Surface Tension and Adsorption Synergism for Solutions of Binary

Equilibrium tension data for aqueous C12E8/SDS (in 0.5 M NaCl) and our own data for C12E5/Triton X-100 were analyzed for tension synergism. Adsorbed ...
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Ind. Eng. Chem. Res. 1996, 35, 3223-3232

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Surface Tension and Adsorption Synergism for Solutions of Binary Surfactants Faisal A. Siddiqui and Elias I. Franses* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283

The phenomenon of synergism in tension or adsorption for binary surfactants in water, for concentrations above and below the cmc, is studied using the nonideal adsorbed solution (NAS) model with one-parameter (βσ) regular solution model for interactions at the interface. Another parameter (βm) is used for intramicellar interactions in the pseudophase separation model. Ranges of concentrations, compositions, and surface tensions are determined for which a mixture is synergistic below or above the cmc. Equilibrium tension data for aqueous C12E8/SDS (in 0.5 M NaCl) and our own data for C12E5/Triton X-100 were analyzed for tension synergism. Adsorbed surface densities were calculated from tension data and “maps” showing areas of tension or adsorption synergism are calculated for the first time. Although tension synergism is predicted when βσ < 0, adsorption synergism is possible even when βσ ) 0. These results can be useful in designing surfactant formulations of controlled tension or surface composition with minimal amounts. 1. Introduction Mixtures of surfactants are often designed to attain lower surface tensions than would have been possible with single surfactants. Net attractive interactions between surfactants at the interface can cause increased adsorption and lowering of surface tension over those of the individual surfactants at the same total concentration. This phenomenon is called synergism. Synergism in tension can be important in foaming, wetting, emulsification, detergency, and flotation (Rosen, 1989). Certain conditions for predicting the existence of synergism have been reported (Rubingh, 1978; Hua and Rosen, 1982; Holland and Rubingh, 1983). These conditions (as detailed later) are restrictive by being surface pressure or concentration dependent. In this paper, composition regions of synergistic surface tension behavior are identified using a realistic equilibrium adsorption and tension nonideal adsorbed solution model (Siddiqui and Franses, 1996). This model (Section 2.1) for equilibrium adsorption and tension is based on the adsorbed solution model of Myers and Prausnitz (1965) and considers differences in molecular sizes or adsorption capacities (Γm,i), equilibrium adsorption constants (KL,i), and intermolecular interactions in the mixed monolayer (as measured by the activity coefficients, ζi). The model extends isotherms of the pure components to describe mixture behavior. In this article we have taken, for simplicity and with supporting data, that both pure components follow the one-component Langmuir isotherm. This isotherm relies on a lattice model where each molecule adsorbs ideally on one (imaginary) surface site. The number of sites depends on the adsorption capacity. The NAS model extends the Langmuir one-component lattice model to molecules of different adsorption capacities. It also considers nonideal mixing which has been modeled with the regular solution theory, for simplicity, and can be generalized to other models. The regular solution theory considers nonzero enthalpy of mixing with zero entropy of mixing, which also implies no molar area change upon mixing (Hildebrand and Scott, 1962). * Phone: (317) 494-4078. Fax: (317) 494-0805. Email: [email protected].

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Other models of binary surfactant tension and adsorption have recently been proposed by Wu¨stneck et al. (1993), Talbot et al. (1994), Lucassen-Reynders (1994), and Jan´czuk et al. (1995). These models differ from the general NAS model by being restricted to the same isotherm equation for each pure component, to the regular solution model, and to no area change upon mixing at the interface. In this article, a single-parameter (βσ) regular solution model is used to describe the activity coefficients in the mixed monolayer; βσ is assumed to be constant for all mole fractions and concentrations. A model for activity coefficients which considers area changes upon mixing is given by Talu et al. (1995). If βσ is a function of surface pressure or concentration, then a more general model may be used as described by Siddiqui and Franses (1996). Adsorbed densities can be calculated using the NAS model based on parameters obtained in a continuous way, by fitting tension data rather than the pointwise method of Gibbs-Hutchinson (Hutchinson, 1948). Calculated surface densities at the air/water interface are therefore indirect, unlike the evidence by radiotracer or surface techniques. This article aims to determine ranges of concentration for which a binary surfactant mixture is synergistic in tension, either below or above the cmc (Section 3). The previous approach of Rosen and co-workers determined that if βσ < 0, there may be some range of concentrations for tension synergism below the cmc, but they did not specify this range, because of lack of a general model. The general model used here is not actually restricted to a one-parameter regular solution model and can be extended to post-cmc concentrations for the first time. Concentration and composition ranges for which synergism in surface adsorption, or total mixture adsorption greater than that of pure components at the same concentration, are discussed in section 4. Surface adsorption at the air/water interface may be particularly important for foam stability and surface viscosity. Analogous synergism in surface adsorption may also be important at the gas/solid or liquid/solid interfaces. For post-cmc concentrations, tension data are fitted with the NAS model by obtaining the micellar inventories from the pseudophase micellar separation model. © 1996 American Chemical Society

3224 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Although no explicit analytical conditions were developed for synergism at micellar concentrations, the concentration/composition range for synergism were evaluated for certain examples. Sufficient conditions for synergism in mixed micelle formation are well known (Hua and Rosen, 1982, 1988; Holland, 1992), for the regular solution model with a parameter βm. Premicellar and post-cmc tension data of a binary mixture are fitted using the NAS model to obtain the parameters βσ and βm. Regions of concentration and composition for synergism can be obtained from these parameters. Analogous concentration and composition regions of synergism for adsorption densities calculated using the NAS model can also be plotted. For illustrative purposes, two sets of tension data were used for predicting tension and adsorption synergism. Data for a limited concentration range and a single-mixture composition were obtained from the literature (Rosen and Zhao, 1983) for the sodium dodecylsulfate and dodecyl octaethylene glycol ether (SDS/C12E8) mixture, in 0.5 M NaCl. Another mixture of dodecyl pentaethylene glycol ether and polyethylene glycol tertoctylphenyl ether (C12E5/Triton X-100) was used to obtain data for over a wider concentration range (Siddiqui and Franses, 1996).

Szyszkowski-Langmuir equation:

Π01 ) Πm,iRT ln(1 + KL,ic0i ) ) -Γm,iRT ln(1 - θ0i ) (4) where the fractional area coverages are

θ0i ≡ Γ0i /Γm,i

Equation 4 is derived from the Gibbs adsorption isotherm and the Langmuir isotherm when the solution is ideal or when it is nonideal, and the activity coefficient is independent of concentration. Otherwise, more elaborate equations should be derived for both the adsorption equation and the surface equation of state (Baikerikar and Hansen, 1991; Chang and Franses, 1995). Equating the chemical potential of component i in the bulk and at the interface for an ideal bulk solution above and below the cmc yields (Myers and Prausnitz, 1965) the following:

yicT ) xiζic0i

Π ) Π01 ) Π02 Γ1 Γ01

Γ2 +

Γ02

)1

(1) (2)

Γi are the Gibbs surface excess densities and Π(≡γ0 γ) is the surface pressure (γ0 is the surface tension of pure solvent) of a mixture with concentration ci. Πi0 is a reference surface pressure corresponding to a pure component concentration of ci0. Γi0 is the Gibbs surface excess density for a pure component concentration of ci0 evaluated using any single-component isotherm. The Langmuir isotherm is used in this article for each component.

KL,ic0i Γ0i ) Γm,i 1 + KL,ic0i

(3)

where Γm,i is the maximum monolayer surface density, and KL,i is the adsorption equilibrium constant. The pure component surface pressure is given by the

(6)

The micellar solution is assumed to be ideal, and the interactions within the micelles are nonideal and modeled using the regular solution theory (section 2.3). Equations 3-6 yield

(

2. Nonideal Adsorbed Solution Model 2.1. Equilibrium Tension and Adsorption Model. The nonideal adsorbed solution (NAS) model for equilibrium tension and adsorption considers nonideal intermolecular interactions. Siddiqui and Franses (1996) have developed this model for surfactants at the air/ water interface and have used it to fit data for nonionic/ nonionic binary surfactants. For ionic/ionic surfactants this model needs to be further developed. However, this model can be used for approximate calculations for anionic/cationic mixtures which typically have βσ < 0 and large |βσ|. A brief summary is given here. The key idea is the formation of a mixed monolayer by mixing single-surfactant monolayers of the same total surface pressures. This model starts from equilibrium between monolayers and bulk solution and can be reduced to the following system of equations for no area change upon mixing (eqs 1-6):

(5)

c1 1 + KL,1 ζ1x1

) ( Γm,1

)

c2 ) 1 + KL,2 ζ2 x2

Γm,2

(7)

and

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

(8)

where cT ≡ c1 + c2, Γt ≡ Γ1 + Γ2, yi ≡ ci/cT, and xi ≡ Γi/Γt. The activity coefficient ζi is calculated from the excess Gibbs free energy of mixing in the monolayer. The one-parameter regular solution model leads to (Hildebrand and Scott, 1962)

ζi ) exp[βσ(1 - xi)2]

(9)

where βσ is the nonideal interface molecular interaction parameter. Negative βσ implies attractive interactions between the adsorbed surfactants. An additional term should be considered in eqs 2 and 8 for area change upon mixing. If the concentrations c1 and c2, pure component parameters Γm,i and KL,i, and nonideal interaction parameter βσ are known, the surface coverages Γ1 and Γ2 can be calculated from eqs 7-9. From Γ1 and Γ2, the surface tension can be calculated as

(

ci γ ) γ0 - Π0i ) γ0 - Γm,iRT ln 1 + KL,i ζixi

)

(10)

2.2. Equation of State: γ(Γ1,Γ2). A system of equations relating surface tension (γ) to coverages (Γi) can be developed using the NAS model. From eqs 1-5 one gets

(1 - θ01)RΓ ) (1 - θ02) θ1 θ01

θ2 +

θ02

)1

(11) (12)

where θi ) Γi/Γm,i and RΓ ≡ Γm,1/Γm,2. Then eq 4 implies

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3225

eled using the NAS model with monomer concentrations, if one assumes that micelles are not surface active. This assumption is reasonable, since normally the tension of pure surfactants remains nearly constant above the cmc. For calculating monomer and micellar concentrations, one needs an equilibrium micellization model. The pseudophase micellization model for nonionic surfactants (Kamrath and Franses, 1983; Hall, 1987) is the simplest model that can be used to obtain realistic estimates of these concentrations, if the micelle size is large and intermicellar interactions are not strong. In this model the surfactant material balance equations are

ci + xi,mcm ) yicT

Figure 1. Surface tension plotted as a function of surface coverage, θ1 ≡ Γ1/Γm,1 and θ2 ≡ Γ2/Γm,1, using the surface equation of state (eqs 11 and 12). The parameters are Γm,1 ) 9.0 × 10-6 mol/m2 and Γm,2 ) Γm,1/3.

that

(13)

For given surface area coverages θ1 and θ2, the reference surface area coverages θ10 and θ20 can be calculated from eqs 11 and 12. The surface tension can then be calculated from eq 13. This yields the equation of state γ(θ1,θ2), or γ(Γ1,Γ2). Surprisingly, in this model the surface equation of state is shown to be independent of the equilibrium adsorption constants (KL,i) and the intermolecular interaction parameter (βσ). Nonetheless, for a given bulk concentration (ci), the parameters Γm,i, KL,i, and βσ do determine the extent of surface coverage and therefore, do influence the surface tension indirectly. As in the pure component surface equation of state (eq 13, for one component only), the surface tension depends only upon the surface coverage and Γm,i. This evidently results directly from the use of the Langmuir isotherm which is based on a lattice model. Perhaps with other isotherms not relying on lattice models or with nonideal entropy of mixing, monolayer interactions may affect the surface equation of state. For a hypothetical system for which component 2 has three times larger Γm,i (Γm,1 ) 9 × 10-6 mol/m2), an example is shown in Figure 1 as a three-dimensional plot. The reduction in surface tension of pure 1 (when θ2 ) 0) with increasing θ1 is less pronounced than for pure 2 (when θ1 ) 0). For the larger molecule, the fractional area coverage (θ2) is greater for the same molecular surface density (mol/m2) at the interface. This causes an increased reduction in surface tension of the larger molecules. As the total surface coverage, θt ) θ1 + θ2, approaches unity (which implies failure of the model at high surface coverages), the calculated surface tension γ tends asymptotically to minus infinity. For avoiding this anomaly, the surface tension is assumed to be zero for θt greater than or equal to 0.999. Figure 1 illustrates how the surface tension depends only on the surface coverages (Γ1 and Γ2) and the relative capacities (Γm,1 and Γm,2) for this system. Such information can be useful in estimating dynamic surface tension from calculated dynamic surface densities and can provide the basis for inferring surface intermolecular interactions. 2.3. Mixed Micellar Model. Tensions and surface densities at concentrations above the cmc can be mod-

(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. Equality of chemical potentials of surfactant i in the bulk solution and in the micellar pseudophase yields (for an ideal bulk solution, as in the adsorption model)

ci ) xi,mζ*ic*i γ ) γ0 + Γm,iRT ln(1 - θ0i )

i ) 1, 2

i ) 1, 2

(15)

where ci* is the individual cmc and ζi* is the activity coefficient in the micelle. ζi* can be evaluated using either the regular solution model (an equation similar to eq 9), with βm as the parameter instead of βσ, or the other models. As before, the critical micelle concentration of the mixtures, c*, can be calculated for the regular micellar solution model from the following equation from the pseudophase separation model (Hua and Rosen, 1982; Holland and Rubingh, 1983): m 2 yic* ) x* ic* i exp[β (1 - x* i) ]

i ) 1, 2

(16)

where xi* is the mole fraction of component i in the mixed micelle at the critical micelle concentration, c*. From eqs 14-16 one finds x1,m, c1, and c2, and cm, for cT g c*. From these and the adsorption model (section 2.1) one can now calculate Γ1, Γ2, and γ at concentrations for which the previous approaches (of Gibbs-Hutchinson and Rosen) do not work, since now a complete model is available. 2.4. Determination of βσ and βm from Data. Previously, the parameter βσ was calculated at a fixed surface pressure Π at which the reference concentrations are c10 and c20 (Rosen et al., 1994). Also, the parameter βm was determined from the mixed cmc’s and the individual cmc’s only (Rubingh, 1978; Holland, 1992; Rosen et al., 1994; Rosen, 1994). Here we use the following procedure. The parameters Γm,i and KL,i are first determined by fitting the Langmuir-Szyszkowski equation (eq 4) to the pure component tension vs concentration data. Then, the parameters βσ and βm are determined from the tension data (γ(cT)) on mixtures at some value of y1. βσ is determined from the pre-cmc data using the NAS model (section 2.1) and minimizing the variance (mean square deviation) between the experimental and calculated tension values. βm is then determined by minimizing the variance in tension for concentrations above the cmc, using the determined value of βσ. With this procedure, the values of βσ and βm most consistent with all the available data are obtained. 3. Premicellar Tension Synergism 3.1. Conditions for Tension Synergism. Hua and Rosen (1982) first explored the conditions for the exist-

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Figure 2. Schematic representation of concentration/composition curves indicating the region of synergism and the point of maximum synergism. The thin line is the boundary for the synergistic region.

ence of synergism with premicellar concentrations. These conditions (βσ < 0 and |βσ| > |ln(c10/c20)|) of synergism were derived by using eq 6 and recognizing that (i) when synergism exists there is a minimum in the cT versus y1 curve (at constant surface tension γ) and (ii) maximum synergism is attained when this curve shows a minimum (Figure 2). It has been shown that for the activity coefficient, ζi modeled with the regular solution theory (eq 9), maximum synergism is attained when x1 ) y1 (Hua and Rosen, 1982). In Appendix A, it is shown that this condition for maximum synergism is generally valid for any solution model used for the activity coefficient. Concentration and composition ranges for which synergism is observed can be evaluated from the cT-y1 curves (at constant Π) calculated using the NAS model. Such curves are schematically shown in Figure 2. Synergism at surface pressure Π is observed for the composition and concentration range for which cT is lower than either c10 or c20 (in this case it is taken that c20 < c10). For surface pressure Π3, the composition range is greater than for that Π2. At Π1, synergism is not observed for any composition range. The thin line joining the points where cT is equal to c20, at the various values of Π, is the boundary for the region of synergism and provides the basis for several calculations described below. Maximum synergism is attained when the cTy1 curve shows a minimum, as indicated on the figure. Rosen and co-workers have derived analytical expressions for the existence of synergism when such a minimum is observed. A line joining all the points of maximum synergism at different Π can also be drawn. Regions of synergism and lines of maximum synergism were calculated for two examples, SDS/C12E8 and C12E5/ Triton X-100. 3.2. Tension Synergism for SDS/C12E8. A binary system of sodium dodecyl sulfate (SDS; component 1) and dodecyl octaethylene glycol ether (C12E8; component 2) in 0.5 M NaCl can be described as a nonionic/nonionic model, since SDS behaves thermodynamically as effectively nonionic at high ionic strength (Hiemenz,

Figure 3. Data fits for SDS (1, b, y1 ) 1) and C12E8 (2, [, y1 ) 0) of Rosen and Zhao (1983) with the Langmuir-Szyszkowski equation. The parameters are Γm,1 ) 6.45 × 10-6 mol/m2, KL,1 ) 33 m3/mol, Γm,2 ) 2.67 × 10-6 mol/m2, and KL,2 ) 5.6 × 103 m3/ mol. Mixture data (y1 ) 0.91, 0) are fitted with the NAS theory. The resulting optimal value of βσ ) -3.2. Solid lines: fits for pure surfactants. Broken line: fit for mixture.

1986). For the data reported by Rosen and Zhao (1983) (Figure 3) the pure component parameters, Γm,i and KL,i, are calculated here from fits of eq 4. Data are available for only one mixture composition (y1 ) 0.91) and a limited concentration range. This system is considered here for illustrative purposes, and it is assumed that βσ fitted for this mole fraction is the same at other mole fractions. The mixture data were fitted with our NAS model to obtain the nonideal interaction parameter βσ ) -3.2. This compares well with the value of -3.1 calculated by Rosen and Zhao (1983) with their constant tension (or constant Π) scheme. The available data show no synergism for the entire concentration range. This shows that a negative βσ does not guarantee synergism at y1 ) 0.91, but at some unspecified ranges of y1 and cT. To specify these ranges for this system we will use the NAS model and βσ ) -3.2 in Figures 4-6. Figure 4 shows calculated concentration-composition curves for γ ) 65, 50, and 35 mN/m. As discussed in the previous section, when synergism exists there is a minimum in the cT-y1 curve and the total concentration (at fixed γ) is lower than the pure component concentration (c10 or c20). At γ ) 65 mN/m, the curve has no minimum, and no synergism is expected. At γ ) 50 mN/ m, synergism is expected for y1 ranging from 0 to 0.12, and for γ ) 35 mN/m, synergism is expected for y1 ranging from 0 to 0.57. No synergism is expected at any concentration for y1 ) 0.91, in agreement with the available data. For γ ) 65 mN/m, the value of |ln(c10/ c20)| ) 36, which is greater than |βσ| ) 3.1, and by Rosen’s criterion no synergism is expected for the entire mole fraction range. For γ ) 50 and 35 mN/m, |ln(c10/ c20)| equals 2.88 and 1.68, respectively. Hence, synergism is expected to be observed at least for some range of y1. This prediction remains to be tested. Using the cT-y1 curves for various surface tensions, one can generate a “map” showing the ranges (areas) of synergism and the lines of maximum synergism (section 3.1). Therefore, using some data for pure components, the data at y1 ) 0.91, and our model can lead to a comprehensive map of predicted synergism at all cT’s and y1’s, provided that the model is good with βσ

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3227

Figure 4. Concentration versus y1 curves for the system in Figure 3 at surface tensions γ ) 35, 50, and 65 mN/m (as indicated). The pure component concentrations at fixed surface tension are indicated as c10 and c20. For γ ) 35 mN/m, the range of synergism (y1 ) 0-0.61) and the point (y1 ) 0.24) of maximum synergism (minimum in cT at fixed γ) are noted. The broken vertical line for y1 ) 0.91 corresponds to Figure 3.

Figure 5. Surface tension-composition map (for each y1 and γ, cT is fixed) of synergism for the system in Figure 3. Solid line separates regions of synergism and no synergism; broken line indicates maximum synergism.

Figure 6. Concentration-composition map (for each y1 and cT, γ is fixed) of synergism for the system in Figure 3; same symbols as in Figure 5.

constant. In Figures 5 and 6, the expected regions of synergism and the points of maximum synergism for the mixtures are shown. The broken line in Figure 5 is the line of maximum synergism. For the example of y1 ) 0.91 (Figure 3), the mixture is theoretically

Figure 7. Data fits for C12E5 (1, 4, y1 ) 1) and Triton X-100 (2, O, y1 ) 0) with the Langmuir-Szyszkowski equation; surfactants dissolved in pure water at 25 °C. The parameters are Γm,1 ) 8.0 × 10-1 mol/m2, KL,1 ) 220 m3/mol, Γm,2 ) 2.8 × 10-6 mol/m2, and KL,2 ) 1.8 × 103 m3/mol. Mixture data (y1 ) 0.75, [) are fitted with the NAS theory. The resultant optimal value for βσ ) -1.6. Solid lines: fits for pure surfactants. Broken line: fit for mixture. Lines of constant tension illustrate existence of synergism or no synergism. See predicted synergism maps in Figures 8 and 9.

expected to be synergistic only for concentrations greater than 2 × 10-4 M (perhaps post-cmc), for which the model has to be augmented (see section 5), or for surface tensions well below 25 mN/m for which no data are available. 3.3. Tension Synergism for C12E5/Triton X-100. Tension data for dodecyl pentaethylene glycol ether (C12E5, component 1), polyethylene tert-octylphenyl ether (TX100, component 2), and a mixture with C12E5 mole fraction y1 ) 0.75 were obtained in our laboratory. The data (at 25 °C) were obtained with the Wilhelmy plate and the bubble method and are detailed elsewhere (Siddiqui and Franses, 1996). Even though TX100 is a mixture of different molecular weight components, it can be modeled by a single-component Langmuir isotherm. Fitted with eq 4, the individual component data yielded the values of Γm,i and KL,i (Figure 7). C12E5 has a larger adsorption capacity (smaller area per molecule) and is less surface active than TX100. At low concentrations, TX100 adsorbs more than C12E5 and reduces tension to a greater extent. At higher concentrations, C12E5 (the smaller-area molecule) adsorbs preferentially for steric reasons and lowers the tension to a greater extent, resulting in the observed crossover (at cT ) 6.54 × 10-6 M, Figure 7), where the pure components have the same surface tension. At high (γ ) 72.8-63 mN/m) and low (γ e 40 mN/m) surface tensions, the curve obtained by fitting the mixture data to the NAS model at y1 ) 0.75 does not show synergism. Synergism is expected for concentrations from 1.56 × 10-6 to 2.03 × 10-5 M. The ranges of tensions and concentrations for which the system is expected to be synergistic are mapped in Figures 8 and 9. The regions of synergism and the line of maximum synergism are quite different, quantitatively and qualitatively, for this mixture than for SDS/C12E8. For y1 ) 0.75 (Figure 7; vertical thin lines in Figures 8 and 9), the mixture is synergistic for surface tensions between 63.5 and 39 mN/m. For a hypothetical mixture of the same individual components but for βσ ) -3.0, the tension range of synergism would have been broader, from γ between 72.8 to about 26 mN/m, as shown in Figure 8 for illustrative purposes. For βσ < -3 the regions of synergism would have been even wider.

3228 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

Figure 8. Map of surface tension and composition for the system in Figure 7. Thick solid lines: synergism boundary for βσ ) -1.6. Broken line: line of maximum synergism for βσ ) -1.6. Thin solid lines: synergism boundary for βσ ) -3.0. Thin vertical line indicates y1 ) 0.75.

Figure 9. Map of concentration and composition for the system in Figure 7. Same symbols as in Figure 8.

For y1 ) 0.75 the synergistic range of concentrations is from 1.6 × 10-6 M to 2.0 × 10-5 M and maximum synergism is expected at cT ) 1.3 × 10-5 M (Figure 9). For such a mixture at y1 ) 0.75, if one had βσ ) -3.0, then synergism would be observed from very low concentrations to 4.5 × 10-5 M. Mixtures with larger negative values of βσ (typically found in anionic/cationic mixtures), may exhibit larger areas of synergism, and the simple criterion of βσ < 0 may then be all that is needed. 4. Conditions for Adsorption Synergism Synergism in adsorption at the air/water interface is defined when the total mixture adsorption (Γt ) Γ1 + Γ2) is larger than the pure component adsorption (Γip). This phenomenon may be important in free surface flows, foam stability, and foam fractionation. By using the NAS model, it is possible to predict individual surface coverages of mixtures from parameters obtained by fitting tension data. Adsorption synergism may become possible when the net attractive interactions between adsorbates cause higher surface coverages at the monolayer than those due to surface/adsorbate interactions. The NAS model, which has been shown to describe well the equilibrium tension data for several mixtures,

Figure 10. Calculated surface densities (Γ1, Γ2, and Γt ) Γ1 + Γ2) versus cT at fixed y1 for the mixture in Figure 7. Points A, B, and C correspond to concentrations 1 × 10-7 M, 1 × 10-6 M, and 1 × 10-5 M for y1 ) 0.75.

can be used to determine the possible ranges of adsorption synergism. The conditions for adsorption synergism are qualitatively different from those of tension synergism, since they are defined at constant concentration, whereas tension synergism is defined at conditions of constant surface pressure (eq 1). This is for convenience, since the condition of constant surface pressure is difficult to describe in an explicit analytical form when dealing with surface adsorption. In an example of such differences (see below), the condition βσ < 0 is not necessarily associated with adsorption synergism. At present only numerically obtained cT- and y1- ranges for adsorption synergism are available in this article. At a fixed constant concentration cT, if Γt is plotted as a function of y1, the pure component surface coverages are Γ1p and Γ2p, when y1 ) 1 or 0, respectively. The presence of a maximum in the Γt-y1 curve (at constant cT) indicates adsorption synergism (mixture adsorption greater than Γip at the same total concentration). Γty1 curves can be plotted (see below) indicating areas of synergism and line of maximum synergism. To illustrate such a phenomenon, for which no direct data have been reported (to our knowledge), we have used the approach outlined above to the the mixture examined in section 3.3. For concentrations below the cmc, calculated surface coverages for the C12E5/TX100 mixture are shown in Figure 10. The model predicts that the total mixture surface coverage Γt (at y1 ) 0.75) is greater than the pure component surface coverage, and adsorption is expected to be synergistic for concentrations from 5.01 × 10-7 to 3.2 × 10-6 M (Figure 10). At lower concentrations, the surface density of TX100 increases to a maximum and then drops. However, increased adsorption of C12E5 (the more surface active component) causes Γt to increase and can cause synergism. At higher concentrations, the density of TX100 decreases, for steric reasons as detailed previously. Γt is no longer greater than the pure component adsorbed density (Γip). In Figure 10, we have marked, for convenience, points A and C in the nonsynergistic region and point B in the synergistic region. The calculated surface densities are plotted in Figure 11 for the same example and for cT ) 10-7, 10-6, and 10-5 M. Synergism is expected for a broad range, y1 ) 0-0.92, at cT ) 10-6 M, and for a narrower range, y1 ) 0-0.48 (see figure caption) for 10-7 M. Maximum synergism at these concentrations is expected at y1 )

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3229

could also be important for adsorption to other interfaces such as gas/solid or liquid/solid, if there are significant negative deviations from ideality. 5. Conditions for Tension Synergism above the CMC

Figure 11. Calculated total surface density (Γt) versus composition at fixed total concentration for the C12E5/Triton X-100 mixture in Figure 7, for βσ ) -1.6. The range of synergism (y1 from 0 to 0.92) and the point (y1 ) 0.5) of maximum synergism (maximum in Γt at fixed cT) are indicated. Points A, B, and C correspond to the points in Figure 10. The broken line represents the hypothetical ideal case with no interactions (βσ ) 0). For c ) 10-7 M, a horizontal line (not shown to avoid crowding) starting from Γ02 (at y1 ) 0) intersects the solid line at y1 ) 0.48.

Figure 12. Concentration/composition adsorption synergism (for βσ ) -1.6) map for the mixture in Figure 7. A, B, and C correspond to those points in Figure 11.

0.5 and y1 ) 0.21, respectively. For y1 ) 0.25, synergism is expected at point B (Figure 10). By using various Γt-y1 curves at different concentrations one obtains the concentration-composition map of expected adsorption synergism shown in Figure 12. For C12E5/TX100 mixture, for which the larger molecule is more surface active, there is usually a nonzero concentration (cT ) 1.6 × 10-6 M, in this example) at which the surface densities of the two pure components are the same, i.e. Γ1p ) Γ2p (the point of intersection of the Γt-cT curve in Figure 10). At this concentration, synergism can be expected for all values of y1, since Γip would be the lowest surface density at that concentration. Surface densities as a function of y1 at fixed cT were calculated for βσ ) 0, at fixed cT (Figure 11). For βσ ) 0 an area of synergism is observed for concentrations about 1.6 × 10-6 M. Adsorption synergism, while plausible, can only be inferred from indirect equilibrium tension data. Direct adsorption density data (with optical, spectroscopic, or radiotracer techniques) would be useful for testing these predictions. Nonetheless, this article points toward the possibility of synergism in surface adsorption, which

5.1. Subsection Mixed Micelle Formation and Post-cmc Tensions. Synergism in mixed micelle formation is defined when the mixed cmc (c*) is lower than either pure cmc (ci*). Conditions for synergism in mixed micelle formation have been determined by Hua and Rosen (1982) for the one-parameter regular solution model and are analogous to the conditions derived for pre-cmc tension synergism: (i) βm < 0 and (ii) |βσ| > |ln(c1*/c2*)|. These necessary and sufficient conditions rest on the assumption that the pseudophase separation model describes micellization adequately and the oneparameter regular solution model describes fairly the mixing in micelles. Previously, the mixed cmc has been used to determine βm from c*, c1*, and c2*, from eqs 15 and 16 (Hua and Rosen, 1982; Holland and Rubingh, 1983; Kamrath and Franses, 1983; Holland, 1992). This method assumes that there is a sharp break in tension at the cmc. However, since both βσ and βm influence tension above the cmc, using all tension data above the cmc in determining the optimal βm may provide better estimates (section 2.4). Post-cmc tensions have not been studied previously for synergism, because (i) it has been assumed that tension remains constant with increasing concentration and (ii) the constant-γ and the Gibbs-Hutchinson approach cannot be applied as they are restricted to precmc concentrations. Zhu and Rosen (1984) have studied conditions for synergism in tension reduction effectiveness at the cmc. Although this information is useful, the conditions are complex: (i) βσ - βm < 0 and (ii) |βσ - βm| > |ln(c10,cmcc2*)/(c20,cmcc1*)| where ci0,cmc is the concentration of individual surfactants which yields a surface tension equal to that of the mixture at cmc and is obtained by extrapolating pure component tension data to the mixture cmc. Since these conditions apply only at the cmc, they cannot describe synergism at postcmc concentrations at which tensions may change and the behavior may remain synergistic or become nonsynergistic. Hence, to probe for possible synergistic behavior, it is important to consider extended concentration ranges above the cmc. 5.2. Sample Calculations. Sample calculations (Figure 13) were done for two hypothetical mixtures with parameters similar to those of a real mixture: Γm,1 ) 5.0 × 10-6 mol/m2, Γm,2 ) Γm,1/3, KL,1 ) 3.0 × 103 m3/ mol, KL,2 ) KL,1/10, c1* ) 10 mM, c2* ) 50 × c1*, and the premicellar-mixing parameter βσ ) -5. The mixtures have micellar-mixing parameters βm ) 0 (broken line) or βm ) -2 (solid line), in order to gauge the effect of βm upon the range of synergism. Mixture compositions of y1 ) 0.25 and 0.75 are considered. These calculations were done to illustrate (i) the effect of βm, (ii) tension variation for post-cmc concentrations, and (iii) the range of concentrations for post-cmc synergism. The mixed cmc for βm ) 0 and -2 are between the pure cmc’s, since the second condition (|βm| > |ln(c1*/ c2*)|) is not satisfied. The cmc for βm ) -2 is lower than for βm ) 0, since a smaller βm implies more favorable micellization. However, when micellization occurs at higher concentrations (at βm ) 0), the tension at the cmc is lower. Since βσ ) -5 . βm in this example, micellization is favored at higher concentrations than with the

3230 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996

concentration of a mixture (βm ) -2) at point C were higher, the tension would increase and no synergism would be expected. Increasingly negative values of βm would indicate a greater tendency for the molecules to form mixed micelles, rather than adsorb at the air/water interface. This would cause a smaller reduction in tension at higher micellar concentrations and a shrinking of the region of synergism. 6. Concluding Remarks

Figure 13. Sample calculations to illustrate regions of micellar synergism. The parameters for this mixture are Γm,1 ) 5.0 × 10-6 mol/m2, KL,1 ) 3.0 × 103 m3/mol, cmc1 ) 10.0 mM, Γm,2 ) Γm,1/3, KL,2 ) KL,1/10, and cmc2 ) 50 × cmc1. Premicellar tension calculations are for βσ ) -5.0; micellar tension calculations for βσ ) 0 or -2.

Figure 14. Map of synergism at premicellar and post-cmc concentrations for the mixture in Figure 13. Regions of synergism are shaded. A, C, D, and E are points which correspond to similar points on Figure 13. Thick solid lines: boundaries of synergism. Dotted line: line of maximum synergism.

pure components, and tensions are lower at the cmc. Mixtures with strong net attractive interactions at the interface and weak interactions within the micelle (i.e. βm very different from βσ) are less likely than mixtures with βm = βσ. For y1 ) 0.25 and βm ) -2, the micellar mixture is synergistic from point A (cT ) 0.03 M) to point C (cT ) 0.63 M). For concentrations higher than that at point C, the surface tension is expected to be higher than that of pure 1, and there is no synergism. Points A and C define the concentration range for synergism at y1 ) 0.25 and βm ) -2. These points are also shown on the synergism “map” in Figure 14. The premicellar nonsynergistic point D and the mixed cmc point E are also shown in Figures 13 and 14. The premicellar concentration/composition ranges for synergism have been calculated with a similar method as in Figures 6 and 9. The micellar concentration/ composition ranges for synergism were obtained by calculating the synergistic range of concentrations for various y1, using curves such as those in Figure 13. As βm decreases, with βσ fixed, the area of micellar synergism decreases (Figure 14). For micellar synergism, there is a upper limit for concentration beyond which the mixture may not be synergistic (Figure 14). If the

General conditions for synergism in adsorption and tension are explored using the NAS model. Equations are developed for pure components following the onecomponent Langmuir isotherm. Molecular interactions at the interface (βσ) and within the micelle (βm) are modeled with the one-parameter regular solution model. Concentrations above and below the cmc are considered for tension synergism. Micellization thermodynamics is modeled using the pseudophase separation model. Conditions for adsorption synergism are analyzed using adsorbed densities calculated with the NAS model from parameters obtained by fitting tension data. Concentration and composition ranges for synergism are analyzed and plotted, for cases where βσ is the same for other y1. By plotting cT-y1 curves at various values of Π, maps showing concentration and composition areas of synergism are drawn. Such maps are useful in formulating mixtures to attain specific surface tensions and surface compositions using minimal amounts of surfactant. Tension data for two mixtures SDS/C12E8 and C12E5/Triton X-100 were obtained and expected regions of tension synergism were plotted. For micellar concentrations, a hypothetical mixture is considered to highlight the effect of βσ and βm. Tensions for micellar concentrations are not assumed to be constant. Concentration and composition ranges for synergism are plotted for the micellar mixture. Although no explicit analytical conditions for adsorption synergism are available, concentration and composition ranges for synergistic behavior are plotted as maps indicating areas of synergistic adsorption. Surface adsorbed densities calculated using the NAS model provide indirect evidence of synergism and give plausible predictions of ranges of synergism. Acknowledgment This research was supported in part by the National Science Foundation (Grants Numbers BCS 91-12154 and CTS 93-04328) and by the Purdue Research Foundation. Nomenclature c* ) cmc of the mixed micelle, mol/L ci ) concentration of i in the mixture, mol/L ci* ) critical micelle concentration (cmc) of pure i, mol/L ci0 ) reference concentration of pure i, mol/L cm ) concentration of the micelle in the mixture, mol/L cT ) total concentration of the mixture, mol/lL KL,i ) equilibrium adsorption constant, L/moL R ) universal gas constant, 8.315 Nm/K RΓ ) relative adsorption capacity, Γm,1/Γm,2 T ) temperature, K xi ) mole fraction of i at the interface, Γi/Γt xi* ) mole fraction of i in the mixed micelle at cmc xi,m ) mole fraction of i in the mixed micelle yi ) mole fraction of i in the bulk solution, ci/cT

Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 3231

Rearranging eq A.5 and using eq A.7 reduces to

Greek Symbols βm

) micellar molecular interaction parameter using the regular solution model βσ ) interface molecular interaction parameter using the regular solution model γ ) surface tension of solution, mN/m γ0 ) surface tension of the pure solvent, mN/m Γi ) surface excess coverage of i in a mixture, mol/m2 Γi0 ) surface excess coverage of pure i at Π0i, mol/m2 Γip ) surface excess coverage of pure i, mol/m2 Γm,i ) maximum monolayer surface coverage of pure i, mol/ m2 Γt ) total surface excess coverage, ∑iΓi, mol/m2 ζi ) activity coefficient at the interface ζi* ) activity coefficient in the micelle θi ) fractional area coverage, Γi/Γm,i θi0 ) fractional area coverage at reference concentration ci0, Γi0/Γm,i Π ) surface pressure, mN/m Π0i ) surface pressure of pure i, mN/m

[

]

dy1 y1(1 - y1) 1 d ln ζ1 ) + dx1 1 - x1 x1 dx1

(A.8)

Equations A.4 and A.8 imply that

y1(1 - y1) ) y1 1 - x1

(A.9)

from which it follows that

y1 ) x1

(A.10)

Therefore for maximum synergism, the surface mole fraction is equal to the bulk mole fraction for any nonideal monolayer solution model. Literature Cited

Appendix A. General Conditions for Maximum Synergism

By using the Gibbs-Duhem equation, it is shown below that maximum synergism is obtained when x1 ) y1 for any nonideal solution model (activity coefficient ζi) and not just for the regular solution model. For synergism to exist, there should be a minimum in the cT versus y1 curve; hence,

dcT )0 dy1

(A.1)

One then takes the logarithm of eq 6, for components i ) 1 and 2,

ln y1 + ln cT - ln x1 - ln ζ1 - ln c01 ) 0 (A.2) ln(1 - y1) + ln cT - ln(1 - x1) - ln ζ2 - ln c02 ) 0 (A.3) and differentiates eq A.2 with respect to y1. Using eq A.1 one gets

[

]

dy1 1 d ln ζ1 ) y1 + dx1 x1 dy1

(A.4)

Subtracting eq A.3 from eq A.2 and differentiating with respect to y1 leads to

[

]

d ln ζ1 d ln ζ2 dx1 1 1 )0 + dx1 dx1 dy1 y1(1 - y1) x1(1 - x1) (A.5) From the Gibbs-Duhem equation for this mixture,

d ln ζ1 d ln ζ2 x1 + x2 )0 dx1 dx2

(A.6)

and since x2 ) 1 - x1, eq A.6 leads to

d ln ζ2 x1 d ln ζ1 ) dx1 1 - x1 dx1

(A.7)

Baikerikar, K. G.; Hansen, R. S. Inference of Surface Excess and Its High Activity Limit in Monolayer Adsorption from Interfacial Tension and Activity Data. Langmuir 1991, 7, 1963. Chang, C. H.; Franses, E. I. Adsorption Dynamics of Surfactants at the Air/Water Interface: A Critical Review of Mathematical Models, Data and Mechanisms. Colloids Surf. A 1995, 100, 1. Franses, E. I.; Siddiqui, F. A.; Ahn, D. J.; Chang, C. H.; Wang, N.-H. L. Thermodynamically Consistent Equilibrium Adsorption Isotherms for Mixtures of Different-Size Molecules. Langmuir 1995, 11, 3177. Hall, D. G. Micellar Effects on Reaction Rates and Acid-Base Equilibria. J. Phys. Chem. 1987, 91, 4287. Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker, Inc.: New York, 1986. Hildebrand, J. H.; Scott, R. L. Regular Solutions; Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1962. Holland, P. M. Modeling Mixed Surfactant Systems. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; American Chemical Society: Washington, D.C., 1992. Holland, P. M.; Rubingh, D. N. Nonideal Multicomponent Mixed Micelle Model. J. Phys. Chem. 1983, 87, 1984. Hua, X. Y.; Rosen, M. J. Synergism in Binary Mixtures of Surfactants. I. Theoretical Analysis. J. Colloid Interface Sci. 1982, 90, 212. Hua, X. Y.; Rosen, M. J. Conditions for Synergism in Surface Tension Reduction Effectiveness in Binary Mixtures of Surfactants. J. Colloid Interface Sci. 1988, 125, 730. Hutchinson, E. Mixed Monolayers. I. Adsorbed Films at Air-Water Surfaces. J. Colloid Sci. 1948, 3, 413. Janczuk, B.; Bruque, J. M.; Gonza´lez-Martin, M. L.; DoradoCalasanz, C. The Properties of Mixtures of Ionic and Nonionic Surfactants in Water at the Water/Air Interface. Colloids Surf. A 1995, 104, 157. Kamrath, R. F.; Franses, E. I. Thermodynamics of Micellization Pseudo-Phase Separation Models. Ind. Eng. Chem. Fundam. 1983, 22, 230. Lucassen-Reynders, E. H. Competitive Adsorption of Emulsifiers 1. Theory for Adsorption of Small and Large Molecules. Colloids Surf. A 1994, 91, 79-88. Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed-Gas Adsorption. AIChE J. 1965, 11, 121. Rosen, M. J. Selection of Surfactant Pairs for Optimization of Interfacial Properties. J. Am. Oil Chem. Soc. 1989, 66, 1840. Rosen, M. J. Predicting Synergism in Binary Mixtures of Surfactants. Progr. Colloid Polym. Sci. 1994, 95, 39. Rosen, M. J.; Zhao, F. Binary Mixtures of Surfactants. The Effect of Structural and Microenvironmental Factors on Molecular Interaction at the Aqueous Solution/Air Interface. J. Colloid Interface Sci. 1983, 95, 443. Rosen, M. J.; Gao, T.; Nakatsuji, Y.; Masuyama, A. Synergism in Binary Mixtures of Surfactants 12. Mixtures Containing Surfactants with Two Hydrophilic and Two or Three Hydrophobic Groups. Colloid Surf. A 1994, 88, 1.

3232 Ind. Eng. Chem. Res., Vol. 35, No. 9, 1996 Rubingh, D. N. Mixed Micelle Solutions. In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1978. Siddiqui, F. A.; Franses, E. I. Equilibrium Adsorption and Tension of Binary Surfactant Mixtures at the Air/Water Interface. Langmuir 1996, 12, 364. Talbot, J.; Jin, X.; Wang, N.-H. L. New Equations for Multicomponent Adsorption Kinetics. Langmuir 1994, 10, 1663. Talu, O.; Li, J.; Myers, A. L. Activity Coefficients of Adsorbed Mixtures. Adsorption 1995, 1, 103. Wu¨stneck, R.; Miller, R.; Kriwanek, J. Quantification of synergistic interaction of surfactants using a generalized FrumkinDamaskin isotherm. Colloid Surf. A 1993, 81, 1.

Zhu, B. Y.; Rosen, M. J. Synergism in Binary Mixtures of Surfactants. II. Effectiveness of Surface Tension Reduction. J. Colloid Interface Sci. 1984, 99, 435.

Received for review January 29, 1996 Revised manuscript received March 20, 1996 Accepted March 22, 1996X IE960044+ X Abstract published in Advance ACS Abstracts, August 15, 1996.