11432
J. Phys. Chem. B 2001, 105, 11432-11438
Simple Method to Estimate Surface Tension of Mixed Surfactant Solutions V. B. Fainerman† and R. Miller*,‡ International Medical Physicochemical Centre, Donetsk Medical UniVersity, 16 Ilych AVenue, Donetsk 83003, Ukraine, and Max-Planck-Institut fu¨ r Kolloid- und Grenzfla¨ chenforschung, Forschungscampus Golm, 14476 Golm, Germany ReceiVed: NoVember 13, 2000; In Final Form: August 8, 2001
A thermodynamic model is derived for the description of the adsorption behavior of mixed surfactant systems. The model requires the surface tensions of the single-surfactant systems or the adsorption isotherms to construct the isotherm of the mixture. The approach does not assume extra interaction parameters between the different compounds. The model is tested with a number of literature data, such as mixed sodium alkyl sulfates, nonionic surfactant mixtures, and an anionic-nonionic mixture. The agreement between experimental data and the theoretical calculations is excellent.
Introduction The equilibrium and dynamic adsorption behavior of surfactants in mixed adsorption layers at liquid-gas and liquid-liquid interfaces was studied in many papers.1-36 In most of them, the authors proposed procedures or models to predict the adsorption behavior (in particular, the interfacial or surface tension) for a surfactant mixture from the known surface characteristics of the single compounds. Therefore, in all equations of state and adsorption isotherms proposed for mixed surfactant systems, the specific parameters for the individual solutions are involved. This approach naturally requires detailed experimental studies of surface tension for individual solutions. In addition, to match the experimental data with theoretical predictions, the theoretical models typically define additional parameters that account for the mutual influence of the surface layer components,1-12,19,20,25,26,34-36 which imposes certain restrictions on the capability of such approaches. In ref 13, a surface equation of state for surfactant mixtures is derived on a molecular basis, which does not require experimentally determined parameters. This theoretical equation allows prediction of the surface tension and surface concentration of the mixed surfactant solution-air interface and the composition as a function of the total surfactant bulk concentration and composition, both below and above the critical micelle concentration of the surfactant mixture. The areas per molecule were calculated from bond lengths and angles for surfactants having compact hydrophilic headgroups and via Monte Carlo simulations for surfactants having flexible, polymer-like hydrophilic heads. However, the theory does not allow calculation of reliable theoretical estimates of the difference between the chemical potentials of the component in the bulk and that at the surface, which determines the adsorption activity of the surfactant. Therefore, to implement the model proposed in ref 13, one has to know experimental surface tension values of individual solutions at one concentration. In the present study, an attempt is made to derive a general but simple approximate expression for the surface tension of a * To whom correspondence should be addressed. † Donetsk Medical University. ‡ Forschungscampus Golm.
surfactant mixture that allows estimation of the characteristics of a mixed solution, without any detailed analysis of the behavior of the individual solutions and/or any account for specific interactions between the mixed species. More precisely, it will be shown that it is possible to predict the surface tension of a mixed solution of two (or even n) surfactants of different natures from the surface tensions for the individual solutions. The present model is based on current theoretical models that describe the adsorption at liquid-gas interfaces, taking into account not only the nonideality of the surface layer constituents, surfactant ionization, and differences in the molar areas of the components, but also processes happening at the interface, such as changes in the state of adsorbed molecules due to aggregation or cluster formation.21,22,37-42 The predictions of the theoretical model are compared with experimental results reported in the literature. Theory 1. Ideal Mixture of Homologues. The equations of state and adsorption isotherms for ideal (both in the bulk and at the surface) mixtures of two homologues (surfactants 1 and 2 with similar partial molar surface area, ω) can be presented in the form1-5,21
Π)-
RT ln(1 - θ1 - θ2) ω
bici )
θi (1 - θ1 - θ2)
(1) (2)
where b is a constant, c is the surfactant concentration in the solution bulk, R is the gas law constant, T is the temperature, θ ) Γω is the monolayer coverage, Γ is the adsorption, Π ) γ0 - γ is the surface pressure, and γ0 and γ are the surface tensions of solvent and solution, respectively. Equation 1 combined with eq 2 can be rewritten in form of the generalized Szyszkowski equation
Π)
(
)
θ1 + θ2 RT RT ln 1 + ) ln(b1c1 + b2c2 + 1) (3) ω 1 - θ 1 - θ2 ω
10.1021/jp004179b CCC: $20.00 © 2001 American Chemical Society Published on Web 10/31/2001
Method to Estimate Surface Tension
J. Phys. Chem. B, Vol. 105, No. 46, 2001 11433
Using the corresponding equations for the individual solutions of surfactants 1 and 2 (in the following the subscript 0 refers to the value for the individual solution),
RT RT Πi ) - ln(1 - θ0i) ) ln(1 + bici) ω ω bici )
Πi ) (4) bici )
θ0i (1 - θ0i)
(5)
and replacing the terms bici by the corresponding products from eqs 4, one can express the equation of state (eq 3) as
h2-1 exp Π h ) exp Π h 1 + exp Π
(6)
where Π h ) Πω/(RT), Π h 1 ) Π1ω/(RT), and Π h 2 ) Π2ω/(RT) are the dimensionless surface pressures of the mixture and individual solutions of components 1 and 2, respectively, at the same surfactant concentrations as in the mixture. Equation 6 is essentially more suitable than eqs 1 and 2 because it enables one to calculate the surface pressure of a mixture from the experimental surface tensions of the individual solutions, requiring no knowledge of the parameters of the surface tension isotherms for the individual solutions. It is known that to calculate the isotherm parameters one should perform extensive experimental measurements in a broad surfactant concentration (surface pressure) range. In contrast, the use of eq 6 for a given mixture composition would require only the knowledge of one pair of experimental values of surface pressure (tension) of the individual solutions at the same concentrations as in the mixture. The only parameter of the isotherm entering eq 6, the molar area, ω, can be either estimated from the surfactant molecule geometry or determined experimentally from the limiting slope (near the CMC or solubility limit) of the γ vs ln c curve (the surface tension isotherm) for the individual surfactants using the Gibbs equation
ω ) -RT/
(d dγln c)
cfCMC
(7)
Moreover, it will be shown later that the calculations according to eq 6 are quite insensitive to the choice of ω. Therefore, as first approximation, one can assume a value typical for the surfactant type considered, for example, (1-2) × 105 m2/mol for fatty alcohols, acids, ethers, and most anionic and cationic surfactants. In the subsequent sections, more complicated mixtures of surfactants, characterized by intermolecular interactions, different molar areas of the components, mixtures of ionic surfactants, and other systems are considered. 2. Nonideal Mixture of Homologues. Equations of state and adsorption isotherms for a nonideal surface layer of two surfactants with similar molar areas are given by1-5,21
Π)-
RT [ln(1 - θ1 - θ2) + a1θ12 + a2θ22 + 2a12θ1θ2] ω (8) bici )
θi (1 - θ1 - θ2)
2, eqs 8 and 9 reduce to the ordinary Frumkin equation of state and isotherm43
exp(-2aiθi - 2a12θj)
(9)
where a1, a2, and a12 are the respective constants of intermolecular interaction. For solutions of individual surfactants 1 and
RT [ln(1 - θ01) + aiθ0i2] ω θ0i (1 - θ0i)
exp(-2aiθ0i)
(10) (11)
The value of Π in eq 8 can be expressed via the terms bici using the expressions for the isotherms of eq 9. These products can in turn be expressed via Πi using eqs 10 and 11. For a12 ) 0, this leads to the (dimensionless) equation of state for the mixed interfacial layer
h 1 + k2 exp Π h 2 - k3 exp Π h ) k1 exp Π
(12)
where k1 ) exp[a1(θ012 - θ12 + 2θ1 - 2θ01) - a2θ22], k2 ) exp[a2(θ022 - θ22 + 2θ2 - 2θ02) - a1θ12], and k3 ) exp(2a1θ1 - a1θ12 - a2θ22) + exp(2a2θ2 - a1θ12 - a2θ22) - exp(-a1θ12 - a2θ22). Assuming small deviations from the ideal behavior, the simple approximation k1 = k2 = k3 = 1 results. Thus, eq 12 derived for nonideal layers transforms into eq 6 valid for ideal mixtures of homologues. Equation 12 and the expressions for ki given above were obtained from eqs 8 and 9 for a12 ) 0. If the more accurate expression a12 ) (a1 + a2)/2 is introduced,5,15 then the resulting equation has also the form of eq 12, but the expressions for ki become more cumbersome and are therefore omitted here. Note that for ideal layers, because a1 ) a2 ) a12 ) 0 and thus k1 ) k2 ) k3 ) 1, eq 12 coincides exactly with eq 6. 3. Mixture of Ionic Homologues. Mixtures of ionic and nonionic surfactants can be treated quite similarly to those considered above because the contribution of the electric double layer (DEL) to the surface pressure for ionic surfactants can be approximated by a nonideality factor.1,2,5 If the individual solutions of ionic surfactants and their mixtures are studied at a fixed ionic strength (which is usually accomplished by the addition of inorganic electrolyte), the surface pressure of the mixed solution, similar to nonionic surfactants, can be calculated from the data obtained for individual solutions according to eq 6. It is known that mixtures of ionic surfactants at any arbitrary ionic strength can be satisfactorily described by the Langmuir or Frumkin isotherms if, instead of the ionic surfactant concentrations, the electroneutral combinations of the corresponding ions are considered.1,2,5,21,26 When a solution contains a mixture of two anionic or cationic surfactants (for example, anionic homologues R1X and R2X with a common counterion X+) with or without addition of inorganic electrolyte XY, the counterion concentration X+ is given by the sum of concentrations of R1X, R2X, and XY. For simplicity, the saturation adsorptions of the two homologues will be taken as equal, i.e., ωR1X ) ωR2X ) ω ) 2ω0 (where ω0 is molar area of solvent5). For ideal surface layers (a1 ) a2 ) a12 ) 0), the following surface pressure isotherm can be derived:5,21
Π)
2RT ln[((b1f1()2cR1XcX- + (b2f2()2cR2XcX-)1/2 + 1] (13) ω
The surface pressure of an individual ionic surfactant solution RiX (with or without addition of inorganic electrolyte XY) obeys the equation
Πi )
2RT ln[((b1f1()2cR1XcX-)1/2 + 1] ω
(14)
11434 J. Phys. Chem. B, Vol. 105, No. 46, 2001
Fainerman and Miller
Combining eqs 13 and 14, one obtains an expression that describes the surface tension of a mixed solution
exp Π h ) ((exp Π h 1 - 1)2 + (exp Π h 2 - 1)2)1/2 + 1
(15)
Unlike the cases of nonionic surfactants or ionic surfactants in the presence of additional electrolyte, as given by eq 16, the concentrations (activities) of individual solutions have to be chosen such that the products of the concentrations of corresponding surface-active ions and counterions in the individual solutions and in the mixture are equal. For example, assuming that the concentrations of R1X and R2X in the surfactant mixture are c1 and c2, the surface pressure in the individual solutions should be taken at concentrations c01 ) [c1(c1 + c2)]1/2 and c02 ) [c2(c1 + c2)]1/2, respectively. This means that the increase of the counterion concentration in the surfactant mixture due to the addition of the second surfactant counterion should be taken into account. In a similar way, the influence of small additions of inorganic electrolyte with a common counterion has to be considered. In contrast, mixtures of anionic and cationic surfactants, in which typically extremely surface-active nonionic surfactants are formed, cannot be described by eqs 6 or 15. The nonideality of the components in the surface layer of mixtures of ionic surfactants does not affect the form of eq 15. Thus, eq 6 for a counterion excess and eq 15 in the absence of inorganic electrolyte (including the correction of the concentrations in the individual solutions as described above) can describe nonideal mixtures of ionic surfactants with similar molar areas. Changes in the adsorbed state of molecules at the interface (aggregation or cluster formation21,22,37-42) also have a minor effect on the form of the equation that relates the surface pressure of a mixed solution with the corresponding values for the individual solutions. This is demonstrated further below for a mixture in which one of the components forms clusters in the adsorption layer. 4. Mixture of Homologues Forming Surface Aggregates. An equation of state and a respective adsorption isotherm for surfactants that form surface aggregates were derived in refs 21 and 37. For large aggregates, these equations can be written in the following form:42
Π1 ) -
RTΓ1c ln[1 - ωΓ1] ωΓ1
b1c1 )
Γ1ω [1 - ωΓ1]Γc/Γ1
(16)
(17)
where Γ1c is the critical adsorption of aggregation. In these equations, the adsorption Γ1 is expressed via the available total adsorption of monomers Γ11 and aggregates Γn1 recalculated as monomers
Γ1 ) Γ11 + nΓ1n
(18)
Equations 16 and 17 assume an ideal behavior of the surface layer in the transcritical adsorption range (Γ1 > Γ1c). This is correct when the contribution to the surface pressure from cluster-cluster and cluster-monomer interactions is considered because for n . 1 the adsorption of clusters as kinetic entities Γ1n is negligible. As in the transcritical adsorption range and for n . 1, the relation Γ11 ) constant ) Γ1c holds; the contribution resulting from the monomers’ nonideality (∼Γ1c2) does not depend on Γ1 and can be approximated by an appropriate correction of Γ1c. For ideal (both in the bulk and at
the surface) mixtures of two surfactants (1 and 2) with similar molar areas of which only one component (say the 1st) is able to form large two-dimensional surface aggregates, the following equations have been derived:39,42,44
Π)-
RT(θc + θ2) ln(1 - θ1 - θ2) ω(θ1 + θ2)
b 1c 1 )
b 2c 2 )
θ1 (1 - θ1 - θ2)(θc+θ2)/(θ1+θ2) θ2 (1 - θ1 - θ2)(θc+θ2)/(θ1+θ2)
(19)
(20)
(21)
Here, θ1 ) ωΓ1 ) ω(Γ11 + nΓ1n) and θ2 ) ωΓ2 are the degrees of the interfacial coverage by component 1 (aggregates and monomers) and component 2. θc ) ωΓ1c is the critical surface layer coverage, which corresponds to the component 1. Note that the eqs 19-21 have been derived under the assumption that only of one component of the mixture forms clusters. This case is in accordance with experimental data, for example, for the system of sodium dodecyl sulfate and dodecanol44 and that of β-lactoglobulin mixed with a phospholipid.45 There are, of course, systems in which the formation of mixed clusters is possible.46 For such systems, the additive models derived here are obviously not applicable. Equation 19 can be transformed into
exp(Π h /R) ) b1c1(1 - θ1 - θ2)R-1 + b2c2(1 - θ1 - θ2)R-1 + 1 (22) where R ) (θc + θ2)/(θ1 + θ2) is the fraction of monomers in the net adsorption of the two surfactants. Similarly to the previous derivation, the values of bici can be again expressed via the corresponding surface pressures of the individual solutions using eqs 16 and 17 for the aggregating surfactant 1 and using the Szyszkowski-Langmuir eq 4 for the solution of surfactant 2, which results in an expression identical to eq 12, however, with the constants defined by k1 ) exp(1/β - 1/R) (1 - θ1 - θ2)R-1/(1 - θ01)β-1, k2 ) exp(-1/R) (1 - θ1 - θ2)R-1 and k3 ) exp(-1/R). θ01 ) ωΓ01 is the degree of surface coverage by component 1 in the individual solution, and β ) Γ01c/Γ01. Because R > β, R < 1, and β < 1, it follows that k1 > 1, k2 < 1, and k3 < 1. As the ki enter the first and second terms of the right-hand side of eq 12 with an opposite influence on the surface pressure, it can be approximately assumed that k1 ) k2 ) 1, while the value of k3 varies from k3 ) 1 at the offset of the phase transition of component 1 to k3 = 0 in the extremely saturated surface layer for relatively low values of Γ1c and Γ2. Therefore, eq 6 can be used to estimate the surface pressure in the mixed solution of two surfactants both in the precritical and in the transcritical adsorption range of the aggregating component. 5. Mixture of Components with Different Molar Areas. This case is of practical importance because mixtures of surfactants with essentially different molar areas, e.g., oxyethylated nonionic surfactant (large ω) and ionic surfactant (low ω), are often used in practice. The equation of state for the ideal mixture of two surfactants with different ωi is expressed by eq 1, provided that the dividing surface is chosen according to Lucassen-Reynders.1,2,21 This allows the contribution of the nonideality of entropy to the surface tension to be neglected. However, the value of ω in
Method to Estimate Surface Tension
J. Phys. Chem. B, Vol. 105, No. 46, 2001 11435
this equation no longer remains constant but depends on the corresponding partial molar areas, ωi, and the adsorptions of components 1 and 2:
ω)
ω1Γ1 + ω2Γ2 Γ 1 + Γ2
(23)
For such mixtures, the adsorption isotherm reads (see ref 21)
bici )
θi (1 - θ1 - θ2)ni
(24)
where ni ) ωi/ω. Following the lines of the preceding sections, one can use eq 24 to substitute the terms bici in eq 1 by the corresponding bici values for the individual solutions. This again results in eq 12 in which the coefficients are now k1 ) (1 - θ1 - θ2)1-n1, k2 ) (1 - θ1 - θ2)1-n2, and k3 ) 1 - (1 - θ1 θ2)1-n1 - (1 - θ1 - θ2)1-n2. If the ni are close to 1, then the coefficients ki are also approximately 1, i.e., eq 6 is also valid for this case. Note that, in contrast to the examples above, the dimensionless pressures in eq 6 are Π h ) Πω/(RT), Π h 1 ) Π1ω1/ (RT) and Π h 2 ) Π2ω2/(RT). To determine the average ω value for the mixture, the same substitution procedure for bici should be used and eq 23 becomes now
ω)
ω1(exp Π h 1 - 1) + ω2(exp Π h 2 - 1)(1 - θ1 - θ2)n2-n1 exp Π h 1 - 1 + (exp Π h 2 - 1)(1 - θ1 - θ2)n2-n1
Π h1+Π h2 ω1Π1 + ω2Π2 ) ω1 Π1 + Π2 Π h1+Π h 2(ω1/ω2)
Π1 + Π2 Π 1 + Π2 ) ω1 Π1/ω1 + Π2/ω2 Π1 + Π2(ω1/ω2)
RT ln(1 + bici) ω ig1
∑
(27)
(28)
It will be shown below that both models provide a quite satisfactory description of surfactant mixtures with remarkably different molar areas of the components. 6. Mixture of n Components. Equation 6 can be easily generalized to mixtures of n different components (ideal or nonideal), including ionic surfactants with the additions of
(29)
one obtains the equation for the mixture of n surfactants n
exp Π h )
exp Π hi+1-n ∑ i)1
(30)
If the molar areas of the components are different, then the average value can be estimated as
(26)
When we assume that the adsorptions characterized by different molar areas in a densely packed layer (i.e., at sufficiently large Π values) are roughly inversely proportional to ωi, cf. eqs 4 and 5, then instead of eq 27 we obtain (model II)
ω)
Π)
ω)
only if ω1 = ω2. However, one can use other expressions for the mean area, which do not involve unknown quantities. When the adsorption is roughly proportional to the surface pressure (for extremely diluted surface layers Π ) RTΓ), the expression for the mean molar area, eq 23, can be presented as (model I)
ω)
electrolytes and surfactants with different molar areas. Starting from the generalized Szyszkowski equation,21
(25)
Estimates made from this equation have shown that if the difference between the ωi is relatively small and the surface coverage is sufficiently low, the term (1 - θ1 - θ2)n2-n1 can be omitted. Therefore, eq 25 can be correctly transformed into the limiting expression
h 1 - 1) + ω2(exp Π h 2 - 1) ω1(exp Π ω) exp Π h 1 + exp Π h2-2
Figure 1. Experimental (from ref 11) and theoretical surface tension isotherms for aqueous solutions of 1-heptanol (2) and 1-octanol (4): dashed lines, Langmuir model with ω ) 1.3 × 105 m2/mol for both surfactants; solid lines, Frumkin model with ω ) 1.6 × 105 m2/mol for both surfactants; a ) 0.86 (1-heptanol) and a ) 1.15 (1-octanol).
∑ωiΠi ∑Πi
(31)
If the solution contains no additional inorganic electrolytes but contains m ionic surfactants with common counterion, then instead of eq 30 one can use n
exp Π h )
∑ i)1
m
exp Π hi+[
(exp Π h i - 1)2]1/2 + 1 - n ∑ i)1
(32)
Using eqs 30-32, one can estimate the surface pressure for a mixture of any number of surfactants from the surface pressures of the individual solutions. Results and Discussion To demonstrate the capacity of the derived equations, we will compare here results calculated from the approximate eqs 6 and 15 with experimental results for mixtures of various surfactants. At first, mixtures of homologues (nonionic and ionic) and then mixtures of surfactants of different nature will be considered using data from literature. Figure 1 shows the surface pressure isotherms for 1-octanol and 1-heptanol at the solution/air interface as given in ref 11. The parameters of the isotherms were calculated from the Szyszkowski and Frumkin equations. One can see that the Frumkin equation provides better agreement with the experiment (solid lines). The experimental results for various mixtures of these two alcohols are shown in Figure 2. The theoretical values
11436 J. Phys. Chem. B, Vol. 105, No. 46, 2001
Figure 2. Dependence of surface tension for an aqueous 1-heptanol solution at various concentrations of 1-octanol: (() 0, (2) 0.2, (4) 0.4, (b) 0.6, (O) 0.9, (×) 1.2, (9) 1.6, (0) 2.0 mmol/l; theoretical curves calculated from eq 6 with ω ) 1.3 × 105 m2/mol; data from ref 11.
Fainerman and Miller
Figure 4. Dependence of surface pressure of aqueous SDeS solutions for various SDS additions: (() 0.5 mmol/l; (9) 1.0 mmol/l; (2) 2.0 mmol/l; (b) 4.0 mmol/l); solid lines calculated from eq 15.
Figure 3. Dependence of surface pressure for aqueous SDeS and SDS solutions without (2) and (O) and with the addition of 0.1 M NaCl (b), and 0.5 M NaCl (×) plotted in the coordinates of eq 14: solid lines calculated from Frumkin’s equation for ionic surfactants5 with ω ) 2.3 × 105 m2/mol and a ) 0.9 for SDS (curve 1) and with ω ) 2.2 × 105 m2/mol and a ) 0.73 for SDeS (curve 2); data from ref 26.
Figure 5. Concentration dependence of surface pressure for the SDS (() and STS (9) solutions and their 3:1 molar fraction mixture (2) in the presence of 0.03 M NaCl; dashed line, theoretical values calculated from eq 6; data from ref 26.
were calculated from eq 6 with ω ) 1.3 × 105 m2/mol, using the experimental Π values for individual solutions at the same concentrations. The calculated curves are given in Figure 2 and demonstrate the perfect agreement between theory and experiment. Note that the adsorption behavior of the components in the individual adsorption layers is by no means ideal. Moreover, to match the values calculated from eqs 8 and 9 to the experimental data, it was shown in ref 11 that an additional parameter, a12, had to be introduced. The results of calculations from eq 6 for mixtures of alcohols, as shown in Figure 2, are rather insensitive to the value of the molar area. When the ω value is varied from 1.0 × 105 to 1.6 × 105 m2/mol, the variations in Π are only some tenths of a millinewton per meter for Π < 10 mN/m and do not exceed 2 mN/m for Π > 25 mN/m. This indicates the applicability of eq 6 even when the ω value cannot be specified accurately. The solutions of sodium decyl sulfate (SDeS) and sodium dodecyl sulfate (SDS), both without and with addition of 0.1 and 0.5 M of NaCl, were studied in ref 26. The experimental results are plotted in Figure 3 using the coordinates of eq 14. One can see that the theoretical model, which assumes an electroneutral compositions of ions, provides for an excellent agreement with the experiment and both the pure system and the system with added electrolyte match the same curve. The
theoretical dependencies shown in Figure 3 correspond quite well to the Frumkin equations (eqs 10 and 11) with the factor 2 at the right-hand site of eq 10. To account for the counterion effect on the adsorption activity of surface active ions, the electroneutral ionic combination [(b1f1()2cRiXcX-]1/2 was introduced into the left-hand side of eq 11 instead of the term bici. The experimental dependencies of the surface pressure of SDeS solutions in the presence of various SDS concentrations are shown in Figure 4 (data from ref 26). The data correspond to different concentrations of the counterion Na+; therefore, the calculations were performed with eq 15. The corresponding concentrations of SDeS and SDS for individual solutions were determined by c01 ) [c1(c1 + c2)]1/2 and c02 ) [c2(c1 + c2)]1/2, respectively. The calculated curves shown in Figure 4 agree very well with the experimental data. Note that in ref 26 agreement between theory and experiment was achieved only with value of a12 = (a1 + a2)/2 in eq 8. The behavior of mixtures of ionized homologues in the presence of added electrolyte is illustrated in Figure 5 (data from ref 36). The individual solutions of SDS, sodium tetradecyl sulfate (STS), and their mixtures at molar fractions of 3:1 were studied in the presence of 0.03 M NaCl. The dependence of surface pressure on surfactant concentration for the individual solutions can be satisfactorily described by the Szyszkowski eq 4 with ω1 ) 1.8 × 105 m2/mol for STS and ω2 ) 2.2 × 105
Method to Estimate Surface Tension
Figure 6. Surface tension isotherms for Triton X-100 (0) and C12EO5 (9): theoretical isotherms calculated from the Szyszkowski equation for Triton X-100 (ω ) 3.4 × 105 m2/mol) and from Frumkin’s equation for C12EO5 (ω ) 1.5 × 105 m2/mol, a ) 0.6); data from ref 35.
Figure 7. Surface tension isotherm for a 1:1 mixture of Triton X-100 and C12EO5: theoretical values calculated from eqs 6 and 27 (solid line); data from ref 35.
m2/mol for SDS. Note that if the dependencies of Π for the individual solutions in Figure 5 are plotted in the coordinates of eq 14, i.e., as dependence on (cSDScNaCl)1/2 or (cSTScNaCl)1/2, then the slope of the curves approximately doubles, which corresponds to the same ωi values as shown in Figure 3 with the factor 2 at the right-hand site of eq 10: ω1 ) 2.3 × 105 m2/mol for SDS and ω2 ) 2.2 × 105 m2/mol for STS. Moreover, in these coordinates, the data presented in Figure 5 for SDS are in a good agreement with those shown in Figure 3. Because the difference between the parameters ωi is quite small, the mean values for ω calculated from eqs 27 and 28 are equal. The surface pressure values for SDS/STS mixtures calculated from eq 6 agree well with the experimental data reported in ref 36. Mixtures of Triton X-100 and C12EO5 studied in ref 35 can be considered as example mixtures of nonionic surfactants with different ωi values. Figures 6 and 7 illustrate the surface tension isotherms for their individual solutions and the 1:1 mixture, repectively. The theoretical isotherms in Figure 6 were calculated from the Szyszkowski equation for Triton X-100 and from the Frumkin equation for C12EO5. For this surfactant type, the reorientation isotherm is most adequate;41 however, the insufficient number of experimental points available and also the low precision of surface tension measurements ((1.5 mN/m, as indicated in ref 35) prevent unambiguous conclusion about the isotherm type. The results of calculations from eq 6, with a
J. Phys. Chem. B, Vol. 105, No. 46, 2001 11437
Figure 8. Surface tension isotherms for the solutions of C12SO3Na (() and its mixtures with C12EO8 in various concentrations: (9) 2.18 × 10-4 mmol/l; (2) 6.24 × 10-4 mmol/l; (0) 6.24 × 10-3 mmol/l; (b) 1.49 × 10-2 mmol/l. Theoretical curve for individual solutions of C12SO3Na was calculated from Frumkin’s eq 10 without factor 2 (ω1 ) 1.5 × 105 m2/mol, a ) 0.6), while for mixtures the calculations were made using eq 6 and the mean value of ω calculated from eqs 27 and 28 (curves); data were taken from ref 10.
Figure 9. Surface tension isotherms for C12EO8 (() solutions and the mixtures with C12SO3Na in different concentrations: (9) 0.316 mmol/ l; (2) 1.26 mmol/l; (b) 3.16 mmol/l. Theoretical curve for individual solutions of C12EO8 was calculated from the reorientation model (for details see ref 41) with ωmax ) 1.3 × 106 m2/mol, ω2 ) ωmin ) 3.9 × 105 m2/mol, and R ) 0.6, while for the mixtures, eq 6 was used with a mean ω value calculated from eqs 27 and 28 (curves); data were taken from ref 10.
mean molar area of the mixture estimated from eq 27, are in a rather good correspondence with the experimental data given in Figure 7. The results reported in ref 10 for the mixtures of a nonionic surfactant (C12EO8) with anionic sodium dodecanesulfonate (C12SO3Na) are shown in Figures 8 and 9. Figure 8 illustrates the surface tension isotherms for pure C12SO3Na and mixtures at various C12EO8 concentrations. Individual solutions of these surfactants are described by the reorientation isotherm for C12EO8 and the Frumkin isotherm for C12SO3Na. The isotherm parameters for C12EO8 are consistent with those reported in ref 41, while for C12SO3Na, the parameters are, quite expectedly, close to those calculated for C12SO4Na (SDS, see Figure 3). Because the molar areas of the mixed surfactants are essentially different from each other (2.6 times without the factor 2 in the equation of state for ionics), the mean values of ω for the mixtures were calculated from the two models, given by eq 27 and eq 28. The differences between the calculated mean values are very small (less than 10%); however, for low and medium Π values, eq 28 yields better results, while for Π > 25 mN/m,
11438 J. Phys. Chem. B, Vol. 105, No. 46, 2001 the values calculated from eq 27 correspond better to the experimental data. Therefore, a value of ω averaged from both models was introduced into eq 6. It is seen that there is good agreement between the measured surface tensions of the mixtures and those calculated from eq 6. This model cannot be used for anionic/cationic mixtures. The interionic interaction leads to the formation of higher surface active molecules in the solution and, at the same time, results in changes in the intermolecular interaction.47-49 Therefore, the calculations made from eq 6 in which this effect is disregarded lead to surface tensions much higher than the measured values. A theoretical model that describes the mixtures of oppositely charged ionic surfactants was discussed elsewhere.50 Conclusions It is shown that for mixtures of different surfactants, which follow different types of adsorption isotherms (e.g., those assuming cluster formation), one can transform the equation of state for a mixed layer into the simple relations eq 6 or 15. To calculate the surface tension of surfactant mixtures, one can use various types of information concerning the surface tension of the individual solutions: either the experimental values at a given concentration in the individual solutions or the parameters of the corresponding adsorption isotherms. Moreover, mixed layers composed of an arbitrary number of nonionic and ionic surfactants can also be described using the simple eq 32. The derived expressions allow description of several surfactant systemssnonionic, ionic, and mixed surfactant onessin which the molar areas of the mixed components and the types of the adsorption isotherm can deviate in a very wide range. The “universal” validity of the model can be ascribed primarily to the fact that many particular features of the adsorption behavior of mixed components (surface nonideality, capability of reorientation, or cluster formation in the surface layer) are accounted for “automatically” because the surface tension of the individual solutions enters eqs 6 and 15. Only the specific interaction between mixed molecules at the surface remains unaccounted for in the proposed model. However, it was shown by a number of authors that this specific interaction is rather weak and can be neglected in most cases. The proposed approach is only of approximate character, and to explain all peculiar features of mixed adsorption layers, more generalized theoretical models are necessary. We believe the presented approach is quite simple but useful, in particular, when the behavior of a mixed system has to be estimated without the performance of a huge number of experiments or complicated mathematical calculations. Acknowledgment. The work was financially supported by the Max Planck Society. References and Notes (1) Lucassen-Reynders, E. H. J. Colloid Sci. 1964, 19, 584. (2) Lucassen-Reynders, E. H. J. Colloid Sci. 1964, 19, 584; J. Phys. Chem. 1966, 70, 1777.
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