Langmuir 1999, 15, 3279-3282
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Langmuir-BET Surface Equation of State in Fluid-Fluid Interfaces J. Gracia-Fadrique Departamento de Fı´sica y Qı´mica Teo´ rica, Facultad de Quı´mica, Universidad Nacional Auto´ noma de Me´ xico, Me´ xico D.F. 04510, Me´ xico 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 equation is presented. This equation describes both monolayer and multilayer adsorption. The emphasis of this paper is on monolayers, for which this new equation only contains thermodynamic parameters. In this form, it describes the characteristic behavior of nonionic surfactants, the relation between maximum surface pressure and critical micelle concentration, and the link between the standard free energy of adsorption and the standard free energy of micellization. It also explains and justifies the fundamental work of Rosen regarding the relative effects of structural factors on the surface tension reduction. The deviation of the model from experimental data is explained in terms of the activity coefficient at the critical micelle concentration, which is proportional to the micelle aggregation molecular weight.
Introduction The Gibbs adsorption isotherm allows us to transform an isotherm to a surface equation of state. The Langmuir isotherm in fluid-fluid interfaces, coupled with the Gibbs equation,1 leads to
( )
x2 dπ θ) ΓmRT dx2
βx2 ) 1 + βx2 T
(1)
where θ is the surface coverage (θ ) Γ/Γm), Γ is the Gibbsian surface concentration, Γm is the maximum or saturation surface concentration, Π is the surface pressure, R is the gas constant, T is the absolute temperature, x2 is the bulk concentration in mole fraction for the surface-active solute, and β is equivalent to the Henry constant in two dimensions. The BET equation and the Gibbs adsorption equation2 give
θ)
( )
x2 dπ ΓmRT dx2
T
)
θf1
(5)
x f xcmc
(6)
π f π(cmc)
(7)
Γ f Γm
(8)
where π(cmc) is the maximum or saturation surface pressure and xcmc is the critical micelle mole fraction. Solving the right-hand side of eq 2 for θ ) 1
βxcmc
βx2 (1 - x2)(1 - x2 + βx2)
(2)
In both cases, the integral form leads to the corresponding surface equation of state:
π ) ΓmRT ln(1 + βx2)
(
of surface tension that show an inflection point.4 The advantage of eq 4 over eq 3 is that an analytical solution can be obtained when the system reaches the saturation condition. In this case,
(3)
)
x2 π ) ΓmRT ln 1 + β 1 - x2
(4)
Discussion By a simple analysis of eq 3, the transformation of the Langmuir isotherm into a surface equation of state is observed. This equation was proposed empirically by Szyszkowski.3 The BET equation is a natural extension of the Langmuir model to the multilayer case; thus, we will call eq 4 the Langmuir-BET (L-B) surface equation of state. This work is devoted to the most common case of monolayer behavior. However, it can be used for systems (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.
(1 - xcmc)(1 - xcmc + βxcmc)
)1
(9)
Or in an explicit form for β we get
β)
(
)
1 - xcmc xcmc
2
(10)
Substitution of eq 10 in eq 4 leads in a final form to a novel L-B surface equation of state containing only measurable parameters:
[ (
π ) ΓmRT ln 1 +
)
]
1 - xcmc 2 x2 xcmc 1 - x2
(11)
For practical purposes, and for a typical micelle concentration (10-7 < xcmc < 10-3), eq 11 reduces to
[
π ) ΓmRT ln 1 +
x2 x2cmc
]
(12)
(4) A Ä guila-Herna´ndez, J.; Herna´ndez, I.; Trejo, A. Int. J. Thermophys. 1995, 16, 45.
10.1021/la981244o CCC: $18.00 © 1999 American Chemical Society Published on Web 04/01/1999
3280 Langmuir, Vol. 15, No. 9, 1999
Gracia-Fadrique
At the critical micelle concentration, eq 11 reduces to
π(cmc) ) -ΓmRT ln xcmc
(13)
This last equation explains the typical behavior of etoxylated nonionic surfactants for which the most hydrophobic members show lower critical micelle concentrations and higher surface pressures and, on the other hand, are more hydrophilic. Surfactants show higher critical micelle concentrations and lower surface pressures (Shick).5-7 A possible mechanism to prove the functionality of eq 13 is to compare the experimental surface pressure at the cmc with the predicted surface pressure when the value of the critical micelle concentration and the maximum surface concentration Γm are known. In the extensive work of Rosen8-10 on the effectiveness of the surfactants in the surface tension reduction, the value of the maximum surface pressure π is compared to the condition for the same surfactant when the surface pressure is 20 mN/m. When the surface pressure is the Rosen pressure π ) 20 mN/m, eq 12 becomes
20 ) ΓmRT ln
[
]
x2cmc + xπ)20 x2cmc
[ ] xπ)20
[ ] xcmc xπ)20
(17)
This equation is the same as that deduced by Rosen and accounts for the effectiveness of the reduction in the surface tension.11 It is important to point out that this reduction is found under the condition proposed in eq 16; eq 17 is valid only near the saturation conditions, and eq 18 confirms the condition of the surface pressure value being around 20 mN/m or higher. For nonionic surfactants the standard free energy of micellization is given by12
∆Gmic ) RT ln xcmc
cmc (eq 11)
cmc (exp)
4 25 48
4.76 4.23 3.43
1514 3874 6626
0.0251 0.0158 0.0124
0.0286 0.0163 0.0121
energy of micellization and the maximum surface pressure π(cmc):
π(cmc) ) -Γm∆Gmic
(19)
The standard free energy of adsorption is defined as
-∆Gads ) RT ln
() π x2
(20)
x2f0
The limiting value at infinite dilution predicted by eq 11 is then
() π x2
) ΓmRT x-2 cmc
(21)
x2f0
2∆Gmic - ∆Gads ) RT ln ΓmRT
The difference between eq 13 and eq 16 is
πcmc - (π ) 20) ) ΓmRT ln
β
(15)
(16)
x2cmc
1010Γm (mol/cm2)
Kronberg et al.13 have suggested that the standard free energies of adsorption and micellization are related by a constant. Combining eqs 19, 21, and 22, we obtain a general relation for the standard free energies of adsorption and micellization.
Thus, eqs 14 and 15 can be expressed as
20 ) ΓmRT ln
temp (°C)
(14)
In the vicinity of the critical micelle concentration
x2cmc + xπ)20 ≈ xπ)20
Table 1. Values of the cmc Predicted with Eq 11 and Experimental Values of the cmc for the System 2-Butoxyethanol + Water14 at Different Temperatures
(18)
(22)
Equation 11 has been tested with our own experimental data, previously published14 for the system 2-butoxyethanol + water at several temperatures (4, 25, 48 °C). Table 1 shows the experimental value of the cmc obtained from the abrupt change in the slope of the surface pressure versus ln(x) plot. The results show a good agreement with the values predicted by eq 11. Γm and β are adjusted parameters from the experimental data. However, the last example is a particular case that fulfills eq 11. For characteristic cmc values (10-7 to 10-3 mole fraction) and surface saturation concentrations (24) × 10-10 gmol/cm2, 25 °C), eq 11 leads to nonrealistic surface pressures values. So, lets call systems L-B those ones that obey eq 11. The departure from this behavior is very closely related to the structure of the chemical potential involved. Both eqs 3 and 4 are deduced using the ideal chemical potential. Ross15 has pointed out the need for considering the activity coefficients for the solute and the solvent in the bulk and surface phase for a complete description of the adsorbed solutes. This lack of information for many systems can be avoided by the use of the symmetric convention for the activity coefficient in the cmc region. Under the symmetrically normalized convention
Then, eq 13 provides a direct relation of the standard free
γi f 1 as xi f 1
(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.
and the BET equation may be expressed in terms of the activity coefficient of the surface-active solute in the bulk
(23)
(13) Kronberg, B.; Lindstro¨m, M.; Stenius, P. Competitive Adsorption of an Anionic and a Nonionic Surfactants on Polystyrene Latex in Phenomena in Mixed Surfactant Systems; Scamehorn, V. F., Ed.; ACS Symposium 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.
Langmuir-BET Surface Equation of State
Langmuir, Vol. 15, No. 9, 1999 3281
Figure 1. Natural logarithm of the activity coefficient at the cmc (γcmc) as a function of the ethylene oxide content (EO) for polyoxyethylene hexadecanols21 (b), dodecanols5 (9), nonylphenols22 (1), and octylphenols23 (2).
Figure 2. Natural logarithm of the critical micelle concentration (cmc) (2), the aggregation number (b), and the activity coefficient at the cmc (γcmc) (9) for nonylphenol ethoxylates, at 25 °C.22
phase γ2 and the activity coefficient for solute at the surface phase Λ as
Table 2. Group Contribution of the cmc and γcmc to the Standard Free Energy of Micellization5,21,22,23 (eq 31)
βγ2x2
ΛΓ ) (1 - x2)(1 - x2 + βγ2x2) ΛmΓm
(24)
NF (25 °C)
γ2 f γcmc
p.t. OF (25 °C)
Γ f Γm Λ f Λm and eq 24 at the cmc is now
βγcmcxcmc (1 - x2)(1 - x2 + βγcmcxcmc) Solving eq 25 for β we get
(
)1
(25)
)
1 1 - xcmc γcmc xcmc
2
(26)
and the L-B equation (eq 11), in terms of the activity coefficient, is
[
π ) ΓmRT ln 1 +
C16Ei (25 °C) C12Ei (55 °C)
In the vicinity of the cmc
β)
homologous series
(
)
]
1 1 - xcmc 2 x2 γcmc xcmc 1 - x2
(27)
but 1 - xcmc = 1, and eq 27 at the cmc is reduced to
π(cmc) ) -ΓmRT ln(γcmcxcmc)
(28)
It is necessary to emphasize that eq 27 is accurate in the vicinity of the cmc, although for many systems only the calculation of one constant (β) is enough for a total description.16-18 Selected values from the literature for π(cmc), xcmc, and Γm allow us to calculate γcmc with eq 28. (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.
∆Gmic (cal/mol)
χHPHOB (cal/mol)
χHPHIL (cal/mol)
RT ln xcmc RT ln γcmc RT ln xcmc RT ln γcmc RT ln xcmc RT ln γcmc RT ln xcmc RT ln γcmc
-652 +540 -812 +630 -550 +350 -550 +385
+33 -166 +15 -80 +27 -56 +53 -222
Figure 1 shows a linear behavior of the logarithm of the activity coefficient versus the ethylene oxide content for several homologous series. The negative contribution to the activity coefficient is common for every one, and the aggregation number of the micelle is proportional to the activity coefficient at the cmc (Figure 2). The same empirical equation for the logarithm of the cmc versus ethylene oxide content has been proposed for the classical behavior of nonionic surfactants. Hsiao et al.19 have proposed that the critical micelle concentration and the ethylene oxide chain length for a homologous series should be related by the equation.
ln xcmc ) A +Bn
(29)
where A and B are empirical constants for a given homologous series of fixed hydrophobic content and n is the ethylene oxide chain length. Equation 29 may be expressed in terms of hydrophilic-hydrophobic contribution groups to the standard free energy of micellization20
∆G°mic ) RT ln xcmc ) mχHPHOB + nχHPHIL (30) If we write eq 30 in terms of the activity of the solute at (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.
3282 Langmuir, Vol. 15, No. 9, 1999
Gracia-Fadrique
the cmc acmc instead of only the mole fraction, then
∆G°mic ) RT ln acmc ) RT ln γcmcx cmc
(31)
From Figure 2, we can appreciate the opposite values for the slope (χHPHIL) and the intercept (mχHPHOB) for the cmc and γcmc. Table 2 shows this contribution for different group contributions of several homologous series. As is expected for χcmc, ethylene oxide groups show a positive contribution to the free energy of micellization (unfavorable to the micelle formation) and the CH2 groups show a negative contribution to the free energy of micellization (favorable to the micelle formation). On the other hand, for γcmc, the ethylene oxide groups show a negative contribution to the free energy of
micellization (favorable to the micelle formation) and the CH2 groups show a positive contribution to the free energy of micellization (favorable to the micelle formation). This activity coefficient behavior suggests a reduction in the hydrophobic character as a result of the association process, where the coupling of the hydrocarbon tails decreases the hydrophobic effect and the neighboring of the ethylene tails promotes a higher water structuring. It seems that this mechanism, together with the number or degree of association of the micelle, plays an important role in avoiding the segregation of the surfactant.
LA981244O
