Equilibrium Adsorption and Tension of Binary Surfactant Mixtures at

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Langmuir 1996, 12, 354-362

Equilibrium Adsorption and Tension of Binary Surfactant Mixtures at the Air/Water Interface Faisal A. Siddiqui and Elias I. Franses* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283 Received July 21, 1995. In Final Form: October 3, 1995X The nonideal adsorbed solution model (NAS) is extended to the air/water interface for determining the surface tension of aqueous binary surfactant mixtures with molecules of different size, surface activity, and nonideal interactions. The regular solution theory is used for modeling the nonideal interactions between the adsorbed molecules and in the micelles. Two mixing parameters βσ and βm are used for fitting data for premicellar and micellar concentrations. This model can also be extended to account for molar area changes upon mixing in the monolayer or more complex nonideal interactions. Synergism in surface tension reduction efficiency can be predicted with this model. The ranges of mole fractions and concentrations for the existence of synergism can be determined. Tension data of two nonionic binary mixtures, C12E8/ SDS (from the literature) and C12E5/Triton X-100, fit this model well. New tension data for C12E5/Triton X-100 are reported, and they are synergistic for a specific range of concentrations. Moreover, the surface coverages and compositions for the two nonionic mixtures are calculated with the NAS model. These calculations show that the larger molecules adsorb at the interface more favorably at low concentrations than at high concentrations.

1. Introduction Many practical surfactant systems involve multicomponent systems of molecules of different molecular size, adsorption capacity, and ionic character (anionic, nonionic, etc.).1,2 It is important to develop equilibrium adsorption isotherms and surface equations of state for realistically describing equilibrium adsorption at the air/water interface and surface and interfacial tensions.3 These isotherms and equations are also needed in modeling the dynamic adsorption and tension of these mixtures.4,5 Such isotherms should (i) reduce to the individual component isotherms, so that single component data can be used as a basis; (ii) account for differences in size or ionic character; (iii) be thermodynamically consistent;6 and (iv) account for surface molecular interactions or thermodynamic nonidealities which are prevalent in most mixtures. Moreover, for a model to be more practically relevant, it should be combined with a mixed micellar model which can also account for nonidealities. Finally, the most realistic models should account for bulk solution nonidealities and micelle-micelle interactions, as in the molecular thermodynamic approaches of Ruckenstein,7,8 Nagarajan,9 and Blankschtein.10 For single component adsorption at the air/water interface, the Langmuir isotherm is a two-parameter * Author to whom correspondence should be addressed. Telephone: (317) 494-4078. Fax: (317) 494-0805. E-mail: franses@ ecn.purdue.edu. X Abstract published in Advance ACS Abstracts, December 15, 1995. (1) Shinoda, K.; Nakagawa, T.; Tamamushi, B.-I.; Isemura, T. Colloidal Surfactants. Some Physicochemical Properties; Academic Press: New York, 1963. (2) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (3) Evans, D. F.; Wennerstrom, H. The Colloidal Domain; VCH Publishers: New York, 1994. (4) Fainerman, V. B.; Miller, R. Colloids Surf. A 1995, 97, 65. (5) Chang, C.-H.; Franses, E. I. Colloids Surf. A 1995, 100, 1. (6) Franses, E. I.; Siddiqui, F. A.; Ahn, D. J.; Chang, C.-H.; Wang, N.-H. L. Langmuir 1995, 11, 3177. (7) Ruckenstein, E.; Rao, I. V. J. Colloid Interface Sci. 1987, 117, 104. (8) Rao, I. V.; Ruckenstein, E. J. Colloid Interface Sci. 1987, 119, 211. (9) Nagarajan, R. Langmuir 1985, 1, 331. (10) Nikas, Y. J.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2680.

0743-7463/96/2412-0354$12.00/0

equation which successfully describes many surfactants, either nonionic or ionic, the latter with semiempirical or empirical parameters. Radke and co-workers have developed more rigorous and complex models for ionic surfactants.11 The Langmuir isotherm parameters, Γm and KL (Γm is the maximum surface density at equilibrium, and KL is the adsorption equilibrium constant), which describe adsorption and tension (Szyszkowski equation), have been determined for many systems and have been reviewed recently.5,12 The Langmuir isotherm has been empirically generalized for mixtures. This equation satisfies only the first criterion above and has been shown to be thermodynamically inconsistent.6,13,14 It only describes ideally-mixed monolayers of point molecules on a fixed lattice and cannot account for size or ionic-character differences. The Langmuir isotherm has been extended to mixtures using a thermodynamically consistent approach, namely the ideal adsorbed solution (IAS) theory of Myers and Prausnitz.6,15 In this article the Langmuir-Szyszkowski equation is extended for the first time to the nonideal adsorbed solution (NAS) theory for binary nonionic/nonionic surfactants and to tensions of mixed micellar solutions. This theory can be extended to multicomponent systems and ionic surfactants. After we briefly develop this model, we apply it in this article to data available in the literature as well as to some data from our laboratory. The literature data are by Lange and Beck16 for C12E8 (dodecyl octaethylene glycol ether) and SDS (sodium dodecyl sulfate) in 0.5 M NaCl. Calculations for tensions and surface densities for concentrations both below and above the cmc were done using the same model. Another nonionic mixture of C12E5 (dodecyl pentaethylene glycol ether) and Triton X-100 (polyethylene glycol tert -octylphenyl ether) was used for obtaining data over a wider concentration range. Even though the latter surfactant is a mixture of different chain (11) Keesom, W. H.; Zelenka, R. L.; Radke, C. J. J. Colloid Interface Sci. 1988, 125, 575. (12) Chang, C.-H.; Wang, N.-H. L.; Franses, E. I. Colloids Surf. 1992, 62, 321. (13) Kemball, C.; Rideal, E. K.; Guggenheim, E. A. Faraday Soc. Trans. 1948, 44, 948. (14) Broughton, D. R. Ind. Eng. Chem. 1948, 40, 1506. (15) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (16) Lange, V. H.; Beck, K. H. Kolloid Z. Z. Polym. 1973, 251, 424.

© 1996 American Chemical Society

Binary Surfactant Mixtures at the Air/Water Interface

length components, it is described well by a single component Langmuir isotherm. The equilibrium tension data for the C12E5/SDS system show evidence of adsorbed solution nonideality, mixed micelle nonideality, and synergism in surface tension reduction.17 2. Nonideal Adsorbed Solution and Nonideal Mixed Micelle Model 2.1. Theory. 2.1.1. General Nonideal Adsorbed Solution Model. The ideal adsorbed solution (IAS) theory and the nonideal adsorbed solution (NAS) theory were first reported by Myers and Prausnitz15 for adsorption equilibria of gaseous mixtures on ideal homogeneous surfaces for which an expression analogous to Raoult’s law is obtained. These models are inherently consistent with thermodynamics and are extended here to adsorption of surfactants at the air/water interface. At equilibrium, the chemical potential of component i (i ) 1, 2, ..., k) in the adsorbed layer is equal to the chemical potential in the bulk solution, yielding

fici ) ζixic0i (Π)

[

For a binary mixture adsorbing at the interface the molar area of the mixture is

a ) a01x1 + a02x2 + am )

]

∂(nGE/(RT)) ∂ni

(2)

T,P,nj

where ni is the number of moles of component i and n ) k ∑i)1 ni. GE is the Gibbs excess free energy of the mixed monolayer relative to an ideal mixture and can be expressed as a power series in x1, as is common in solution thermodynamics:18 E

G ) β0 + β1x1 + β2x12 + ... x1x2RT

(3)

with β0, β1, β2, ... being parameters specific to a particular surfactant mixture. These parameters may be constants (section 2.1.3) or may depend upon the surface pressure (section 2.1.2). (17) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 90, 212. (18) Smith, J. M.; Van Ness, H. C. Introduction to Chemical Engineering Thermodynamics, 4th ed.; McGraw-Hill: New York, 1987.

x1

x2 + + am 0 0 Γ1 Γ2

(4)

where a is the total molar area after mixing components 1 and 2, and a01 ) 1/Γ01 and a02 ) 1/Γ02 are the molar areas of pure components 1 and 2 corresponding to bulk concentrations c01(Π01) and c02(Π02), Γ01 and Γ02 are the surface densities associated with the reference concentrations c01 and c02, and am is the (nonideal) molar area change upon mixing, which is given by18

[

]

∂(ln ζ1) am ) x1 RT ∂Π

T,x1,x2

[

+ x2

]

∂(ln ζ2) ∂Π

T,x1,x2

(5)

Hence, am * 0 if GE or ζi depends on surface pressure. For the specific case of single component Langmuir isotherms,

(1)

where ci is the bulk concentration, xi is the mole fraction on the surface layer, fi is the activity coefficient in the bulk, and ζi is the activity coefficient of component i in the adsorbed monolayer. In this article, we only consider cases where the bulk solution is ideal (fi ) 1 ). A central point of the ideal adsorbed solution theory is that monolayer mixing is chosen to occur at constant surface pressure Π (Π ≡ γ0 - γ, where γ0 is the surface tension of the pure solvent and γ is the surface tension of the solution), i.e., Π ) Π01 ) Π02, c0i is the concentration of pure component i that corresponds to a reference surface pressure of Π0i , and Γi is the adsorbed solute density at equilibrium or more precisely the Gibbs surface excess density relative to the solvent. The surface mole fractions are defined as k Γj, and the bulk mole fractions are yi ≡ ci/ xi ≡ Γi/∑j)1 k ∑j)1 cj. For a binary mixture, x1 + x2 ) 1 and y1 + y2 ) 1. At sufficiently low surface coverage, one would expect that ζi ) 1. However, when the surface coverage exceeds a substantial fraction of the surface area the adsorbed solution phase may be nonideal, ζi * 1. Such interactions are at the core of synergism in surface tension reduction17 and other mixture phenomena. The activity coefficient ζi is related to the partial molar excess Gibbs free energy as follows:

ln ζi )

Langmuir, Vol. 12, No. 2, 1996 355

Γ0i

KL,ic0i

) Γm,i 1 + KL,ic0i

i ) 1, 2

Π0i (c0i ) ) RTΓm,i ln(1 + KL,ic0i )

i ) 1, 2

(6) (7)

Equation 7 is the Langmuir-Szyszkowski (LS) equation and is applicable to nonionic surfactants. For ionic surfactants which may dissociate in aqueous solutions eq 7 needs to be suitably modified.6 Many pure surfactants at the air/water interface follow the single component Langmuir isotherm, and data for this isotherm are extensively available in the literature.12 Other single component isotherms such as the Freundlich, the Langmuir, and the modified Langmuir isotherms can be applied along with the nonideal adsorbed solution theory for mixtures. The modeling method is not restricted to cases when both isotherms need to be identical. In other words, one could use this method when one of the components follows the Langmuir and the other follows the Freundlich isotherm, or other combinations. The nonideal adsorbed solution theory can predict surface densities and tensions for molecules of different sizes and equilibrium adsorption isotherms. With this theory, it is possible to model the interactions between the adsorbate molecules using the regular solution theory with constant parameters (section 2.1.3) or other specific models which may account for area changes upon mixing (section 2.1.2). Normally, one needs to find Γ1, Γ2, and Π as a function of c1 and c2. Since Π01(c01) ) Π02(c02) and c0i ) ci/(ζixi) (assuming a dilute bulk solution, fi ) 1 ), one finds from eqs 4 and 7, the following compact formulas for the Langmuir isotherm case:

(

) (

c1 1 + KL,1 ζ1x1

Γm,1

c2 ) 1 + KL,2 ζ2x2

)

Γm,2

x1(ζ1x1 + KL,1c1) x2(ζ2x2 + KL,2c2) 1 + + am ) Γm,1KL,1c1 Γm,2KL,2c2 Γt

(8)

(9)

where Γt ) Γ1 + Γ2. In the above equations am and ζi have to be determined on the basis of a specific model, e.g., the models given in section 2.1.2 or 2.1.3. The surface densities can be determined from Γi ) xiΓt, and the surface tension γ can be determined from a modified form of eq 7, if xi and ζi are known:

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(

γ ) γ0 - RTΓm,i ln 1 +

Siddiqui and Franses

)

KL,ici ζixi

(10)

The above equations are new and quite general. 2.1.2. Regular Solution Model with Area Change upon Mixing. If one uses eq 3 with a single parameter, βσ ) β0 (which implies GE/RT ) βσx1x2 ), then the well-known regular solution model for the activity coefficients is obtained. The regular solution model for binary solutions yields

ζi ) exp[βσ(1 - xi)2]

i ) 1, 2

(11)

If β1 ) β2 ) ... ) 0 in eq 3, β0 ≡ βσ, and if βσ depends on the surface pressure Π, then the area change upon mixing am * 0. If βσ is a simple polynomial function of Π,

βσ(Π) ) βσ0 + βσ1Π + βσ2Π2 + ...

(12)

Then eq 5 yields

am ) x1(1 - x1)(βσ1 + 2βσ2Π + 3βσ3Π2 + ...) RT

(13)

One may use one or two terms in the series in eq 13. This method involves fitting tension data for the constants, βσi. This can be done by determining several βσ values at various surface pressures and then fitting these values to eq 12. Equations 8, 9, and 13 can be solved with an iterative procedure. To calculate the area change upon mixing, eqs 10, 11, and 13 have to be solved simultaneously. 2.1.3. Regular Solution Model with No Area Change upon Mixing. If βσ is not a function of surface pressure, Π, then from eq 5 there is no area change upon mixing, am ) 0. This model is more realistic for systems at low surface coverages or when βσ does not change with surface pressure or composition. At high surface coverages, the interactions between the adsorbed molecules are more complex, and the model may need improvements or additional terms. The solution procedure for this model is simple. In this case, eq 11 can be used along with eqs 8 and 9 to solve for Γ1, Γ2, and Π. Due to its simplicity and ease of application this model has been used for subsequent calculations and curve fitting for the binary surfactant systems studied in sections 2.2, 2.3, and 2.5. 2.1.4. Mixed Micellar Model. For modeling adsorption and tension above the cmc one can use either the pseudophase micellization model for nonionic surfactants or the pseudo-phase separation model for ionic surfactants,19 if needed, or mass-action models of mixed micellization.20 For both systems considered in this article we use the first model because at high ionic strength the ionic surfactant can be represented thermodynamically by a nonionic surfactant model. The surfactant material balance equations are

ci + xi,mcm ) yicT

i ) 1, 2

(14)

where xi,m is the mole fraction of component i in the micelle and cm is the concentration of the surfactant in a micellar state (sometimes called the micellar “inventory”21 ). Equality of chemical potentials of surfactant i in the bulk (19) Kamrath, R. F.; Franses, E. I. Ind. Eng. Chem. Fundam. 1983, 22, 230. (20) Franses, E. I.; Bidner, M. S.; Scriven, L. E. In Micellization, Solubilization and Microemulsions; Mittal, K. L., Ed.; Plenum Press: New York, 1977; Vol. 2. (21) Hall, D. G. J. Phys. Chem. 1987, 91, 4287.

solution and in the micellar pseudo-phase yields (for an ideal bulk solution)

ci ) xi,mζ*i c*i

i ) 1, 2

(15)

where c* i is the individual cmc and ζ* i is the activity coefficient in the micelle. This equation is analogous to eq 1. Hence, ζ*i can be evaluated using either the regular solution theory (eq 11), with βm as the parameter instead of βσ, or other models. From these equations one finds x1,m, c1, and c2. From these and the adsorption model (section 2.1.3) one calculates Γ1, Γ2, and γ above the cmc. The critical micelle concentration of the mixtures, c*, can be calculated for the regular micellar solution model using the following equation (follows from eqs 15, the pseudo-phase separation model assumptions),

yic* ) x*i c*i exp[βm(1 - x*i )2]

i ) 1, 2

(16)

where x*i is the mole fraction of component i in the micelle at the critical micelle concentration, c*. Calculations have been made in section 3 for comparing the mixed cmc’s calculated by this model and those reported in the literature earlier. 2.1.5. Method of Fitting Tension Data to Model. For a single component following the Langmuir-Szyszkowski isotherm, the tension characteristics are defined by the parameters Γm,i and KL,i and the cmc c* i. Individual component data are fitted using eq 7, with a program which uses the least-squares method for finding the best fit and determining Γm,i and KL,i. For the individual components the cmc c*i is normally determined visually from the slopes of the data around the cmc. In this paper, the mixture data were fitted using the model in section 2.1.3 with no area change upon mixing. The activity coefficients are calculated using the regular solution model (eqs 11 and 16). The additional nonideality parameters βσ and βm were used for fitting equilibrium tension data of binary mixtures; βσ was used, along with predetermined Γm,i and KL,i, for fitting premicellar concentration data; βm was used for concentrations above the cmc. This method uses βσ and βm to fit the data for the entire concentration range at each (fixed) mixture composition y1. A computer program which minimizes the variance between the experimental and calculated tension values was used for finding the nonideality parameters. The mixed cmc c* was calculated (eq 16) and compared to the visually determined cmc. 2.2. Sample Calculations. Adsorption and tension calculations were done for a set of hypothetical binary systems for studying the effect of increased interactions between the molecules at the interface and in the mixed micelle. The simplest nonideal model in section 2.1.3 was used in these calculations. Results for a binary system for which the adsorption capacities are different (RΓ ≡ Γm,1/Γm,2 ) 3), the surface activities are the same (KL,1 ) KL,2), the cmc’s differ by 100-fold, and y1 ) 0.6 are presented in Figures 1-3. Other calculations for two specific real systems are also reported in later sections. In Figure 1, y1 ) 1 and y1 ) 0 represent pure components 1 and 2, respectively. Component 1 by convention is the smaller molecule (Γm,1 > Γm,2 ), which has the lower cmc in this example. If, in addition KL,1 ) KL,2, then the smaller molecule is more efficient in lowering the surface tension at the conditions used. For the pseudo-phase model used, the surface tension for pure components 1 and 2 remains unchanged above the cmc. Increasingly negative values of βσ and βm imply more attractive interactions between the molecules. Normally, nonionic surfactants have weak interactions and the βσ


Figure 1. Tension calculations for the nonideal adsorbed solution model at different values of βσ and βm. Curve 1, βσ ) βm ) 0; curve 2, βσ ) 0 and βm ) -5; curve 3, βσ ) -5 and βm ) 0; curve 4, βσ ) -5 and βm ) -5; curve 5, βσ ) -5 and βm ) -15; curve 6, βσ ) -15 and βm ) -5. For all calculations, Γm,1 ) 5 × 10-6 mol/m2, Γm,2 ) Γm,1/3, KL,1 ) KL,2 ) 3 m3/mol, c*1 ) 1 M and c* 2 ) c* 1/100.

Figure 2. Calculated surface coverage predictions θ1 for the same cases as in Figure 1.

and βm values usually range from 0 to -4.22,23 Nonionic/ anionic or nonionic/cationic mixtures have βσ and βm ranging from ca. -3 to -8. The strongest attractive interactions are for anionic/cationic systems for which βσ and βm range from -10 to -20. Repulsive interactions are observed for mixed hydrocarbon-fluorocarbon surfactants, where βσ and βm can range from 0 to 2.24 These parameters were determined using individual mixed cmc’s or tensions of mixed surfactants at one surface pressure.17 Even though βσ and βm are usually comparable, some calculations were made with βσ * βm to gauge the effect of their differences. Ideal mixing at the interface and within the micelle (βσ ) βm ) 0) is represented by case 1 in Figures 1-3. The tension for this mixture falls between those of the pure components for concentrations both below and above the (22) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1988. (23) Holland, P. M. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501: American Chemical Society: Washington, DC, 1992. (24) Zhao, G. X.; Zhu, B. Y. In Phenomena in Mixed Surfactants Systems; Scamehorn, J. F., Ed.; ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986.

Langmuir, Vol. 12, No. 2, 1996 357

Figure 3. Calculated surface coverage predictions θ2 for the same cases as in Figure 1.

cmc. This indicates that there is no synergism for ideal mixing. Synergism in surface tension reduction efficiency is defined when the surface tension of the mixture is less than that of either pure component at the same total concentration. Similarly, synergism in mixed micelle formation is defined when the cmc of the mixed micelle is lower than both the pure component cmc’s. In this example, there is a tension minimum at the cmc, as discussed elsewhere,20 since the surfactant with the lower cmc has a lower surface tension at the cmc. This occurs because the surface coverage θ1 shows a maximum at the cmc (Figure 2). At very low concentrations and surface coverages, both θ1 and θ2 increase with concentration. As the surface coverage increases, for steric and entropic reasons,6 it is more favorable for the smaller molecules to adsorb at the surface5 (since Π increases faster with c0i for larger Γm; see eq 7). The surface coverage for component 2 (the larger molecule) goes through a maximum, below the cmc (Figure 3). At micellar concentrations, more of component 1 enters the micelles, and the monomer concentration of component 1 drops (Figure 2). This allows for more of the larger molecules (2) to be adsorbed at the surface after micellization (Figure 3), and θ2 shows a minimum. For studying the effect of increased interactions within the micelles, a system which is ideal at the interface but has attractive interactions in the micelle (βm ) -5 ) is considered in case 2. Below the cmc the tension and area coverage are the same, as in the ideal case since βσ ) 0. However, since the molecules have attractive interactions within the micelles, the cmc is lower than for the ideal case. A lower cmc and a lower absolute value of |βσ| than |βm| indicate in this case a preference for mixed micellization over mixed adsorption. The minimum at the cmc is more pronounced than in case 1, and the surface tension at higher micellar concentrations is higher than that of either pure component. This indicates negative synergism in surface tension reduction effectiveness (higher minimum tension), because at concentrations above the cmc the surface coverage θ1 of the more efficient component is lower than for the ideal case. Another system which has net attractive interactions at the interface (βσ ) -5 ) but not in the micelle (βm ) 0) is considered as case 3, in which mixed adsorption is favored over mixed micellization. Case 3 is qualitatively plausible if the molecular density is higher in the mixed monolayer than in the mixed micelle. For premicellar concentrations, synergism is observed at low concentrations. Even though βσ < 0, and one would expect synergism

358


on the basis of Rosen’s condition,17 synergism is observed only above a certain concentration. This is the first model prediction of a concentration range for synergism. In fact, data for certain systems which are discussed later show evidence of such behavior. In case 3, the tension at the cmc is lower than that of either pure component. This is synergism in surface tension reduction effectiveness. A necessary conditions for this kind of synergism has been determined elsewhere.25 It has been shown that this synergism can exist when βσ < βm, and this is supported by our model. In this case, because of attractive interactions on the surface layer, more of component 2 (and less of component 1) is adsorbed at the interface than when βσ ) 0. The total surface coverage for any concentration increases as βσ increases. This causes a reduction in surface tension below that of either pure component and leads to synergism. Case 4 is for βσ ) βm ) -5, for which the molecules at the interface and in the micelles have similar interactions. For concentrations below the cmc, the tension and surface coverages are the same as for case 3. The cmc is lower than that in case 3, since βm is more negative. No synergism in surface tension reduction effectiveness (higher minimum tension) is observed (the surface tension at micellar concentrations is between those of the pure components). However, since there are stronger molecular interactions within the micelles, more of component 1 enters the micelles, thus reducing the monomer concentration of component 1 and causing more of component 2 to be adsorbed at the interface. For βσ ) -5 and βm ) -15 (case 5), molecular interactions within the micelles are much stronger than at the interface, mixed micellization is highly favored over mixed adsorption, and the cmc is much lower than that of either component. Since it is more favorable for the molecules to enter the micelles, the total surface coverage after micellization is less than for cases 3 and 4. Hence, the tension after micellization is much higher than that of either pure component, indicating strong negative synergism in surface tension reduction effectiveness. For case 6, βσ ) -15 < βm ) -5, the strongest synergism and the lowest minimum tension are observed, consistent with Rosen’s criterion.17,30 The surface coverages for both components 1 and 2 are high, because attractive interactions cause cooperative adsorption. The mixed micellar σ m cmc, c*, is also lower than c*1 and c* 2. If β ) β ) -15, minimum tension is as low as in case 6 and the mixed cmc is the same as in case 5. However, since βσ is not smaller than βm, no synergism is observed in the minimum tension. After gaining some insight into the effects of βσ and βm in this section, we now apply this model to two real systems below. 3. Application of the Model to C12E8/SDS Data An extensive search of the literature revealed few sets of tension data for binary mixtures of nonionic surfactants which included pure component data at the same conditions as the mixture. The option of getting missing data at the same conditions as the mixture was considered but rejected, because then one might not reproduce well the conditions of these literature sources. Data of Lange and Beck16 for SDS and C12E8 in 0.5 M NaCl were chosen for (25) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1988, 125, 730. (26) Carless, J. E.; Challis, R. A.; Mulley, B. A. J. Colloid Sci. 1964, 19, 201. (27) Lin, S. Y.; McKeigue, K.; Maldarelli, C. AIChE J. 1990, 36, 1785. (28) Chang, C.-H.; Franses, E. I. J. Colloid Interface Sci. 1994, 164, 107. (29) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (30) Zhu, B. Y.; Rosen, M. J. J. Colloid Interface Sci. 1984, 99, 435.


Figure 4. Data fits for SDS (1,9) and C12E8 (2,b) of Lange and Beck16 with the Langmuir-Szyszkowski equation. The parameters are Γm,1 ) 4.8 × 10-6 mol/m2, KL,1 ) 7.8 m3/mol, c* 1 ) 4.09 mM, Γm,2 ) 3.17 × 10-6 mol/m2, KL,2 ) 2.2 × 102 m3/mol, and c* 2 ) 0.55 mM. Table 1. Calculated βσ and βm Parameters for the C12E8/SDS Data16 c*, mM, from

b

y1

βσ

βm

a

0 0.2 0.4 0.6 0.8 0.9 0.96 1.0

N/A -2.6 -2.6 -2.5 -3.0 -3.1 -3.5 N/A

N/A -2.5 -2.4 -2.2 -2.6 -2.6 -3.1 N/A

0.55 0.52 0.58 0.75 1.04 1.34 1.67 4.09

b N/A 0.54 0.61 0.76 0.96 1.30 1.69 N/A

a Found from visual inspection of the data (Figures 5 and 6). Found from a fit of all the data to the model (see Section 2.1.5).

testing the model, even though they were available only for a limited concentration range around the cmc. The data were read from a plot of tension vs log c. The SDS surfactant behaves thermodynamically as effectively nonionic at high ionic strengths (0.5 M NaCl). Data for pure SDS and C12E8 have been found to fit well to the Langmuir isotherm.26,27 The mixture data have been fit for the first time for the entire concentration range using the model of section 2.1.3. The pure component data for SDS (component 1) and C12E8 (component 2) were fitted with the LangmuirSzyszkowski equation. The parameters obtained for SDS are Γm,1 ) 4.85 × 10-6 mol/m2, KL,1 ) 7.8 m3/mol, and c*1 ) 4.09 mM; for C12E8 we use Γm,2 ) 3.17 × 10-6 mol/m2, KL,2 ) 2.2 × 102 m3/mol, and c* 2 ) 0.55 mM (Figure 4). At these conditions, SDS is the smaller molecule and is less surface active than C12E8 for any given premicellar concentration (C12E8 is more efficient in reducing surface tension). Tension data at lower concentrations would be required for more accurate determination of these parameters. Mixture data for the SDS/C12E8 system are plotted and fitted using the nonideal adsorbed solution model (Table 1). βσ ranges from -2.5 to -3.5, and βm ranges from -2.2 to -3.1. The cmc’s determined from βσ and βm were compared to the visually determined cmc and are in agreement to within 7%. For clarity, the mixture data for y1 ) 0.2, 0.4, and 0.6 are plotted in Figure 5 and for y1 ) 0.8, 0.9, and 0.96 in Figure 6. For y1 ) 0.2 and 0.4, a slight synergistic effect in the surface tension reduction efficiency


Figure 5. Data and fits for the SDS/C12E8 mixture16 for y1 ) 0 (b), y1 ) 0.2 (]), y1 ) 0.4 (0), y1 ) 0.6 (O), and y1 ) 1.0 (9). The fitted parameters are given in Table 1. Results for higher mole fractions are given in Figure 6. S indicates synergism and NS no synergism.

Langmuir, Vol. 12, No. 2, 1996 359

Figure 7. Calculated surface coverages, θ1, of SDS, for the results of Figures 5 and 6.

Figure 8. Calculated surface coverages, θ2, of C12E8, for the results of Figures 5 and 6. Figure 6. Same as Figure 5 but for higher mole fractions. y1 ) 0 (b), y1 ) 0.8 (0), y1 ) 0.9 (O), y1 ) 0.96 (]), and y1 ) 1.0 (9).

is observed at concentrations slightly lower than the cmc (indicated by S in the figure). If this model is extrapolated to lower concentrations (for which no data are available), these mixtures show no synergism. Similarly for mixed micelle formation, even though βm < 0 there is no synergism. This indicates that negative values of βσ and βm may not always guarantee synergistic behavior at given values of y1 and cT, i.e., the conditions βσ < 0 and βm < 0 are necessary for synergism but not sufficient. With the NAS model and the parameters given in Table 1, it is possible to calculate surface coverages for the entire concentration range (Figures 7 and 8). As y1 increases, the monomer concentration of SDS in the bulk solution increases, causing an increase in the surface coverage. Since SDS is less surface active than C12E8 (at these conditions), one finds θ2 > θ1 even when the bulk solution is predominantly SDS (y1 ) 0.96 ). The mole fractions of SDS at the interface, x1, and in the micelle, x1,m, are shown in Figure 9. For increasing premicellar concentration, there is little change in the mole fraction of SDS at the interface. After micellization, however, the mole fraction of SDS increases along with the mole fraction of SDS within the micelles. Micellization occurs at concentrations when the surface coverages are high and the interactions in the mixed monolayer are more significant. This favors

Figure 9. Calculated surface and micellar mole fractions of SDS, for the results of Figures 5 and 6.

adsorption of the smaller molecules (SDS). Similarly, the mole fraction of SDS within the micelle is higher than that of C12E8. The monomer concentration of C12E8 decreases due to its inclusion within the micelle, allowing for higher mole fractions of SDS at the interface. Although no direct data on surface coverages and surface compositions are available for testing the predictions of the model,

360



Figure 10. Tension data for C12E5 (y1 ) 1.0, O) and Triton X-100 (y1 ) 0, 0) and fitted curves based on the LangmuirSzyszkowski equation.

Figure 11. Tension data for mixed C12E5/Triton X-100 and a fitted curve based on the nonideal adsorbed solution (NAS) model; y1 ) 0.25 (b), y1 ) 0.5 (9), and y1 ) 0.75 (2). βσ and βm for different y1 are given in Table 3.

Table 2. C12E5 and Triton X-100 Parameters for the Szyszkowski-Langmuir Equation (Eq 7) C12E5 param

Triton X-100 lit.26,34

exptl

Γm,i, 8.0 ((0.7) × mol/m2 KL,i, 2.2 ((0.3) × 102 m3/mol c*, 4 × 10-5 mol/L

10-6

7.5 ×

10-6

1.9 × 102

lit.27

exptl 2.8 ((0.2) ×

10-6

1.8 ((0.3) × 103

3.5 × 10-5 3.3 × 10-4

2.9 × 10-6 1.5 × 103 2.9 × 10-4

the predictions have thermodynamic significance (based on the Gibbs isotherm), to the extent of course that the equilibrium data are accurate. 4. Materials and Experimental Methods C12E5(CH3(CH2)11(OCH2CH2)5OH) and Triton X-100 ((CH3)3CCH2C(CH3)2C6H4(OCH2CH2)9.5OH) were obtained from Sigma Chemicals and from Fluka Chemicals, respectively, and were used as received. Ultrapure water from Millipore’s four-stage ion-exchange filtration Milli-Q system was used for making all solutions. Tension was measured using the Wilhelmy plate method with a Kru¨ss interfacial tensiometer with a roughened platinum plate attached to a precision torsion balance. The tensiometer allows for temperature control with a circulating water bath. All tension measurements were made at 25 °C. These tension data were reproducible on the tensiometer to within 1 mN/m and agreed with Triton X-100 data from the pulsating bubble surfactometer28 and the inverted bubble method27 to within 2 mN/m.

Figure 12. Calculated tensions for the data shown in Figure 11. The curves are shown here again to better highlight cases of synergism. S indicates synergism and NS no synergism. Table 3. Parameters As Calculated from the NAS Model and by Using Rosen’s Method for Constant Surface Pressure y1

βσ

γ, mN/m

βσ (γ)35

βm

βm (c*)23

0.25

-0.2((0.4)

65 50 35

0.3 0.3 -0.1

-0.6((0.2)

-0.9

0.50

-0.2((0.4)

65 50 35

-0.6 -0.3 2.0

-0.5((0.3)

1.7

0.75

-1.6((0.8)

65 50 35

-4.6 -7.9 1.2

-1.5((0.3)

-7.1

5. Results: Data and Fitting to Model The tension data for pure C12E5 and Triton X-100 are plotted in Figure 10, and good fits can be obtained using the Szyszkowski-Langmuir equation (eq 7). The parameters obtained are given in Table 2 and compare favorably (within 7%) to the parameters determined in the literature.27 C12E5 has a lower cmc, and the surface tensions of C12E5 and Triton X-100 above the cmc’s are about the same. These surfactants have an interesting tension behavior. Whereas at low concentrations Triton X-100 is more efficient in reducing surface tension, at high concentration it produces higher tensions and a crossover of the tension-concentration curves. C12E5 is smaller (larger Γm,1, Table 2) and less surface active (lower KL,1 ) than Triton X-100. Mixture data for C12E5/Triton X-100, for y1 ) 0.25, 0.5, and 0.75 are plotted in Figure 11. The nonideal adsorbed solution (NAS) model describes these data well. Mixture parameters βσ and βm, obtained for these data (Table 3),

are small for this nonionic mixture, as expected. These parameters were found to vary with y1. This variation may be due in part to the use of a model (section 2.1.3) in which no area change upon mixing is assumed. At higher concentrations, the higher the mole fraction y1 the lower the tension, indicating that the presence of C12E5 is more important for reducing surface tension. The calculated mixture tensions are compared to those of the pure components in Figure 12. For y1 ) 0.75, this figure shows more clearly a limited concentration range for synergism from ca. 1.5 × 10-6 to 2.0 × 10-5 M. No


Langmuir, Vol. 12, No. 2, 1996 361

Figure 13. Calculated surface coverages, θ1, of C12E5, for the results of Figure 12.

Figure 16. Monomer concentration (c1) and micellar inventory (cm) of C12E5, for the results of Figure 12.

Figure 14. Calculated surface coverages, θ2, of Triton X-100, for the results of Figure 12.

(Γm,i) become more important than adsorption free energies (KL,i).6 The mole fraction of C12E5, x1, at the interface increases as the concentration increases (Figure 15). For micellar (post-cmc) concentrations, this mole fraction decreases as the monomer concentration of C12E5 decreases. Although the mole fraction of C12E5 at the interface and within the micelles decreases with the total concentration, cT, the monomer concentration continues to decrease because of the rapid increase in the amount of micelles (Figure 16). The monomer concentration of C12E5 decreases more rapidly for y1 ) 0.25 than for y1 ) 0.75. For a mixture with y1 ) 0.75, more C12E5 molecules are available for either adsorption at the interface or formation of micelles. The regular solution model parameters were also determined at constant surface pressure (βσ(γ)) and from the visually determined mixed cmc (βm(c*)), according to the procedures suggested by Rosen.17,30 These parameters are reported in Table 3, where βσ(γ) has been calculated at γ ) 65, 50, and 35 mN/m, and βm for the mixed cmc reported in Table 2. The values of βσ(γ) calculated by this method vary with surface tension. A mixture that is evaluated to be synergistic at a particular surface pressure may not be so at other surface pressures. Evaluating the synergism of a mixture only at a particular surface pressure may not imply synergism at other surface pressures or concentrations. The parameter βm(c*) calculated with this method relies on the visually determined mixed cmc and may not fit well tension data above the cmc. We think that an improved method is to fit the postcmc tension data using the NAS model and then calculate the mixed cmc using eq 16.19 The NAS model used provides good fits for the tension data of C12E5/Triton X-100 and can be used for calculating surface coverages of individual surfactants in a mixture. Data for individual surface densities would surely be useful for additional testing or refinements of the model.

Figure 15. Calculated surface and micellar mole fractions of C12E5, for the results of Figure 12.

synergism is observed for y1 ) 0.25 and 0.5, even though βσ is negative. Figures 13-15 show the calculated surface coverages and mole fractions for the data in Figure 12. The surface coverages θ2 are greater for Triton X-100 than for C12E5. At high concentrations, the surface coverage of Triton X-100 shows a maximum, as steric and entropic factors

6. Discussion In this article, two regular solution model parameters, βσ and βm, are determined by fitting the pre-cmc and the post-cmc tension data. Although Rosen,17,25 Holland,23,29 and others have also used such parameters for describing synergism in surface tension and mixed micelle formation, they have not used the NAS model or the LangmuirSzyszkowski equation and have calculated βσ at constant Π (or constant ln(c01/c02)) and βm only from the mixed cmc.

362



These values of βσ may vary with Π. The values of βm may not represent the tension behavior of post-cmc concentrations, since they are based only on the visually determined mixed cmc, c*. In our approach, the values of βm are determined from the entire range of concentrations above the cmc, since tensions for mixtures above the cmc can vary with concentration. The parameters calculated with our approach may vary slightly with y1 (Tables 1 and 3), perhaps because the model used neglects the area change upon mixing. The area change upon mixing should not be important at low Π’s and low total surface coverages. The estimated values of βσ and βm are nearly equal, since these parameters are a measure of intermolecular interactions, which should not be grossly different at the interface layer than in the micelle. The NAS model can be readily extended to multicomponent systems and even to ionic/nonionic mixtures, both below and above the cmc. Equations of state (eqs 8 and 9) can be used for relating surface coverages to surface tension in dynamic adsorption calculations.5 The NAS model can effectively account for (i) molecular size differences, (ii) nonideal interactions between molecules, and (iii) area changes upon mixing. Other models which can account for molecular size differences are the Frumkin-Damaskin (FD)31 and the scaled particle theory (SPT).32 Although both these isotherms can account for molecular size differences, only the FD isotherm considers nonideal interactions between molecules at the interface. The FD isotherm, unlike the NAS model, has not yet been generalized to consider area changes upon mixing. Finally, adsorption of different sized surfactants at the oil/ water interface has been reported, but it is restricted to the use of the Butler equation.33

7. Conclusions A new general approach for determining nonideal interactions in adsorption and their effects on tension has been developed. The regular solution model parameters are determined for the simplest model. With these parameters it is possible to calculate the mixed cmc, the surface coverages, and the tension using a single set of equations, unlike in the cases of the Gibbs-Hutchinson or Rosen’s method. The NAS model is fairly flexible and can be used with other models, besides the regular solution theory for nonideal interactions. Moreover other isotherms for the individual components adsorption can be used, as dictated by the data. The nonideal adsorbed solution model can be used effectively for understanding equilibrium adsorption of surfactants of different sizes. Larger molecules adsorb favorably at low concentrations when the total surface coverages are low. At higher surface coverages, adsorption of the larger molecules is hindered relative to the smaller molecules for steric and entropic reasons.6 However, as attractive interactions between molecules increase (βσ becomes more negative), more of the larger molecules can be adsorbed. The new model can also account for synergism and can help determine the concentration and mole fraction ranges for synergism. Two binary mixtures (C12E8/SDS and C12E5/Triton X-100) were used for fitting the tension data, and the parameters βσ and βm were determined at each value of y1 from γ(cT) data. Good fits were obtained for the tension data using this model. The C12E5/Triton X-100 mixture exhibits a range of synergism which supports the model predictions. Area changes upon mixing may need to be considered for better fits, especially at the higher surface coverages.

(31) Damaskin, B. B. Elektrochimija 1969, 5, 346. (32) Talbot, J.; Jin, X.; Wang, N.-H. L. Langmuir 1994, 10, 1663. (33) Lucassen-Reynders, E. H. Colloids Surf. A 1994, 91, 79. (34) Sagitani, H.; Hayashi, Y.; Ochiai, M. J. Am. Oil Chem. Soc. 1989, 66, 146. (35) Rosen, M. J.; Gao, T.; Nakatsuji, Y.; Masuyama, A. Colloids Surf. A 1994, 88, 1.

Acknowledgment. This research was supported in part by the National Science Foundation (Grants #BCS 91-12154, and CTS 93-04328) and by the Purdue Research Foundation. LA9506032

Equilibrium Adsorption and Tension of Binary Surfactant Mixtures at

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