Langmuir 1992,8, 26W2697
2690
Prediction of Critical Micelle Concentrations of Nonideal Binary Surfactant Mixtures Claudia Sarmoria, Sudhakar Puwada,+ and Daniel Blankschtein' Department of Chemical Engineering and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received June 8,1992 We have simplifieda recently developedmolecular-thermodynamictheory of mixed surfactantsolutions to obtain a "working model" which can be utilized to predict critical micelle concentration6 (cmc's) of nonideal binary surfactant mixtures where at least one of the surfactant species is ionic. The required inputs of the model are (i) the cmc's of the pure surfactants, (ii) the overall surfactant Composition and other solution conditions such as temperature and salt type and concentration, and (iii) the chemical structuresof the hydrophobicand hydrophilic moietiesof both surfactantspecies. We find that the model yields reasonable quantitativepredictionsfor anionic-nonionic and cationic-nonionicsurfactantmixtures of single-tailed surfactants having linear hydrocarbon chains or h e a r phenyl hydrocarbon chains. We view the simplified"working model" for the prediction of mixture cmc's as a valuablepreliminary screening tool in the design and selection of nonideal surfactant mixtures of practical importance.
I. Introduction In many practical applications involving surface-active compounds,one often mixes different types of surfactants in order to conveniently tune and control the desired properties of the system. It is well-known that mixtures of nonionic surfactants tend to mix while mixtures of ionic-nonionic, anioniecationic, and hydrocarbon-based-fluorobon-based surfactants can exhibit significant departures from the ideal behavior.'+ Deviations from ideality result from either synergistic (attractive) or antagonistic (repulsive) specific interactions between the different surfactant species.' For example, a surfactant mixture that exhibits synergism can have a critical micelle concentration (cmc)which is considerably lower than the cmc of each constituent pure surfactant, a feature that can be very advantageous in many practical applications.'P2 In this paper we concentrate on predicting cmc's of nonideal binary surfactant mixtures in aqueous solution, where at least one of the surfactant speciesis ionic. Various theoretical formulations have been developed3-12to dewribe mised micellizationin nonidealsurfactant mixtures. Of particular relevance to this paper are regular-solution theories which have been successfully utilized2p7J2to describe and correlate the observed nonidealities in the cmc's of avariety of binary surfactant mixtures. However, these theories are not predictive, since they rely on an experimentally fitted empirical interaction parameter which is mixture dependent.2*7J2Molecular-thermody-
* To whom correspondence should be addressed.
t Current address: Center for Bio/Molecular Science
and Engineering, Naval Research Laboratory,Washington, DC 20376-5000. (1) Scamehom, J. F. In Phenomena in Mixed Surfactant Systems; Scamehorn, J. F., Ed.;ACS Symposium Series 311; American Chemical Society: Washington, DC, 1986. (2) Roeen, M. J. Surfactants and Interfacial Phenomena; Wiley: New York, 1989; Chapter 11. (3) Nagarajan,R. Langmuir 1985,1,331, and references cited therein. (4) Roeen, M. J.; Hua, X. Y. J. Am. Oil Chem. SOC.1982,59, 584. ( 5 ) Mukerjee, P.; Handa, T. J . Phys. Chem. 1981,85, 2298. (6) Handa, T.;Mukerjee, P. J. Phys. Chem. 1981,85, 3916. (7) Rubingh, D. N.in Solution Chemistry of Surfactants; Mittal, K . L., Ed.;Plenum: New York, 1979; Vol. 1. (8) Lange, V. H. Kolloid 2. 1953,96, 131. (9) Clint, J. H. J . Chem. SOC.,Faraday Trans. 1 1975, 71, 1327. (10)Shinoda, K.J . Phys. Chem. 1964,58, 541. (11) Holland, P. M.; Rubingh, D. N. J . Phys. Chem. 1983,87, 1984. (12) Holland, P. M. In Aduances in Colloid and Interface Science; Elsevier: Amsterdam, 1986; Vol. 26, p 111.
namic theories, which incorporate explicitly the detailed chemical structures of the various surfactant species, the mixture composition, and other solution conditions,have also been devel~ped.~J~-'~ Although these theories have proven to be successful in predicting a broad spectrum of mixed micellar solution properties, they may be somewhat too detailed and computationally involved if one is interested solely in predicting a single mixed micellar Solution property such as the mixture cmc. In this paper we implement a simplified version of a recently d e ~ e l o p e d ' ~ molecular-thermodynamic '~ theory of mixed surfactant solutions in order to predict critical micelle concentrations of nonideal binary surfactant mixtures in aqueous solution. The simplified "working model" presented here takes advantage of some of the insights gained from the more detailed molecular-thermodynamic theory in order to reduce the number of required inputs, as well as simplify the complexity of the calculations. More specifically, the only required inputs are (i) the cmc's of the pure surfactants,'s (ii) the overall surfactant composition and other solution conditionssuch as temperature and salt type and concentration, and (iii) the chemicalstructures of the hydrophobicand hydrophilic moieties of both surfactant species. Using this readily available information, we show below that it is possible to obtain reasonableagreement with experimental cmc data. Consequently, we view the simplified 'working model" for the prediction of mixture cmc's presented in this paper as a valuable preliminary screeningtool in the design and selection of nonideal surfactant mixtures of practical importance. The remainder of this paper is organized as follows. Section I1 describes the simplified theoretical "working model" to predict mixture cmc's. Section I11presenta the theoretical results and a comparison with experiments. Finally, some conclusions are presented in section IV. (13) Puwada, S.;Blankschtein, D. J. Phys. Chem. 1992, 96,5667. (14) Puwada, S.;Blankachtein, D. J. Phys. Chem. 1992,96,5679. (15) Puwada, S.;Blankschtein, D. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds., ACS Sympoeium Seriw 501; American Chemical Society: Washington, DC, 1992; p 96. (16) Nota that instead of utilizing experimental CMC valuen, it in actually poesible to implement the detailed version of the molecularthermodynamic theory of single-surfactant S O ~ U ~ ~ OtoMpredict the cmc's of the pure surfactanta (seePuwada, 5.;Blankachtein,D.J . Chem. Phye. 1990,92,3710). However,thecalculationeinthiscawbe"econeiderably more involved, thus reducing the simplicity of the approach proposed in thia paper.
0143-1463/92/2408-2690$03.00/00 1992 American Chemical Society
Rediction of Critical Micelle Concentrations
Langmuir, Vol. 8, No.11, 1992 2691
11. Theory As stated in section I, a molecular-thermodynamic theory of mixed surfactant solutions capable of describing and predicting micellization, phase behavior, and phase separation of mixed micellar solutions has recently been devel~ped.~"'~The theoretical formulation consists of combining a thermodynamic theory of mixed micellar solutions, which incorporate^'^ (i) the formation of the various solution components, (ii) the solution entropy of mixing, and (iii) interactions between the various components, with a molecular model of mixed micellization which captures" the essential driving forces for micelle formation at the molecular level. The resulting molecularthermodynamic theory has been very successful in a broad spectrum of micellar and phase behavior properties of aqueous solutions of mixed surfactants, with particular emphasis on n-alkyl poly(ethy1ene oxide), CiEj, nonionic surfactants. Using the molecular-thermodynamictheory, we derived13an expression for the mixture cmc which is identical to the well-known expression derived7 in the context of the pseudophase separation model using the regularsolution theory with an empirical interaction parameter. Specifically, we have shown13 that the cmc of a binary mixture of surfactants A and B can be expressed as a function of the cmc's of the constituent pure surfactants as follows
where CMC-, CMCA,and CMCBare the critical micelle concentrations of the mixture, pure surfactantA, and pure surfactant B, respectively, a1 is the solution monomer composition (note that near the mixture CMC one has a1 a,d = overall surfactant composition),and the variables f~and fB are the micellar activity coefficientsof surfactants A and B, respectively. The micellar activity coefficients, f~ and f ~can , be computed from13 E;:
(3)
where @ABis a parameter that reflects specific interactions between surfactants A and B, a* is the optimal micellar composition, namely, the composition at which the free energy of mixed micellization attains its minimum value, k is the Boltzmann constant, and T is the absolute temperature. Note that the micellar composition, a, is defined with respect to surfactant A, namely, a = 1 corresponds to pure surfactant A. Unlike regular-solution theories, where the parameter @ABis empirical in the sense that its evaluation requires fitting to experimental cmc data,2s7J2the molecularthermodynamic theorypr~videsl~ a way to actuallypredict @AB.Specifically, we have shown13that @ABreflects two main contributions to the free energy of mixed micellization. These are (1) a free-energy contribution, associated with interactions between the hydrophobic moieties of surfactants A and B in the micellar core, and (2) an electrostatic contribution, &=, associated with electrostatic interactions between the charged hydrophilic moieties of surfactants A and B. In other words
based) surfactants' but is greater than zero for a binary mixture of hydrocarbon-based and fluorocarbon-baeed surfactants due to repulsive interactions in the micellar core.68 In this paper we concentrate on binary mixtures of hydrocarbon-based surfactants, and, consequently, one can set = o in eq 4. From simple electrostatic considerations one expects17 that, to leading order, the total electrostatic contribution to the free energy of mixed micellization, gel=, should be proportional to q2, where q = aezA + (1 - a)ezB is the average charge per surfactant molecule (ZA and ZB are the valencies of surfactantsA and B, respectively, and e is the electronic char e). In view of this, one can write13g b = dw + (1-a)& + a(l -a)&, where&',= &&A2 and gk = Kel&B2 are the electrostatic contributions aseociated with pure surfactants A and B, respectively, and = -&&A - z B ) ~reflects electrostatic interactions between the charged hydrophilic moieties of eurfactanta A and B. Note that K e h is a numerical prefactor that can be evaluated from electrostatic theories (see below) and depends on micelle shape and size, as well as on salt type and concentration. Usin the expression for given above, and assuming that g& = 0 (as explained earlier), one can write eq 4 as
em
eh
e
@AB
* -&w(zA - 2,)
2
(5)
Equation 5 indicates that for nonionic-nonionic ( Z A = ZB = 0) hydrocarbon-based surfactant mixtures, as well as for hydrocarbon-basedmixtures of two ionic surfactanta having equal charges ( Z A = ZB), the interaction parameter @AB 0. In such cases, the surfactant mixtures should behave quite ideally, an observation which is substantiated by several experiments.'"J2 To compute Kelw, we first relate it to geh given above. In particular, at the optimal micellar composition, a*,one 0btainsl3 E;:
gel= = &= ((2, - ZBI2(a*I2 + 2(zA - z&Ba* + Z B ~ ) (6) The value of a* appearing in eqs 2, 3, and 6 can be obtained from the molecular-thermodynamic theory. Specifically, we have shown13that by minimizing the free energy of mixed micellization with respect to the micellar composition, a,the followingrelation between a*,al, @AB, CMCA,and CMCB is obtained -@AB (1-2a*)+ln(-)=ln(~""".)
kT
1-a*
1- al CMCA
(7)
where, as stated earlier, near the mixture CMC one can = assume that, to a very good approximation, a1 overall surfactant composition. In order to compute the mixture CMC using eqs 1-7, we first need to obtain an expression for gel= (see eq 6). Although very detailed electrostatic theories are currently available to compute electrostatic free energies,lg21these formulations often involve complex numerical solutions. Since the aim of this paper is to provide a relatively simple and fast 'working modell to predict mixture cmc's, we have chosen to describe the micellar surface charge distribution using an approximate analytical solution to E;:
em,
em
(4)
It ia noteworthy that is typically equal to zero for a mixture of two hydrocarbon-baaed (or fluorocarbon-
(17) Bockrii, J. O'M.; Reddy, A. K. N. Modern Electrochemintry; Plenum: New York, 1977. (18) Haydon, D.A. In Recent Propene in Surface Science; Danirlli, J. F.; Pankhumt, K. G. A, Riddiford, A. C., %.;Academic Preen: New York, lW, and references cited therein. (19) Bell, G. M.;Levine, S. Trans. Faraday SOC.1967,53,143. (20) Knrpe, P.; Ruckenstein, E. J . Colloid Interface Sci. 1991, 141,
534. (21) Spamany, M.J. Recl. Trau. Chim. Payn-Ban 1968, 77,872.
Sarmoria et al.
2692 Langmuir, Vol. 8, No. 11, 1992 the Poisson-Boltzmann equation, from which an approximate analytical expression for gel, can be obtained.22*23 Specifically, for charges distributed over a spherical surface of radius R, resulting from a single surfactant species, one obtainsnv23
gel, ____ =
(1 + (s/2)2)'/2 2kT [In (42 + (1+ ( ~ / 2 ) ~ ) '-/ ~ ) SI2
-1
1-
with
where COis the bulk ionic concentration, a is the available surface area per charge, Q is the dielectric constant of the solvent (water in our case), z is the valence of the ionic surfactant, s is a convenient dimensionlessparameter, and K - ~ is the well-known Debye screening length." The first term in eq 8 corresponds to the electrostatic free energy of a planar double layer, and the second term corresponds to a correction due to the curvature (l/R) of the spherical surface. Equation 8 is only valid for symmetric ionic surfactants, for which the valence of the fully dissociated ionic hydrophilic moiety is equal to that of the counterion. Below we will apply eq 8 to the case of 1-1ionic surfactants with or without added 1-1 inorganic salts. The expression in eq 8 is derived from an analytical approximation to the nonlinear Poisson-Bolt" equation22that is very precise as long as the product id3 > 0.5 (the maximum deviation between the full numerical solution and the approximateanalytical expression is about 5% for KR = 0.5). Note that eq 8suffers from the sameinherent limitations of the Poisson-Boltzmann formalism.18*21Specifically,the ions are treated as point charges, dielectric saturation effects are not accounted for, ion polarization and selfatmosphere effects of the counterions are neglected, and specific ion effects are not accounted for. The inclusion of some of these effects in the Poisson-Boltzmann equation, through suitable approximations, has been shown18 to lead to small corrections in many cases of practical interest. In particular, the corrections are expected to be of the order of 3% and are particularly sensitive to the ionic volume of the counterion.18 It is noteworthy that eq 6, relating gel= to Kelec,is valid for micelles of any composition, while the expression for gelecin eq 8 is only valid for pure ionic surfactant micelles (a = 1 or a = 0). Since according to the molecularthermodynamic theory,13 to leading order, the value of Kdw should be weakly dependent on micellar composition, we can equate eq 8 with eq 6, at a* = 1if pure surfactant A is ionic, or at a* = 0 if pure Surfactant B is ionic, in order to find KelW. If surfactants A and B are both ionic, then two values of Kelecare computed, one for a* = 1and the other for a* = 0, and their arithmetic average is taken as an estimate of KelM. The calculation of gel, utilizing eqs 8 and 9, from which one can then compute Kel=(a* = 1)or Kelec(a*= 0)using the other constants eq 6, requires knowledge of R, a, and CO, being readily available. Regarding the radius of the surface of charge, R, we assume that it is equal to the radius of the micellar core, Rwre, plus the distance of the surface of Mitchell, D. J.; Ninham, B. W . J. Phys. Chem. 1983,87, 2996. (23) Evans, D. F.; Ninham, B. W . J. Phys. Chem. 1983,87, 5025. (22)
Table I. Values of D,,-,S Including the Length of the CHI Group Adjacent to the Ionic Hydrophilic Moiety (about 1.27 A) ionic hydrophilic moiety Dcha. (A) 3.77 3.27 3.27 5.59
charge from the micellar corewater interface,D-e, is
that
(10) R = R,, + Dctlwe The value of R , in eq 10 is estimated by assuming that it is approximatelyequal to the fully-extended (all trans) length of the linear hydrocarbon chain of the ionic surfactant which comprises n, carbon atoms. As in previous studies, we assume14that the CH2 group adjacent to the hydrophilic moiety lies within the domain of hydration of the hydrophilic moiety and, therefore, does not contribute to the hydrophobicity of the hydrocarbon chain." Accordingly,we utilize (nc- 1)in the computation of R,,, which yields"
R,, = 1.5 + 1.265(nc- 1) (in A) (11) Note that if SurfactantsA and B are both ionic, we estimate Rc, by taking n, = (ncA + n,~)/2in eq 11,where ncA and ncB are the number of carbon atoms in the hydrocarbon chains of surfactants A and B, respectively. The value of D , h e in eq 10 is estimated from the known chemical structure of the ionic surfactant hydrophilic moiety (note that, in view of eq 11,D , h e should also include the length of the CH2 group adjacent to the hydrophilic moiety (about 1.27 A)). Values of D c 9 e 2 Scorresponding to the various ionic surfactants considered in this paper are presented in Table I. The available surface area per charge, a , at the spherical surface of charge having radius R is simply given by a = 47rR2/N,, (12) where N 8 dis the total number of ionic surfactants in the micelle. Making the conventional assumption24that the micellar core is 'dry", that is, it has a uniform density given by that of bulk liquid hydrocarbon, simple geometric considerations indicate that for a spherical micellar core having radius R,re one has
where" V , = 27.4 + 26.9(nc - 1)(in A3) is the volume of the hydrocarbon chains that make up the micellar core. The bulk ionic concentration, CO,is given by
Co CMCionic+ Cdt (14) since, at the conditions of interest, the bulk surfactant monomer concentration is approximately equal to the CMC of the ionic surfactant. Using the values of R , a, and CO,obtained from eqs 1014, in eqs 8 and 9, one can calculate gelec,which can then be used in eq 6 (with a* = 1or a* = 0)to compute Kelw. Once K e h is known, eq 5 can be utilized to predict DAB. Using the @ABvalue so deduced, along with the values of CMCa, CMCB,and a1 E a B din , eq 7, one can compute a*. Using the values of @AB and a* obtained in this way, one (24) Tnnford, C . The Hydrophobb Effect; Wiley: New York, 1980. (25) Wells, A. F.StructuralInorganic Chemistry; Oxford University
Prese: New York, 1984.
Langmuir, Vol. 8, No.11, 1992 2693
Prediction of Critical Micelle Concentrations 10
0.2
0.6
0.4
2' 0
1
0.8
1
0.2
0.4
asurf
Figure 1. Predicted critical micelle concentration,CMC-,
as
a function of overall surfactant composition, a d , for a mixture of SDS ( a d = 1) and C12E8 (and= 0) in aqueous solution with 0.5 M NaCl at 25 OC: (-) prediction with flg= -1.41 kT; (. ideal-mixtureprediction with flm = 0. The experimental CMC- values ( 0 )are from ref 33,and (- - -) denotes the beat fit to the experimental data with g"& = -2.61 kT.
-
0.8
0.6
a)
Figure 2. Predicted critical micelle concentration,C M L , as a function of overall surfactant composition,w,for a mixture of SDS (ad= 1) and C&Z (= 0) in aqueous solutionwith no added salt at 26 O C : (-) prediction with /3g= -4.03 kT; (. ideal-mixture prediction with flm = 0. The experimental CMCh values (0)are from ref 33,and (- -1 denotes the beat fit to the experimentaldata with = -4.36 kT.
-
a)
a
can predict fA, fB, and CMCmixas a function of the overd surfactant composition, aSd,using eqs 2, 3, and 1, respectively. In the next section we utilize the formalism developed in this section topredict BAB aswell as CMC- as a function of the overall surfactant composition, as&, for anionicnonionic, cationic-nonionic, and anionic-cationic surfactant mixtures.
t
l
111. Theoretical Predictions and Comparison with
Experiments 0
We have collected experimental CMC- data from the literature1'**% cuvering anionic-nonionic, cationic-nonionic, and anionic-cationic surfactantmixtures in aqueous solutionswith and without added inorganicsalts. We have applied the simplified "working model" presented in section I1 to each of these mixtures and have compared the predicted values of CMC,t at various surfactant compositions, a,d, with the reported experimental values of CMC- (see Figures 1-7). We have also compared the with predicted values of Om, denoted hereafter as the experimentally-deduced optimal values, (see Table I1 for anionic-nonionic surfactant mixtures, and Table 111for cationic-nonionic surfactantmixtures). Note that P& corresponds to the flm value which best fits the experimental CMC- versus a , d data in a least-squares fit sense (for details, see the Appendix). Note that all the predicted data correspond to cases where the product KR > 0.5,since, as stated earlier, eq 8 is not expected to be a good approximation when KR < 0.5.22 Indeed, we have found that for mixtures where KR < 0.5 the simplified model yields particularly poor predictions. Note that by
Be,
~~
1 9 a , 81,207. (30) Lucassen-Reyndere, E. H.; Lucaeeen, J.; Gilee, D. J . Colloid Interface Sci. 1981,81, 160. (31)Stellner, K. L.; Scamehom, J. F. J. Am. Oil Chem. SOC. 1986,63,
566. (32) Carri6n Fit4, F. J. Tenside Deterg. 1985,22, 226. (33) Lange, V. H.; Beck, K. H.Kolloid 2. 2.Polym. 1973, 261, 424. (34)Bowel, M.;Bernard, D.; Graciaa, A. Tenside Deterg. ISM, 22, 311.
0.2
0
0.6
0.4
0.8
1
asurf
Figure 3. Predicted critical micelle concentration,CMCd, as
a function of overall surfactant compoeition, a d , for a mixture ( a d = 0) in aqueous solutionwith prediction with = -4.02 kT; (. *) ideal-mixtureprediction with flm = 0. The experimental CMC- values ( 0 )are from ref 26,and (- - -) denotesthe beat fit to the experimentaldata with = -3.86 kT. Nota that the predicted (using fie) and fitted (using curves are almoet indietinguishable in this case. of SDS (and= 1) and C& no added salt at 25 O C : (-)
e
.
a)
"*""
0.25
-f BE
I
4
1
0.20 0.15
0
5
0.10 0.05
~
(26) Treiner, C.; Amar Khodja, A.; Fromon, M. J. Colloid Interface Sci. 1989, 128, 416. (27) Yoesting, 0. E.; Scamehom, J. F. Colloid Polym. Sci. 1986,264, 148. (28) Nguyen, C. M.; Rathman, J. F.; Scamehorn, J. F. J . Colloid Interface Sci. 1986, 112,438. (29) Hey, M. J.; MacTaggart, J. W. J. Chem. Soc., Faraday Trans. 1
(35) Rubingh, D. N. Private communication.
1
asurf
0.00
'
0
I 0.2
0.4
0.6
0.8
1
asurf Figure 4. Predicted critical micelle concentration,CMCA, an a function of overall surfactant composition, w,for a mixture of SDS (and= 1) and GC&-E10 (- = 0) in aqueous solution with 0.66 M NaCl at 30 OC: (-1 prediction with = -1.28 kT; (. .) ideal-mixturepredictionwithflm= 0. Theexperimental CMC- values ( 0 )are from ref 31,and (- - -1 denotes the beat fit to the experimentaldata with p d = -1.62 kT.
utilizing the full numerical solution to the PoissonBoltzmann equation, it is possible to relax the constraint KR > 0.5. However, the calculations in this case become
2694 Langmuir, Vol. 8,No. 11, 1992
-=g,
Sarmoria et al.
O 0.15 a2OI
I
0.00 I 0
0.4
0.2
0.6
1
0.8
asurt
Figure 6. Predicted critical micelle concentration, C M L , as a function of overall surfactant compoeition, u,for a mixture of Cla-Cat-SOaa. (w= 1) and G C f i - 4 0 (CY.& = 0). in
aqueous solution mth 0.15 M NaCl at 38 O C : (-) predicbon with = -2.21 kT; .) ideal-mixture prediction with /3m = 0. The experimental CMCd values ( 0 )are from ref 28, and (- - -) denotes the beat fit to the experimental data with = -2.24 kT. (a
50
-
t
value, fJ,which is found by calculatingbm at each value of a.d for which a measured value of CMC- is reported, and then taking the arithmetic average of all the values so deduced (for details, see the Appendix). We have adopted this procedure of deducing @mbecause it hae been implemented by some researchers to interpret their measured CMC- data.82 Although we have found that the difference between the experimentally-deduced values of p& and ,$Jis small in most cases (about 0.1 kT), in about 20% of the mixtures that we have examined a difference of about 1 kT was found. This finding is significant because it is quite common to fiid reported literature values of @AB where the method utilized to deduce the "best" @ABvalue is not clearly ~pecified.11*%*81*~ This, in turn, implies that caution should be exercised when utilizing reported literature values of @AB, since it is quite possible that the reported @mvalue is about 1 kT off from the "best" value. As shown in Tables I1and 111,Mm,the differencebetween and @&,is also of the order of 1 kT for several muttures, thus making the error in the theoretical predictions comparable to that resulting from the different w a y of fitting the experimental CMC- data to deduce
@e
om.
In the remainder of the paper we use an abbreviated notation for the various surfactante considered (for an explanation of the abbreviations, see Tables 11and III). An examination of Table I1 (for anionic-nonionic surfactant mixtures) reveals that, in most cases, the simplified "working model" yields very reasonable predictions. The best result in Table I1 (mixture 6, SDS-CgC&-Ea) corresponds to = = -4.10 kT. The worst reeulta in Table I1 (mixture 1,SDS-C&, and mixture 14,SDSC1&) correspondto A& = - pg values of 1.63 kT and -1.64 kT,respectively. In about half of the mixtures in Table 11,the A@- values are lese than 0.5 kT,and, with the exception of mixtures 1 and 14 described earlier, no A@mvalue exceeds 1.4 kT. As shown below, within this range of M m values, it is poesible to obtain good predictions of CMC- versus as& (see Figures 1-5). The aimpWied "working model" is also able to capture the effect of added inorganic salta, as reflected in the corresponding values of in Table I1 (see also Figures 1 and 5). In particular, mixtures 2 to 5 in Table I1 (SDSCg-C&-Elo) illustrate the effect of adding NaCl. In general, one expects that an increase in the d t concentration will lead to a more effective screening of electrostatic interactions, thus bringing the mixture closer to ideality. In other words, the @ABvalues should get closer to zero (recallthat @AB= 0 correspond to an ideal mixture) as the salt concentration increases. This is precieely what the simplified "working model" predicte: = -3.03 kT with 0.03 M NaC1, and = -1.13 kT with 0.9 M NaCl, where M denotes mol/L. The experimentally-deduced values of p& reflect a similar trend: pg = -2.75 kT with 0.03 M NaCl, and p& = -1.68 kT with 0.9 M NaCl. Predictions of CMC- as a function of the overall surfactant composition, aSd,for five representative anionic-nonionic surfactant mixtures taken from Table 11 are presented in Figures 1-5. In each w e , the full line correspondsto predictions made by using b e , the dotted line correspond to the ideal-mixture prediction (using BAB = 0),the dashed line corresponds to the best fit to the experimental data using @& (see the Appendix), the circles correspond to the experimentalCMC- values, and alld= 1 corresponds to the anionic surfactant.
@ea
0.2
0
0.6
0.4
0.8
1
asurt
Figure 6. Predicted critical micelle concentration, CMCd, as a function of overall surfactant composition, a d ,for a mixture = 0) in aqueous solution of CloTMAB (w= 1) and C& (u with 0.05 M NaBr at 23 OC: (-) prediction with = -2.22 k T; ideal-mixture prediction with /3m = 0. The experimental CMCd values (0)are from ref 11, and (- - -) denotes the best fit to the experimental data with = -1.76 kT. (0
0)
at
@e
B e
5.
0.14
--
0.12
--
I
E 3
Y
E
Y 0
0.02
'
0.2
0
0.4
0.6
0.8
I
1
asurt
Figure 7. Predicted critical micelle concentration, CMCd, as
a function of overall surfactant compoeition, a,&, for a mixture = 1) and G C A - E l o ( a d = 0) in aqueous of CleTMAC (u solution with 0.03 M NaCl at 30 OC: (-) prediction with = -3.49 kT; (- -)ideal-mixture prediction with / 3 =~0. The experimental CMCd values ( 0 )are from ref 28, and (- - -1 denotes the best fit to the experimental data with = -1.56
e
-
k T. considerably more involved, thus reducing the simplicity and usefulness of the approach proposed in this paper. We would like to emphasize that, for each mixture considered, we have ale0deduced an average experimental
@e
@e
Prediction of Critical Micelle Concentrations
@e, a,and A
Table 11. Valuer of 1
2 3 4
5 6
7 8 9 10 11
mixture SDS C& SDS GC&-Eio SDS GW-Eio SDS GC&-Eio SDS GC&-Eio SDS GCBHI-Es SDS GC&-Eio SDS GCBHI-Eis SDS GC&-E20 SDS CeEs SDS
salt (M) NaBr 0.05 NaCl 0.03 NaCl 0.4 NaCl 0.65 NaCl 0.9
temp (OC) 25
30 30 30 30 25 25
Langmuir, Vol. 8, No. 11,1992 2696 h=
@e- a for Various Anionic-Nonionic Surfactant Mistum.
CMCA(mM) and CMCB(mM) 17.9 6.30 2.12 0.0525 0.40 0.0286 0.284 0.027 0.20 0.0277 7.24 0.055 7.24
7.24 0.107 7.24 0.173 8.00 8.60 8.00 9.00 8.00 7.00 0.41 0.06 8.00 0.098 8.10 0.065 8.10 0.045 7.96 7.17 0.79 0.08 0.379 0.0286 0.2175 0.04788
25 25 25 25
SDS C&
13
14 15 16 17
SDS Cl& SDS Cl& SDS C1& SDS Cl2& SDS C&
18
SDS
19
CPC&-E7.6 SDS
20
& (kT)
Mm(kT)
ref
-2.35
-3.98
1.63
11
-3.03
-2.75
4.28
31
-1.51
-1.60
0.09
31
-1.28
-1.52
0.29
31
-1.13
-1.68
0.55
31
-4.10
-4.10
0
32
-4.10
-3.60
-0.5
32
-4.10
-4.92
0.82
32
-4.10
-4.09
-0.01
32
-4.03
-3.99
4.04
33
-4.03
-4.36
0.33
33
-4.03
-3.07
-0.96
33
-1.41
-2.61
1.2
33
-4.03
-2.39
-1.64
33
-4.02
-2.63
-1.39
26
-4.02
-3.86
-0.16
26
-4.03
-2.90
-1.13
29
-2.05
-2.76
0.71
27
-1.51
-1.40
-0.11
28
-2.21
-2.24
0.03
28
0.0758
ca12
12
& (kT)
25 NaCl 0.5
25 25 25 25 25
NaCl 0.15 NaCl 0.4 NaCl 0.15
30 30 38
a In thia Table, SDS denotes sodium dodecyl sulfate, and n-alkyl poly(ethy1ene oxide) or n-alkylphenyl poly(ethy1ene oxide) surfactants with i alkyl carbons and j ethylene oxide unita are denoted as CiEj or C i - W - E j , respectively.
Table 111. Valuer of 1
2 3 4
5 6
mixture CloTMAB C& CieTMAC Cl& CieTMAC C rW - E i o cU(CsH$J)c1 GCBHI-Eio CirTMAC
@e, a,and
salt (MI NaBr 0.05 NaCl 0.1 NaCl 0.03 NaCl 0.03
A@AB=
temp ("(2) 23
25 30 30 25
cia6
(CdDMBAC
25
cia6
& - a for Variour Cationic-Nonionic Surfactant Mixtures.
CMCA(mM) and CMCB(mM) 48.51 6.20 0.068 0.087 0.15 0.052 0.0915 0.0531 5.78 0.76 19.71 0.761
& (kT)
MAE (kT)
-1.76
-0.46
ref 11
-2.64
-0.92
-1.72
33
-3.49
-1.56
-1.93
28
-3.17
-1.30
-1.87
28
-4.63
-1.44
-3.19
35
-1.30
-1.52
0.22
35
(kT) -2.22
In this table, TMAB denotes trimethylammonium bromide, TMAC d e n o h trimethylammonium chloride, DMBAC denotee dimethylbenzylammonium chloride, CBHI ie a phenyl group, Ci is a segment with i alkyl carbons,and Ej is a segment with j ethylene oxide units.
Figure 1 corresponds to a surfactant mixture of SDS and Cl2Es in aqueous solution with 0.5 M NaCl at 25 OC (mixture 13 in Table I1 with = -1.41 kT and p$ = -2.61 kT). In spite of the fact that A@ABis not small (1.2 kT), the overall CMC- versus a.d predictions are reasonably good. The largest errors occur for a n d> 0.8 and are of the order of 30%. This is a considerable improvementover the ideal-mixture prediction (BAB= 0), which yields errors of the order of 90% for a , d > 0.8. Figure 2 corresponde to a surfactant mixture of SDS and C&Z in aqueous solution with no added salt at 25 OC = -4.03 kT and g"g= (mixture 11 in Table I1 with
/3e
-4.36 kT). Although this surfactant mixture departa considerably from ideality, both the CMC,i, versus a , d predictions and the predicted value of are very good. The largest error occurs at a n d= 0.05, where the prediction is off by 18% (note that the fitted curve using p$ is off by 11% at that composition). All other errors m the predicted CMC- values are smaller than 5%. Figure 3 corresponds to a surfactant mixture of SDS and ClzE4 in aqueous solution with no added salt at 25 OC = -4.02 kT and p& = (mixture 16 in Table I1 with -3.86 kT). In this case, A/3m = -0.16 kT,and the C M C h versus a n d predictions are excellent. Note that the
/3e
2696 Langmuir, Vol. 8,No. 11, 1992
@g)
Sarmoria et al.
$g)
methylammonium surfactants, but do not seem to be predicted (using and fitted (using curves are present or are greatly reduced in anionic alkyl sulfate practically indistinguishable,and no error in the predicted surfactants.ss The closer association of the cationic CMC- values is larger than 5%. hydrophilic moiety with its hydrated counterion results Even though the simplified “working model” presented in a more effective screening of the electrostatic interacin section I1was tailored for surfactants containinga linear tions, thus reducing the magnitude of geh. Thia leads, in hydrocarbon chain (see eqs 10 and 11) and the accomturn, to a smaller synergism in the cationic-nonionic case panying discussion), it can also be implemented for as compared to the anionic-nonionic case. Note that, as surfactants which possess a phenyl group adjacent to the stated earlier, specific-ion effects of the type described hydrophilic moiety. For surfactants of this type, we have above are not accounted for by the Poisson-Boltzmann assumed that the effective size of the phenyl group is equation, or its simplifed version, eq 8,which, in practice, comparable to that of three consecutive alkyl carbons in is incapable of distinguishing between cationic and anionic the evaluation of R,. In other words, eq 11 is now written surfactants on this basis. As a result, the value of g e h as R , e 1.5 1.265(nc+ 3 - 1) = 1.5 + 1.265(nc + 2) (in predicted in the cationic case using eq 8 is larger than the A). Figures 4 and 5 show two representative cases of this actualgel, value. In view of eqs 6and 5, a larger predicted type taken from Table 11. Figure 4 corresponds to a value of gelecleads to larger predicted negative values of surfactant mixture of SDS and Cg-CsH4-Elo in aqueous as compared to the experimentally-deduced g”& solution with 0.65 M NaCl at 30 OC (mixture 4 in Table values. Consequently, is is plausible that the observed = -1.28 kT and g”g= -1.52 kT). Figure 5 I1 with negative values of A/3m in the cationic-nonioniccase reflect corresponds to a surfactant mixture of Cl2-C&SO3Na the inability of the Poisson-Boltzmann equation (or its and Cg-C&-Elo in aqueous solution with 0.15 M NaCl at simplified version, eq 8) to capture some of the subtle 38 OC (mixture 20 in Table I1 with = -2.21 kT and specific-ioneffecta associatedwith the cationic surfactants. g”& = -2.24 kT). In both cases, the predictions of Figures 6 and 7 show two representative cases taken and CMC- versus as& are very good. The predicted from Table 111, where the notation is identical to that and fitted (using g”&) curves are very close, (using used in Figures 1-5. Note that a,& = 1 corresponds to and no error in the predicted CMCmi, values is larger than the cationic surfactant. Figure 6 corresponds to a sur8%. factant mixture of CloTMAB and C& in aqueoussolution The simplified model has also been utilized to predict with 0.05 M NaBr at 23 OC (mixture 1 in Table I11 with values of @ip and CMC- versus aB& for several cat= -2.22 kT and g”$ = -1.76 kT). Figure 7 correionic-nonionic surfactant mixtures11*30*s*36 (see Table I11 sponds to a surfactant mixture of ClsTMAC and C&& and Figures 6 and 7). An examinationof Table I11reveals El0 in aqueoussolution with 0.03 M NaCl at 30 OC (mixture and g”& is that the overall agreement between 3 in Table I11 with = -3.49 kT and g”& = -1.56 k!l‘,). reasonable (althoughnot as good as in the anionic-nonionic The figures show that the predictions are reasonable. The case, see the discussion below). Note that in all the largest errors in Figure 6 are of the order of 10%. The mixtures listed in Table 111, the product KR > 0.5 (as errors in Figure 7 are somewhat larger, of the order of explained earlier). Since we have fewer mixtures than in 35% ,which, nevertheless, represents a considerable imthe anionic-nonionic case, it is more difficult to draw provementover the ideal-mixture prediction which yields general conclusions. With this in mind, the best result in errors of the order of 50%. Table I11 (mixture 6, (Cs)2DMBAC-C&s) corresponds Regarding binary mixtures of anionic and cationic to A@m= 0.22 kT,the worst result (mixture 5, CMTMACsurfactants, we were only able to find three sets of C&s) corresponds to A@m= -3.19 kT,and all remaining published CMCmi, versus a,& data.11@*34One of these mixtures have A&m values smaller than 2 kT. We have mixturesM does not fulfii the requirement that the product found that reasonable CMC,i, versus a,& predictions are KR > 0.5. The simplified “working model” yields a fair obtained using these @ABvalues (except for mixture 5, prediction (BGd = -8.98 kT and g”& = -12.65 kT) for the which exhibits the largest A@mvalue), as illustrated in second mixture” (SDSand CloTMAB in aqueoussolution Figures 6 and 7. with 0.05 M NaBr a t 23 “C) and a poor prediction An examination of Tables I1 and I11 indicates that = -15.53 kT and g”& = -22.64 k T ) for the third anionic-nonionic surfactant mixtures exhibit larger negmlxturegO (SDS and C12TMAB in aqueous solution with ative deviations from ideality (more synergism) than no added salt at 25 OC). Clearly,these data are insufficient cationic-nonionic ones, as reflected in the more negative to draw any meaningful conclusions about the applicability in the former case. This result is in line with values of of our approach in the case of anionic-cationic surfactant the well-accepted view that synergism increases in the mixtures. sequence nonionic-nonionic, cationic-nonionic, anionicnonionic, and anioniccationic surfactant mixture~.~v~J~ IV. Conclusions Table I11 also indicates that, except for mixture 6, the We have simplified a recently developed molecularsimplified “working model” consistently predicts larger thermodynamic theory of mixed surfactant solutions to negative deviations from ideality than those observed obtain a “workingmodel”that can be utilized with relative experimentally, as reflected in the more negative values ease to predict cmc’s of nonideal binary surfactant of Bg* as compared to those of g”&. One can speculate mixtures, as well as to predict values of the specific about the origin of the observed trend in the @ABvalues interaction parameter, The required inputs of the in the cationic-nonionic case, namely A@m< 0, as follows. model are the cmc’s of the pure surfactants, the overall Water 170relaxation-rate measurements seem to suggest surfactant composition as well as other solution conditions the existence of specific interactions between the water of such aa temperature and salt type and concentration, and counterion hydration and the water of hydration associated the chemical structures of the hydrophobicand hydrophilic with the cationic surfactant hydrophilic moiety.= These moieties of both surfactant species. We find that the model specific interactions are present in the cationic alkyltriyields reasonable quantitative predictions for anionicnonionic and cationic-nonionic mixtures of single-tailed (36)Halle, B.; CarktrBm, G. J. Phys. Chem. 1981, 85, 2142, and references cited therein. surfactants having linear hydrocarbon chains or linear
+
@e,
B e
@e
@e
@g)
@g
@e
@e
(@c
g”g
@e.
Rediction of Critical Micelle Concentrations phenyl hydrocarbon chains. We do not have sufficient experimentaldata on anionic-cationic surfactantmixtures to assess the predictive capabilities of the model for these mixtures. We hope that the simplied "working model" for the prediction of mixture cmc's presented in this paper may serve as a valuable preliminary screening tool in the design and selection of nonideal surfactant mixtures of practical importance.
Acknowledgment. This research was supported in part by the National Science Foundation (NSF) Presidential Young Investigator (PYI)Award to Daniel Blankschtein and an NSF Grant (No. DMR-84-18778) administered by the Center for Materials Science and Engineering at MIT. Daniel Blankschtein is grateful for the support of the Texaco-Mangelsdorf Career Development Professorship at MIT and is also grateful to the following companies for providing PYI matching funds: BASF, Kodak,and Unilever. We also thank D. N. Rubingh for making accessible to us some of his unpublished data on cmc's of cationic-nonionic surfactant mixtures (see mixtures 5 and 6 in Table 111). Appendix A Deduction of fl3and 0% from the Experimental CMC,i, Data If the critical micelle concentration of a mixture, CMC-, having composition,a d ,is known,then, utilizing
Langmuir, Vol. 8, No.11, 1992 2697 eqe 1,2,3, and 7, one obtains @m -kT -
[(~~CMC-)/(~*CMCA)I (1- a*)2
(Al)
where a1 = as&. An average value PJ can be found by applying eq A1 to each measured CMC- value and subsequently taking the arithmetic average, that is
In order to obtain a more statistically-representative value, Pg,which corresponds to the value of @mthat best fits the experimental CMC- versus a n ddata in a leastsquares fit sense, we defiie a function F such that
where a1 = as&, fA and fB are given in eqs 2 and 3, and (CMCZ)j is the ith measured value of CMC-. Minimization of F with respect to @AB(note that a*,which appears in the expressions for fA and fB, is also a function of @m)yields the optimal value P&.
