J. Phys. Chem. 1996, 100, 7669-7675
7669
Adsorption Isotherm and Surface Tension Equation for a Surfactant with Changing Partial Molar Area. 1. Ideal Surface Layer V. B. Fainerman,†,§ R. Miller,*,† R. Wu1 stneck,‡ and A. V. Makievski§ Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Rudower Chaussee 5, D-12489 Berlin-Adlershof, Germany, Institute of Solid Physics, UniVersity of Potsdam, Rudower Chaussee 5, D-12489 Berlin-Adlershof, Germany, and Institute of Technical Ecology, 25 blV. SheVchenko, Donetsk 340017, Ukraine ReceiVed: January 16, 1996X
An adsorption model is proposed which assumes a different partial molar surface ω for adsorbed molecules depending on the degree of saturation of the adsorption layer. Equations for an adsorption isotherm and a surface tension isotherm are derived for the adsorption of one single surfactant or of a mixture of surfactants. The model assumes that the molecules may adsorb in two different states, i.e. a large ω at smaller surface pressures Π and a smaller ω at large surface pressures Π. Adsorption data of (N-n-hexadecyl-N,Ndimethylammonio)acetic acid bromide at the air/solution interface are used to demonstrate the application of the theory. The experimental results are in very good agreement with the model.
Introduction The description of experimental data by the Langmuir adsorption isotherm and the von Szyskowski surface tension isotherm often shows significant deviations. These deviation can be reduced when additional interactions between the adsorbed molecules are taken into account.1-11 The equations of Gibbs, Buttler,12 or Pethica13 are basic equations for deriving adsorption isotherms or equations of state of an adsorption layer. For nonideal behavior, different approximations are used in the literature based on the theory of regular solutions. Sufficiently good agreement between theory and experiment has been found when each kind of interaction between the components in the solution was considered.14,15 This, however, restricts considerably the possibilities for predicting the properties of mixtures of two or more surfactant components. However, the interaction parameters calculated from experimental data do not always correlate with the properties of the surfactants. The interaction parameters usually increase with increasing alkyl chain length. When an increment for the interaction energy of a methyl group is calculated, it varies for different surfactant series. Furthermore, the interaction energy does not always agree with the van der Waals energy of a nonideal gas. There is more evidence that the behavior of adsorption layers is not solely caused by molecular interaction. The change of orientation of surfactant molecules at the interface with increasing surface saturation may lead to a deviation from the Langmuir isotherm. Recently, it was shown16,17 that a “superdiffusion” kinetics of octylphenyl poly(ethylene glycol) ethers may be described by a model assuming two different orientations of the poly(ethylene glycol) chain, i.e. flat and normal oriented molecules depending on the interfacial pressure. It was found that rearrangement processes during adsorption lead to increasing surface tension depression when the number of EO units (n) increases, starting with n ) 4 up to 40.17 The results also show that the hydrophobic part of the ethers prefers a flat orientation at small surface pressures. Very fast kinetics of interfacial tension depression for Triton X-100 at the interface * To whom all correspondence should be addressed. † Max-Planck-Institut fu ¨ r Kolloid- und Grenzfla¨chenforschung. ‡ University of Potsdam. § Institute of Technical Ecology. X Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(96)00148-7 CCC: $12.00
with heptane at short adsorption times was explained by an initial flat orientation of the surfactant molecules.18 There are other experimental results which lead to the same assumption of reorientation of surfactant molecules with increasing interfacial saturation, for instance from interfacial potential19,20 and X-ray reflection studies.21,22 At present there have been several theoretical attempts to describe the adsorption equilibrium of surfactant monolayers assuming different area per molecule depending on the coverage of the surface by surfactant molecules.23-26 In the present paper a new approach to describe the adsorption of nonionic surfactants considering reorientation of adsorbed molecules at higher surface pressure is given. It is a model which was already suggested by Langmuir. Recently published results27 will be used to demonstrate the capability of the new model. The derived equations will be compared to the Langmuir and Frumkin isotherms to demonstrate that interfacial interactions mimic molecular reorientations. Theory Using the Buttler equation, the chemical potential of the component i in the interfacial layer µsi is given by2-4,12
µis ) µi0s + RT ln f isxis - γωi
(1)
and in the bulk phase
µiR ) µi0R + RT ln f iRxiR
(2)
Here x is the molar fraction, f is the activity coefficient, ω is the partial molar area, γ is the surface tension, R is the gas constant, T is the temperature, and “s” and “R” indicate surface and bulk, respectively. Assuming constant chemical potential in the bulk and at the surface eqs 1 and 2 yield
µi0s + RT ln f isxis - γωi ) µi0R + RT ln f iRxiR
(3)
Now the standard state has to be formulated; for the solvent (i ) 0) usually a pure component is assumed. This means © 1996 American Chemical Society
7670 J. Phys. Chem., Vol. 100, No. 18, 1996
Fainerman et al.
xs0 ) 1, f s0 ) 1, xR0 ) 1, f R0 ) 1, and γ ) γ0. From eq 3 the following is obtained: 0R µ0s 0 - γ0ω0 ) µ0
(4)
Adding xs0 from (7) and xs1 and xs2 from (8), the following relationship is obtained:
(
The standard state for the i surface active components is an infinitely diluted solution. Therefore, xRi f 0, f Ri ) 1, f si ) 1, γ ) γ0, and eq 3 leads to
µi0s + RT ln xis|xiRf0 - γ0ωi ) µi0R + RT ln xiR|xiRf0
(5)
and
µi0R - µi0s ) -γ0ωi + RT ln Ki
(6)
Ki ) (xsi /xRi )xRi f0 is the distribution coefficient for infinite dilution. From (3) and (4) results
(γ0 - γ)ω0 f s0xs0 ln R R ) RT f 0 x0
(7)
and from (3) and (6) we get
(γ0 - γ)ωi f isxis )ln R R RT Ki f i xi
RT RT ln xs0 ) - ln(1 - xs1 - xs2) ω0 ω0
γ0 - γ ) -
(9)
The equation for the adsorption isotherm is obtained from (7) and (8) by excluding γ0 - γ:
xs0 xs1 xs2 1 1 1 ) ln ) ln ln ω0 xR ω1 K xR ω2 K xR 0
1 1
(10)
2 2
Therefore, we have
KjxjR
)
xjs (1 - xs1 - xs2)ωj/ω0
(11)
(
)
Π(ω2 - ω1) xs1K2 ) exp sK RT x 1 2
where Π ) γ0 - γ is the surface pressure.
(12)
)
(
)
RT π ) - ln[1 - (θ1 + θ2)] ω0 b1c )
b2c )
(14)
θ1
(15)
[1 - (θ1 + θ2)]ω1/ω0 θ2
(16)
[1 - (θ1 + θ2)]ω2/ω0
(
)
Π(ω2 - ω1) Γ1b2 ) exp Γ2b1 RT 1)
(
) [ (
)
(17)
(
)]
Πω0 Πω1 Πω2 + c b1 exp + b2 exp RT RT RT
exp -
(18)
b1 and b2 are constants of the adsorption equilibrium of states 1 and 2. For practical use of (14)-(18) it is necessary to determine the following: proper choice of ω0; the average partial molar surface area ωΣ; the surface molar fractions through the definitions (a) or (b); relationship between b1 and b2. Analysis of (14)-(18) shows that the system is invariant with respect to different definitions of θj (Γjωj or ΓjωΣ) and therefore also to ω0. For instance ω0 ) ωΣ can be used. This may be easily shown by introducing θ1 + θ2 from (14) into (15) and (16) and deriving the values of b1c and b2c from (18). In all cases (18) transforms into the same final result. However, the use of a realistic surface area for ω0 (approximately 10 Å2 for a H2O molecule) contradicts experimental data. For ω1 ) ω2 ) ω, (18) transforms into
Π)
with j ) 1, 2. When the surfactant can adsorb with a different molar surface area (ω1 for small γ0 - γ and ω2 for bigger γ0 - γ, with ω1 > ω2), eq 8 leads to
(
This equation is the generalized form of the surface tension equation derived by Joos28 for a surfactant mixture with different values of ω for each component. For further derivations it is necessary to substitute the molar ratio of surfactants at the surface, xsj , in (9)-(12) by their real Gibbs adsorption values, which are related to their respective surface tension values. For this we introduce the degree of saturation of the surface layer, i.e. (a) θj ) Γjωj or (b) θj ) ΓjωΣ. Γj is the adsorption value of the surfactant in the state j, and ωΣ is the average of the partial molar surface areas of all possible states (here just two states). Replacing xsj in (9)-(12) by θj and introduction of the surfactant bulk concentration c in (11) and (13) instead of xRi , we get the following system of equations:
(8)
By taking f Ri ) 1 and defining activity coefficients of the components at the interface from the interaction energy of all components, nearly all known equations of state of the interface and adsorption isotherms can be derived from eqs 7 and 8. In this paper only ideal interfaces are considered, i.e., f si ) f s0 ) 1. In agreement with the aim of the paper we extend the meaning of the index i used for each component. Now we discuss only the case of one surfactant, and the index corresponds to the two different states of adsorbed molecules occupying a surface area of ωi, i.e. ω1 and ω2. The equations for the general case of a system of i different surfactants, which can all occur with j different states at the interface, are given in Appendix A. From (7) the generalized equation of state for an ideal surface layer results,
)
Πω0 Πω1 + K1xR1 exp + RT RT Πω2 (13) K2xR2 exp RT
xs0 + xs1 + xs2 ) 1 ) exp -
RT ln(1 + b*c) ω0
(19)
with b* ) b1 + b2. This equation differs from the von Szyskowski equation when ω0 is used instead of the real surface area of a surfactant molecule, ω > ω0. This contradiction was excellently solved by Lucassen-Reynders.2,3 She positioned the dividing surface in the plane ω0 ) ω such that the total adsorption of all components, including H2O, is equal the saturation adsorption, i.e. ∑n0Γi ) Γ∞ ) 1/ω.
Adsorption Isotherm and Surface Tension Equation
J. Phys. Chem., Vol. 100, No. 18, 1996 7671
Obviously for strongly surface active components this position does not change the adsorption values from those obtained for the Gibbs dividing surface (Γ0 ) 0). For ω . ω0 the dividing surface defined by Lucassen-Reynders is located only by some parts of the diameter of a water molecule below the Gibbs dividing surface. Lucassen-Reynders3 and Joos and co-workers29,30 also proposed equations for estimating an average value of the molecular area for all surfactant components in a mixture taking into account different ωi. The contribution of each component to ωΣ is determined by their adsorptions. Here, we assume ωΣ ) ω0 and obtain
ω0 ) ωΣ )
ω1Γ1 + ω2Γ2 Γ1 + Γ2
(20)
Using (17), the following results from (20):
ω0 ) ωΣ )
[
]
Π(ω2 - ω1) RT (ω2 - ω1) 1 + (b1/b2)exp Π RT
ω2 + (b1/b2)ω1 exp
[
]
(21)
For the next step it is necessary to assume a dependence of the constants bj on ωj, for example
b1 ) b2(ω1/ω2)R
(22)
where the constant R g 0. The case R ) 0 means independence of adsorption activity from molecular surface area, while for R > 0 the orientation with the larger molecular surface area is more strongly surface active. There is an analogue to the wellknown experimentally confirmed fact of the dependence of the constant b on the number of carbon atoms in the alkyl chain of a homologous series of a surfactant,
bn ) b0 exp(βn)
(23)
where b0 and β are constants. It is worth noticing that (23) contains the balance between hydrophobic and hydrophilic groups in the molecule, whereas (22) points only to a dependence on orientation. Substituting (20) and (22) into (14)(18) and (21), the following system of equations result which describe a solution of one surfactant with two different adsorption states at the interface with ω1 > ω2. The average partial molar surface area is given by
() [ () [
ω2 + ω1 ωΣ )
ω1 ω2
ω1 1+ ω2
R
R
exp
]
Π(ω2 - ω1) RT
]
Π(ω2 - ω1) exp RT
RT ln[1 - (Γ1 + Γ2)ωΣ] ωΣ
Γ2ωΣ [1 - (Γ1 + Γ2)ωΣ]ω2/ωΣ ω2 ω1
R
Γ1ωΣ [1 - (Γ1 + Γ2)ωΣ]ω1/ωΣ
R
Πω1 Πω2 + exp RT RT
exp -
)
( )
ΠωΣ (27) RT
1 - exp -
The principle of Braun-Le Chatelier31 is expressed by the relation
() [
ω1 Γ1 ) Γ2 ω2
R
exp
]
Π(ω2 - ω1) RT
Discussion The main objective now is to demonstrate the effect of the new assumption of two different surface areas on the surface pressure and its dependence on concentration. To compare the results obtained by the newly derived relationships with classical models, for example the Langmuir or Frumkin isotherms, some model calculations are presented here. It will be shown that the interaction of adsorbed molecules, which is additionally assumed in the Frumkin isotherm for better agreement with experimental data, can be mimicked by a change in the surface orientation of adsorbed molecules or vice versa. Experimental results are usually given as a function of γ ) γ(c) as represented for example by the surface tension equation (27). An equivalent equation of easier structure would be (B7). This equation has very severe limitations due to the simplifications made during its derivation; for example for θ f 1 and θ , 1 it transforms into the Langmuir-Szyskowski or Henry equation, respectively. In the range of intermediate Π value, ω1 and ω2K(Π) are comparable and (B7) produces higher values of Π compared to the Langmuir equation. This effect suggests the use of a Frumkin isotherm with a positive interaction constant aF to describe the data:
bc )
(25)
(26)
(28)
The corresponding equations for a mixture of i surfactants with j states at the interface are summarized in Appendix A. The derivation of (25)-(28) was performed on the basis of the Buttler equation (1) and the model equations (20) and (22). An additional possibility for deriving an adsorption isotherm and a surface tension equation depends on solving (14) and using the Gibbs equation. This has been attempted previously.6,7 This derivation, however, needs several simplifications until finally some relationships are obtained. Examples of these attempts are given in Appendix B.
(24)
)
()
ω1 ω2
RT [ln(1 - θ) + aFθ2] ω
and the adsorption isotherm reads
b 2c )
[( ) ( ) ( )]
b2c
Π)-
The equation of state of the interfacial layer is consequently
Π)-
The interfacial tension equation finally results in the form
θ exp(-2aFθ) 1-θ
(29) (30)
These equations are often used for data interpretation of γ ) γ(c) assuming nonideal surface layer behavior. Figure 1 shows the influence of aF on the shape of γ ) γ(c). The case aF ) 0 coincides with the von Szyskowski equation. The curves in Figure 1 are calculated for b ) 4101/mol using (B9), whereas the results of (30) were normalized to Π ) 30 mN/m. A theoretical shortcoming of (B7) is the simultaneous use of (14) and (B1) in the derivation. For these two relationships the dividing surfaces are defined in different ways. Definitely from this theoretical point of view, the generalized equation (27) seems to be more convincing, although it is more complicated and has a nontraditional form. Therefore, we will use this equation in the discussion and want to suggest its general
7672 J. Phys. Chem., Vol. 100, No. 18, 1996
Figure 1. Π(c)-dependencies calculated for a Frumkin isotherm using ω ) 50 Å2/molecule and different values of aF: aF ) 1 (]), aF ) 0 (0), aF ) -1 (4).
Figure 2. Π(c)-dependencies calculated from (27) for ω2 ) 50 Å2/ molecule, R ) 0, and different values for ω1; ω1 ) 70 (9), ω1 ) 100 (4), ω1 ) 200 (]), ω1 ) 300 (O) Å2/molecule; bold line, Langmuir isotherm.
use in the description of experimental data for systems where changes in the orientation of adsorbed molecules can be expected. Equation 27 contains four parameters describing the adsorption of one surfactant in two different states while (29) and (30) contain only three parameters. However, the only empirical parameter which has to be approximated is R, and we will be able to show that its influence on the dependence γ ) γ(c) is not very strong. For high values of Π the first term on the left side of (27) becomes negligible and the equation transforms into the von Szyskowski equation with b ) b2 and ωΣ ) ω2. Therefore the two parameters, ω2 and b2, which correspond to the minimum surface area of a surfactant molecule, have a clear physical meaning and may be easily determined from experiments. The parameter ω1 can be estimated easily with a sufficient accuracy using the known molecular geometry of a surfactant. For molecules with a complicated asymmetrical structure the error in estimating ω1 can be as high as 10-20%. As a first approximation for the coefficient R we can take R ) 0. Results of dynamic surface tension measurements of Tritons with different ethoxylane chain length16,17 show that the best agreement with the diffusion kinetics model is achieved when Π* (cf. (B8) for definition) is 5-10 mN/m. In this case we obtain R = 1 using (28). This value is understandable and reflects the nature of the adsorption process. In the second part of this paper we will show that the effect of the coefficient R can be described in a different way, namely, by the assumption of nonideal entropy of the adsorption layer. The consideration of
Fainerman et al.
Figure 3. Π(c)-dependencies calculated from (27) from ω1 ) 200 Å2/molecule, ω2 ) 50 Å2/molecule, and different R; R ) 0 (O), R ) 1 (0), R ) 2 (4); bold line, Langmuir isotherm.
Figure 4. Π(c)-dependencies calculated from (27) for ω1 ) 70 Å2/ molecule, ω2 ) 50 Å2/molecule, and different R: R ) 0 (O), R ) 1 (0), R ) 2 (4); bold line, Langmuir isotherm
a nonideal entropy at Π f 0 leads to Γ1/Γ2 ) exp(ω1 - ω2)/ ωΣ, which coincides with (28) at R ) 1. The results of model calculations of (27) are shown in Figures 2-4. They show the influence of the different parameters of (27) on the isotherm compared with the results of the von Szyskowski equation. Analogous to Figure 1, all curves were normalized such that they cross to Π ) 30 mN/m. The strongest effect on the shape of the curves is obtained by varying the ratio ω1/ω2 (Figure 2). For ω1 > 2ω2 the curves approach the Frumkin isotherm for RF > 0. An increase of R at ω1 > 2ω2 strongly influences the course of γ ) γ(c) (Figure 3). The curves then degenerate into two straight lines with a strong kink point. The flat part of the curve is lower than the curve of the ideal approach, while for intermediate Π it is remarkably higher. This behavior clearly differs from that of the Frumkin isotherm (Figure 1). The shape of the curves in Figures 2 and 3 for ω1 > 2ω2 resembles experimental results shown in some papers of Lunkenheimer.24,32,33 In contrast, for ω1 ≈ ω2 an increase of R decreases Π (Figure 4), which would simulate the effect of a negative value of the interaction parameter aF in (29) and (30). For large Π the exponential part on the right-hand side of (27) with the term ωΣ becomes zero. Therefore to estimate ωΣ, the same approximation can be used as was done for small Π where the exponents in (24) may be neglected:
ωΣ )
ω2 + ω1(ω1/ω2)R 1 + (ω1/ω2)R
(31)
Adsorption Isotherm and Surface Tension Equation
J. Phys. Chem., Vol. 100, No. 18, 1996 7673
TABLE 1: Surface Area of (N-n-Hexadecyl-N,N-dimethylammonio)acetic Acid Bromide at pH ) 7 Calculated from the van der Waals Diameter and the Resulting Adsorption Parameters and Standard Deviation Using ω1 ) ωmax van der Waals diameter a (Å)
b (Å)
c (Å)
ωmax (Å2)
ωmin (Å2/m)
ω2 (Å2/m)
ωΣ|Π)0 (Å2/m)
b2 (1/M)
s (mN/m)
all trans
7.5
6.6
28.2
49.3
one gauche at the head
7.0
8.0
27.4
two gauche, one at the head, one at arbitrary position average area
7.0
10.6
24.6
211.4 185.8 218.2 191.7 262.1 171.2 206.7
44.4 43.4 44.6 43.6 45.8 42.7 44.2
127.9 114.6 131.4 117.7 153.9 106.9 125.5
2.47 × 106 2.31 × 106 2.51 × 106 2.35 × 106 2.72 × 106 2.20 × 106 2.45 × 106
0.3258 0.3832 0.3234 0.3482 0.3457 0.3927 0.285
configuration
Figure 5. γ/log c-plot of (N-n-hexadecyl-N,N-dimethylammonio)acetic acid bromide at pH ) 7: (]) experimental points; (solid line) approximated isotherm.
Results Experimental results for a surfactant published recently will be interpreted now using the surface tension equation (27) with R ) 1. The average surface area ωΣ is calculated from (31). The experimental surface tension data used belong to N-nhexadecyl-N,N-dimethylammonio acetic acid bromide (at pH ) 7 predominantly betains) having the structure27 CH3 N
74.0 59.7
Figure 6. Standard deviation between the experimental points (Figure 5) and the fitted isotherm as a function of the parameter ω1.
equal to 0.409 mN/m, which is on the order of experimental accuracy. Figure 6 shows the minimum standard deviation between the experimental results and the calculated curves for different ω1. The best fit is achieved for values of ω1 in the range 206-207 Å2/molecule. The resulting value for ω2 is 44 Å2/molecule, which is in good agreement with the expected values (a*b) for the all-trans configuration of the molecule. Conclusions
+
R
55.8
CH2
COO–
CH3
The values ω1 (partial molar surface area for “flat lying” molecules) were calculated using the van der Waals diameter of the atoms and assuming that the hydrophobic part of the molecule is positioned at the air/solution interface. The molecules were placed into boxes having the edge lengths a, b, and c. Two different areas can be occupied by “flat lying” molecules when the possible configurations and the resulting positions of the box, a*c or b*c, in the plane of the interface are taken into account. The numerical calculations were performed with a molecular dynamics model. This model takes into consideration the effect of thermooscillations of the atoms on the van der Waals diameter. The time used for the calculations was 104 times longer than the characteristic time of thermooscillations of the atoms. For the present calculations we used only three possible conformations of the molecule. Table 1 shows the van der Waals diameters for these configurations and the resulting minimum and maximum areas. As can be seen, these area values are very different. For all these data the resulting adsorption parameters were calculated keeping the values of ω1 fixed and R ) 1. The average value of ω1 equals 206.7 Å2/molecule. Figure 5 shows the experimental results (points) and the best fit (line) achieved by (27). The standard deviation from this best fit using the Langmuir approach was found to be
Experimental surface tension data for solutions of a nonionic surfactant may be sufficiently well described by using a simple adsorption model which takes into account different orientations of the molecules at the interface. The isotherm was derived on the basis of the Buttler and Gibbs equations and includes four free parameters. In some cases the parameters governing the adsorption isotherm leads to an outstanding behavior of the predicted γ ) γ(c)-dependencies which differ remarkably from those of the Langmuir and Frumkin approaches. The given examples of data interpretation for some experimental results show very good agreement with the given approach of a surface tension equation and adsorption isotherm. Further directions of this approach are the description of the adsorption behavior of polyelectrolytes, especially of proteins.34 The equations given in Appendix A allow a more precise description of the systems. Nevertheless, in some cases other model equations are necessary or helpful. Therefore, more general relationships have to be worked out in a later paper. A generalization of the model for ionic surfactants and nonideal interfaces seems to be also possible. Acknowledgment. The authors want to express their thankfulness to Prof. P. Joos from the University of Antwerp for many helpful discussions. We also thank Dr. Knochenhauer for his kind support in calculating the geometric data of the betain molecules. The work was financially supported by the INTAS Project 93-2463 of the European Union.
7674 J. Phys. Chem., Vol. 100, No. 18, 1996
Fainerman et al.
Appendix A For solutions containing i different surfactants which exist in j different states at the interface the following system of equations may be formulated on the basis of the Buttler equation (1) and eqs 20 and 22:
ωΣ )
∑i Γijωij ∑ωij(ωij/ωik) i,j ,j
Ri
)
Γij ∑ i,j
[
[
(ωij/ωik)R exp ∑ i,j i
RT
]
]
Π(ωik - ωij) RT
RT ln[1 - ωΣ∑Γij] ωΣ i,j
∑i
[ ( ) ( )] j
ωik
Πωij RT
(
)
with θ ) θ1 + θ2, Γ ) Γ1 + Γ2, θ1 ) Γ1ω1, and θ2 ) Γ2ω2. Equation B3 has an analytical solution in the range of small θ, when
ln(1 - θ) θ = - ) -1 θ θ
d ln c )
( ) ΠωΣ
) 1 - exp -
(B4)
RT
]
ωij Ri Π(ωik - ωij) Γij ) exp Γik ωik RT
dθ + d ln Γ - d ln θ θ(1 - θ)
(A5)
Appendix B The Gibbs equation for the present system is
-dγ ) RTΓ1 d ln c + RTΓ2 d ln c ) RT(Γ1 + Γ2) d ln c (B1) Index 1 and 2 indicate the adsorption states of the surfactant at the interface.
(B5)
By integration of (B5), the adsorption isotherm results:
Bc )
(A4)
The sum in (A4) is taken over all states j of the component i and, after this, over all components i of the system. The condition of standardization bi in (A4) for j ) 1 is obviously ∑ibi ) b. As the number of terms on the left-hand side of (A4) depends on j, an additional standardization bij is taken. When one of the states k has a ωk which is remarkably smaller than all the others, the standardization in (A4) is the same as for j ) 1. Equation 28, which presents the principle of Braun-Le Chatelier, holds without any changes. Taking into account the system of indices, the following equation results:
( ) [
d ln Γ d ln θ dθ ln(1 - θ) (B3) + θ θ θ(1 - θ)
(A3)
[1 - ωΣ∑Γij]ωij/ωΣ
exp -
dΓ2 ) Γ1
It is interesting, however, the range of all possible values of θ, i.e., 0-1. Taking the approximation given by (B4) yields
ΓijωΣ
Ri
∂γ ∂Γ2
× (Γ1ω1 + Γ2ω2)2 ln(1 - Γ1ω1 - Γ2ω2) (B2)
d ln c )
(A2)
and the surface tension isotherm (generalized Joos equation)
ωij
Γ2
( )
(dΓ1 + dΓ2)(Γ1ω1 + Γ2ω2) - (Γ1 + Γ2)(dΓ1ω1 + dΓ2ω2)
i,j
bikci∑
dΓ1 -
Substituting dγ in (B1) by (B2) yields the following differential equation:
the equation of the adsorption isotherm
bijci )
∂γ ∂Γ1
(A1)
where Ri are constants for each surfactant (Ri g 0); index k expresses the fixed state, for instance a ω value of the component i within a minimum surface demand. The sum in (A1) is taken from all components and from all states. The equation of state of the interface becomes
Π)-
( )
-dγ ) RT
Π(ωik - ωij)
exp
Differentiating (14) for one surfactant, using different Γ1 and Γ2, and taking into account (20) results in
Γ 1-θ
(B6)
Inserting this relation into (14) and accounting for (17) and (18) lead to the surface tension equation for one surfactant adsorbing in two states:
[ (
)]
ω1 + ω2K(Π) 1 + K(Π) ln 1 + Bc Π ) RT ω1 + ω2K(Π) 1 + K(Π)
(B7)
with
K(Π) )
[
]
Π(ω1 - ω2) b1 exp b2 RT
The relation for K(Π) may be rewritten as
K(Π) ) exp[(Π - Π*)(ω1 - ω2)/RT]
(B8)
Here Π* is the surface pressure for the case Γ1 ) Γ2. For determining the constant B in (B6) and (B7) we consider large Π where K(Π) . 1. In this case and also at ω1 ) ω2, (B7) transforms into the von Szyskowski equation
Π)
RT ln(1 + bc) ω2
where b ) Bω2, i.e. B ) b/ω2.
(B9)
Adsorption Isotherm and Surface Tension Equation
J. Phys. Chem., Vol. 100, No. 18, 1996 7675
Considering (17) and (B7) the adsorption isotherm (23) changes into
Bc )
(
)
1 + K(Π) θ 1 - θ ω1 + ω2K(Π)
(B10)
When ω1 ) ω2 ) ω or K(Π) . 1 from (B10), the Langmuir adsorption isotherm results:
bc )
θ 1-θ
(B11)
The equations of isotherms of surface tension (B7) may be approximately dissolved for the case of surfactant mixtures when considering additivity and Γi ≈ bici. For a solution of two surfactants (1 and 2) and permitting the two species to adsorb with two different values of partial molar area, the following equation arises.
Π ) RT
[
1 + K1 + (B2c2/B1c1)(1 + K2)
]
ω11 + ω12K1 + (B2c2/B1c1)(ω21 + ω22)K2 ω11 + ω12K1 ω21 + ω22K2 ln 1 + B1c1 + B2c2 1 + K1 1 + K2
[
(
)
(
×
)]
(B12)
with the first index indicating different surfactants, but the second, different states of interface:
K1 ) exp[(Π - Π* 1)(ω11 - ω12)/RT]
(B13)
K2 ) exp[(Π - Π* 2)(ω21 - ω22)/RT]
(B14)
References and Notes (1) Frumkin, A. N. Z. Phys. Chem. 1925, 116, 466. (2) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777. (3) Lucassen-Reynders, E. H. J. Colloid Interface Sci. 1972, 91, 156; 1982, 85, 178.
(4) Lucassen-Reynders, E. H. Colloids Surf. A 1994, 91, 79. (5) Tedoradze, J. A.; Arakeljan, R. A.; Belokolos, E. D. Electrokhimija 1966, 2, 563. (6) Damaskin, B. B. Electrokhimija 1969, 5, 249. (7) Damaskin, B. B.; Frumkin, A. N.; Borovaja, N. A. Elektrokhimija 1972, 8, 807. (8) Rosen, M. J.; Xua, X. Y. J. Colloid Interface Sci. 1982, 86, 164. (9) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 87, 469; 1982, 90, 212. (10) Fainerman, V. B.; Lylyk, S. V. Koll. Zh. USSR 1983, 45, 500. (11) Fainerman, V. B. Zh. Fiz. Khim. 1986, 60, 681. (12) Buttler, J. A. V. Proc. R. Soc. Ser. A 1932, 138, 348. (13) Pethica, B. A. Trans. Faraday Soc. 1952, 48, 1052. (14) Wu¨stneck, R.; Miller, R.; Kriwanek, J.; Holzbauer, H.-R. Langmuir 1994, 10, 3738. (15) Wu¨stneck, R.; Miller, R.; Kriwanek, J. Colloids Surf. 1993, 81, 1. (16) Fainerman, V. B.; Makievski, A. V.; Joos, P. Colloids Surf. A 1994, 90, 213. (17) Fainerman, V. B.; Miller, R.; Makievski, A. V. Langmuir 1995, 11, 3054. (18) Liggieri, L.; Ravera, F.; Passerone, A. J. Colloid Interface Sci. 1995, 169, 226. (19) Geeraerts, J.; Joos, P.; Ville, F. Colloids Surf. A 1993, 75, 243. (20) Kretzschmar, G.; Vollhardt, D. Monatsber. Dtsch. Akad. Wiss. Berlin 1968, 10, 206. (21) Bo¨hm, Ch.; Leveiller, F.; Jacquemain, D.; Mo¨hwald, M.; Kjaer, K.; Als-Nielsen, J.; Weissbuch, W.; Lieserowitz, L. Langmuir 1994, 10, 880. (22) Brezesinski, G.; Bo¨hm, C.; Dietrich, A.; Mo¨hwald, H. Physika B 1994, 198, 146. (23) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1991, 7, 1055. (24) Lunkenheimer, K.; Hirte, R. J. Phys. Chem. 1992, 96, 8683. (25) Lin, S. Y.; McKeigue, K.; Maldarelli, C. Langmuir 1994, 10, 3442. (26) Lin, S. Y.; Tsay, R. Y.; Hwang, W. B. Colloids Surf. A, in press. (27) Fiedler, H.; Wu¨stneck, R.; Weiland, B.; Miller, R.; Haage, K. Langmuir 1994, 10, 3959. (28) Joos, P. Bull. Soc. Chim. Belg. 1967, 76, 591. (29) van den Bogaert, P.; Joos, P. J. Phys. Chem. 1980, 84, 190. (30) Fang, J. P.; Joos, P. Colloids Surf. A 1994, 83, 63. (31) Joos, P.; Serrien, G. J. Colloid Interface Sci. 1991, 145, 291. (32) Lunkenheimer, K.; Holzbauer, H.-R.; Hirte, R. Progr. Colloid Polym. Sci. 1994, 97, 116. (33) Lunkenheimer, K.; Czichocki, G.; Hirte, R.; Barzyk, W. Colloids Surf. A 1995, 101, 187. (34) Fainerman, V. B.; Miller, R.; Wu¨stneck, R. Submitted for publication to J. Colloid Interface Sci.
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