaminess1), can be measured directly by regulating the bubble barbotage rate to maintain the complete surface coverage in the bubble monolayer. The difference between Τ and τ is that Τ characterizes the internal foam stability, whereas τ is (7) Krotov, V. V.; Nekrasov, A. G.; Rusanov, A. I. Mendeleev Commun. 1996, No. 6, 220. (8) Rusanov, A. I.; Krotov, V. V.; Nekrasov A. G. J. Colloid Interface Sci. 1998, 206, 392. (9) Go¨tte, E. Kolloid-Z. 1933, 64, 327. (10) Krotov, V. V.; Nekrasov, A. G.; Rusanov, A. I. Mendeleev Commun. 1996, No. 5, 178.
10.1021/la0358623 CCC: $27.50 © 2004 American Chemical Society Published on Web 01/22/2004
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shall first consider completely or partially closed films where the Gibbs elasticity and the elasticity related to disjoining pressure, act in parallel. The theory will be shown to be able to explain the peaks of foaminess and foam stability near the cmc as a result of behavior of the elasticity modulus. Second, we shall consider an open equilibrium film in the absence of Gibbs elasticity. The elastic behavior of such a film is governed by the transversal elasticity, which is an important component of the total film elasticity. For transversal elasticity, theory also predicts the existence of a maximum near the cmc, although with another mechanism. We also present the first measurements of the transversal elasticity modulus and experimental data to confirm the prediction of the theory. Figure 2. Concentration dependence of the foaminess of the SDS solution.
related to the outer stability. Indeed, mostly those films burst that are in contact with the surroundings in the case of the bubble monolayer (which is also always present in real foam as a peripheral bubble layer). Figure 2 shows the plot of τ vs the SDS concentration. Foaminess turns to be still more sensitive characteristic and exhibits a sharp peak near the cmc. In addition to the above properties, the evolution time and foaminess, we shall show in the experimental part of this paper that the film thickness in equilibrium foam also passes through a maximum near the cmc. Thus, already three properties behave in a similar manner. The possible line of explanation is the analysis of the foam film elasticity, which is the principal factor of the foam stability. Up to the present time, however, the theory of film elasticity has been formulated differently for thin and thick films (both present in foams) and needs further development. The distinction between thin and thick films is attributed to the account or neglect of the overlap of the film surface layers, respectively. The overlapping surface layers interact and create the disjoining pressure, which is a specific feature of thin films, whereas thick films have no disjoining pressure. Thick films are only capable of “Gibbs elasticity” related to the redistribution of the components of a mixed film between the film surface and the bulk during the film stretching. Although the Gibbs elasticity mechanism is universal and equally important for thick and thin films, this kind of elasticity is traditionally referred to thick films for which it is the only and, therefore, easily recognizable mechanism. Gibbs was first to formulate the theory of elasticity of thick films, which, however, turned to be insufficient, was essentially improved in the 1960s and 1970s and later reviewed.11 As for thin films, the effect of disjoining pressure has been in the highlight, although disjoining pressure creating the second kind of elasticity. There is no general theory embracing both the kinds of elasticity up to the present time. This work purposes filling up this gap. A film is called closed if the total amounts of all film species are constant (the terms “partially closed” or “partially open” imply a certain part of the film species to be capable of exchange with the surrounding medium). As is known, the Gibbs elasticity is realized if a film is closed at least with respect to one of the film species. We (11) Rusanov, A. I.; Krotov, V. V. Gibbs Elasticity of Films, Threads, and Foams. In Progress in Surface and Membrane Science; Cadenhead, D. A., Danielli, J. F., Eds.; Academic Press: New York, 1979; Vol. 13, pp 415-524.
Thermodynamics of Elasticity of a Thin Film A film is called elastic if it increases its tension γ at stretching. This kind of elasticity is quantitatively characterized by the film elasticity modulus E, which Gibbs defined as
E≡
dγ d(ln A)
(1)
where A is the film area. This definition is of general character and may be applied to any kind of elasticity. We consider a multicomponent flat symmetrical thin film at a fixed temperature. Because of its small thickness, the film does not contain a bulk phase in the interior, but, in principle, it could be in equilibrium with its mother phase having the same values of the temperature and chemical potentials as in the film. The state of the mother phase is governed by the Gibbs-Duhem equation
dp )
∑i ci dµi
(2)
where p is the pressure, c is the concentration, and µ is the chemical potential, subscript i referring to the species of which the film is composed. The form of a thermodynamic fundamental equation for the film itself depends on the method of description. Following the method of Gibbs’ surface thermodynamics with using a single dividing surface, the film fundamental equation is of the form of the Gibbs adsorption isotherm
∑i Gi dµi
dγ ) -
(3)
where Gi is the excess (with respect to the surrounding medium) amount of the ith species per unit film area. We assume the content of the film species in the surroundings to be negligible, and we understand Gi as the total amount of the ith species per unit film area. In this case, Gi does not depend on the dividing surface location. So it is impossible to convert one of the quantities Gi to zero by shifting the dividing surface, contrary to the case of the ordinary Gibbs adsorption equation. The total amount of the ith species in the film is GiA and is constant if the film is closed. Then we have the condition for a closed film
d(ln A) ) -d(ln Gi)
(4)
Putting eqs 3 and 4 into eq 1, we obtain the expression
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eqs 6 and 8, we can exclude the chemical potential of a solvent from eq 6 to obtain
dγ ) -2 Figure 3. Method of two dividing surfaces in the theory of thin films: R is the imaginary bulk mother phase of a thin film, β is the surrounding medium, and h is the dividing surfaces separation.
for the elasticity modulus of a completely closed film
E)
dµi
∑i Gi2dG
(5)
i
Each term is positive on the right-hand side of eq 5 since the derivatives are positive according to thermodynamic stability conditions. In accordance with the reduced Le Chatelier-Brown principle,12 eq 5 exhibits a decrease of the elasticity modulus with increasing the number of open species. If a species becomes open, the constancy of its mass is replaced by the constancy of its chemical potential, and the corresponding term disappears on the right-hand side of eq 5. When the film becomes entirely open, the whole right-hand side of eq 5 is reduced to zero, and the film elasticity vanishes. In this case, the thermodynamic state of the film is fixed. When stretched, the film increases its area, but not tension. Typically for Gibbs’ formalism, eq 3 does not include such important characteristics of a thin film as the film thickness and disjoining pressure. To incorporate them in thermodynamic equations, one has to use the method of two dividing surfaces,13 which is formulated as follows. Similarly to the case of a thick film, we introduce two dividing surfaces on both the sides of the film (Figure 3) and imaginary fill in the space between them with the film mother phase (R) possessing the same values of temperature and chemical potentials as in the real thing film. Referring eq 2 to phase R and subtracting it from eq 3, we transform the fundamental equation of a thin film to the form
dγ ) -2
∑i Γi dµi + h dΠ
(6)
where Γi is the adsorption (with respect to the mother phase) of the ith species, h is the separation between the dividing surfaces, and Π is the disjoining pressure defined as
Π ≡ pβ - pR
(7)
Equation 7 shows that, in the case of a thin film, the pressure in the mother phase remains variable at constant external pressure, which is quite impossible in a thick film. Equation 2 can be separately written for phases R and β. Taking the difference of these equations and using eq 7, we obtain for the disjoining pressure
dΠ )
∑i
(cβi
-
cRi )
dµi
(8)
which is an additional relationship to eq 6. Solving together (12) Rusanov, A. I. Phasengleichgewichte und Grenzflaechenerscheinungen; Akademie-Verlag: Berlin, 1978; Chapter 2.
∑i
(
) (
cβi - cRi
)
2Γj Γi - Γj dµi + h dΠ cβj - cRj cβj - cRj (9)
where subscript j refers to the solvent. The coefficient of dµi in eq 9 is well-known to be the relative adsorption of the ith species in Gibbs’ surface thermodynamics. Both the coefficients of dµi and of dΠ are invariant with respect to shifting the dividing surfaces. As for their numerical values, these coefficients have a very simple interpretation as the adsorption and the separation, respectively, referring to the particular location of the dividing surfaces (called an equimolecular surface) where the solvent adsorption is zero. Indeed, setting Γj ) 0 in eq 9, we come back to eq 6 provided we understand Γi as relative adsorptions, h as the separation between two equimolecular surfaces, and the summation as including all solutes (e.g., surfactants) but no solvent. The quantity h defined in this way is often used as the film thickness. A relative difference between h and the real film thickness ht is negligible for thick films, but can be essential for thin films, so we below discriminate h from ht. Putting eq 6 into eq 1 yields the description of the film elasticity modulus. However, the result depends on the external conditions, imposed on the film to determine the degree of exchange of matter between the film and the surroundings. There are three main cases of such conditions to be considered below: a completely closed film, a partially closed (open) film, and a completely open film. Completely Closed Film For the sake of simplicity, we consider a closed film of a solution of a single surfactant. In this case, eq 6 becomes
dγ ) -2Γ dµ + h dΠ
(10)
where Γ and µ are the relative adsorption and the chemical potential of the surfactant, respectively, and h is the distance between the two equimolecular surfaces of the film. The external pressure is assumed to be constant. Correspondingly, dΠ ) -dp (p ≡ pR; we omit superscript R after this point) according to eq 7. Since the surfactant chemical potential is a function of the pressure p and the surfactant concentration c in the mother phase, the expression holds
dµ ) -v dΠ + (∂µ/∂c)dc
(11)
where v is the partial molar volume of the surfactant. Putting eq 11 into eq 10 yields
dγ ) -2Γ(∂µ/∂c) dc + ht dΠ
(12)
ht ≡ h + 2Γv
(13)
where
is the total film thickness with accounting for the surfactant adsorbed layers (a negligible difference between the v values in eqs 11 and 13 can be caused by compressibility). (13) Rusanov, A. I. Method of Two Dividing Surfaces in Thermodynamics of Thin Films. In Surface Forces and Boundary Layers of Liquids; Deryagin, B. V., Ed.; Nauka: Moscow, 1983 (in Russian); p 152.
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In addition to eq 12, we have to take into account the relations following from the constancy of amounts of all species in a completely closed film. Neglecting compressibility, we also may consider the film volume Vt ) Aht as a constant. This yields the relationship
d(ln A) ) -d(ln ht)
(14)
In this case, the balance equations hold
d(2AΓ + Vc) ) 0
(15)
dVt ) d(V + 2ΓAv) ) 0
(16)
where V ) Ah is the volume of the space between the two equimolecular dividing surfaces (the relation between Vt and V is easily established by multiplication of eq 13 by area A). As mentioned above, we neglect the variation of v, so that eq 16 yields
dV ) -2v d(ΓA)
(17)
Putting eq 17 into eq 15 leads to the relationship
d(ln A) ) -d(ln Γ) -
h dc 2Γ(1 - φ)
(18)
where φ ≡ vc is the packing fraction of the surfactant in the film mother phase (φ is usually small as compared with unity). We now divide eq 12 by dln A and use eqs 18 and 14 when dividing the first and second terms on the righthand side of eq 12, respectively, to obtain the final expression for the film elasticity modulus
E)
2Γ2∂µ/∂c + htE⊥ dΓ/dc + h/2(1 - φ)
(19)
Here we have introduced the transversal elasticity modulus E⊥
E⊥ ≡ -
dΠ d(ln ht)
(20)
as a specific characteristic of a thin film related to the disjoining pressure isotherm. Equation 19 exhibits the additivity of contributions from the Gibbs elasticity and the specific elasticity to the elasticity modulus of a thin film. The second term on the right-hand side of eq 19 is absent for a thick film whose elasticity is determined exclusively by the Gibbs mechanism. In contrast, the first term disappears for a thin film of a pure solvent (Γ ) 0), and the film elasticity is determined only by disjoining pressure. Both the mechanisms operate in parallel in the general case of a thin film. Let us apply eq 19 to the vicinity of the cmc where the surfactant aggregation develops. Since the cmc is typically small, we can assume that the surfactant chemical potential in the mother phase is determined by the surfactant monomeric concentration c1 to write
dµ ) RT d(ln c1)
(21)
where R is the gas constant and T is the temperature. Using eq 21, eq 19 can be rearranged to the form 2
E)
∂c1 2RTΓ /c1 + htE⊥ dΓ/dc + h/2(1 - φ) ∂c
(22)
There are two factors in the first term on the righthand side of eq 22. Bearing in mind an adsorption isotherm of the Langmuir type, it is easy to see that the first factor increases with concentration. The second factor, the derivative ∂c1/∂c, is practically unity below the cmc, but falls off dramatically almost to zero when entering the narrow cmc region. It is well-known that the saturation of the adsorbed layer is attained still before the cmc, so that we have first increasing and then decreasing of the elasticity modulus. Thus, we can say theory predicts the existence of a maximum of the film elasticity near the cmc. Importantly, the prediction is of general character for all types of surfactants. Since the film elasticity is an important factor of the foam stability, we can say the above theory also predicts a maximum of the film stability near the cmc. As was already noted and illustrated by Figures 1 and 2 in the Introduction, a number of such indications have been reported in the literature. The peak of foaminess in Figure 2 is very steep and about 40 times exceeds the background value. In our lab we observed even higher peaks (about 80 times as much as foaminess values far from the cmc) for nonionic surfactants. Thus, we may conclude that the effect predicted by the above theory is very important for practice. Film with a Single Closed Species We now consider the case when a film is open with respect to all species except one (for example, a film of a surfactant aqueous solution in a chamber saturated with water vapor). In this case there is only one term with the chemical potential µ not only in eqs 3 and 6, but also in eq 2. If a film was thick, the remaining chemical potential µ would be also constant due to the constancy of the external pressure, and there would be no film (Gibbs) elasticity. However, the realization of Gibbs elasticity is possible in a thin film with only one closed component. As was mentioned above, Gibbs elasticity is usually attributed to thick films. We now consider a unique example when, by contrast, Gibbs elasticity is impossible for a thick film and is realized in a thin film. In accordance with eq 7, we again have dΠ ) -dp at a constant external pressure. Then eq 2 for the film mother phase can be written as
dΠ ) -c dµ
(23)
Using eq 23, one can formulate the theory either in terms of disjoining pressure or in terms of chemical potential with transition to Gibbs elasticity. Choosing the second route, we turn to eq 5 that is now reduced to the simple expression
G2 dµ dµ ) ‚ E ) G2 dG dG/dc dc
(24)
The total derivative of the chemical potential implies a variable pressure. To avoid this complication, we pass to the partial derivative by putting eq 23 in eq 11. This yields
dµ 1 ∂µ ) dc 1 - φ ∂c
(25)
where the partial derivative is taken at constant pressure. The behavior of the partial derivative of the chemical potential is well-known: it falls off almost to zero when passing to micellization. Before the micellization when a
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concentration c and the film thickness ht: Π ) Π(c, ht). We rewrite the definition expressed in eq 29 in the form
surfactant solution almost ideal, we may set
∂µ RT ) ∂c c
Since the surfactant packing fraction φ is typically negligible below the cmc, we may apply this result also to the total derivative of the chemical potential, which changes eq 24 to the form
G2/c E ≈ RT dG/dc
(27)
According to eq 27, the film elasticity modulus is zero at zero concentration (G also becomes zero, but the ratio G/c remains a finite nonzero value as c f 0) and is positive at a finite concentration, so that the elasticity modulus increases with concentration. As is seen from eq 25, the total derivative dµ/dc reproduces the behavior of the partial derivative and, hence, steeply decreases when passing the cmc. Correspondingly, the elasticity modulus decreases, so that eq 24 again predicts a maximum of the film elasticity near the cmc, similarly to the case of a completely closed film. Completely Open Film All chemical potentials are fixed in a completely open film, and, therefore, also pressure and all other state parameters of the film mother phase are fixed. In particular, the surfactant concentration and adsorption are also constant, and this means that the Gibbs mechanism of elasticity becomes impossible. However, the film disjoining pressure (as well as the film tension) is capable of variation by changing the external pressure pβ as follows from eq 7. In this case, eq 6 is reduced to
dγ ) h dΠ
(28)
If the film area is also fixed, the definition of the elasticity modulus expressed in eq 1 fails. Then we have to turn to the transversal elasticity modulus, eq 20, since the disjoining pressure always reacts to a change in the film thickness. For a completely open film, the transversal elasticity modulus is defined as
(
E⊥ ≡ -
)
dΠ d(ln ht)
(29)
T,µi
This quantity can be easily measured in the equilibrium foam. In this case, it is enough to measure the foam film thickness since the disjoining pressure is determined by the height H of the foam column and the solution density F6
Π ) FgH
( )
(26)
(30)
where g is the acceleration due to gravity. According to eq 30, the transversal elasticity modulus is referred to a certain value of disjoining pressure by measuring it at a certain height. As stated above, the surfactant concentration does not change in the course of the experiment. However, the experiment can be carried out for various given concentrations, and in this way, the concentration dependence of the transversal elasticity modulus can be determined. We discuss this dependence below with a special reference to the cmc. Let us consider a thin film containing a single surfactant and assume the film disjoining pressure to be a function of two variables at a fixed temperature, the surfactant
E⊥ ≡ - ht
dΠ dht
(31)
c
to differentiate it with respect to concentration at a given disjoining pressure:
( ) ∂E⊥ ∂c
( )( ) ( ) ∂ht ∂c
)-
Π
Π
∂Π ∂Π ) ∂ht c ∂c
ht
(32)
Remarkably, the resulting eq 32 shows that the concentration dependence of the transversal elasticity modulus at a given disjoining pressure coincides with the concentration dependence of disjoining pressure at a given film thickness. The latter dependence is expected to pass through a maximum of the disjoining pressure in the case of an ionic surfactant. Indeed, there are two opposing tendencies in this case, the growth of charge and the decrease of the surface layer thickness. The former enhances and the latter weakens the disjoining pressure because of decreasing the surface layer overlapping with increasing ionic strength. This can be illustrated by the formula for the disjoining pressure of a film of a solution of a symmetrical electrolyte14
Π ) 64ckTΦ exp(-κh/2) - 4AH/27πh3
(33)
where Φ ) tan(hψze/2kT), ψ is surface potential, ze is the ionic charge, κ is the reciprocal Debye length, AH is the Hamaker constant, and kT is of usual meaning. According to eq 33, the disjoining pressure depends on the concentration as c exp(- xc), and this function has a maximum. The effect is enhanced for colloidal surfactants possessing the cmc. Below the cmc, almost the whole amount of a surfactant added incomes to the surface layers of a film. By contrast, the surfactant mainly enters the interior of the film above the cmc, which creates the screening effect and leads to a steeper decrease of the disjoining pressure after the maximum. Similar speculations can be found in the literature.5 Accounting for eq 32, we now can say that the behavior of the transversal elasticity modulus should be the same. In other words, we expect the existence of a maximum of the transversal elasticity modulus close to the cmc. We now consider how the film thickness depends on the surfactant concentration if the disjoining pressure of the film is fixed. The latter condition is expressed by the equation
dΠ )
∂Π dh + ( ) (∂Π ∂h ) ∂c c
h
dc ) 0
(34)
from which we obtain
(∂h∂c )
)Π
∂Π (∂Π ∂c ) /( ∂h ) h
c
(35)
Since the derivative ∂Π/∂h is made negative by the stability condition of a thin film, it follows from eq 35 that extremes of the disjoining pressure and of the film thickness should occur simultaneously, at the same concentration. Since we know that the transversal elasticity modulus duplicates the behavior of the disjoining pressure (see eq 32), we can also say that extremes of the transversal elasticity (14) Shah, D. O.; Djabbarah N. F.; Wasan D. T. Colloid Polym. Sci. 1978, 256, 1001.
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Figure 4. Concentration dependence of the film transversal elasticity modulus in the foam of the SDS solution.
Figure 5. Dependence of the foam film thickness on the concentration of SDS.
modulus and of the film thickness should coincide in concentration.
although the mechanism of elasticity is different for closed and completely open films. These similar locations of the maxima may be interpreted as giving evidence of the leading role of the cmc in both cases. In conclusion we would like to state the following. The above analysis is based on thermodynamics, which yields a generality advantage. In particular, the prediction of maxima of the film elasticity and, as a consequence, of the foam stability is equally applicable to ionic and nonionic surfactant systems. On the other side, generality suffers from an insufficiency of details. Naturally, there are many other factors (e.g., the ionic strength mentioned above), besides the elasticity modulus, that influence the foam stability and that proved to be outside of the scope of this paper. However, a thermodynamic consideration always is a good beginning for further investigation.
Experimental Section The experiments were carried out with an aqueous solution of SDS (Fluka, >99% grade). The foam with the cell radius of about 1 mm was produced by the air flux through a porous membrane. The foam film thickness was measured optically using the setup and experimental technique described earlier.7,8 To evaluate the transversal elasticity modulus, the film thickness was measured in the foam column at heights H1 ) 50 mm and H2 ) 79 mm, and the following formula was used
E⊥ )
Fg(H2 - H1)[h(H1) + h(H2)] 2[h(H1) - h(H2)]
(36)
The concentration dependence of the film thickness was determined at a height of 60 mm. The data obtained are presented in Figures 4 and 5. In both figures, well-pronounced maxima are seen near the cmc as predicted by the above theory. It is of interest that the maxima are located approximately at the same place on the concentration axis as the maxima of foaminess and of the foam stability,
Acknowledgment. This work was financially supported by the Russian Foundation for Basic Research (Grants 01-03-32334 and 01-03-32322), by the Presidential program of support of leading Russian scientific schools (Grant 789.2003.03), and by the program “Scientific researches of the high school on priority trends in science and engineering” (Grant 203.06.06.035). LA0358623
