Extremes of Some Foam Properties and Elasticity of ThinFoam 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 isshown to be additive with respect to contributions from Gibbs elasticity and disjoining pressure. A detailedexpression for the film elasticity modulus explains the pronounced maxima of foaminess and foam stabilitynear the critical micelle concentration observed earlier in many experiments. A theory of transversalelasticity of thin foam films is formulated under conditions excluding the action of Gibbs elasticity. Nearthe critical micelle concentration, the theory predicts maxima of the transversal elasticity modulus andof the films thickness as functions of concentration at a given disjoining pressure. The prediction has beenverified experimentally by measuring the film thickness in equilibrium foam as a function of height.

IntroductionThe motivation of this paper is rooted both in experi-

mental and theoretical reasons related to foams and foamfilms. Experimentally, there are some facts (extremes ofa number of the foam and foam film properties) whoseorigin remains unclear up to the present. This can becaused by the insufficient state of the theory of foam films,which needs further development.

Among the foam properties, the foam stability looksthe most important. Numerous methods of quantitativecharacterization of the foam stability are well describedand have been reviewed in the literature.1-6 The authorscontributed to this area by investigating the evolutiontime for the foam cells at a given horizontal level. Theevolution time is introduced as follows. The internalstability of foam depends on the lifetime of internal filmsseparating foam cells. Every breakup of an internal filmmeans the coalescence of neighboring cells with thecorresponding enlargement of a foam cell. The resultingstate of polydispersity is characterized by the cell volumeratios close to the ratios of integer numbers. Reasonablyassuming the number of bursting films per unit time tobe proportional to the total film number, the process ofcell growingshouldproceedexponentially.Sowemaywrite

where Rt and R0 are the average cell radius and its initialvalue, respectively, t is time, and Τ is a coefficient that wecalled the evolution time and used as a characteristic ofthe foam stability. The evolution time is a time needed forthe enlargement of a foam bubble by e times. Since theprobability of film bursting practically does not dependon the film area, Τ is very close to being a constant, even

in polydisperse foam. Using an optical technique formeasuring the transmission factor and the theory ofmultiple light scattering in foams, the foam cell radius isdetermined as a function of time. In this way, the evolutiontime can be found from experiment. The experimentalsetup and results have been described elsewhere.7,8 Theinvestigation discovered a maximum of the evolution timenear the critical micelle concentration (cmc) for the foamprepared from the sodium dodecyl sulfate (SDS) solution(Figure 1). Similar observations of peaks of the foamstability were still earlier reported in the literature1,9 asobtained by traditional measuring the foam-column heightdecrease in time. However, the explanation has not beengiven so far and remains to be a challenge to theory.

Additional data were obtained8,10 from studying thedense bubble monolayer on the surface of the SDS solution.The bubble monolayer can be regarded as a kind of two-dimensional foam, but, in any case, is an object of a specificinterest in colloid science. In this case, the average bubblelifetime, τ (that we called foaminess1), can be measureddirectly by regulating the bubble barbotage rate tomaintain the complete surface coverage in the bubblemonolayer. The difference between Τ and τ is that Τcharacterizes the internal foam stability, whereas τ is

* Corresponding author. E-mail: rusanov@rus.usr.pu.ru.(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 In-

dustry; 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 ofCapillary Systems; Imperial College Press: London, 1999; p 372.

(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 InterfaceSci. 1998, 206, 392.

(9) Gotte, E. Kolloid-Z. 1933, 64, 327.(10) Krotov,V.V.;Nekrasov,A.G.;Rusanov,A. I. MendeleevCommun.

1996, No. 5, 178.

Figure 1. Concentration dependence of the evolution time forthe foam of the SDS solution.

Rt ) R0 exp (t/Τ)

1511Langmuir 2004, 20, 1511-1516

10.1021/la0358623 CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 01/22/2004

related to the outer stability. Indeed, mostly those filmsburst that are in contact with the surroundings in thecase of the bubble monolayer (which is also always presentin real foam as a peripheral bubble layer). Figure 2 showsthe plot of τ vs the SDS concentration. Foaminess turnsto be still more sensitive characteristic and exhibits a sharppeak near the cmc.

In addition to the above properties, the evolution timeand foaminess, we shall show in the experimental part ofthis paper that the film thickness in equilibrium foamalso passes through a maximum near the cmc. Thus,already three properties behave in a similar manner. Thepossible line of explanation is the analysis of the foamfilm elasticity, which is the principal factor of the foamstability. Up to the present time, however, the theory offilm elasticity has been formulated differently for thinand thick films (both present in foams) and needs furtherdevelopment.

The distinction between thin and thick films is at-tributed to the account or neglect of the overlap of the filmsurface layers, respectively.Theoverlappingsurface layersinteract and create the disjoining pressure, which is aspecific feature of thin films, whereas thick films have nodisjoining pressure. Thick films are only capable of “Gibbselasticity” related to the redistribution of the componentsof a mixed film between the film surface and the bulkduring the film stretching. Although the Gibbs elasticitymechanism is universal and equally important for thickand thin films, this kind of elasticity is traditionallyreferred to thick films for which it is the only and, therefore,easily recognizable mechanism. Gibbs was first to for-mulate the theory of elasticity of thick films, which,however, turned to be insufficient, was essentially im-proved in the 1960s and 1970s and later reviewed.11 Asfor thin films, the effect of disjoining pressure has beenin the highlight, although disjoining pressure creatingthe second kind of elasticity. There is no general theoryembracing both the kinds of elasticity up to the presenttime. This work purposes filling up this gap.

A film is called closed if the total amounts of all filmspecies are constant (the terms “partially closed” or“partially open” imply a certain part of the film speciesto be capable of exchange with the surrounding medium).As is known, the Gibbs elasticity is realized if a film isclosed at least with respect to one of the film species. We

shall first consider completely or partially closed filmswhere the Gibbs elasticity and the elasticity related todisjoining pressure, act in parallel. The theory will beshown to be able to explain the peaks of foaminess andfoam stability near the cmc as a result of behavior of theelasticity modulus. Second, we shall consider an openequilibrium film in the absence of Gibbs elasticity. Theelastic behavior of such a film is governed by thetransversal elasticity, which is an important componentof the total film elasticity. For transversal elasticity, theoryalso predicts the existence of a maximum near the cmc,although with another mechanism. We also present thefirst measurements of the transversal elasticity modulusand experimental data to confirm the prediction of thetheory.

Thermodynamics of Elasticity of a Thin Film

A film is called elastic if it increases its tension γ atstretching. This kind of elasticity is quantitatively char-acterized by the film elasticity modulus E, which Gibbsdefined as

where A is the film area. This definition is of generalcharacter and may be applied to any kind of elasticity.

We consider a multicomponent flat symmetrical thinfilm 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 motherphase having the same values of the temperature andchemical potentials as in the film. The state of the motherphase is governed by the Gibbs-Duhem equation

where p is the pressure, c is the concentration, and µ isthe chemical potential, subscript i referring to the speciesof which the film is composed. The form of a thermody-namic fundamental equation for the film itself dependson the method of description. Following the method ofGibbs’ surface thermodynamics with using a singledividing surface, the film fundamental equation is of theform of the Gibbs adsorption isotherm

where Gi is the excess (with respect to the surroundingmedium) amount of the ith species per unit film area. Weassume the content of the film species in the surroundingsto be negligible, and we understand Gi as the total amountof the ith species per unit film area. In this case, Gi doesnot depend on the dividing surface location. So it isimpossible to convert one of the quantities Gi to zero byshifting the dividing surface, contrary to the case of theordinary Gibbs adsorption equation.

The total amount of the ith species in the film is GiAand is constant if the film is closed. Then we have thecondition for a closed film

Putting eqs 3 and 4 into eq 1, we obtain the expression

(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.

Figure 2. Concentration dependence of the foaminess of theSDS solution.

E ≡ dγd(ln A)

(1)

dp ) ∑i

ci dµi (2)

dγ ) -∑i

Gi dµi (3)

d(ln A) ) -d(ln Gi) (4)

1512 Langmuir, Vol. 20, No. 4, 2004 Rusanov et al.

for the elasticity modulus of a completely closed film

Each term is positive on the right-hand side of eq 5since the derivatives are positive according to thermo-dynamic stability conditions. In accordance with thereduced Le Chatelier-Brown principle,12 eq 5 exhibits adecrease of the elasticity modulus with increasing thenumber of open species. If a species becomes open, theconstancy of its mass is replaced by the constancy of itschemical potential, and the corresponding term disappearson the right-hand side of eq 5. When the film becomesentirely open, the whole right-hand side of eq 5 is reducedto zero, and the film elasticity vanishes. In this case, thethermodynamic state of the film is fixed. When stretched,the film increases its area, but not tension.

Typically for Gibbs’ formalism, eq 3 does not includesuch important characteristics of a thin film as the filmthickness and disjoining pressure. To incorporate themin thermodynamic equations, one has to use the methodof two dividing surfaces,13 which is formulated as follows.Similarly to the case of a thick film, we introduce twodividing surfaces on both the sides of the film (Figure 3)and imaginary fill in the space between them with thefilm mother phase (R) possessing the same values oftemperature and chemical potentials as in the real thingfilm. Referring eq 2 to phase R and subtracting it from eq3, we transform the fundamental equation of a thin filmto the form

where Γi is the adsorption (with respect to the motherphase) of the ith species, h is the separation between thedividing surfaces, and Π is the disjoining pressure definedas

Equation 7 shows that, in the case of a thin film, thepressure in the mother phase remains variable at constantexternal pressure, which is quite impossible in a thickfilm.

Equation 2 can be separately written for phases R andâ. Taking the difference of these equations and using eq7, we obtain for the disjoining pressure

which is an additional relationship to eq 6. Solving together

eqs 6 and 8, we can exclude the chemical potential of asolvent from eq 6 to obtain

where subscript j refers to the solvent. The coefficient ofdµi in eq 9 is well-known to be the relative adsorption ofthe ith species in Gibbs’ surface thermodynamics. Boththe coefficients of dµi and of dΠ are invariant with respectto shifting the dividing surfaces. As for their numericalvalues, these coefficients have a very simple interpretationas the adsorption and the separation, respectively, refer-ring to the particular location of the dividing surfaces(called an equimolecular surface) where the solventadsorption is zero. Indeed, setting Γj ) 0 in eq 9, we comeback to eq 6 provided we understand Γi as relativeadsorptions, h as the separation between two equimo-lecular surfaces, and the summation as including allsolutes (e.g., surfactants) but no solvent. The quantity hdefined in this way is often used as the film thickness. Arelative difference between h and the real film thicknessht is negligible for thick films, but can be essential for thinfilms, so we below discriminate h from ht.

Putting eq 6 into eq 1 yields the description of the filmelasticity modulus. However, the result depends on theexternal conditions, imposed on the film to determine thedegree of exchange of matter between the film and thesurroundings. There are three main cases of such condi-tions to be considered below: a completely closed film, apartially closed (open) film, and a completely open film.

Completely Closed FilmFor the sake of simplicity, we consider a closed film of

a solution of a single surfactant. In this case, eq 6 becomes

where Γ and µ are the relative adsorption and the chemicalpotential of the surfactant, respectively, and h is thedistance between the two equimolecular surfaces of thefilm. The external pressure is assumed to be constant.Correspondingly, dΠ ) -dp (p ≡ pR; we omit superscriptR after this point) according to eq 7. Since the surfactantchemical potential is a function of the pressure p and thesurfactant concentration c in the mother phase, theexpression holds

where v is the partial molar volume of the surfactant.Putting eq 11 into eq 10 yields

where

is the total film thickness with accounting for thesurfactant adsorbed layers (a negligible difference betweenthe v values in eqs 11 and 13 can be caused by compress-ibility).

(12) Rusanov, A. I. Phasengleichgewichte und Grenzflaechenerschei-nungen; Akademie-Verlag: Berlin, 1978; Chapter 2.

(13) Rusanov, A. I. Method of Two Dividing Surfaces in Thermo-dynamics of Thin Films. In Surface Forces and Boundary Layers ofLiquids; Deryagin, B. V., Ed.; Nauka: Moscow, 1983 (in Russian);p 152.

Figure 3. Method of two dividing surfaces in the theory ofthin films: R is the imaginary bulk mother phase of a thin film,â is the surrounding medium, and h is the dividing surfacesseparation.

E ) ∑i

Gi2dµi

dGi

(5)

dγ ) -2 ∑i

Γi dµi + h dΠ (6)

Π ≡ pâ - pR (7)

dΠ ) ∑i

(ciâ - ci

R) dµi (8)

dγ ) -2 ∑i (Γi - Γj

ciâ - ci

R

cjâ - cj

R) dµi + (h -2Γj

cjâ - cj

R)dΠ

(9)

dγ ) -2Γ dµ + h dΠ (10)

dµ ) -v dΠ + (∂µ/∂c)dc (11)

dγ ) -2Γ(∂µ/∂c) dc + ht dΠ (12)

ht ≡ h + 2Γv (13)

Elasticity of Thin Foam Films Langmuir, Vol. 20, No. 4, 2004 1513

In addition to eq 12, we have to take into account therelations following from the constancy of amounts of allspecies in a completely closed film. Neglecting compress-ibility, we also may consider the film volume Vt ) Aht asa constant. This yields the relationship

In this case, the balance equations hold

where V ) Ah is the volume of the space between the twoequimolecular dividing surfaces (the relation between Vtand V is easily established by multiplication of eq 13 byarea A). As mentioned above, we neglect the variation ofv, so that eq 16 yields

Putting eq 17 into eq 15 leads to the relationship

where æ ≡ vc is the packing fraction of the surfactant inthe film mother phase (æ is usually small as comparedwith unity).

We now divide eq 12 by dln A and use eqs 18 and 14when dividing the first and second terms on the right-hand side of eq 12, respectively, to obtain the finalexpression for the film elasticity modulus

Here we have introduced the transversal elasticitymodulus E⊥

as a specific characteristic of a thin film related to thedisjoining pressure isotherm. Equation 19 exhibits theadditivity of contributions from the Gibbs elasticity andthe specific elasticity to the elasticity modulus of a thinfilm. The second term on the right-hand side of eq 19 isabsent for a thick film whose elasticity is determinedexclusively by the Gibbs mechanism. In contrast, the firstterm disappears for a thin film of a pure solvent (Γ ) 0),and the film elasticity is determined only by disjoiningpressure. Both the mechanisms operate in parallel in thegeneral case of a thin film.

Let us apply eq 19 to the vicinity of the cmc where thesurfactant aggregation develops. Since the cmc is typicallysmall, we can assume that the surfactant chemicalpotential in the mother phase is determined by thesurfactant monomeric concentration c1 to write

where R is the gas constant and T is the temperature.Using eq 21, eq 19 can be rearranged to the form

There are two factors in the first term on the right-hand side of eq 22. Bearing in mind an adsorption isothermof the Langmuir type, it is easy to see that the first factorincreases with concentration. The second factor, thederivative ∂c1/∂c, is practically unity below the cmc, butfalls off dramatically almost to zero when entering thenarrow cmc region. It is well-known that the saturationof the adsorbed layer is attained still before the cmc, sothat we have first increasing and then decreasing of theelasticity modulus. Thus, we can say theory predicts theexistence of a maximum of the film elasticity near thecmc. Importantly, the prediction is of general characterfor all types of surfactants.

Since the film elasticity is an important factor of thefoam stability, we can say the above theory also predictsa maximum of the film stability near the cmc. As wasalready noted and illustrated by Figures 1 and 2 in theIntroduction, a number of such indications have beenreported in the literature. The peak of foaminess in Figure2 is very steep and about 40 times exceeds the backgroundvalue. In our lab we observed even higher peaks (about80 times as much as foaminess values far from the cmc)for nonionic surfactants. Thus, we may conclude that theeffect predicted by the above theory is very important forpractice.

Film with a Single Closed Species

We now consider the case when a film is open withrespect to all species except one (for example, a film of asurfactant aqueous solution in a chamber saturated withwater vapor). In this case there is only one term with thechemical potential µ not only in eqs 3 and 6, but also ineq 2. If a film was thick, the remaining chemical potentialµ would be also constant due to the constancy of theexternal pressure, and there would be no film (Gibbs)elasticity. However, the realization of Gibbs elasticity ispossible in a thin film with only one closed component. Aswas mentioned above, Gibbs elasticity is usually attributedto thick films. We now consider a unique example when,by contrast, Gibbs elasticity is impossible for a thick filmand is realized in a thin film.

In accordance with eq 7, we again have dΠ ) -dp ata constant external pressure. Then eq 2 for the film motherphase can be written as

Using eq 23, one can formulate the theory either interms of disjoining pressure or in terms of chemicalpotential with transition to Gibbs elasticity. Choosing thesecond route, we turn to eq 5 that is now reduced to thesimple expression

The total derivative of the chemical potential implies avariable pressure. To avoid this complication, we pass tothe partial derivative by putting eq 23 in eq 11. This yields

where the partial derivative is taken at constant pressure.The behavior of the partial derivative of the chemicalpotential is well-known: it falls off almost to zero whenpassing to micellization. Before the micellization when a

dΠ ) -c dµ (23)

E ) G2dµdG

) G2

dG/dc‚ dµ

dc(24)

dµdc

) 11 - æ

∂µ∂c

(25)

d(ln A) ) -d(ln ht) (14)

d(2AΓ + Vc) ) 0 (15)

dVt ) d(V + 2ΓAv) ) 0 (16)

dV ) -2v d(ΓA) (17)

d(ln A) ) -d(ln Γ) - h2Γ(1 - æ)

dc (18)

E ) 2Γ2∂µ/∂c

dΓ/dc + h/2(1 - æ)+ htE⊥ (19)

E⊥ ≡ - dΠd(ln ht)

(20)

dµ ) RT d(ln c1) (21)

E )2RTΓ2/c1

dΓ/dc + h/2(1 - æ)

∂c1

∂c+ htE⊥ (22)

1514 Langmuir, Vol. 20, No. 4, 2004 Rusanov et al.

surfactant solution almost ideal, we may set

Since the surfactant packing fraction æ is typicallynegligible below the cmc, we may apply this result alsoto the total derivative of the chemical potential, whichchanges eq 24 to the form

According to eq 27, the film elasticity modulus is zero atzero concentration (G also becomes zero, but the ratio G/cremains a finite nonzero value as c f 0) and is positiveat a finite concentration, so that the elasticity modulusincreases with concentration. As is seen from eq 25, thetotalderivativedµ/dc reproduces thebehaviorof thepartialderivative and, hence, steeply decreases when passingthe cmc. Correspondingly, the elasticity modulus de-creases, so that eq 24 again predicts a maximum of thefilm elasticity near the cmc, similarly to the case of acompletely closed film.

Completely Open FilmAll chemical potentials are fixed in a completely open

film, and, therefore, also pressure and all other stateparameters of the film mother phase are fixed. Inparticular, the surfactant concentration and adsorptionare also constant, and this means that the Gibbs mech-anism of elasticity becomes impossible. However, the filmdisjoining pressure (as well as the film tension) is capableof variation by changing the external pressurepâ as followsfrom eq 7. In this case, eq 6 is reduced to

If the film area is also fixed, the definition of the elasticitymodulus expressed in eq 1 fails. Then we have to turn tothe transversal elasticity modulus, eq 20, since thedisjoining pressure always reacts to a change in the filmthickness. For a completely open film, the transversalelasticity modulus is defined as

This quantity can be easily measured in the equilibriumfoam. In this case, it is enough to measure the foam filmthickness since the disjoining pressure is determinedby the height H of the foam column and the solution den-sity F6

where g is the acceleration due to gravity. According toeq 30, the transversal elasticity modulus is referred to acertain value of disjoining pressure by measuring it at acertain height. As stated above, the surfactant concentra-tion does not change in the course of the experiment.However, the experiment can be carried out for variousgiven concentrations, and in this way, the concentrationdependence of the transversal elasticity modulus can bedetermined. We discuss this dependence below with aspecial reference to the cmc.

Let us consider a thin film containing a single surfactantand assume the film disjoining pressure to be a functionof two variables at a fixed temperature, the surfactant

concentration c and the film thickness ht: Π ) Π(c, ht).We rewrite the definition expressed in eq 29 in the form

to differentiate it with respect to concentration at a givendisjoining pressure:

Remarkably, the resulting eq 32 shows that the concen-tration dependence of the transversal elasticity modulusat a given disjoining pressure coincides with the concen-tration dependence of disjoining pressure at a given filmthickness. The latter dependence is expected to passthrough a maximum of the disjoining pressure in the caseof an ionic surfactant. Indeed, there are two opposingtendencies in this case, the growth of charge and thedecrease of the surface layer thickness. The formerenhances and the latter weakens the disjoining pressurebecause of decreasing the surface layer overlapping withincreasing ionic strength.

This can be illustrated by the formula for the disjoiningpressureofa filmofasolutionofasymmetrical electrolyte14

where Φ ) tan(hψze/2kT), ψ is surface potential, ze is theionic charge, κ is the reciprocal Debye length, AH is theHamaker constant, and kT is of usual meaning. Accordingto eq 33, the disjoining pressure depends on the concen-tration as c exp(- xc), and this function has a maximum.The effect is enhanced for colloidal surfactants possessingthe cmc. Below the cmc, almost the whole amount of asurfactant added incomes to the surface layers of a film.By contrast, the surfactant mainly enters the interior ofthe film above the cmc, which creates the screening effectand leads to a steeper decrease of the disjoining pressureafter the maximum. Similar speculations can be found inthe literature.5 Accounting for eq 32, we now can say thatthe behavior of the transversal elasticity modulus shouldbe the same. In other words, we expect the existence ofa maximum of the transversal elasticity modulus close tothe cmc.

We now consider how the film thickness depends on thesurfactant concentration if the disjoining pressure of thefilm is fixed. The latter condition is expressed by theequation

from which we obtain

Since the derivative ∂Π/∂h is made negative by the stabilitycondition of a thin film, it follows from eq 35 that extremesof the disjoining pressure and of the film thickness shouldoccur simultaneously, at the same concentration. Sincewe know that the transversal elasticity modulus duplicatesthe behavior of the disjoining pressure (see eq 32), we canalso say that extremes of the transversal elasticity

(14) Shah, D. O.; Djabbarah N. F.; Wasan D. T. Colloid Polym. Sci.1978, 256, 1001.

∂µ∂c

) RTc

(26)

E ≈ RT G2/cdG/dc

(27)

dγ ) h dΠ (28)

E⊥ ≡ -( dΠd(ln ht))T,µi

(29)

Π ) FgH (30)

E⊥ ≡ - ht(dΠdht

)c

(31)

(∂E⊥

∂c )Π

) -(∂ht

∂c )Π(∂Π∂ht

)c

) (∂Π∂c )

ht

(32)

Π ) 64ckTΦ exp(-κh/2) - 4AH/27πh3 (33)

dΠ ) (∂Π∂h )c

dh + (∂Π∂c )h

dc ) 0 (34)

(∂h∂c)Π

) - (∂Π∂c )h/(∂Π

∂h )c(35)

Elasticity of Thin Foam Films Langmuir, Vol. 20, No. 4, 2004 1515

modulus and of the film thickness should coincide inconcentration.

Experimental SectionThe experiments were carried out with an aqueous solution

of SDS (Fluka, >99% grade). The foam with the cell radius ofabout 1 mm was produced by the air flux through a porousmembrane. The foam film thickness was measured optically usingthe setup and experimental technique described earlier.7,8 Toevaluate the transversal elasticity modulus, the film thicknesswas measured in the foam column at heights H1 ) 50 mm andH2 ) 79 mm, and the following formula was used

The concentration dependence of the film thickness was deter-mined at a height of 60 mm.

The data obtained are presented in Figures 4 and 5. In bothfigures, well-pronounced maxima are seen near the cmc aspredicted by the above theory. It is of interest that the maximaare located approximately at the same place on the concentrationaxis as the maxima of foaminess and of the foam stability,

although the mechanism of elasticity is different for closed andcompletely open films. These similar locations of the maximamay be interpreted as giving evidence of the leading role of thecmc in both cases.

In conclusion we would like to state the following. The aboveanalysis is based on thermodynamics, which yields a generalityadvantage. In particular, the prediction of maxima of the filmelasticity and, as a consequence, of the foam stability is equallyapplicable to ionic and nonionic surfactant systems. On the otherside, generality suffers from an insufficiency of details. Naturally,there are many other factors (e.g., the ionic strength mentionedabove), besides the elasticity modulus, that influence the foamstability and that proved to be outside of the scope of this paper.However, a thermodynamic consideration always is a goodbeginning for further investigation.

Acknowledgment. This work was financially sup-ported by the Russian Foundation for Basic Research(Grants 01-03-32334 and 01-03-32322), by the Presidentialprogram of support of leading Russian scientific schools(Grant 789.2003.03), and by the program “Scientificresearches of the high school on priority trends in scienceand engineering” (Grant 203.06.06.035).

LA0358623

Figure 4. Concentration dependence of the film transversalelasticity modulus in the foam of the SDS solution.

E⊥ )Fg(H2 - H1)[h(H1) + h(H2)]

2[h(H1) - h(H2)](36)

Figure 5. Dependence of the foam film thickness on theconcentration of SDS.

1516 Langmuir, Vol. 20, No. 4, 2004 Rusanov et al.