Langmuir-BET Surface Equation of State in Fluid-FluidInterfaces

J. Gracia-Fadrique

Departamento de Fısica y Qımica Teorica, Facultad de Quımica, Universidad NacionalAutonoma de Mexico, Mexico D.F. 04510, Mexico

Received September 14, 1998. In Final Form: January 21, 1999

A novel surface equation of state obtained from the BET adsorption isotherm and the Gibbs equationis presented. This equation describes both monolayer and multilayer adsorption. The emphasis of thispaper is on monolayers, for which this new equation only contains thermodynamic parameters. In thisform, it describes the characteristic behavior of nonionic surfactants, the relation between maximumsurface pressure and critical micelle concentration, and the link between the standard free energy ofadsorption and the standard free energy of micellization. It also explains and justifies the fundamentalwork of Rosen regarding the relative effects of structural factors on the surface tension reduction. Thedeviation of the model from experimental data is explained in terms of the activity coefficient at the criticalmicelle concentration, which is proportional to the micelle aggregation molecular weight.

IntroductionThe Gibbs adsorption isotherm allows us to transform

an isotherm to a surface equation of state. The Langmuirisotherm in fluid-fluid interfaces, coupled with the Gibbsequation,1 leads to

where θ is the surface coverage (θ ) Γ/Γm), Γ is the Gibbsiansurface concentration, Γm is the maximum or saturationsurface concentration, Π is the surface pressure, R is thegas constant, T is the absolute temperature, x2 is the bulkconcentration in mole fraction for the surface-active solute,and â is equivalent to the Henry constant in twodimensions. The BET equation and the Gibbs adsorptionequation2 give

In both cases, the integral form leads to the correspondingsurface equation of state:

DiscussionBy a simple analysis of eq 3, the transformation of the

Langmuir isotherm into a surface equation of state isobserved. This equation was proposed empirically bySzyszkowski.3 The BET equation is a natural extensionof the Langmuir model to the multilayer case; thus, wewill call eq 4 the Langmuir-BET (L-B) surface equationof state. This work is devoted to the most common caseof monolayer behavior. However, it can be used for systems

of surface tension that show an inflection point.4 Theadvantage of eq 4 over eq 3 is that an analytical solutioncan be obtained when the system reaches the saturationcondition. In this case,

where π(cmc) is the maximum or saturation surfacepressure and xcmc is the critical micelle mole fraction.Solving the right-hand side of eq 2 for θ ) 1

Or in an explicit form for â we get

Substitution of eq 10 in eq 4 leads in a final form to a novelL-B surface equation of state containing only measurableparameters:

For practical purposes, and for a typical micelle concen-tration (10-7 < xcmc < 10-3), eq 11 reduces to

(1) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848.(2) Brunauer, S.; Emmettand, P. H.; Teller, E. T. J. Am. Chem. Soc.

1938, 60, 309.(3) Von Szyszkowski, B. Z. Phys. Chem. 1908, 64, 385.

(4) AÄ guila-Hernandez, J.; Hernandez, I.; Trejo, A. Int. J. Thermophys.1995, 16, 45.

θ )x2

ΓmRT(dπdx2

)T

)âx2

1 + âx2(1)

θ )x2

ΓmRT(dπdx2

)T

)âx2

(1 - x2)(1 - x2 + âx2)(2)

π ) ΓmRT ln(1 + âx2) (3)

π ) ΓmRT ln(1 + âx2

1 - x2) (4)

θ f 1 (5)

x f xcmc (6)

π f π(cmc) (7)

Γ f Γm (8)

âxcmc

(1 - xcmc)(1 - xcmc + âxcmc)) 1 (9)

â ) (1 - xcmc

xcmc)2

(10)

π ) ΓmRT ln[1 + (1 - xcmc

xcmc)2 x2

1 - x2] (11)

π ) ΓmRT ln[1 +x2

xcmc2 ] (12)

3279Langmuir 1999, 15, 3279-3282

10.1021/la981244o CCC: $18.00 © 1999 American Chemical SocietyPublished on Web 04/01/1999

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At the critical micelle concentration, eq 11 reduces to

This last equation explains the typical behavior ofetoxylated nonionic surfactants for which the mosthydrophobic members show lower critical micelle con-centrations and higher surface pressures and, on the otherhand, are more hydrophilic. Surfactants show highercritical micelle concentrations and lower surface pressures(Shick).5-7 A possible mechanism to prove the functionalityof eq 13 is to compare the experimental surface pressureat the cmc with the predicted surface pressure when thevalue of the critical micelle concentration and the maxi-mum surface concentration Γm are known.

In the extensive work of Rosen8-10 on the effectivenessof the surfactants in the surface tension reduction, thevalue of the maximum surface pressure π is compared tothe condition for the same surfactant when the surfacepressure is 20 mN/m.

When the surface pressure is the Rosen pressure π )20 mN/m, eq 12 becomes

In the vicinity of the critical micelle concentration

Thus, eqs 14 and 15 can be expressed as

The difference between eq 13 and eq 16 is

This equation is the same as that deduced by Rosen andaccounts for theeffectivenessof thereduction in thesurfacetension.11 It is important to point out that this reductionis found under the condition proposed in eq 16; eq 17 isvalid only near the saturation conditions, and eq 18confirms the condition of the surface pressure value beingaround 20 mN/m or higher.

For nonionic surfactants the standard free energy ofmicellization is given by12

Then, eq 13 provides a direct relation of the standard free

energy of micellization and the maximum surface pressureπ(cmc):

The standard free energy of adsorption is defined as

The limiting value at infinite dilution predicted by eq 11is then

Kronberg et al.13 have suggested that the standard freeenergies of adsorption and micellization are related by aconstant.Combiningeqs19,21,and22,weobtainageneralrelation for the standard free energies of adsorption andmicellization.

Equation 11 has been tested with our own experimentaldata, previously published14 for the system 2-butoxyetha-nol + water at several temperatures (4, 25, 48 °C). Table1 shows the experimental value of the cmc obtained fromthe abrupt change in the slope of the surface pressureversus ln(x) plot. The results show a good agreement withthe values predicted by eq 11. Γm and â are adjustedparameters from the experimental data.

However, the last example is a particular case thatfulfills eq 11. For characteristic cmc values (10-7 to 10-3

mole fraction) and surface saturation concentrations (2-4) × 10-10 gmol/cm2, 25 °C), eq 11 leads to nonrealisticsurface pressures values.

So, lets call systems L-B those ones that obey eq 11.The departure from this behavior is very closely relatedto the structure of the chemical potential involved. Botheqs 3 and 4 are deduced using the ideal chemical potential.

Ross15 has pointed out the need for considering theactivity coefficients for the solute and the solvent in thebulk and surface phase for a complete description of theadsorbed solutes. This lack of information for manysystems can be avoided by the use of the symmetricconvention for the activity coefficient in the cmc region.Under the symmetrically normalized convention

and the BET equation may be expressed in terms of theactivity coefficient of the surface-active solute in the bulk

(5) Schick, M. J. J. Colloid Sci. 1962, 17, 801.(6) Schick, M. J.; Atlas, S. M.; Eirich, R. J. Phys. Chem. 1962, 66,

1326.(7) Meguro, K.; Takasawa, Y.; Kawahashi, N.; Tabata, Y.; Ueno, M.

J. Colloid Interface Sci. 1981, 83, 50.(8) Rosen, M. J. J. Am. Oil Chem. Soc. 1972, 49, 293.(9) Rosen, M. J. J. Am. Oil Chem. Soc. 1974, 51, 461.(10) Rosen, M. J. J. Colloid Interface Sci. 1976, 56, 320.(11) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed;

John Wiley & Sons: New York, 1989; Chapter 5, p 217, eq 5.1.(12) Molyneux, P.; Rhodes, C. T.; Swarbrick, J. Trans. Faraday Soc.

1965, 61, 1043.

(13) Kronberg, B.; Lindstrom, M.; Stenius, P. Competitive Adsorptionof an Anionic and a Nonionic Surfactants on Polystyrene Latex inPhenomena in Mixed Surfactant Systems; Scamehorn, V. F., Ed.; ACSSymposium Series No. 311; American Chemical Society: Washington,DC, 1986; Chapter 17.

(14) Elizalde, F.; Gracia, J.; Costas, M. J. Phys. Chem. 1988, 92,1032.

(15) Ross, S. Colloids Surf. 1983, 7, 121.

π(cmc) ) -ΓmRT ln xcmc (13)

20 ) ΓmRT ln[xcmc2 + xπ)20

xcmc2 ] (14)

xcmc2 + xπ)20 ≈ xπ)20 (15)

20 ) ΓmRT ln[xπ)20

xcmc2 ] (16)

πcmc - (π ) 20) ) ΓmRT ln[ xcmc

xπ)20] (17)

∆Gmic ) RT ln xcmc (18)

Table 1. Values of the cmc Predicted with Eq 11 andExperimental Values of the cmc for the System

2-Butoxyethanol + Water14 at Different Temperatures

temp(°C)

1010Γm(mol/cm2) â

cmc(eq 11)

cmc(exp)

4 4.76 1514 0.0251 0.028625 4.23 3874 0.0158 0.016348 3.43 6626 0.0124 0.0121

π(cmc) ) -Γm∆Gmic (19)

-∆Gads ) RT ln(πx2

)x2f0

(20)

(πx2

)x2f0

) ΓmRT xcmc-2 (21)

2∆Gmic - ∆Gads ) RT ln ΓmRT (22)

γi f 1 as xi f 1 (23)

3280 Langmuir, Vol. 15, No. 9, 1999 Gracia-Fadrique

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phase γ2 and the activity coefficient for solute at the surfacephase Λ as

In the vicinity of the cmc

and eq 24 at the cmc is now

Solving eq 25 for â we get

and the L-B equation (eq 11), in terms of the activitycoefficient, is

but 1 - xcmc = 1, and eq 27 at the cmc is reduced to

It is necessary to emphasize that eq 27 is accurate in thevicinity of the cmc, although for many systems only thecalculation of one constant (â) is enough for a totaldescription.16-18 Selected values from the literature forπ(cmc), xcmc, and Γm allow us to calculate γcmc with eq 28.

Figure 1 shows a linear behavior of the logarithm of theactivity coefficient versus the ethylene oxide content forseveral homologous series. The negative contribution tothe activity coefficient is common for every one, and theaggregation number of the micelle is proportional to theactivity coefficient at the cmc (Figure 2). The sameempirical equation for the logarithm of the cmc versusethylene oxide content has been proposed for the classicalbehavior of nonionic surfactants. Hsiao et al.19 haveproposed that the critical micelle concentration and theethylene oxide chain length for a homologous series shouldbe related by the equation.

where A and B are empirical constants for a givenhomologous series of fixed hydrophobic content and n isthe ethylene oxide chain length. Equation 29 may beexpressed in terms of hydrophilic-hydrophobic contribu-tion groups to the standard free energy of micellization20

If we write eq 30 in terms of the activity of the solute at

(16) Van hunsel, J.; Joos, P. Langmuir 1987, 3, 1069.(17) Lucassen-Reynders, E. H.; Van den Tempel, M. Proceedings of

the 4th International Congress on Surface Active Substances (Brussels,1964); Gordon & Breach, New York, 1964; Vol. II, p 779.

(18) Van Hunsel, J.; Joss, P. J. Colloid Interface Sci. 1989, 129, 286.

(19) Hsiao, L.; Dunnin, H. N.; Lorenz, P. B. J. Phys. Chem. 1956, 60,657.

(20) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.;John Wiley & Sons: New York, 1989; p 151.

(21) Elworthg, P. H.; MacFarlene, C. B. J. Pharm. Pharmacol. 1962,14, 100.

(22) Shick, M. J. J. Colloid Interface Sci. 1965, 20, 464.(23) Crook, E. H. J. Phys. Chem. 1963, 67, 1987.(24) Crook, E. H. J. Phys. Chem. 1964, 68, 3592.

Figure 1. Natural logarithm of the activity coefficient at thecmc (γcmc) as a function of the ethylene oxide content (EO) forpolyoxyethylene hexadecanols21 (b), dodecanols5 (9), nonylphe-nols22 (1), and octylphenols23 (2).

âγ2x2

(1 - x2)(1 - x2 + âγ2x2)) ΛΓ

ΛmΓm(24)

γ2 f γcmc

Γ f Γm

Λ f Λm

âγcmcxcmc

(1 - x2)(1 - x2 + âγcmcxcmc)) 1 (25)

â ) 1γcmc

(1 - xcmc

xcmc)2

(26)

π ) ΓmRT ln[1 + 1γcmc

(1 - xcmc

xcmc)2 x2

1 - x2] (27)

π(cmc) ) -ΓmRT ln(γcmcxcmc) (28)

Figure 2. Natural logarithm of the critical micelle concentra-tion (cmc) (2), the aggregation number (b), and the activitycoefficient at the cmc (γcmc) (9) for nonylphenol ethoxylates, at25 °C.22

Table 2. Group Contribution of the cmc and γcmc to theStandard Free Energy of Micellization5,21,22,23 (eq 31)

homologousseries

∆Gmic(cal/mol)

øHPHOB(cal/mol)

øHPHIL(cal/mol)

C16Ei (25 °C) RT ln xcmc -652 +33RT ln γcmc +540 -166

C12Ei (55 °C) RT ln xcmc -812 +15RT ln γcmc +630 -80

NF (25 °C) RT ln xcmc -550 +27RT ln γcmc +350 -56

p.t. OF (25 °C) RT ln xcmc -550 +53RT ln γcmc +385 -222

ln xcmc ) A +Bn (29)

∆G°mic ) RT ln xcmc ) møHPHOB + nøHPHIL (30)

Langmuir-BET Surface Equation of State Langmuir, Vol. 15, No. 9, 1999 3281

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the cmc acmc instead of only the mole fraction, then

From Figure 2, we can appreciate the opposite values forthe slope (øHPHIL) and the intercept (møHPHOB) for the cmcand γcmc. Table 2 shows this contribution for different groupcontributions of several homologous series. As is expectedfor øcmc, ethylene oxide groups show a positive contributionto the free energy of micellization (unfavorable to themicelle formation) and the CH2 groups show a negativecontribution to the free energy of micellization (favorableto the micelle formation).

On the other hand, for γcmc, the ethylene oxide groupsshow a negative contribution to the free energy of

micellization (favorable to the micelle formation) and theCH2 groups show a positive contribution to the free energyof micellization (favorable to the micelle formation).

This activity coefficient behavior suggests a reductionin the hydrophobic character as a result of the associationprocess, where the coupling of the hydrocarbon tailsdecreases the hydrophobic effect and the neighboring ofthe ethylene tails promotes a higher water structuring.It seems that this mechanism, together with the numberor degree of association of the micelle, plays an importantrole in avoiding the segregation of the surfactant.

LA981244O

∆G°mic ) RT ln acmc ) RT ln γcmcx cmc (31)

3282 Langmuir, Vol. 15, No. 9, 1999 Gracia-Fadrique

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