The Properties of a Binary Mixture of Nonionic Surfactants in Waterat the Water/Air Interface

Katarzyna Szymczyk and Bronisław Jan´czuk*

Department of Interfacial Phenomena, Faculty of Chemistry, Maria Curie-Skłodowska UniVersity, MariaCurie-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 densitymeasurements. The obtained results of the surface tension measurements were compared with those calculated fromthe relations derived by Joos, Miller, and co-workers. From the comparison, it appeared that by using these twoapproaches the adsorption behavior of TX100 and TX165 mixtures at different mole fractions can be predicted. Thenegative deviation from the linear relationship between the surface tension and composition of TX100 and TX165mixtures in the concentration range corresponding to that of the saturated monolayer at the interface, the values ofthe parameters of molecular interaction, the activity coefficients, as well as the excess Gibbs energy of mixed monolayerformation calculated on the basis of Rosen and Motomura approaches proved that there is synergism in the reductionof 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 ofBlankschtein indicate that there is also synergism in the micelle formation for TX100 and TX165 mixture. It was alsofound that the values of the standard Gibbs energy of adsorption and micellization for the mixture of these twosurfactants, which confirm the synergetic effect, can be predicted on the basis of the proposed equations, which includethe 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 manyapplications, from laundering to tertiary oil recovery. In general,nonionic surfactants composed of a poly(ethylene oxide) chainto which a hydrophobic part is attached have widespread industrialand technological applications.1-4 There are several reasons fortheir frequent use: (i) the properties of each compound (e.g.,solubility) which can be modified considerably by changing thelength of the polyoxyethylene group; (ii) these surfactants arevery effective in sterically stabilizing emulsions and dispersion;(iii) nonionic surfactants are suited for mixing with othersurfactants, among other things. The last point seems to be veryimportant, because in practical applications, nonionic surfactantmixtures are often used because they are usually more effectivethan a single surfactant.5-7 The effectiveness of mixed surfactantsystems is related to specific interactions between molecules(ions) of different surfactants, which can enhance or deterioratethe action of a mixture with respect to some property of thesesystems. On the other hand, some effects that are not expectedin single systems can take place in aqueous solution containingmixed surfactants, for example, synergism in surface tensionreduction. The most frequently used surfactant pairs that show

synergism are mixtures of ionic and nonionic surfactants.8-10

Synergism in this case results from interaction between differenthead groups. The determination of critical micelle concentrations(CMC) and the compositions of mixed micelles and adsorptionlayers at various interfaces seem to also be important for modelingthe structure and properties of these systems, as well as variousprocesses (adsorption, wetting, solubilization, micellar catalysis,etc.).

Our earlier studies showed that, even for mixtures of twoanionic surfactants, having different hydrophilic heads and adifferent length of hydrophobic alkyl tails,11and also for cationic/nonionic surfactant mixtures,12 there is no linear relationshipamong surface tension, critical micelle concentration andwettability of hydrophobic low energetic solids, and thecomposition of the mixtures. For those mixtures, the deviationfrom a linear relationship between the concentration excess atwater-air and hydrophobic solid-water interfaces and theircomposition is also observed. It was interesting regardless ofwhether the deviation from a linear relationship between above-mentioned parameters and composition of the mixture of twononionic surfactants with a different number of poly(ethyleneoxide) chains will take place. Thus, the purpose of our studieswas to determine the adsorption behavior of mixed layers basedon the equations of Gibbs, Joos, Miller, and co-workers,13-17 aswell as the interaction between two nonionic surfactants in the* To whom correspondence should be addressed. Phone (48-81) 537-

5649, fax (48-81) 533-3348, e-mail Bronek@hermes.umcs.lublin.pl.(1) Blin, J. L.; Leonard, A.; Su, B. L.J. Phys. Chem. 2001, 105, 6070.(2) Perez-Arevalo, J. F.; Dominguez, J. M.; Terre´s, E.; Rojas-Herna´ndez, A.;

Miki, M. Langmuir2002, 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., A2005,

1, 153.(7) Shiloach, A.; Blankschtein, D.Langmuir1998, 14, 2965.

(8) McCarroll, M.; Toerne, K.; Wandruszka, Rv.Langmuir1998, 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.

Langmuir2000, 16, 1508.(11) Jan´czuk, B.; Zdziennicka, A.; Wo´jcik, W. Colloids Surf., A2003, 220,

61.(12) Szymczyk, K.; Jan´czuk, B.Colloids Surf., A,in press.(13) Rosen, J. M.Surfactants and Interfacial Phenomena; Wiley-Inter-

science: New York, 2004.(14) Joos, P.Bull. Soc. Chim. Belg. 1967, 76, 591.

4972 Langmuir2007,23, 4972-4981

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

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

2. Experimental Section

2.1. Materials. Triton X-100 (TX100),p-(1,1,3,3-tetramethyl-butyl)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 solutionsof individual surfactants and TX100 and TX165 mixtures at differentratios of TX100 to TX165 were prepared using double-distilled anddeionized water (Destamat Bi18E). The surface tension of waterwas always controlled before the solution preparation.

2.2. Methods.2.2.1. Surface Tension Measurements.Surfacetension measurements were made at 293 K with a Kru¨ss K9tensiometer under atmospheric pressure by the ring method. Theplatinum ring was thoroughly cleaned and the flame dried beforeeach measurement. The measurements were done in such a way thatthe vertically hung ring was dipped into the liquid to measure itssurface tension. It was then subsequently pulled out. The maximumforce needed to pull the ring through the interface was then expressedas the surface tension,γLV (mN/m). Measurements of the surfacetension of pure water at 293 K were performed to calibrate thetensiometer and to check the cleanliness of the glassware. In allcases, more than ten successive measurements were carried out, andthe standard deviation did not exceed(0.2 mN/m. The temperaturewas controlled within(0.1 K.

2.2.2. Density Measurements.We have measured the densities ofwater and aqueous solutions of the individual surfactants as well asTX100 and TX165 mixtures using a vibrating tube densimeter byAnton Paar, model DMA 5000. The accuracy of the thermometerand the density measurements were(0.01 K and(0.005 kg/m3,respectively. The precision of the density and temperature measure-ments given by the manufacturer was(0.001 kg/m3 and(0.001 K.

The densimeter was calibrated regularly with distilled anddeionized water. After measuring the density of water, more thanthree measurements of density were carried out at constanttemperature of 293 K.

2.2.3. EValuation of the Surface Excess Concentration of Surfactantat Interface.The surface excess concentration of surfactants at thewater-air interface can be determined on the basis of the adsorptionisotherms using the Gibbs equation.13

For dilute solution (10-2 mol/dm3 or less) containing nonionicsurfactant, the Gibbs equation can be written in the form

whereC represents the concentration of surfactant andγLV its surfacetension.

The concentration of each surfactant at the interface can becalculated from the slope ofγLV-log C plot. It is convenient if thedependence between the surface tension and the concentration ofthe aqueous surfactant solution can be expressed by the knownmathematical function.

2.2.4. EValuation of the Molecular Interaction Parameters andMiscibility of Surfactants in the Adsorbed Film and Micelle.Forsurfactant mixtures, the characteristic phenomena are the formationof mixed monolayers at the interface and mixed micelles in the bulksolution. Most of the theories are based on the regular solutiontheory, and they have been applied to the phase separation modelfor the micelles and to the monolayer model for the adsorbed filmsin order to estimate the interaction parameterâ in various binarysurfactant systems.13 The molecular interaction parameter,â, for

monolayer can be evaluated, among other things, by using theequation derived by Rubingh and Rosen13,18,19

whereR is the mole fraction of surfactant 1 in the mixture of twosurfactants,X1 is the mole fraction of surfactant 1 in the mixedmonolayer,C 1

0 andC12 are the molar concentrations in the bulk ofsurfactant 1 and of the mixture of surfactants 1 and 2, respectively,required to produce a given surface tension value.X1 can be obtainedfrom

whereC20 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,âM, from the relation of Rubinghand Rosen in the form18,19

whereC1M, andC12

M are the critical micelle concentrations (CMC)of the individual surfactant 1 and the mixture of surfactants 1 and2, respectively, andX1

M is the mole fraction of surfactant 1 in themixed micelle.

X1M can be evaluated from the equation

whereC2M is the CMC of the individual surfactant 2.

With the knowledge of the interaction parameters for the mixedmonolayer and micelles, it is possible to determine the activitycoefficient of the surfactants in the mixtures. From the nonidealsolution theory, the activity coefficients of surfactants 1 and 2 in themixed film (f1 and f1) and mixed micelle (f 1

M and f 2M) fulfill the

respective conditions

and

From this theory, the following equations also result

and

wheregandgM are the excess Gibbs energy of the mixed monolayerand micelle formation, respectively.(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.

(18) Hua, X. Y.; Rosen, M. J.J. Colloid Interface Sci. 1982, 87, 469.(19) Rubingh, D. N. InSolution Chemistry of Surfactants; Mittal, K., Ed.;

Plenum Press: New York, 1979; p 337.

Γ ) - C dγRTdC

) - 1RT

dγd ln C

) - 12.303RT

dγd logC

(1)

âσ )ln(RC12/X1C1

0)

(1 - X1)2

(2)

(X1)2 ln(RC12/X1C1

0)

(1 - X1)2 ln[(1 - R)C12/(1 - X1)C1

0]) 1 (3)

âM )ln(RC12

M /X1MC1

M)

(1 - X1M)2

(4)

(X1M)2 ln(RC12

M /X1C1M)

(1 - X1M)2 ln[(1 - R)C12

M /(1 - X1M)C2

M]) 1 (5)

ln f1 ) âσ(1 - X1)2 (6)

ln f2 ) âσ(X1)2 (7)

ln f1M ) âM(1 - X1

M)2 (8)

ln f2M ) âM(X1

M)2 (9)

g ) RT(X1ln f1+ X2 ln f2) (10)

gM ) RT(X1M ln f 1

M + X2M ln f 2

M) (11)

Properties of Binary Mixture Langmuir, Vol. 23, No. 9, 20074973

Villeneuve et al.20,21claimed that the treatment of intermolecularinteraction parameters by Rubingh and Rosen18,19is not appropriatein the sense that they do not take into account the presence of thesolvent,20 and the physical significance of the parameterâ is notclear when the excess entropy of mixing is not zero.21They proposeda thermodynamic strategy to examine the miscibility of surfactantsin adsorbed films and micelles by new concentration variables. Themiscibility of surfactants in adsorbed films may be examined by theequation22,23

whereXh1 and Xh2 for the mixture of two nonionic surfactants aredefined20,23asXh1 ) m1/mj andXh2 ) m2/mj , respectively, andmj fulfilsthe condition given by the equation

wherem1 andm2 are the molalities of nonionic surfactants 1 and 2,respectively.

The magnitude ofXh 2His defined here by

whereΓ 1H andΓ 2

H are defined with respect to two dividing planeschosen so as to make the excess numbers of moles water and airzero.

On the basis ofXh 1H andXh 2

H determined from eq 12, it is possibleto calculate the activity coefficients of surfactant 1,fh1

H, andsurfactant 2,fh2

H, in the mixed monolayer at the interface as well asthe excess Gibbs energy in this monolayer,gjH,E. These magnitudes,which allow us to draw conclusions about molecular interaction inthe mixed adsorbed film, can be estimated from the followingequations:22,23

and

Symbols mj 10 and mj 2

0 refer to the molality of the individualsurfactants 1 and 2, respectively, at a givenγLV.

The composition of the surfactants in the micelle can be estimatedfrom the following expression:22,23

whereCh is equal tomj at CMC.The magnitude ofXh 2

M in the mixed micelle of two nonionicsurfactants is defined by22

whereN1M and N2

M are the excess numbers of molecules of thenonionic surfactants 1 and 2, respectively, per micelle particle ofwhich the dividing surface between the bulk solution is defined soas to make the excess number of water zero.22,23

The chemical potentials and the activity coefficients in the micelleare defined similarly to those in the adsorbed film.22,23 Thus

and

SymbolgjM,E refers to the excess Gibbs energy of micelle formationper mole of the surfactant mixture, andCh 1

0 ) mj 10 and Ch 2

0 ) mj 20,

respectively, at CMC.On the basis of the above-mentioned equations, it is possible to

establish the composition of the adsorbed mixed monolayer at theinterface and the mixed micelle, as well as the molecular interactionsin monolayer and micelle.

2.2.5. Equation of State Describing Mixed Adsorption BehaVior.Using the adsorption isotherm derived by Joos14 and modified byus11 for the systems including two nonionic surfactants, the surfaceadsorption behavior of the mixture of these surfactants can bepredicted in a quite accurate way. The equation for the mixture ofTX100 and TX165 can be written in the form11,14

if their activity is close toC (for C < 10-2 M), whereΓ 0∞, Γ 1

∞, andΓ 2

∞ are the maxima of the solvent adsorption and surfactants 1 and2, respectively.Π is the surface pressure. The parametersa1 anda2

can be expressed as

whereµS is the chemical potential in the surface under standardconditions,µB is the chemical potential in the bulk under standardconditions, andω is the number of molecules of water per liter.

Assuming thatC2/C1 ) b ) constant, andCtot ) C1 + C2 ) C1(1+ b), we obtain

Miller et al.,15-17taking into account the assumption that for an idealmixture of homologuesa1 ) a2 ) a12 ) 0 andω ) ω1 ) ω2, aswell as Πh ) Πω/RT, Πh 1 ) Π1ω/RT, and Πh 2 ) Π2ω/RT, havederived the equation of state which relates the surface pressure ofa surfactant mixture with the surface pressure of individual solutions.This equation can be expressed in the form

wherea1, a2, anda12are the constants of intermolecular interactions;ω1 andω2 are the partial molar surface areas of surfactant 1 and 2,respectively; andΠ1,Π2, andΠ are the surface pressures of solutionsof the individual surfactants and their mixture, respectively, equal

(20) Villeneuve, M.; Sakamoto, H.; Minamizawa, H.; Aratono, M.J. ColloidInterface Sci. 1997, 194, 301.

(21) Motomura, K.; Aratono, M. InMixed 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 InterfaceSci.1990, 139, 188.

(23) Aratono, M.; Villeneuve, M.; Takiue, T.; Ikeda, N.; Iyota, H.J. ColloidInterface Sci.1998, 200, 161.

Xh 2H ) Xh2 - (Xh1Xh2

mj )(∂mj∂Xh2

)T,p,γ

(12)

mj ) m1 + m2 (13)

Xh 2H )

Γ 2H

Γ 1H + Γ 2

H(14)

Xh1( mjmj 1

0)2) (fh 1

H)2Xh 1H (15)

Xh2( mjmj 2

0)2) (fh 2

H)2Xh 2H (16)

gjH,E ) RT(Xh 1H ln fh 1

H + Xh 2H ln fh 2

H) (17)

Xh 2M ) Xh2 - (Xh1Xh2

Ch )( ∂Ch∂Xh2

)T,p

(18)

Xh 2M )

N2M

N1M + N2

M(19)

Xh1( ChCh 1

0)2) (fh 1

M)2Xh 1M (20)

Xh2( ChCh 2

0)2) (fh 2

M)2Xh 2M (21)

gjM,E ) RT(Xh 1M ln fh 1

M + Xh 2M ln fh 2

M) (22)

exp( -ΠRTΓ 0

∞) + exp( -ΠRTΓ 1

∞) C1

a1+ exp( -Π

RTΓ 2∞) C2

a2) 1 (23)

a1 ) exp(µ 1S - µ 1

B

RT )ω a2 ) exp(µ 2S - µ 2

B

RT )ω (24)

exp( -ΠRTΓ 0

∞) + [exp( -ΠRTΓ 1

∞) 1a1

+ exp( -ΠRTΓ 2

∞) ba2] Ctot

1 + b) 1

(25)

expΠh ) expΠh 1 + expΠh 2 - 1 (26)

4974 Langmuir, Vol. 23, No. 9, 2007 Szymczyk and Jan´czuk

to the difference between the surface tension of the solvent andsolution (γ LV

0 - γLV).

3. Results and Discussion

3.1. Adsorption Isotherms.The measured values of the surfacetension (γLV) of aqueous solutions of TX100 and TX165 andtheir mixtures are presented in Figure 1. This figure shows thedependence betweenγLV and logC (C represents the concentra-tions of TX100, TX165, and their mixtures at a givenR) foraqueous solution of TX165 (curve 1) and TX100 (curve 6) andtheir mixtures (curves 2-4). From this figure, it appears that theshape of curve 1 is somewhat different than those of curve 6 andother curves; however, for both surfactants and their mixturesa linear dependence exists betweenγLV and logCnear the criticalmicelle concentration (CMC) (Table 1). Because of thesedifferences, it is possible that “efficiency” and “effectiveness”of the adsorption of TX100 are different than those of TX165and their mixtures at a givenR.

The surface excess concentration at the surface saturation,Γm,is a useful measure of the adsorption effectiveness of the surfactantat the water-air interface, since it is the maximum valueadsorption can attain, whereas the efficiency of the adsorptionis related to the negative logarithm of the concentration ofsurfactants or their mixture in the bulk phase required to producea 20 mN/m reduction in the surface tension of the solvent,pC20.13

On the basis of the adsorption isotherms (Figure 1), the surfaceexcess concentration at water-air saturation was calculated byusing the Gibbs equation of adsorption (eq 1) and then the minimalarea (Am) per TX100 and TX165 and their mixture molecule at

the interface from the relation

whereN is Avogadro’s number. The values ofΓm andpC20 arepresented in Table 1.

From Table 1, it is seen that the values ofΓm are arranged inthe following direction:

and those ofpC20

From the above series, we see that nonionic surfactant TX165has the lowest efficiency and effectiveness in reduction of thesurface tension of water and that the values ofΓm and pC20

strongly depend on the composition of the mixture of TX100and TX165, because the highest effectiveness is shown by themixture atR ) 0.8, and the highest efficiency atR ) 0.6.

To show the influence of the composition of mixtures on thewater surface tension in Figure 2, the dependence between surfacetension and monomer mole fraction of TX100,R, in the mixtureis plotted. From this figure, it is seen that only at a very lowconcentration of surfactant mixtures is there an almost lineardependence between the surface tension and mole fraction ofTX100 in the mixture. However, at concentration close to 10-5

M and higher, there is a negative deviation from the linearrelationship betweenγLV andR. In other words, at concentrationscorresponding to the beginning of the saturation monolayerformation, nonideal mixing of surfactants is evident. It suggeststhat the composition of the saturated monolayer at the water-airinterface should be different from that of the surfactant in thebulk phase. This suggestion is confirmed by the data in Figure3, which presents the relationship between the mole fraction ofTX100 in the mixed monolayer for eachR calculated from eq3, X1, and the surface tension of the solution. From this figure,it is evident that the direction of change ofX1 for all R is the sameasγLV decreases, and all values ofX1 grow as the surface tensionbecomes smaller. AtR equal 0.2, 0.4, and 0.6 atγLV less than60 mN/m, the mole fraction of TX100 in the mixed monolayeris bigger than that in the bulk phase. In the whole range of the

Figure 1. Dependence of the surface tension of aqueous TX165(curve 1) and TX100 (curve 6) solutions and their mixture at monomermole fraction of TX100) 0.2, 0.4, 0.6, and 0.8 on logC.

Table 1. Values of the Critical Micelle Concentrations (CMC),Negative Logarithm of the Concentration of Surfactants orTheir 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 5.41× 10-4 4.4962 2.22× 10-6 0.74800.2 4.35× 10-4 4.7292 2.53× 10-6 0.65620.4 3.42× 10-4 4.8267 2.67× 10-6 0.62180.6 2.73× 10-4 4.8544 2.70× 10-6 0.61450.8 2.72× 10-4 4.8530 2.73× 10-6 0.60821 2.90× 10-4 4.7254 2.83× 10-6 0.5870

Figure 2. Dependence of the surface tension of aqueous solutionsof 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.

Am ) 1NΓm

(27)

TX165 < 0.2< 0.4< 0.6< 0.8< TX100

TX165 < TX100 < 0.2< 0.4< 0.8< 0.6

Properties of Binary Mixture Langmuir, Vol. 23, No. 9, 20074975

presented surface tension atR ) 0.8, the values ofX1 are smallerthan in the bulk phase. As mentioned above, the composition ofthe mixed monolayer can also be calculated from the eq 12proposed by Motomura at all.20-23 The values ofXh 1

H obtainedfrom this equation are smaller than those ofX1 at about 0.02,which indicates that entropy plays an important role in the processof surfactant mixing in the monolayer.

Because we proved that the composition of the saturatedmonolayer at the water-air interface calculated on the basis ofRosen and Motomura’s models is significantly different than thecomposition in the bulk phase, it seems to be necessary to calculatethe values ofΓm for the investigated mixtures on the basis ofX1

andXh 1H values from the relation

whereΓm1 andΓm2 are the maximal excess values of a singlesurfactant concentration at the water-air interface (Table 1).

Figure 4 shows the values ofΓm determined in different ways.Curve 1 in this figure represents the values calculated on thebasis of an adsorption isotherm mixture of surfactants (Table 1)and curve 2 the values obtained from the relation

Curves 3 and 4 represent the values calculated from eqs 28 and29.

As is seen in Figure 4, there are essential differences betweenthe values of surfactant concentration in mixed monolayerscalculated on the basis of the adsorption isotherms (curve 1) andcomposition of the mixture in the bulk phase (curve 2) andmonolayer (curves 3 and 4). However, there is a good agreementbetween the values ofΓm calculated on the basis of the monomermole fraction of surfactants in mixed monolayers obtained fromeqs 3 and 12. This fact proves that mutual interactions betweenmolecules or ions of surfactants in the mixed monolayer play animportant role in the composition and packing of surfactants inthis monolayer, and this composition has a valid influence onthe properties of the monolayer but does not give an answerabout synergistic or antagonistic properties of the mixture in

reduction of the surface tension of water. To explain these effects,the interaction parameter,âσ, should be calculated from eq 2.The calculated values of this parameter are presented in Figure5. From this figure, it is seen that for allR this parameter hasa negative value which changes with surface tension decreaseof the aqueous solution of the surfactant mixtures. The smallestvalues ofâσ exist for R ) 0.8 at a lower value of the surfacetension that is at higher concentrations of the solutions, whichmay confirm a clear minimum in Figure 2. The negative valuesof theâσ parameter suggest that there is synergism in the surfacetension reduction efficiency. However, the second condition forthe existence of negative synergism must be fulfilled. Thiscondition is that the absolute value of theâσ parameter shouldbe greater than|ln(C1

0/C20)|, whereC1

0and C20 are the concen-

trations of a single surfactant at a given surface tension. Fromthe comparison of these two values (Table 2), it is evident thatfor each value of the surface tension corresponding to the mixedsaturated monolayer at water-air and for surfactant mixtures ateach composition there is a negative synergism in the surfacetension reduction.

By knowing theX1 andâσ values, it is possible to calculatethe activity coefficients of surfactants 1 and 2 in the mixed film

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

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

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

Γm ) Γm1X1 + Γm2

X2 (28)

Γm ) Γm1Xh 1

H + Γm2Xh 2

H (29)

Γm ) Γ1R + Γ2(1 - R) (30)

4976 Langmuir, Vol. 23, No. 9, 2007 Szymczyk and Jan´czuk

(f1 andf1) from eqs 6 and 7 and the excess Gibbs energy in thisfilm from eq 10, which allows us to describe the interactionbetween surfactants in mixed monolayer and confirm thesynergistic effect. From calculation of the activity coefficients,it appears that all their values are smaller than 1, which, accordingto Motomura’s model, indicates that interactions between TX100andTX165moleculesarestronger thanbetweensinglesurfactants.This fact also confirms the values of the excess Gibbs energyof mixing presented in Figure 6. As it appears from this figure,the values ofg for eachR are smaller than zero.

In Figure 7, there is a comparison between the values of theactivity coefficient calculated from eqs 6, 7, 15, and 16 forγLV

) 45 mN/m. The results show that there is a small differencebetween 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 excessGibbs energy of mixing,g andgjH,E, calculated from eqs 10 and17, and the monomer mole fraction of TX100,R. The values ofgjH,E (curve 2) are significantly lower than those ofg (curve 1)for γLV ) 45 mN/m, which indicates that the excess Gibbs energyof mixing plays an important role in mixing monolayer formation.

3.2. Adsorption Isotherms of Joos and Miller et al.It isinteresting to discover whether, on the basis of the theoreticalisotherms of adsorption, it is possible to predict the surface tensionfor the mixtures of TX100 and TX165 for which synergism inreduction of the surface tension was proven in the whole rangeof concentrations at the saturated monolayer. Therefore, in Figure9, the isotherms of Joos and Miller et al. are presented forR )0.2.

Line 2 in Figure 9 reflects the function ofγLV vs logCcalculatedfrom eq 26. The value ofω used for calculations in this equation,at first approximation, was assumed to be 2× 105 m2/mol.17

Line 3 in this figure reflects the dependence of the surfacetension (γLV) of an aqueous solution of TX100 and TX165mixtures on the total concentration of surfactants (logC) for R(R is the mole fraction of TX100) 0.2) calculated from eq 25using the values ofΓ∞ anda of the individual components.

The values ofΓ0∞, Γ1

∞, Γ2∞, a1, and a2 used in eq 25 were

determined from eq 23 from the data for individual surfactants(TX100 and TX165) on the assumption thatC1 ) 0 or C2 ) 0,which are listed in Table 3. In all cases, it was assumed that thearea occupied by water is close to 0.10 nm2, and thus,Γ0

∞ ) 16.6× 10-6 mol/m2.

The results presented in Figure 9 and those calculated forother mixtures show that the changes ofγLV as a function of logC, for a givenR, have the same shape. Of course, near CMCthere is a linear dependence ofγLV on logC. From Figure 9 andother calculations, it also appears that at low concentrations ofsurfactant mixtures there is a good agreement between the valuesof thesurface tensionof thesolutionmeasuredand thosecalculatedfrom eq 26 (curve 2) and from eq 25 (curve 3). Practically, theexperimental points (points 1) are on the theoretical curves (curves2 and 3) for all values ofR. These facts indicate that by usingthe equation of state derived by Joos,14 and next modified by

Table 2. Values of the|âσ| and |ln(C10/C2

0)| for TX100 andTX165 Mixtures

γLV (mN/m)|âσ|0.2

|âσ|0.4

|âσ|0.6

|âσ|0.8 |ln(C1

0/C20)|

65 1.5716 2.2505 0.9669 1.1619 0.966260 1.9735 2.1469 1.7696 2.0745 0.371755 1.7525 1.9926 1.9980 2.2112 0.287252.8 1.8392 2.1129 2.0765 2.3366 0.529450 1.8786 2.1535 2.3331 2.4873 0.755945 1.6176 1.9471 2.2990 2.5966 1.0657

Figure 6. Dependence of the excess Gibbs energy in mixedmonolayer,g, calculated from eq 10 on the surface tension of anaqueous solution of surfactant mixtures at different monomer molefractions of TX100,R.

Figure 7. Dependence of the activity coefficient in the mixedmonolayer of TX100,f1 (eq 6), andfh1

H (eq 15), and TX165,f2 (eq7), andfh2

H (eq 16) on the monomer fraction of TX100,R, atγLV )45 mN/m.

Figure 8. Dependence of the excess Gibbs energy in mixedmonolayer,g (eq 10), andgjH,E (eq 17) on the monomer fraction ofTX100, R, at γLV ) 45 mN/m.

Properties of Binary Mixture Langmuir, Vol. 23, No. 9, 20074977

us,11 and the simple model proposed by Miller15-17 it is possibleto predict the surface tension of an aqueous solution of TX100and TX165 mixtures in the whole range of their concentrationsfrom 0 to CMC. Of course, it is impossible to predict the surfacetension of the solution of TX100 and TX165 at the concentrationclose to CMC or higher than CMC.

3.3. The Standard Gibbs Energy of Adsorption.On thebasis of the differences between the slopes of the linear parts ofthe curves (Figure 1), which represent the dependences betweenγLV and logC, and the differences between the valuesγLV at thesame concentration for TX100 and TX165, respectively, weconcluded that “efficiency” and “effectiveness” of the adsorptionof TX100, TX165, and their mixtures were different. Theadsorption efficiency is also related to the standard Gibbs energyof the adsorption,∆Gad

o . The standard Gibbs energy of adsorp-tion, ∆Gad

o , can be determined by different methods; amongothers, from the equation derived by Rosen and Aronson.13,24Ifthe surfactant concentration corresponding to the saturatedmonolayer at interface is lower than 1× 10-2 M, the Rosen andAronson equation can be expressed in the form

whereω is the number of water moles per decimeter cubed, andπ is the surface pressure corresponding to the surfactantconcentration,C, at whichAm is achieved.

From eq 31, the values of the standard Gibbs energy of TX100,TX165, and also for their mixtures were calculated. For thesecalculations, the values ofC at γLV ) 45 mN/m were used andthose ofAm 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∆Gad

o

calculated from the relations, respectively

and

where∆Gad1o and∆Gad2

o are the standard Gibbs energy of TX100and TX165, respectively.

From this figure, it appears that∆Gado values calculated from

eqs 32 and 33, that is with respect to monomer mole fraction ofsurfactants in mixed monolayer, are considerably smaller thatthose calculated from eq 31; and by their minimum, they confirmthe synergistic effect in surface tension reduction, which is largestat R ) 0.8. Moreover, the direction of changes of∆Gad

o valuescalculated from these two equations is in agreement with thedirection of changes ofpC20, that confirms the efficiency ofadsorption proven bypC20.

3.4. CMC. Another characteristic property of surfactants istheir ability to form micelles. The concentration at which themicellization process takes place is called critical micelleconcentration (CMC).

In our studies, the values of CMC for TX100, TX165, andtheir mixtures were determined from the adsorption isotherms(Figure 1) and from the density measurements which are presentedin Figure 11. This figure shows the CMC values of the individualsurfactants and their mixtures determined on the basis of thesurface tension (curve 1) and density measurements (curve 2)as a function of monomer mole fraction of TX100 in aqueoussolutions.

Determination of the value of CMC from density measurementswas carried out through a change in the slope when the densityversus the surfactant concentration for surfactant solutions wasplotted. The determined values of CMC for the individualsurfactants, TX100 and TX165, are close to those obtained byother researchers and equal to (TX100) 2.90× 10-4mol/dm3 25,26

and (TX165) 5.41× 10-4 mol/dm3.27 Curve 3 in Figure 11presents the values of CMC calculated on the basis of themolecular thermodynamic theory of mixed surfactant

(24) Rosen, J. M.; Aronson, S.Colloids Surf. 1981, 3, 201.

(25) Musselman, S. W.; Chander, S.J. Colloid Interface Sci. 2002, 256, 91.(26) Ruiz, C. C.; Molina-Bolivar, J. A.; Aguiar, J.Langmuir2001, 17, 6831.(27) Ghzaou, E.; Fabregue, E.; Cassanas, G.; Fulconis, J. M.; Delagrange, J.

Colloid Polym. Sci. 2000, 278, 321.

Figure 9. Dependence of the surface tension of aqueous solutionsof TX100 and TX165 surfactant mixture on logC for R ) 0.2.Points 1 represent the measured values of the surface tension, andcurves 2 and 3 represent the values of the surface tension calculatedfrom eqs 26 and 25, respectively.

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

substance Γ∞ (mol/m2) a (mol/L)

water 16.6× 10-6

TX100 3.15× 10-6 1.95× 10-6

TX165 2.40× 10-6 1.98× 10-4

∆Gado ) 2.303RT log

Cω

- NπAm (31)

Figure 10. Dependence between the Gibbs energy of adsorption,∆Gad

0 , calculated from eq 31 (curve 1), eq 32 (curve 2), and eq 33(curve 3) and monomer mole fraction of TX100,R.

∆Gado ) X1∆Gad1

o + X2∆Gad2o + g (32)

∆Gado ) Xh 1

H∆Gad1o + Xh 2

H∆Gad2o + gjH,E (33)

4978 Langmuir, Vol. 23, No. 9, 2007 Szymczyk and Jan´czuk

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

where CMC12, CMC1, and CMC2 are the critical micelleconcentrations of the mixture, pure surfactant 1, and puresurfactant 2, respectively,R is the solution monomer composition,and the variablesf 1

/ andf 2/ are the micellar activity coefficients

which can be computed from

where â12 is the parameter that reflects specific interactionsbetween surfactants 1 and 2,R* is the optimal micellarcomposition, i.e., the composition at which the Gibbs energy ofmixed micellization attains its minimal value,k is the Boltzmannconstant, andT is the absolute temperature.

The value of R* can be obtained from the molecularthermodynamic theory from the relation

From Figure 11, we can see that there is a good agreement betweenthe values of CMC determined experimentally and theoretically.On curves 1 and 3, a clear minimum exists in the range of themixture composition from 0.6 to 0.8. It means that the theoreticaland experimental minimal values of CMC for the mixture ofTX100 and TX165 appears at the same composition of the mixtureat which the maximal reduction of the surface tension of solutionstakes place (Figure 2). The negative deviation of the CMC valuesfrom linear dependence suggests that there are differences in the

efficiency of surfactant mixtures in the micelization process andsynergistic effect. A useful measure of the micellization efficiencyof the surfactant isΠCMC, which is the surface pressure at CMC.The values ofΠCMC for the single surfactants and their mixtureappear as follows:

The above series is different from the changes of CMC of TX100,TX165, and their mixture, and together with the minimum inFigure 11, may suggest the synergistic effect in the micellizationprocesses. To prove this effect, similarly to the adsorption process,the interaction parameter in the mixed micelle,âM, and the mixedmicelle composition,X1

M, should be evaluated from eqs 4 and5, respectively.13 The calculated values ofX1

M and âM arepresented in Table 4 together with the values off1

M and f2M

calculated from eqs 8 and 9. As can be seen, the values of themole fraction of TX100 in the mixed micelle,X1

M, are biggerthan in the bulk phase, andâM is negative. Because the valuesof âM for all mixtures are negative and their absolute values arehigher than|ln(C1

M/C2M)|, we can state that synergism exists in

the mixed micelle formation in the solution of all examinedmixtures. However, if we take into account the lowest value ofâM, the best synergism exists atR ) 0.6. In Figure 12, the valuesof fh1

M andfh2M calculated from eqs 20 and 21, respectively, as well

as the values off1M andf2

M from Table 4 are presented. From thisfigure, it is seen that apart from the value off2

M at R ) 0.2 and0.4 the others are smaller, pointing to stronger interactions betweenmolecules of TX100 and TX165 than between single surfactantsand confirming the synergistic effect which was shown on thebasis ofâM parameters. The synergistic effect is also evidentfrom the values of the excess Gibbs energy of micelle formationcalculated on the basis of Rosen and Motomura approaches,which are presented in Figure 13. From this figure, it appears

(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.Langmiur1997, 13, 3968.

Figure 11. Dependence of the critical micelle concentration (CMC)determined from the adsorption isotherms (Figure 1) (curve 1), thedensity measurements (curve 2), and calculated from eq 34 (curve3) on the monomer mole fraction of TX100,R.

1CMC12

) Rf1/CMC1

+(1 - R)

f2/CMC2

(34)

f1/ ) exp[â12(1 - R*)2

kT ] (35)

f2/ ) exp[â12(R

*)2

kT ] (36)

â12

kT(1 - 2R*) + ln( R*

1 - R* ) ) ln( R1 - R

CMC2

CMCÅ1) (37)

Figure 12. Dependence of the activity coefficient in the mixedmicelle of TX100,f1

M (eq 8), andfh1M (eq 20), and TX165,f2

M (eq 9),and fh2

M (eq 21) on the monomer fraction of TX100,R.

Table 4. Values of the Mole Fraction of Surfactant 1 in theMixed Micelle, X1

M, Molecular Interaction Parameter in theMixed Micelle, âM, Activity Coefficients of

the Surfactants 1 and 2 in Mixed Micelle (f 1M and f 2

M)

X1M â 1

M f 1M f 2

M

0.2 0.3375 -0.2692 0.8885 0.96980.4 0.5412 -0.6480 0.8725 0.82710.6 0.6550 -1.2544 0.8612 0.58380.8 0.7900 -1.1821 0.9492 0.4782

0.8< TX165 < 0.6< 0.4< 0.2< TX100

Properties of Binary Mixture Langmuir, Vol. 23, No. 9, 20074979

that there is a very good agreement between the values ofgM andgjM,E calculated froms eq 11 and 22 atR in the range 0.2-0.6.

It means that in contrast to the mixed monolayer formationthe entropy contribution in the mixing process during the mixedmicelle formation is small, particularly in the range of the molefraction of TX100 in the mixture from 0 to 0.6.

3.5. The Standard Gibbs Energy of Micellization. Thetendency of surfactants to form micelles can be established onthe basis of the standard Gibbs energy of micelization (∆Gmic

0 ).In the literature, there are many different ways to determine thisenergy. Maeda31,32 has proposed a new approach of standardGibbs energy determination for mixed micelles involving ionicspecies. In this approach,∆Gmic

0 for mixtures of two surfactantsincluding one nonionic and one ionic are given as a function ofthe ionic surfactant in the mixed micelle by

whereB0 is the independent term related to CMC of nonionicsurfactant byB0 ) ln C2. The other parameter,B1, is related tothe standard Gibbs energy change upon replacement of a nonionicmonomer in the nonionic pure micelle with an ionic monomerandB2 which is equivalent toâM calculated from eq 4. Finally,the parametersB1 andB2 are related to the CMC values of puresystems by the equation

The calculated values ofB1 are negative and equal:-0.8936,-0.6244,-1.8799, and-1.8065 forR ) 0.2, 0.4, 0.6, and 0.8,respectively. On the basis of the values ofB1,B2, and the molecularinteraction parameter in the mixed micelle,âM, we determinedthe values of the standard Gibbs energy of micellization of TX100and TX165 and their mixtures from eq 38, which are presentedin Figure 14 (curve 4). The points in curve 1 (Figure 14)corresponding to the values of the standard Gibbs energy ofmicellization of individual surfactants and their mixtures weredetermined from the following equation:31

In Figure 14, there are also presented the values of standardGibbs energy of micellization of TX100 and TX165 mixturesestimated by two other different ways from the followingequations:

In Figure 14 were also presented the values of the standard Gibbsenergy calculated from eq 41 that are identical to those determinedfrom eq 38 and somewhat higher than those determined from eq42 (curve 3). The three relationships presented above showminimum values of∆Gmic at R ) 0.6.

These values and our results of∆Gmic0 calculation for

mixtures of ionic-nonionic surfactants12 indicate that by usingeq 41 it is possible to obtain in a simple way the values of thestandard Gibbs energy of micellization for mixtures of twosurfactants identical to those proposed by Maeda.31,32

4. Conclusions

The measurements of the surface tension and calculations ofthe interaction parameters in the mixed monolayer and micelleof aqueous solution of TX100 and TX165 indicate the following:

(a) Surface tension depends on the concentration and com-position of the aqueous solutions of TX100 and TX165 mixtures,and no linear relationship exists betweenγLV andR in the wholerange of the investigated concentrations.

(b) It is possible to predict the surface tension of an aqueoussolution of TX100 and TX165 mixtures in the whole range oftheir concentrations from 0 to CMC using the modified equationof state derived by Joos and the equation of Miller et al.

(c) Negative values of intermolecular interactions betweensurfactants in the mixed monolayer and micelle and conditionsfor existing synergism or antagonism confirm that there issynergism in the surface tension reduction and micelle formationin the whole composition range.

(d) Entropy contribution in the excess Gibbs energy of mixedmonolayer formation by TX100 and TX165 surfactants is higherthan in the formation of mixed micelle.

(31) Ruiz, C. C.; Aguiar, J.Langmuir2000, 21, 7946.(32) Maeda, H. J.J. Colloid Interface Sci. 1995, 172, 98.

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

∆Gmic0

RT) B0 + B1x1 + B2x1

2 (38)

ln(C1

C2) ) B1 + B2 (39)

Figure 14. Dependence of the Gibbs energy of micellization∆Gmic0

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.

∆Gmic0 ) RT ln CMC (40)

∆Gmic0 ) X1

M∆Gmic10 + X2

M∆Gmic20 + gM (41)

∆Gmic0 ) Xh 1

M∆Gmic10 + Xh 2

M∆Gmic20 + jgM,E (42)

4980 Langmuir, Vol. 23, No. 9, 2007 Szymczyk and Jan´czuk

(e) The CMC values of TX100 and TX165 mixtures can bepredicted satisfactorily on the basis of the MT theory ofBlankschtein.

(f) The values of the standard Gibbs energy of adsorption andmicellization for TX100 and TX165 mixtures can be predictedon the basis of equations including the values of the mole fraction

of surfactant and excess Gibbs energy in the mixed monolayerand micelle formation.

Acknowledgment. The financial support from Ministry ofEducation and Science (MNiSW), grant no. 3 T09A 036 29, isgratefully acknowledged.

LA063554+

Properties of Binary Mixture Langmuir, Vol. 23, No. 9, 20074981