The Properties of a Binary Mixture of Nonionic Surfactants in Water at

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Langmuir 2007, 23, 4972-4981

The Properties of a Binary Mixture of Nonionic Surfactants in Water at the Water/Air Interface Katarzyna Szymczyk and Bronisław Janćzuk* Department of Interfacial Phenomena, Faculty of Chemistry, Maria Curie-Skłodowska UniVersity, Maria Curie-Skłodowska Sq. 3, 20-031 Lublin, Poland ReceiVed December 8, 2006. In Final Form: February 2, 2007

The behavior of mixed nonionic/nonionic surfactant solutions, that is, p-(1,1,3,3-tetramethylbutyl)phenoxy poly(ethylene glycol)s Triton X-100 (TX100) and Triton X-165 (TX165) have been studied by surface tension and density measurements. The obtained results of the surface tension measurements were compared with those calculated from the relations derived by Joos, Miller, and co-workers. From the comparison, it appeared that by using these two approaches the adsorption behavior of TX100 and TX165 mixtures at different mole fractions can be predicted. The negative deviation from the linear relationship between the surface tension and composition of TX100 and TX165 mixtures in the concentration range corresponding to that of the saturated monolayer at the interface, the values of the parameters of molecular interaction, the activity coefficients, as well as the excess Gibbs energy of mixed monolayer formation calculated on the basis of Rosen and Motomura approaches proved that there is synergism in the reduction of the surface tension of aqueous solutions of TX100 and TX165 mixture when saturation of the monolayer is achieved. The negative parameters of intermolecular interaction in the mixed micelle and calculations based on MT theory of Blankschtein indicate that there is also synergism in the micelle formation for TX100 and TX165 mixture. It was also found that the values of the standard Gibbs energy of adsorption and micellization for the mixture of these two surfactants, which confirm the synergetic effect, can be predicted on the basis of the proposed equations, which include the values of the mole fraction of surfactant and excess Gibbs energy TX100 and TX165 in the monolayer and micelle.

1. Introduction The choice of surfactant systems is a critical step for many applications, from laundering to tertiary oil recovery. In general, nonionic surfactants composed of a poly(ethylene oxide) chain to which a hydrophobic part is attached have widespread industrial and technological applications.1-4 There are several reasons for their frequent use: (i) the properties of each compound (e.g., solubility) which can be modified considerably by changing the length of the polyoxyethylene group; (ii) these surfactants are very effective in sterically stabilizing emulsions and dispersion; (iii) nonionic surfactants are suited for mixing with other surfactants, among other things. The last point seems to be very important, because in practical applications, nonionic surfactant mixtures are often used because they are usually more effective than a single surfactant.5-7 The effectiveness of mixed surfactant systems is related to specific interactions between molecules (ions) of different surfactants, which can enhance or deteriorate the action of a mixture with respect to some property of these systems. On the other hand, some effects that are not expected in single systems can take place in aqueous solution containing mixed surfactants, for example, synergism in surface tension reduction. The most frequently used surfactant pairs that show * To whom correspondence should be addressed. Phone (48-81) 5375649, fax (48-81) 533-3348, e-mail [email protected]. (1) Blin, J. L.; Leónard, A.; Su, B. L. J. Phys. Chem. 2001, 105, 6070. (2) Pe´rez-Are´valo, J. F.; Dominguez, J. M.; Terre´s, E.; Rojas-Hernańdez, A.; Miki, M. Langmuir 2002, 18, 961. (3) Myers, D. Surfactant Science and Technology, 2nd ed.; VCH Publishers, Inc.: New York, 1992. (4) Vincent, B.; Edwards, J.; Emmett, S.; Jones, A. Colloid Surf. 1986, 18, 261. (5) Desai, T. R.; Dixit, S. G. J. Colloid Interface Sci. 1996, 177, 471. (6) Lopez-Diaz, D.; Garcia-Mateos, I.; Velaques, M. M. Colloid Surf., A 2005, 1, 153. (7) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 2965.

synergism are mixtures of ionic and nonionic surfactants.8-10 Synergism in this case results from interaction between different head groups. The determination of critical micelle concentrations (CMC) and the compositions of mixed micelles and adsorption layers at various interfaces seem to also be important for modeling the structure and properties of these systems, as well as various processes (adsorption, wetting, solubilization, micellar catalysis, etc.). Our earlier studies showed that, even for mixtures of two anionic surfactants, having different hydrophilic heads and a different length of hydrophobic alkyl tails,11 and also for cationic/ nonionic surfactant mixtures,12 there is no linear relationship among surface tension, critical micelle concentration and wettability of hydrophobic low energetic solids, and the composition of the mixtures. For those mixtures, the deviation from a linear relationship between the concentration excess at water-air and hydrophobic solid-water interfaces and their composition is also observed. It was interesting regardless of whether the deviation from a linear relationship between abovementioned parameters and composition of the mixture of two nonionic surfactants with a different number of poly(ethylene oxide) chains will take place. Thus, the purpose of our studies was to determine the adsorption behavior of mixed layers based on the equations of Gibbs, Joos, Miller, and co-workers,13-17 as well as the interaction between two nonionic surfactants in the (8) McCarroll, M.; Toerne, K.; Wandruszka, Rv. Langmuir 1998, 14, 7166. (9) Griffiths, P. C.; Whatton, M. L.; Abbott, R. J. J. Colloid Interface Sci. 1999, 215, 114. (10) Okano, T.; Tamura, T.; Abe, Y.; Tsuchida, t.; Lee, S.; Sugichara, G. Langmuir 2000, 16, 1508. (11) Janćzuk, B.; Zdziennicka, A.; Wo´jcik, W. Colloids Surf., A 2003, 220, 61. (12) Szymczyk, K.; Janćzuk, B. Colloids Surf., A, in press. (13) Rosen, J. M. Surfactants and Interfacial Phenomena; Wiley-Interscience: New York, 2004. (14) Joos, P. Bull. Soc. Chim. Belg. 1967, 76, 591.

10.1021/la063554+ CCC: $37.00 © 2007 American Chemical Society Published on Web 03/31/2007

Properties of Binary Mixture

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surface layers and micelles. For this purpose, the surface tension and density of aqueous solutions of p-(1,1,3,3-tetramethylbutyl)phenoxy poly(ethylene glycol)s Triton X-100 (TX100) and Triton X-165 (TX165) mixtures were measured.

monolayer can be evaluated, among other things, by using the equation derived by Rubingh and Rosen13,18,19 βσ )

2. Experimental Section 2.1. Materials. Triton X-100 (TX100), p-(1,1,3,3-tetramethylbutyl)phenoxy poly(ethylene glycol) (Fluka), and Triton X-165, p-(1,1,3,3-tetramethylbutyl)phenoxy poly(ethylene glycol) (Fluka), were used for preparation of aqueous solutions. Aqueous solutions of individual surfactants and TX100 and TX165 mixtures at different ratios of TX100 to TX165 were prepared using double-distilled and deionized water (Destamat Bi18E). The surface tension of water was always controlled before the solution preparation. 2.2. Methods. 2.2.1. Surface Tension Measurements. Surface tension measurements were made at 293 K with a Kru¨ss K9 tensiometer under atmospheric pressure by the ring method. The platinum ring was thoroughly cleaned and the flame dried before each measurement. The measurements were done in such a way that the vertically hung ring was dipped into the liquid to measure its surface tension. It was then subsequently pulled out. The maximum force needed to pull the ring through the interface was then expressed as the surface tension, γLV (mN/m). Measurements of the surface tension of pure water at 293 K were performed to calibrate the tensiometer and to check the cleanliness of the glassware. In all cases, more than ten successive measurements were carried out, and the standard deviation did not exceed (0.2 mN/m. The temperature was controlled within (0.1 K. 2.2.2. Density Measurements. We have measured the densities of water and aqueous solutions of the individual surfactants as well as TX100 and TX165 mixtures using a vibrating tube densimeter by Anton Paar, model DMA 5000. The accuracy of the thermometer and the density measurements were (0.01 K and (0.005 kg/m3, respectively. The precision of the density and temperature measurements given by the manufacturer was (0.001 kg/m3 and (0.001 K. The densimeter was calibrated regularly with distilled and deionized water. After measuring the density of water, more than three measurements of density were carried out at constant temperature of 293 K. 2.2.3. EValuation of the Surface Excess Concentration of Surfactant at Interface. The surface excess concentration of surfactants at the water-air interface can be determined on the basis of the adsorption isotherms using the Gibbs equation.13 For dilute solution (10-2 mol/dm3 or less) containing nonionic surfactant, the Gibbs equation can be written in the form C dγ 1 dγ 1 dγ Γ)))RT dC RT d ln C 2.303RT d log C

ln(RC12/X1C 01)

(2)

(1 - X1)2

where R is the mole fraction of surfactant 1 in the mixture of two surfactants, X1 is the mole fraction of surfactant 1 in the mixed monolayer, C 01 and C12 are the molar concentrations in the bulk of surfactant 1 and of the mixture of surfactants 1 and 2, respectively, required to produce a given surface tension value. X1 can be obtained from (X1)2 ln(RC12/X1C 01) (1 - X1)2 ln[(1 - R)C12/(1 - X1)C 01]

)1

(3)

where C 02 is the molecular concentration of surfactant 2 in the bulk required to produce a given surface tension. In the case of mixed micelles, it is possible to calculate the molecular interaction parameter, βM, from the relation of Rubingh and Rosen in the form18,19 βM )

M M ln(RC M 12/X 1 C 1 )

(4)

2 (1 - X M 1)

M where C M 1 , and C 12 are the critical micelle concentrations (CMC) of the individual surfactant 1 and the mixture of surfactants 1 and 2, respectively, and X M 1 is the mole fraction of surfactant 1 in the mixed micelle. XM 1 can be evaluated from the equation M M 2 (X M 1 ) ln(RC 12/X1C 1 ) M M M 2 (1 - X M 1 ) ln[(1 - R)C 12/(1 - X 1 )C 2 ]

)1

(5)

where C M 2 is the CMC of the individual surfactant 2. With the knowledge of the interaction parameters for the mixed monolayer and micelles, it is possible to determine the activity coefficient of the surfactants in the mixtures. From the nonideal solution theory, the activity coefficients of surfactants 1 and 2 in the M mixed film (f1 and f1) and mixed micelle (f M 1 and f 2 ) fulfill the respective conditions ln f1 ) βσ(1 - X1)2

(6)

ln f2 ) βσ(X1)2

(7)

M 2 M ln f M 1 ) β (1 - X 1 )

(8)

M 2 M ln f M 2 ) β (X 1 )

(9)

(1)

where C represents the concentration of surfactant and γLV its surface tension. The concentration of each surfactant at the interface can be calculated from the slope of γLV-log C plot. It is convenient if the dependence between the surface tension and the concentration of the aqueous surfactant solution can be expressed by the known mathematical function. 2.2.4. EValuation of the Molecular Interaction Parameters and Miscibility of Surfactants in the Adsorbed Film and Micelle. For surfactant mixtures, the characteristic phenomena are the formation of mixed monolayers at the interface and mixed micelles in the bulk solution. Most of the theories are based on the regular solution theory, and they have been applied to the phase separation model for the micelles and to the monolayer model for the adsorbed films in order to estimate the interaction parameter β in various binary surfactant systems.13 The molecular interaction parameter, β, for (15) Miller, R.; Fainerman, V. B.; Makievski, A. V.; Czichocki, G. Tenside, Surfactants, Deterg. 2001, 38, 3. (16) Fainerman, V. B.; Miller, R. J. Phys. Chem. B 2001, 105, 11432. (17) Fainerman, V. B.; Miller, R.; Aksenenko, E. V. AdV. Colloid Interface Sci. 2002, 96, 339.

and

From this theory, the following equations also result g ) RT(X1ln f1+ X2 ln f2)

(10)

M M M gM ) RT(X M 1 ln f 1 + X 2 ln f 2 )

(11)

and

where g and gM are the excess Gibbs energy of the mixed monolayer and micelle formation, respectively. (18) Hua, X. Y.; Rosen, M. J. J. Colloid Interface Sci. 1982, 87, 469. (19) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K., Ed.; Plenum Press: New York, 1979; p 337.

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Szymczyk and Jan´ czuk

Villeneuve et al.20,21 claimed that the treatment of intermolecular interaction parameters by Rubingh and Rosen18,19 is not appropriate in the sense that they do not take into account the presence of the solvent,20 and the physical significance of the parameter β is not clear when the excess entropy of mixing is not zero.21 They proposed a thermodynamic strategy to examine the miscibility of surfactants in adsorbed films and micelles by new concentration variables. The miscibility of surfactants in adsorbed films may be examined by the equation22,23

( )( ) X h 1X h 2 ∂m j m j ∂X h2

X h H2 ) X h2 -

X hM 2 )

() ()

(12)

where X h 1 and X h 2 for the mixture of two nonionic surfactants are defined20,23 as X h 1 ) m1/m j and X h 2 ) m2/m j , respectively, and m j fulfils the condition given by the equation (13)

where m1 and m2 are the molalities of nonionic surfactants 1 and 2, respectively. The magnitude of X h H2 is defined here by X h H2 )

Γ H2

(14)

Γ H1 + Γ H2

where Γ H1 and Γ H2 are defined with respect to two dividing planes chosen so as to make the excess numbers of moles water and air zero. On the basis of X h H1 and X h H2 determined from eq 12, it is possible to calculate the activity coefficients of surfactant 1, hf H1 , and surfactant 2, hf H2 , in the mixed monolayer at the interface as well as the excess Gibbs energy in this monolayer, gjH,E. These magnitudes, which allow us to draw conclusions about molecular interaction in the mixed adsorbed film, can be estimated from the following equations:22,23

( ) ( )

X h1

m j m j 01

2

) (fh H1 )2X h H1

(15)

X h2

m j m j 02

2

) (fh H2 )2X h H2

(16)

gj

)

RT(X h H1

ln

+

X h H2 ln

hf H2 )

(17)

Symbols m j 01 and m j 02 refer to the molality of the individual surfactants 1 and 2, respectively, at a given γLV. The composition of the surfactants in the micelle can be estimated from the following expression:22,23 X hM h2 2 )X

( )( ) X h 1X h 2 ∂C h C h ∂X h2

(18)

T,p

where C h is equal to m j at CMC. The magnitude of X hM 2 in the mixed micelle of two nonionic surfactants is defined by22 (20) Villeneuve, M.; Sakamoto, H.; Minamizawa, H.; Aratono, M. J. Colloid Interface Sci. 1997, 194, 301. (21) Motomura, K.; Aratono, M. In Mixed Surfactant System; Ogino, K., Abe, M., Eds.; Marcel Dekker: New York, 1993; p 99. (22) Motomura, K.; Ando, N.; Matsuki, H.; Aratono, M. J. Colloid Interface Sci. 1990, 139, 188. (23) Aratono, M.; Villeneuve, M.; Takiue, T.; Ikeda, N.; Iyota, H. J. Colloid Interface Sci. 1998, 200, 161.

C h C h 01

2

2 M ) (fh M h1 1) X

(20)

X h2

C h C h 02

2

2 M ) (fh M h2 2) X

(21)

gjM,E ) RT(X hM fM hM fM 1 ln h 1 +X 2 ln h 2)

(22)

Symbol gjM,E refers to the excess Gibbs energy of micelle formation per mole of the surfactant mixture, and C h 01 ) m j 01 and C h 02 ) m j 02, respectively, at CMC. On the basis of the above-mentioned equations, it is possible to establish the composition of the adsorbed mixed monolayer at the interface and the mixed micelle, as well as the molecular interactions in monolayer and micelle. 2.2.5. Equation of State Describing Mixed Adsorption BehaVior. Using the adsorption isotherm derived by Joos14 and modified by us11 for the systems including two nonionic surfactants, the surface adsorption behavior of the mixture of these surfactants can be predicted in a quite accurate way. The equation for the mixture of TX100 and TX165 can be written in the form11,14

( ) ( )

exp

( )

-Π -Π C1 -Π C2 + exp + exp )1 ∞ ∞ a RTΓ 0 RTΓ 1 1 RTΓ ∞2 a2

(23)

if their activity is close to C (for C < 10-2 M), where Γ ∞0 , Γ ∞1 , and Γ ∞2 are the maxima of the solvent adsorption and surfactants 1 and 2, respectively. Π is the surface pressure. The parameters a1 and a2 can be expressed as a1 ) exp

hf H1

X h1

and

and H,E

(19)

+ NM 2

M where N M 1 and N 2 are the excess numbers of molecules of the nonionic surfactants 1 and 2, respectively, per micelle particle of which the dividing surface between the bulk solution is defined so as to make the excess number of water zero.22,23 The chemical potentials and the activity coefficients in the micelle are defined similarly to those in the adsorbed film.22,23 Thus

T,p,γ

m j ) m1 + m2

NM 2 NM 1

(

)

µ S1 - µ B1 ω RT

a2 ) exp

(

)

µ S2 - µ B2 ω RT

(24)

where µS is the chemical potential in the surface under standard conditions, µB is the chemical potential in the bulk under standard conditions, and ω is the number of molecules of water per liter. Assuming that C2/C1 ) b ) constant, and Ctot ) C1 + C2 ) C1(1 + b), we obtain

( ) [ ( )

exp

( )]

-Π -Π 1 -Π b Ctot + exp )1 + exp ∞ ∞ a RTΓ 0 RTΓ 1 1 RTΓ ∞2 a2 1 + b (25)

Miller et al.,15-17 taking into account the assumption that for an ideal mixture of homologues a1 ) a2 ) a12 ) 0 and ω ) ω1 ) ω2, as well as Π h ) Πω/RT, Π h 1 ) Π1ω/RT, and Π h 2 ) Π2ω/RT, have derived the equation of state which relates the surface pressure of a surfactant mixture with the surface pressure of individual solutions. This equation can be expressed in the form exp Π h ) exp Π h 1 + exp Π h2-1

(26)

where a1, a2, and a12 are the constants of intermolecular interactions; ω1 and ω2 are the partial molar surface areas of surfactant 1 and 2, respectively; and Π1, Π2, and Π are the surface pressures of solutions of the individual surfactants and their mixture, respectively, equal


Langmuir, Vol. 23, No. 9, 2007 4975

Figure 1. Dependence of the surface tension of aqueous TX165 (curve 1) and TX100 (curve 6) solutions and their mixture at monomer mole fraction of TX100 ) 0.2, 0.4, 0.6, and 0.8 on log C. Table 1. Values of the Critical Micelle Concentrations (CMC), Negative Logarithm of the Concentration of Surfactants or Their Mixture in the Bulk Phase Required to Produce a 20 mN/m Reduction in the Surface Tension of the Solvent, pC20, Maximal Excess of Surfactant Concentration at Water-Air Interface, Γm, and Minimal Area Per Molecule, Am, for TX100, TX165, and Their Mixtures R

CMC (mol/dm3)

pC20

Γm (mol/m2)

Am (nm2)

0 0.2 0.4 0.6 0.8 1

5.41 × 10-4 4.35 × 10-4 3.42 × 10-4 2.73 × 10-4 2.72 × 10-4 2.90 × 10-4

4.4962 4.7292 4.8267 4.8544 4.8530 4.7254

2.22 × 10-6 2.53 × 10-6 2.67 × 10-6 2.70 × 10-6 2.73 × 10-6 2.83 × 10-6

0.7480 0.6562 0.6218 0.6145 0.6082 0.5870

to the difference between the surface tension of the solvent and 0 - γLV). solution (γ LV

3. Results and Discussion 3.1. Adsorption Isotherms. The measured values of the surface tension (γLV) of aqueous solutions of TX100 and TX165 and their mixtures are presented in Figure 1. This figure shows the dependence between γLV and log C (C represents the concentrations of TX100, TX165, and their mixtures at a given R) for aqueous solution of TX165 (curve 1) and TX100 (curve 6) and their mixtures (curves 2-4). From this figure, it appears that the shape of curve 1 is somewhat different than those of curve 6 and other curves; however, for both surfactants and their mixtures a linear dependence exists between γLV and log C near the critical micelle concentration (CMC) (Table 1). Because of these differences, it is possible that “efficiency” and “effectiveness” of the adsorption of TX100 are different than those of TX165 and their mixtures at a given R. The surface excess concentration at the surface saturation, Γm, is a useful measure of the adsorption effectiveness of the surfactant at the water-air interface, since it is the maximum value adsorption can attain, whereas the efficiency of the adsorption is related to the negative logarithm of the concentration of surfactants or their mixture in the bulk phase required to produce a 20 mN/m reduction in the surface tension of the solvent, pC20.13 On the basis of the adsorption isotherms (Figure 1), the surface excess concentration at water-air saturation was calculated by using the Gibbs equation of adsorption (eq 1) and then the minimal area (Am) per TX100 and TX165 and their mixture molecule at

Figure 2. Dependence of the surface tension of aqueous solutions of TX100 and TX165 mixtures on the monomer fraction of TX100, R, at total concentration of surfactants, C, ) 10-7, 10-6, 10-5, 10-4, and 2 × 10-4 M, respectively.

the interface from the relation

Am )

1 NΓm

(27)

where N is Avogadro’s number. The values of Γm and pC20 are presented in Table 1. From Table 1, it is seen that the values of Γm are arranged in the following direction:

TX165 < 0.2 < 0.4 < 0.6 < 0.8 < TX100 and those of pC20

TX165 < TX100 < 0.2 < 0.4 < 0.8 < 0.6 From the above series, we see that nonionic surfactant TX165 has the lowest efficiency and effectiveness in reduction of the surface tension of water and that the values of Γm and pC20 strongly depend on the composition of the mixture of TX100 and TX165, because the highest effectiveness is shown by the mixture at R ) 0.8, and the highest efficiency at R ) 0.6. To show the influence of the composition of mixtures on the water surface tension in Figure 2, the dependence between surface tension and monomer mole fraction of TX100, R, in the mixture is plotted. From this figure, it is seen that only at a very low concentration of surfactant mixtures is there an almost linear dependence between the surface tension and mole fraction of TX100 in the mixture. However, at concentration close to 10-5 M and higher, there is a negative deviation from the linear relationship between γLV and R. In other words, at concentrations corresponding to the beginning of the saturation monolayer formation, nonideal mixing of surfactants is evident. It suggests that the composition of the saturated monolayer at the water-air interface should be different from that of the surfactant in the bulk phase. This suggestion is confirmed by the data in Figure 3, which presents the relationship between the mole fraction of TX100 in the mixed monolayer for each R calculated from eq 3, X1, and the surface tension of the solution. From this figure, it is evident that the direction of change of X1 for all R is the same as γLV decreases, and all values of X1 grow as the surface tension becomes smaller. At R equal 0.2, 0.4, and 0.6 at γLV less than 60 mN/m, the mole fraction of TX100 in the mixed monolayer is bigger than that in the bulk phase. In the whole range of the

4976 Langmuir, Vol. 23, No. 9, 2007


Figure 3. Dependence of the monomer mole fraction of TX100 in the mixed monolayer, X1, on the surface tension of an aqueous solution of surfactant mixtures at different monomer mole fractions of TX100, R.

Figure 4. Dependence between the surface excess concentration at the surface saturation, Γm, calculated from the adsorption isotherms (curve 1), eq 30 (curve 2), eq 28 (curve 3), and eq 29 (curve 4) on the monomer fraction of TX100, R.

presented surface tension at R ) 0.8, the values of X1 are smaller than in the bulk phase. As mentioned above, the composition of the mixed monolayer can also be calculated from the eq 12 h H1 obtained proposed by Motomura at all.20-23 The values of X from this equation are smaller than those of X1 at about 0.02, which indicates that entropy plays an important role in the process of surfactant mixing in the monolayer. Because we proved that the composition of the saturated monolayer at the water-air interface calculated on the basis of Rosen and Motomura’s models is significantly different than the composition in the bulk phase, it seems to be necessary to calculate the values of Γm for the investigated mixtures on the basis of X1 and X h H1 values from the relation

Γm ) Γm1X1 + Γm2X2

(28)

h H1 + Γm2 X h H2 Γm ) Γm1X

(29)

where Γm1 and Γm2 are the maximal excess values of a single surfactant concentration at the water-air interface (Table 1). Figure 4 shows the values of Γm determined in different ways. Curve 1 in this figure represents the values calculated on the basis of an adsorption isotherm mixture of surfactants (Table 1) and curve 2 the values obtained from the relation

Γm ) Γ1R + Γ2(1 - R)

(30)

Curves 3 and 4 represent the values calculated from eqs 28 and 29. As is seen in Figure 4, there are essential differences between the values of surfactant concentration in mixed monolayers calculated on the basis of the adsorption isotherms (curve 1) and composition of the mixture in the bulk phase (curve 2) and monolayer (curves 3 and 4). However, there is a good agreement between the values of Γm calculated on the basis of the monomer mole fraction of surfactants in mixed monolayers obtained from eqs 3 and 12. This fact proves that mutual interactions between molecules or ions of surfactants in the mixed monolayer play an important role in the composition and packing of surfactants in this monolayer, and this composition has a valid influence on the properties of the monolayer but does not give an answer about synergistic or antagonistic properties of the mixture in

Figure 5. Dependence of the molecular interaction parameter, βσ, calculated from eq 4 on the surface tension of an aqueous solution of surfactant mixtures at different monomer mole fractions of TX100, R.

reduction of the surface tension of water. To explain these effects, the interaction parameter, βσ, should be calculated from eq 2. The calculated values of this parameter are presented in Figure 5. From this figure, it is seen that for all R this parameter has a negative value which changes with surface tension decrease of the aqueous solution of the surfactant mixtures. The smallest values of βσ exist for R ) 0.8 at a lower value of the surface tension that is at higher concentrations of the solutions, which may confirm a clear minimum in Figure 2. The negative values of the βσ parameter suggest that there is synergism in the surface tension reduction efficiency. However, the second condition for the existence of negative synergism must be fulfilled. This condition is that the absolute value of the βσ parameter should be greater than |ln(C 01/C 02)|, where C 01and C 02 are the concentrations of a single surfactant at a given surface tension. From the comparison of these two values (Table 2), it is evident that for each value of the surface tension corresponding to the mixed saturated monolayer at water-air and for surfactant mixtures at each composition there is a negative synergism in the surface tension reduction. By knowing the X1 and βσ values, it is possible to calculate the activity coefficients of surfactants 1 and 2 in the mixed film


Langmuir, Vol. 23, No. 9, 2007 4977

Table 2. Values of the |βσ| and |ln(C 01/C 02)| for TX100 and TX165 Mixtures γLV (mN/m)

|βσ| 0.2

|βσ| 0.4

|βσ| 0.6

|βσ| 0.8

|ln(C 01/C 02)|

65 60 55 52.8 50 45

1.5716 1.9735 1.7525 1.8392 1.8786 1.6176

2.2505 2.1469 1.9926 2.1129 2.1535 1.9471

0.9669 1.7696 1.9980 2.0765 2.3331 2.2990

1.1619 2.0745 2.2112 2.3366 2.4873 2.5966

0.9662 0.3717 0.2872 0.5294 0.7559 1.0657

Figure 7. Dependence of the activity coefficient in the mixed monolayer of TX100, f1 (eq 6), and hf H1 (eq 15), and TX165, f2 (eq 7), and hf H2 (eq 16) on the monomer fraction of TX100, R, at γLV ) 45 mN/m.

Figure 6. Dependence of the excess Gibbs energy in mixed monolayer, g, calculated from eq 10 on the surface tension of an aqueous solution of surfactant mixtures at different monomer mole fractions of TX100, R.

(f1 and f1) from eqs 6 and 7 and the excess Gibbs energy in this film from eq 10, which allows us to describe the interaction between surfactants in mixed monolayer and confirm the synergistic effect. From calculation of the activity coefficients, it appears that all their values are smaller than 1, which, according to Motomura’s model, indicates that interactions between TX100 and TX165 molecules are stronger than between single surfactants. This fact also confirms the values of the excess Gibbs energy of mixing presented in Figure 6. As it appears from this figure, the values of g for each R are smaller than zero. In Figure 7, there is a comparison between the values of the activity coefficient calculated from eqs 6, 7, 15, and 16 for γLV ) 45 mN/m. The results show that there is a small difference between the values calculated on the basis of these two approaches, which equal about 0.02. Bigger differences exist in Figure 8, which represents the relationship between the values of excess Gibbs energy of mixing, g and gjH,E, calculated from eqs 10 and 17, and the monomer mole fraction of TX100, R. The values of gjH,E (curve 2) are significantly lower than those of g (curve 1) for γLV ) 45 mN/m, which indicates that the excess Gibbs energy of mixing plays an important role in mixing monolayer formation. 3.2. Adsorption Isotherms of Joos and Miller et al. It is interesting to discover whether, on the basis of the theoretical isotherms of adsorption, it is possible to predict the surface tension for the mixtures of TX100 and TX165 for which synergism in reduction of the surface tension was proven in the whole range of concentrations at the saturated monolayer. Therefore, in Figure 9, the isotherms of Joos and Miller et al. are presented for R ) 0.2. Line 2 in Figure 9 reflects the function of γLV vs log C calculated from eq 26. The value of ω used for calculations in this equation, at first approximation, was assumed to be 2 × 105 m2/mol.17

Figure 8. Dependence of the excess Gibbs energy in mixed monolayer, g (eq 10), and gjH,E (eq 17) on the monomer fraction of TX100, R, at γLV ) 45 mN/m.

Line 3 in this figure reflects the dependence of the surface tension (γLV) of an aqueous solution of TX100 and TX165 mixtures on the total concentration of surfactants (log C) for R (R is the mole fraction of TX100 ) 0.2) calculated from eq 25 using the values of Γ∞ and a of the individual components. The values of Γ ∞0 , Γ ∞1 , Γ ∞2 , a1, and a2 used in eq 25 were determined from eq 23 from the data for individual surfactants (TX100 and TX165) on the assumption that C1 ) 0 or C2 ) 0, which are listed in Table 3. In all cases, it was assumed that the area occupied by water is close to 0.10 nm2, and thus, Γ ∞0 ) 16.6 × 10-6 mol/m2. The results presented in Figure 9 and those calculated for other mixtures show that the changes of γLV as a function of log C, for a given R, have the same shape. Of course, near CMC there is a linear dependence of γLV on log C. From Figure 9 and other calculations, it also appears that at low concentrations of surfactant mixtures there is a good agreement between the values of the surface tension of the solution measured and those calculated from eq 26 (curve 2) and from eq 25 (curve 3). Practically, the experimental points (points 1) are on the theoretical curves (curves 2 and 3) for all values of R. These facts indicate that by using the equation of state derived by Joos,14 and next modified by

4978 Langmuir, Vol. 23, No. 9, 2007


Figure 9. Dependence of the surface tension of aqueous solutions of TX100 and TX165 surfactant mixture on log C for R ) 0.2. Points 1 represent the measured values of the surface tension, and curves 2 and 3 represent the values of the surface tension calculated from eqs 26 and 25, respectively.

Figure 10. Dependence between the Gibbs energy of adsorption, ∆G 0ad, calculated from eq 31 (curve 1), eq 32 (curve 2), and eq 33 (curve 3) and monomer mole fraction of TX100, R.

Table 3. Parameters in Eq 25 for Water, TX100, and TX165

∆G oad ) X1∆G oad1 + X2∆G oad2 + g

(32)

∆G oad ) X h H1 ∆G oad1 + X h H2 ∆G oad2 + gjH,E

(33)

substance water TX100 TX165

Γ∞

(mol/m2)

16.6 × 10-6 3.15 × 10-6 2.40 × 10-6

a (mol/L)

and 1.95 × 1.98 × 10-4 10-6

us,11 and the simple model proposed by Miller15-17 it is possible to predict the surface tension of an aqueous solution of TX100 and TX165 mixtures in the whole range of their concentrations from 0 to CMC. Of course, it is impossible to predict the surface tension of the solution of TX100 and TX165 at the concentration close to CMC or higher than CMC. 3.3. The Standard Gibbs Energy of Adsorption. On the basis of the differences between the slopes of the linear parts of the curves (Figure 1), which represent the dependences between γLV and log C, and the differences between the values γLV at the same concentration for TX100 and TX165, respectively, we concluded that “efficiency” and “effectiveness” of the adsorption of TX100, TX165, and their mixtures were different. The adsorption efficiency is also related to the standard Gibbs energy of the adsorption, ∆G oad. The standard Gibbs energy of adsorption, ∆G oad, can be determined by different methods; among others, from the equation derived by Rosen and Aronson.13,24 If the surfactant concentration corresponding to the saturated monolayer at interface is lower than 1 × 10-2 M, the Rosen and Aronson equation can be expressed in the form

∆G oad ) 2.303RT log

calculated from the relations, respectively

C - NπAm ω

(31)

where ω is the number of water moles per decimeter cubed, and π is the surface pressure corresponding to the surfactant concentration, C, at which Am is achieved. From eq 31, the values of the standard Gibbs energy of TX100, TX165, and also for their mixtures were calculated. For these calculations, the values of C at γLV ) 45 mN/m were used and those of Am from Table 1. The calculated values of the standard Gibbs energy of TX100, TX165, and their mixtures are presented in Figure 10 (curve 1). Curves 2 and 3 in this figure represent the values of ∆G oad (24) Rosen, J. M.; Aronson, S. Colloids Surf. 1981, 3, 201.

where ∆G oad1 and ∆G oad2 are the standard Gibbs energy of TX100 and TX165, respectively. From this figure, it appears that ∆G oad values calculated from eqs 32 and 33, that is with respect to monomer mole fraction of surfactants in mixed monolayer, are considerably smaller that those calculated from eq 31; and by their minimum, they confirm the synergistic effect in surface tension reduction, which is largest at R ) 0.8. Moreover, the direction of changes of ∆G oad values calculated from these two equations is in agreement with the direction of changes of pC20, that confirms the efficiency of adsorption proven by pC20. 3.4. CMC. Another characteristic property of surfactants is their ability to form micelles. The concentration at which the micellization process takes place is called critical micelle concentration (CMC). In our studies, the values of CMC for TX100, TX165, and their mixtures were determined from the adsorption isotherms (Figure 1) and from the density measurements which are presented in Figure 11. This figure shows the CMC values of the individual surfactants and their mixtures determined on the basis of the surface tension (curve 1) and density measurements (curve 2) as a function of monomer mole fraction of TX100 in aqueous solutions. Determination of the value of CMC from density measurements was carried out through a change in the slope when the density versus the surfactant concentration for surfactant solutions was plotted. The determined values of CMC for the individual surfactants, TX100 and TX165, are close to those obtained by other researchers and equal to (TX100) 2.90 × 10-4 mol/dm3 25,26 and (TX165) 5.41 × 10-4 mol/dm3.27 Curve 3 in Figure 11 presents the values of CMC calculated on the basis of the molecular thermodynamic theory of mixed surfactant (25) Musselman, S. W.; Chander, S. J. Colloid Interface Sci. 2002, 256, 91. (26) Ruiz, C. C.; Molina-Bolivar, J. A.; Aguiar, J. Langmuir 2001, 17, 6831. (27) Ghzaou, E.; Fabregue, E.; Cassanas, G.; Fulconis, J. M.; Delagrange, J. Colloid Polym. Sci. 2000, 278, 321.


Langmuir, Vol. 23, No. 9, 2007 4979

Figure 11. Dependence of the critical micelle concentration (CMC) determined from the adsorption isotherms (Figure 1) (curve 1), the density measurements (curve 2), and calculated from eq 34 (curve 3) on the monomer mole fraction of TX100, R.

Figure 12. Dependence of the activity coefficient in the mixed M fM micelle of TX100, f M 1 (eq 8), and h 1 (eq 20), and TX165, f 2 (eq 9), M and hf 2 (eq 21) on the monomer fraction of TX100, R.

solutions.28-30 This theory expresses the CMC of a binary mixture of surfactants 1 and 2 as a function of the CMCs of the constituent pure surfactants as follows:

Table 4. Values of the Mole Fraction of Surfactant 1 in the Mixed Micelle, X M 1 , Molecular Interaction Parameter in the Mixed Micelle, βM, Activity Coefficients of M the Surfactants 1 and 2 in Mixed Micelle (f M 1 and f 2 )

(1 - R) 1 R ) / + / CMC12 f CMC f 2CMC2 1 1

(34)

where CMC12, CMC1, and CMC2 are the critical micelle concentrations of the mixture, pure surfactant 1, and pure surfactant 2, respectively, R is the solution monomer composition, and the variables f /1 and f /2 are the micellar activity coefficients which can be computed from

f /1 )

[

]

β12(1 - R*)2 exp kT

f /2 ) exp

[

]

(35)

β12(R*)2 kT

(36)

where β12 is the parameter that reflects specific interactions between surfactants 1 and 2, R* is the optimal micellar composition, i.e., the composition at which the Gibbs energy of mixed micellization attains its minimal value, k is the Boltzmann constant, and T is the absolute temperature. The value of R* can be obtained from the molecular thermodynamic theory from the relation

(

) (

β12 R* R CMC2 (1 - 2R*) + ln ) ln kT 1 - R CMCÅ1 1 - R*

)

(37)

From Figure 11, we can see that there is a good agreement between the values of CMC determined experimentally and theoretically. On curves 1 and 3, a clear minimum exists in the range of the mixture composition from 0.6 to 0.8. It means that the theoretical and experimental minimal values of CMC for the mixture of TX100 and TX165 appears at the same composition of the mixture at which the maximal reduction of the surface tension of solutions takes place (Figure 2). The negative deviation of the CMC values from linear dependence suggests that there are differences in the (28) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567. (29) Sarmoria, C.; Puvvada, S.; Blankschtein, D. Langmuir 1992, 8, 2690. (30) Shiloach, A.; Blankschtein, D. Langmiur 1997, 13, 3968.

0.2 0.4 0.6 0.8

XM 1

βM 1

fM 1

fM 2

0.3375 0.5412 0.6550 0.7900

-0.2692 -0.6480 -1.2544 -1.1821

0.8885 0.8725 0.8612 0.9492

0.9698 0.8271 0.5838 0.4782

efficiency of surfactant mixtures in the micelization process and synergistic effect. A useful measure of the micellization efficiency of the surfactant is ΠCMC, which is the surface pressure at CMC. The values of ΠCMC for the single surfactants and their mixture appear as follows:

0.8 < TX165 < 0.6 < 0.4 < 0.2 < TX100 The above series is different from the changes of CMC of TX100, TX165, and their mixture, and together with the minimum in Figure 11, may suggest the synergistic effect in the micellization processes. To prove this effect, similarly to the adsorption process, the interaction parameter in the mixed micelle, βM, and the mixed micelle composition, X M 1 , should be evaluated from eqs 4 and M are 5, respectively.13 The calculated values of X M 1 and β M presented in Table 4 together with the values of f 1 and f M 2 calculated from eqs 8 and 9. As can be seen, the values of the mole fraction of TX100 in the mixed micelle, X M 1 , are bigger than in the bulk phase, and βM is negative. Because the values of βM for all mixtures are negative and their absolute values are M higher than |ln(C M 1 /C 2 )|, we can state that synergism exists in the mixed micelle formation in the solution of all examined mixtures. However, if we take into account the lowest value of βM, the best synergism exists at R ) 0.6. In Figure 12, the values fM of hf M 1 and h 2 calculated from eqs 20 and 21, respectively, as well M as the values of f M 1 and f 2 from Table 4 are presented. From this figure, it is seen that apart from the value of f M 2 at R ) 0.2 and 0.4 the others are smaller, pointing to stronger interactions between molecules of TX100 and TX165 than between single surfactants and confirming the synergistic effect which was shown on the basis of βM parameters. The synergistic effect is also evident from the values of the excess Gibbs energy of micelle formation calculated on the basis of Rosen and Motomura approaches, which are presented in Figure 13. From this figure, it appears

4980 Langmuir, Vol. 23, No. 9, 2007


Figure 13. Dependence of the excess Gibbs energy in mixed micelle, gM (eq 11), and gjM,E (eq 22) on the monomer fraction of TX100, R.

Figure 14. Dependence of the Gibbs energy of micellization ∆G 0mic determined from eq 40 (curve 1), eq 41 (curve 2), eq 42 (curve 3), and eq 38 on the monomer mole fraction of TX100, R.

∆G 0mic ) RT ln CMC

(40)

that there is a very good agreement between the values of gM and

gjM,E calculated froms eq 11 and 22 at R in the range 0.2-0.6. It means that in contrast to the mixed monolayer formation the entropy contribution in the mixing process during the mixed micelle formation is small, particularly in the range of the mole fraction of TX100 in the mixture from 0 to 0.6. 3.5. The Standard Gibbs Energy of Micellization. The tendency of surfactants to form micelles can be established on the basis of the standard Gibbs energy of micelization (∆G 0mic). In the literature, there are many different ways to determine this energy. Maeda31,32 has proposed a new approach of standard Gibbs energy determination for mixed micelles involving ionic species. In this approach, ∆G 0mic for mixtures of two surfactants including one nonionic and one ionic are given as a function of the ionic surfactant in the mixed micelle by

∆G 0mic ) B0 + B1x1 + B2x12 RT

(38)

where B0 is the independent term related to CMC of nonionic surfactant by B0 ) ln C2. The other parameter, B1, is related to the standard Gibbs energy change upon replacement of a nonionic monomer in the nonionic pure micelle with an ionic monomer and B2 which is equivalent to βM calculated from eq 4. Finally, the parameters B1 and B2 are related to the CMC values of pure systems by the equation

ln

()

C1 ) B1 + B 2 C2

(39)

The calculated values of B1 are negative and equal: -0.8936, -0.6244, -1.8799, and -1.8065 for R ) 0.2, 0.4, 0.6, and 0.8, respectively. On the basis of the values of B1, B2, and the molecular interaction parameter in the mixed micelle, βM, we determined the values of the standard Gibbs energy of micellization of TX100 and TX165 and their mixtures from eq 38, which are presented in Figure 14 (curve 4). The points in curve 1 (Figure 14) corresponding to the values of the standard Gibbs energy of micellization of individual surfactants and their mixtures were determined from the following equation:31 (31) Ruiz, C. C.; Aguiar, J. Langmuir 2000, 21, 7946. (32) Maeda, H. J. J. Colloid Interface Sci. 1995, 172, 98.

In Figure 14, there are also presented the values of standard Gibbs energy of micellization of TX100 and TX165 mixtures estimated by two other different ways from the following equations: 0 M 0 M ∆G 0mic ) X M 1 ∆G mic1 + X 2 ∆G mic2 + g

(41)

0 0 M,E hM hM ∆G 0mic ) X 1 ∆G mic1 + X 2 ∆G mic2 + jg

(42)

In Figure 14 were also presented the values of the standard Gibbs energy calculated from eq 41 that are identical to those determined from eq 38 and somewhat higher than those determined from eq 42 (curve 3). The three relationships presented above show minimum values of ∆Gmic at R ) 0.6. These values and our results of ∆G 0mic calculation for mixtures of ionic-nonionic surfactants12 indicate that by using eq 41 it is possible to obtain in a simple way the values of the standard Gibbs energy of micellization for mixtures of two surfactants identical to those proposed by Maeda.31,32

4. Conclusions The measurements of the surface tension and calculations of the interaction parameters in the mixed monolayer and micelle of aqueous solution of TX100 and TX165 indicate the following: (a) Surface tension depends on the concentration and composition of the aqueous solutions of TX100 and TX165 mixtures, and no linear relationship exists between γLV and R in the whole range of the investigated concentrations. (b) It is possible to predict the surface tension of an aqueous solution of TX100 and TX165 mixtures in the whole range of their concentrations from 0 to CMC using the modified equation of state derived by Joos and the equation of Miller et al. (c) Negative values of intermolecular interactions between surfactants in the mixed monolayer and micelle and conditions for existing synergism or antagonism confirm that there is synergism in the surface tension reduction and micelle formation in the whole composition range. (d) Entropy contribution in the excess Gibbs energy of mixed monolayer formation by TX100 and TX165 surfactants is higher than in the formation of mixed micelle.


(e) The CMC values of TX100 and TX165 mixtures can be predicted satisfactorily on the basis of the MT theory of Blankschtein. (f) The values of the standard Gibbs energy of adsorption and micellization for TX100 and TX165 mixtures can be predicted on the basis of equations including the values of the mole fraction

Langmuir, Vol. 23, No. 9, 2007 4981

of surfactant and excess Gibbs energy in the mixed monolayer and micelle formation. Acknowledgment. The financial support from Ministry of Education and Science (MNiSW), grant no. 3 T09A 036 29, is gratefully acknowledged. LA063554+

The Properties of a Binary Mixture of Nonionic Surfactants in Water at

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