1642
J . Phys. Chem. 1984, 88, 1642-1648
mospheric moisture, the free energy of activation was not significantly different from the value presented here. Vibrational spectroscopic studies of the adduct, SbCl3-DMA, have clearly established that adduct formation occurs at the oxygen atom of the carbonyl group. A decrease in the CO stretch frequency and an increase in the CN stretch and O C N deformation frequencies have been observed on complex for ma ti or^.^^'^^' Although the presence of a small amount of adduct with coordination a t the nitrogen cannot be completely ruled out, there is no evidence which indicates the presence of this type of adduct. The Lewis acids, AlBr, and GaCl,, have been observed to form complexes with N,N-disubstituted amides in which two molecules of the acceptor are complexed to one molecule of the amide.30 Although the site of second acceptor could not be unambiguously determined, it was concluded that the center of coordination of the second acceptor molecule is not the nitrogen atom. In the molecular orbital description of the amide bond, the planar ground-state configuration is stabilized by mixing of the nitrogen lone-pair orbital with the carbonyl a bond, giving rise to a barrier to internal rotation.31 Adduct formation of Lewis acids with the oxygen of the carbonyl group should further stabilize the ground state. However, a decrease in the rotational barrier would occur if adduct formation had a greater stabilizing effect on the transition state. A rotation of 90° about the CN bond would place the nitrogen lone-pair and the oxygen lone-pair electrons in the same plane, and SbC1, could coordinate to both oxygen and nitrogen simultaneously. SbCI, does form five-coordination complexes with oxygen,10~1z-14*32 nitrogen,32and sulfur donor^.^^-^^ Also, the amide nitrogen would be a much stronger (30) L. A. Ganyuskin, I. P. Romm, E. N. Gur’yanova, and R.R. Shifrina, J . Gen. Chem. USSR (Engl. Transl.), 50, 1739 (1980). (31) J. F. Yan, F. A. Momany, R. Hoffmann, and H. A. Scheraga, J . Phys. Chem., 74, 420 (1970). (32) M. J. Gallagher, D. P. Graddon, and A. R. Sheikh, Thermochim. Acta, 27, 269 (1978). (33) G. Kiel and R. Engler, Chem. Ber., 107, 3444 (1974).
donor in the transition state, where conjugation with the carbonyl group is not possible. A coordination of SbC1, to both oxygen and nitrogen atoms could stabilize the transition state sufficiently to reduce the barrier to internal rotation. Another possible structure for the transition state which would produce the same effect is a cyclic structure in which the oxygen atom is coordinated to the antimony atom and the nitrogen coordinated to SbC1, through one of the chlorine atoms. In a recent calorimetric study of the reactions of antimony(II1) halides with Lewis bases, 3 mol of aliphatic amines was observed to react almost quantitatively with 1 mol of SbX3 in 1,2-dichloroethane solution.32 The evidence indicated that, in these adducts, the three molecules of base are not coordinated with the antimony atom but rather one to each halogen. This type of adduct formation was observed, however, only with the strongest proton bases and was not observed with heterocyclic bases, aromatic amines, oxygen donors, or phosphines. In the nonplanar amide molecule, the nitrogen may be sufficiently basic for this type of interaction to occur. The decrease in the barrier to internal rotation of D M A on adduct formation with SbC1, is apparently due to the greater stabilizing effect in the transition state. This effect may also be present in the other amide adducts in which a decrease in the rotational barrier was o b ~ e r v e d . ~T~h ~e *type of structures suggested for the transition state would not be expected for most amide complexes because of steric or electronic considerations.
Acknowledgment. W e wish to thank Prof. L. W. Reeves of the University of Waterloo, Waterloo, Ontario, Canada, for providing the computer program. The financial support of FAPESP is gratefully acknowledged. Registry No. SbC13-DMA,40325-80-2. (34) G. C. Pellacani, G. Peyronel, W. Malavasi, and L. Menabue, J. Inorg. Nucl. Chem., 39, 1855 (1977). (35) V. M. Schmidt, R. Bender, and C. Bruschka, Z . Anorg. Allg. Chem., 454, 160 (1979).
Mass-Action Model of Mixed Micellization R. F. Kamrath and E. I. Franses* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907 (Received: May 12, 1983)
We develop the single-micelle-sizemodel for dilute aqueous binary ionic surfactants with the same polar group and counterion. We determine the mixed micelle composition(s) by minimizing the total Gibbs free energy. We calculate first and second critical micellization concentrations (cmc’s), monomer, micelle, and counterion concentrations, and micelle mole fraction(s) x . The parameters are total surfactant concentration, mole ratio of surfactants, salt concentration, individual cmc ratio, degree of counterion binding 0,micelle aggregation number N , and the excess free energy function w ( x ) of mixing of the surfactant chains. For dilute solutions with no intermicellar interactions we develop general conditions of azeotropic mixed micellization and of demixing of partially miscible micelles. The results for = 0 apply to binary nonionic surfactants. We present numerical results for w ( x ) = wo, Le., the strictly regular solution model. For that model the necessary condition for demixing is wo > 2 and for azeotropy is lwol 2 (((1 + @ ) N - 1)/N) In (c2*/c1*). The mass-action model (MAM) should be preferred over the simpler pseudophase separation model (PSM) for N less than about 50.
Introduction Surfactants are used extensively to control the bulk and interfacial properties of solutions. Surfactants in solution can self-associate to form equilibrium, closed, colloidal aggregates called micelles. In aqueous solutions, micelles usually consist of 20 to several hundred molecules, which are associated so that the surfactant polar groups form a closed shell surrounding the hydrophobic surfactant hydrocarbon chains.’ The micelle interior region, being fluid-hydrocarbon-like, can solubilize hydrocarbons.* (1) Mukerjee, P. Adu. Colloid Interface Sci. 1967, I , 241.
Micelles indirectly affect interfacial properties. Their formation determines monomer activities and hence surfactant adsorption a t ,interfaces. The formation of mixed micelles, which contain more than one type of surfactant, is especially important since most commercially available surfactants are in fact mixtures. W e have developed general models of micellization of binary nonionic-nonionic and ionic-ionic surfactant mixtures in the pseudophase separation limit.3%4W e have considered the mixed (2) Roberts, R. T.; Chachaty, C. Chem. Phys. Letf. 1973, 22, 348. (3) Kamrath, R. F. M.S. Thesis, Purdue University, West Lafayette, IN, 1981.
0022-3654/84/2088-1642$01.50/00 1984 American Chemical Society
The Journal of Physical Chemistry, Vol. 88, No. 8, 1984 1643
Mass-Action Model of Mixed Micellization micelles to be a pseudophase with an effectively infinite aggregation number. In this paper we examine effects of finite micelle sizes by describing mixed micellization with a Bury-Hartley mass-action m ~ d e l . ~In , ~this model, all micelles are assumed to have the same fixed number of surfactant molecules. In reality, micelle sizes are distributed; they depend on the total surfactant concentration and on the micelle composition. The micelle size of ionic surfactants also increases with increasing ionic strength of the solution.’ In this paper, we ignore size distributions for simplicity and because detailed information on sizes in mixed systems is unavailable. Although multiple-equilibrium models are surely more realistic, the single-equilibrium approach is reportedly capable of matching experimental results, such as the concentration dependence of N M R chemical shifts of solutions of ionic Surfactants in organic solvents.8 For mixed nonionics, the mass-action model has previously been derived for the case of ideal mixing in the micelle^,^ but no calculations have been presented. We generalize this model to ionic systems, accounting for the effect of the surfactant counterions, and to’general nonideal micellar mixing behavior. For micellar mixing described by the strictly regular solution model, we report sample calculations to illustrate the effect of micelle size and to compare the mass-hction model (MAM) results to those for the pseudophase separation model (PSM).
Theory One Type of Mixed Micelle. The micellization equilibrium for the Bury-Hartley mass-action model for mixed binary ionic surfactant with a common counterion is
where R1- and R2- are monomers of surfactants 1 and 2, M+ is the surfactant counterion, (RiR2Mp)N is a mixed micelle composed of N surfactant monomers and pN counterions, and x is the fraction of monomers of surfactant 1 in the mixed micelle. For nonionic mixtures, the surfactant monomers and micelles are of course uncharged and M+ does not enter the equations. Then the model is a limiting case of the ionic model with p = 0. The surfactant and counterion material balance equations are C, XNC, = CYCT (2)
+
~
(+~ - x ) N c , = ( I - o c ) c T CM+ pNC, = CT C,
2
+
(3)
+
(4) where cl, c2, and cM+are the molal concentrations of R1-, R2-, and M+, c, is the concentration of mixed micelles, cT and c, are the total surfactant and salt concentrations, and a is the mole fraction of component 1 in the overall binary mixture. We will take a fixed as cT is increased to simulate what happens with practical surfactants whose overall composition is fixed. W e now minimize the total Gibbs free energy of the system per 1000 g of ~ o l v e n t .Since ~ N is taken to be fixed, G = G(c,,x). Since activity coefficient effects in dilute solution are often secondary,I0 we will ignore them in this paper. We will focus instead on activity coefficient (nonideal mixing) effects in the mixed micelles. The first condition of equilibrium, dG/ac, = 0, yields the usual equation P m = XNPI+ (1 - X)NPZ+ PNPM+ The second condition of equilibrium, aG/ax = 0, yields
(5)
where pmo is the reference chemical potential of mixed micelles of mole fraction x. This equation represents the key difference between the pseudophase separation model (PSM) and the mass-action model (MAM). For the PSM we can define chemical potentials of components 1 and 2 in the micellar pseudophase. For the MAM the free energy of mixing of the chains is contained in the left-hand side of eq 6, which provides the dependence of chemical potential on the aggregation number N. In the limit as N m, eq 5 and 6 reduce to the usual equations of equilibrium between two different phases3s9
-
PI = PI,,
P2
=
W2,m
where the subscript m indicates the component in the micellar pseudophase. We relate the standard free energy of the mixed micelles at infinite dilution to the standard free energies of pure micelles of surfactants 1 and 2 as
CLmO=N
+ ( l - x)kNo N N R q [ x In x (1 - x) In (1 - x)]
+
XPINo
+
+ x(1 - x)w(x)) (7)
Equation 1 can be expressed as the sum of the individual micellization equilibria of surfactants l and 2 and a micelle mixing equilibri~m.~ This fact and the standard definitions of equilibrium constants for mixed micelles (K,) and pure micelles ( K , and K 2 ) readily imply that
K, = K1xK2(1-x)~-*N(1 - x ) - ( ] - ~ exp[-Nx(1 )~ - x)w(x)]
(8)
Notice that the same N was used for pure micelles of 1 and 2 and for mixed micelles. If N1 and N2 are taken as different, then N = N(x) and a more complicated model will result. Using eq 6-8 we now obtain an equation for the equilibrium micelle composition in terms of the monomer concentrations of the two surfactants.
From eq 2-4, 8, and 9 one can calculate the monomer, micelle, and counterion concentrations and the micelle composition for given a , p, N, cs, and cT if the equilibrium constants K , and K2 are known. Normally the equilibrium constants cannot be determined directly. What is directly measurable is the critical micellization concentration, or cmc. Experimentally, the cmc is the narrow range of concentrations over which the slope of a plot of a colligative property of the solution vs. surfactant concentration changes abruptly.” This range is often narrow enough to be taken as a single point. Mathematically, one should select a definition of the cmc that is consistent with this n o t i ~ n . ’ ~In~ the ’ ~ PSM, micelles are taken to constitute a distinct phase and micellization is modeled as a first-order phase transition. Hence, the cmc is defined as that total surfactant concentration below which there are no micelles. In contrast, in the MAM a finite number of micelles is present at essentially all surfactant concentrations. Below a certain concentration the molar concentration of micelles is, however, negligible. For the MAM we have thus chosen to define the crnc as the total surfactant concentration at which a small fraction 6 of the surfactant molecules is in micellar form. Hence, at the cmc, cT = c* when Nc, = cCT n
c
C,
=
CT
- NC, = (1 - C)CT
(10)
1=1
(4) Kamrath, R. F.; Franses, E. I. Ind. Eng. Chem. Fundam. 1983, 22, 230.
(5) Jones, E. R.; Bury, C. R. Philos. Mag. 1927, 4 , 841. (6) Murray, R. C.; Hartley, G. S. Trans. Faraday SOC.1935, 31, 183. (7) Missel, P. J.; Mazer, N. A,; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980, 84, 1044. (8) Muller, N. J . Phys. Chem. 1975, 79, 287. (9) Franses, E. I.; Bidner, M. S.; Scriven, L. E. In ”Micellization, Solubilization, and Microemulsions”; Mittal, K. L., Ed.; Plenum Press: New York, 1977; Vol. 2, p 855. (10) Tanford, C. ‘The Hydrophobic Effect”; Wiley: New York, 1980.
This definition can be used for single or mixed surfactants. For our calculations we have taken E = 0.02. The value of c can of course be varied to account for the physics of the system, for (1 1) Mukerjee, P.; Mysels, K. J. “Critical Micellization Concentration of Aqueous Surfactant Systems”; Superintendent of Documents, US.Government Printing Office: Washington, DC, 1971; NSRDS-NBS-36. (12) Corrin, M. L. J . Colloid Sci. 1948, 3, 333. (13) Hall, D. G. J . Chem. Soc., Faraday Trans. 2, 1972, 68, 668.
1644 The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
example, the width of the cmc range. As N approaches infinity and E approaches zero, the MAM cmc approaches the PSM cmc3 The individual equilibrium constants K1 and K2 can then be expressed in terms of the individual cmc’s of cl* and c2* as
The reader should note that here cI* is the cmc in the absence of salt. The salt concentration affects the cmc but of course does not affect the equilibrium constant. Then eq 9 becomes
- X- 1 --x
Kamrath and Franses The micelle composition at the cmc is independent of the amount of added salt. The cmc decreases, however, with the addition of salt. For the regular solution model, w(x) = wo = constant, we use eq 12-19, except we take awlax = 0. Coexistence of Two Types of Mixed Micelles. If there are severe positive deviations from ideal mixing of the surfactant chains in the mixed micelles, the micelles can become thermodynamically unstable and demix into two stable coexisting types of mixed micelles with different compositions, which are called binodals. Evidence or inference of demixing has been reported for mixtures of fluorocarbon and hydrocarbon s u r f a c t a n t ~ . l ~Using - ~ ~ bulk solution thermodynamic stability criteria,16 we have derived a general sufficient condition for micelle stability as d2g/ax2 >
-
‘IN-’[
TI”[$1
+ Q[ xc2*
x)’Nw(x)
(l+@)N-l
;p[
-(I -
- ~ ( -l
(1 - x)c2* x2(l-x)Ne])
ax c2* + @Q[ ““’
$I”[
(1-x)N
(1 -c2x)c2*]
= (1 - a ) ,CT (14) c2
[ $1
PN
=
7+ 7(15)
Q = t / ( ( l - ~ ) ~ -( @E)@? 1
(16)
exp[-x(1
- x)Nw(x)]
cT
c8
c2
c2
where
Equations 12-16 are the working equations for determining cl, c2, C M t , and x if there is only one type of mixed micelle. Then
c, can be determined from eq 2 or 3. We calculate the mixed cmc as follows. At the mixed cmc, the material balance eq 2-4 and 10 yield CI c2
=
(cy
- ex*)c*
= [(l - a ) - E(1 - x * ) ] c * CM+
= (1 - pE)c*
+ c,
(17)
These and eq 12 imply that the micelle composition x* at the mixed cmc is given by
--X *
1-x*
-
[(l
(20a)
or
This key expression reduces to the corresponding equation for the PSM for N m and to that for nonionic surfactants for @ = 0. Using eq 8 and 11 and the material balance equations, we obtain after some straightforward algebraic manipulations 2[1 c2 *
o
- a ) - E(l - x * ) l L c l . ] 2x*)w(x*) - x * ( 1 - x * ) (
1
r=Y*
Then the mixed cmc at salt concentration c, is found from the equation
a2
Y[X(l
ax2
1
- x)w(x)]
+>o x(1 - x )
where g is the Gibbs free energy per mole of mixed micelles. Since we are modeling only the nonideality of mixing of the surfactant chains in the micelles, the counterions and the ionic surfactant head groups are in a sense taken not to affect the micelle stability. Hence, stability condition 20 is the same for nonionic surfactants as well. Furthermore, although the micelle composition depends on the micelle aggregation number N, the stability condition does not explicitly depend on N. In ref 4 we present equations for the binodals and the spinodals, which are the limits between metastable and unstable compositions. These equations are the same for the PSM and the MAM models. Demixing can occur only if wo > 2 for the regular solution model. Conditions for the spinodals and binodals when the mixing in micelles is described by the Redlich-Kister model17 have been derived.’* Micelle demixing occurs when the micelle composition is between the binodal compositions. For the PSM, the total surfactant concentration for which demixing first occurs for a given overall surfactant composition is called the second cmc, or micelle demixing concentrati~n.~.~ This term is not to be confused with the “second cmc” of single surfactants. Since, in the PSM, there are no micelles below the cmc, demixing cannot occur for total concentrations less than the cmc and hence the second cmc is always greater than or equal to the cmc. For the MAM, however, since micelles are present at all finite total surfactant concentrations, demixing may occur for total concentrations less than the cmc. Hence, if the second cmc were defined on the basis of the demixing criterion alone, it could be below the first cmc. To resolve this issue and to make the MAM second cmc analogous to that for the PSM, we have defined the second cmc as the total concentration which is greater than or equal to the cmc and for which demixing first occurs. When demixing occurs, the monomer and counterion inventories are given by eq 13-15, but with xA(or xB)instead of x . Once the monomer concentrations have been calculated, the inventories of the two types of micelles can be obtained from properly modified material balancesn3Once demixing has occurred, as cT is raised for a fixed a one of the micelle inventories may become zero. Mysels19 first recognized this possibility and termed the concentration at which one micelle concentration becomes zero the critical demicellization concentration, or cdc. Our models can account for such a phenomenon. We give a more detailed discussion of demicellization elsewhere.’* Azeotrope Micellization. We have defined azeotrope micellization to occur when the monomer and micelle compositions are the ~ a m e . Then ~ , ~ the ratio of the monomer concentrations remains (14) Mukerjee, P.; Yang, A. Y. S . J . Phys. Chem. 1976, 80, 1388. (15) Funasaki, N.; Hada, S . J . Phys. Chem. 1980, 84, 736. (16) Prigogine, I.; Defay, R. “Chemical Thermodynamics”; Longmans: London, 1954. (17) Prausnitz, J. M. “Molecular Thermodynamics of Fluid-Phase Equilibria”; Prentice-Hall: Englewood Cliffs, NJ, 1969. (18) Karnrath, R. F.; Franses, E. I. In “Surfactants in Solution”; Mittal, K. L., Ed.; Plenum Press: New York, in press (19) Mysels, K. J. J . Colloid Interface Sci. 1978, 66, 331.
The Journal of Physical Chemistry, Vol. 88, No. 8, 1984 1645
Mass-Action Model of Mixed Micellization fixed as the total concentration is increased. Hence, there could be applications of azeotrope micellization when, for example, an optimum and fixed ratio of monomer activities is desired over a wide range of concentrations. The condition for azeotrope micellization for the mixed ionic MAM is [ c ~ * / c ~ * ] ~ ( ~ + B ) ~ -exp[(l ~I~~
-
.25
- 201)w(01)+ 01(1 - 01) x (aw/ax),,,l
1.5
= 1 (21)
As N m, eq 21 becomes the same as that derived for the mixed ionic PSM.4 For w ( x ) = wo # 0, eq 2 yields
1
L
ii
Since aAzmust of course be between 0 and 1, wo must satisfy Iwol 2
[
(1
+ (3)N- 1
3 [ $1 In
(23)
For wo > 2, since the micelles can demix, the stability criterion 20 must also be satisfied for stable azeotrope micelles to form. For cl* C c2* the necessary condition for azeotropy is
wo>
[ y(1 +
1 1
In (cz*/cl*)
> wo[l - 2xB(wo)]
(24)
Sample Calculations. In this section and in the supplementary material (SM) (see paragraph a t end of text regarding supplementary material) we will demonstrate certain important features of the MAM, compare calculated results with those of the PSM, and apply it to certain experimental results from the literature. In Figure SM1 (supplementary material) we present results for a pure nonionic surfactant. All inventories are normalized for the pure systems with respect to the pure cmc and for the mixed systems with respect to the larger of the two pure cmc's. We have used the general program for a mixed ionic surfactant using 01 = 0.0 and (3 = 0.0 as a test case for our program with a model of known behavior.zOJ The results are as expected. The micelle inventories (Nc,) are finite at all concentrations for the MAM and become zero below the cmc only in the PSM ( N a). The surfactant monomer activity remains constant above the cmc only m. For N = 50, the activity changes by only 10% at for N cT/cl* ranging from 1 to 8 and for N = 10 it changes by over 70%. These changes, especially the latter, cannot be ignored. Hence, for aggregation numbers of the order of 10 or less, the MAM may be necessary to accurately calculate the inventories. In practice of course the variation of the size (or size distribution) and the solution nonideality with concentration should probably be included in the model. In figure S M 2 (supplementary material) we show results calculated for a single ionic surfactant with a typical value of (3 = O.S. As is well-known, the monomer concentration curve has a maximum because of the continuously increasing counterion concentration.z2-24 In the PSM limit there is a cusp maximum occurring at the cmc (cT/cI* = 1). For the MAM ( N finite), the maximum is round and occurs a little above the cmc. The general condition for a maximum to exist is3 cs/Cl* (cT/c1*)(1 - (3)((3N- 1) (25) If c,/cI* = 0, this condition simplifies to N > l/@,which is almost always satisfied for (3 # 0. For salt concentrations c, >> cT, condition 25 may not be satisfied and then there should be no maximum. In this sense, when c, >> cT, ionic surfactants behave similarly to nonionic surfactants. The counterion concentration increases continuously above the cmc but with a slope (3 < 1 . Note that in this model interionic Debye-Huckel effects have been ) ~ / ~ above the ignored. The average ion activity ( c ~ c ~ +increases
-
(20) (21) (22) (23) 593.
-
Stainsby, G.; Alexander, A. E. Trans. Faraday Soc. 1950, 46, 587. Matijevic, E.; Pethica, B. A. Trans. Faraday Soc. 1958, 54, 587. Overbeek, J. Th. G. Chem. Weekbl. 1958, 54, 687. Kale, K. M.; Cussler, E. L.; Evans. D. F. J . Phys. Chem. 1980, 84,
(24) Lindman, B.; Wennerstrom, H. Top. Curr. Chem. 1980, 87, 3.
I .o
2.0
.O
6.0
8 .O
CT/C2S Figure 1. Mixed micelle compositions and normalized monomer (c, and cz), counterion, and micelle inventories for mixed nonionic ( p = 0) surfactants with a = 0.3, w o = 0.0, c2*/cI* = 3.0, and N = 10, 50, and a. The numbers denote micelle aggregation numbers N for the mass-action model (MAM); m denotes the pseudophase separation model (PSM, broken lines).
cmc for every N.3 In the PSM a different average, however, namely remains constant above the cmc. In the MAM it increases, although more slowly than ( c ~ c ~ + ) ~ / ~ , In Figure 1 we show typical curves of monomer and micelle inventories and micelle compositions for mixed nonionic surfactants. One monomer concentration always decreases above the cmc. The monomer concentration maximum can result in an adsorption maximum or a tension m i r ~ i m u m . ~ The ~ ~ ~cusp ,~~ maximum for the PSM becomes rounded for N = 50, almost disappears for N = 10, and should disappear for some smaller aggregation number. Clint's experimental resultsz5 are most consistent with the calculated results for aggregation number of order 50. As N decreases, the micelle mole fraction at the cmc decreases; above the cmc it approaches a more slowly. The sum cI cz of the monomer concentrations (not shown in the figure) increases continuously above the cmc even in the PSM.4 This increase is, therefore, clearly a mixture effect which becomes more pronounced as N decreases. Typical results for mixed ionic surfactants are shown in Figure 2. Because the maximum in the first monomer concentration is due to both mixture and ionic effects, it is more pronounced than those of the single ionic or the mixed nonionic case. Moreover, because of the counterion effect there is also a maximum in the second monomer concentration and in the sum c1 + c2. The maximum in the c1/c2* curve is more pronounced and the maximum in the cz/cz* is less pronounced at wo= -2 compared to wo= 0 (see Figure SM3 (supplementary material)). In the former case, x* and the range of x's are smaller. The opposite effects are observed for positive deviations from ideality (wo= 1.8; see Figure S M 4 (supplementary material)). If there are sufficiently strong positive deviations from ideal mixing, wo >2, micelle demixing may result. An example of demixing of mixed nonionic micelles is shown in Figure 3. In the PSM, once demixing occurs the monomer concentrations are fixed. For the MAM, even after demixing, both monomer concentrations increase continuously. Once demixing occurs and there is a finite number of both types of micelles present, the micelle
+
(25) Clint, J. H. J . Chem. Soc., Faraday Trans. 1 1975, 73, 1327. (26) Trogus, F. J.; Sophany, T.; Schechter, R. S.; Wade, W. H. SOC.Pet. Eng. J . 1977, 17, 337.
1646 The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
,
.75
Kamrath and Franses I .o
I
XA
.25
‘
I
I
I
1.5 1.5
1 .o 1.o
.5 5
n .”
.o
Y.0
2.0
6.0
CT/C2S Figure 2. Inventories and micelle compositions for mixed ionic surfactants with 6 = 0.5, a = 0.3, wo = 0.0, c2*/cI* = 3.0, and c,/c2* = 0.0, for N = 10, 50, and m. /O
n “.O
8.0
1
12.0
W.0
c*c**
6.0
E.0
CT/C2S
Figure 4. Same as Figure 3, except for mixed ionic surfactants with p = 0.5 and no added salt. For clarity we have omitted the plots of cI + c2 and cM+.
50
6
I
,
0 ’ 2.0
1
1.5
I v)
.o
Cl.0
I
.2
.Lt
\ 0
1
1
.6
.8
I 1.0
ALPHA Figure 5. Mixed cmc’s of binary ionic surfactants for N = m (broken curve), 50, and 10 (farthest from broken curve), p = 0.5, c2*/cl* = 3.0, and c,/c2 = 0.0. The numbers on the figure designate values of wo.
5
n
-.o
M/
/ f 2 . 0
C*C**
,
I 4.0
6.0
8.0
CT/C2S
Figure 3. Inventories and micelle compositions for mixed nonionic surfactants with a = 0.3, w, = 3.0, and c2*/c1* = 3.0. Demixing occurs and two types of micelles, labeled A and B, with inventories Nc,* and NcmB
and compositions xA and xB are formed.
calculated mixed cmc’s of ionic surfactants depend strongly on the concentration of added salt with a common ion (Figure SM6 (supplementary material)). The addition of salt lowers the cmc consistently with experimental result^.^ The effect of size N is more pronounced the higher the salt concentration. Figure 6 contains typical plots of the first cmc and the second cmc vs. the overall surfactant composition. For the PSM, we have called such a diagram a micellar pseudophase” diagram.3s4 The broken curves in Figure 6 are for the PSM, and they depict regions where there are no micelles (region l), micelles of one composition only (regions I1 and 111), or micelles of two different compositions (region IV). For the MAM, since micelles are present at all finite total surfactant concentrations, a diagram of this type has a slightly different significance. The first cmc curve separates regions of substantial aggregation from regions where few micelles are present. The second cmc curve divides the region of substantial aggregation into regions where one or two types of mixed micelles coexist. Typical pldts of the first and second cmc’s for the MAM with N = 30 are shown in Figure 6 (the solid curves). Even for N = 30 there is little difference between the predictions of the two models. In Table I we present calculated w o values, for each value of cy and with various values of N and @, for two binary surfactant systems for which mixed cmc data are a~ai1able.l~ We have chosen two typical values of /3 to test the sensitivity of the model. L(
compositions are fixed and equal to the binodals. For the mixed ionic case, however, even though the micelle compositions are fixed in the coexistence region, both monomer concentration curves exhibit maxima because of the counterion effect (Figure 4). The maximum in the second monomer concentration curve is cusplike for finite as well as infinite N , because of the way we have defined the onset of demixing and the second cmc (see Theory section). In Figure 5 we show the effects of w, and N on the mixed cmc of ionic surfactants. For mixed nonionic surfactants, the curves are qualitatively the same. Such figures can be used to estimated w, if 0 and N are k n ~ w n . ~ , The ~ ’ calculated cmc’s, and hence the estimated w,’s are quite sensitive to the value of p (see Figure SM5 (supplementary material)). The sensitivity of the cmc’s to N varies with @ and is most pronounced around = 0.5. The (27) Rubingh, D. N. In “Solution Chemistry of Surfactants”; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 1, p 337.
The Journal of Physical Chemistry, Vol. 88, No. 8, 1984 1647
Mass-Action Model of Mixed Micellization TABLE I: Calculated w o Values for Various Aggregation Numbers N p = 0.53 N=5
c*/c,*
olSp~O'
0.101 0.246 0.491 0.748
1.04b 1.15 1.29 1.12
N=lO
N=30
N=50
p = 0.645
N=mC
N=5
N=10
Sodium Decyl Sulfate-Sodium Perfluorooctanoate 0.80 0.85 0.86 0.91 0.81 0.89 1.55 1.68 1.71 1.76 1.58 1.78 1.55 1.62 1.63 1.66 1.56 1.67 1.41 1.53 1.55 1.70 1.44 1.62
0.73 1.38 1.44 1.26
Sodium Laurate-Sodium Perfluorooctanoate 0.90 1.01 0.97 1.10 0.95 0.78 0.88 0.91b 0.170 1.69 1.48 1.61 1.72 1.45 1.58 1.04 1.28 0.339 1.54 1.69 1.68 1.62 1.64 1.39 1.52 1.16 0.49 0 1.65 1.77 1.73 1.75 1.63 1.71 1.23 1.52 0.570 1.93 2.12 2.14 2.02 2.05 1.90 1.23 1.73 0.666 1.78 1.60 1.71 1.79 1.69 1.41 1.57 1.09 0.835 a SPFO = Sodium perfluorooctanoate. Values determined conductimetrically by Mukerjee and Yang a t the pseudophase separation model in ref 4.
,
I
I'
' I I
8 .O
C m6.0
cu
IV
II
0 \ F
011.0
0
.2
.Li
.6
.8
1.0
ALPHA Figure 6. Micellar pseudophase diagram for binary mixed ionic surfactants with N = 30 and m (broken curves), (3 = 0.5, w o = 3.0, c2*/cI*= 3, and c,/c2* = 0.0. Curve AMB is the first cmc to regions I1 and 111 of one type of micelles. Curve CMD is the second cmc to region IV of two types of coexisting mixed micelles. We also indicate the position of the binodals and the spinodals. Moreover, the variation of p with the technique and the assumptions used in the interpretation of results does not allow smaller uncertainties in /3 than about 0.1. We notice that, even though wodepends on N a n d p, one can conclude that w ovaries significantly with a. Hence, the one-parameter regular solution model cannot fit the data adequately. We also notice that the values of w o for N = 30 and 50 differ little. This indicates that the results are not strongly sensitive to narrow size distributions, although to make this point rigorous one would have to start with a multiple-equilibrium model.
Discussion and Conclusions We have developed mass-action models for micellization of binary nonionic and binary ionic surfactants. In these models, the micelle aggregation number is taken to be concentration independent and the same for all micelles. Realistically, however, the micelle sizes should be distributed and depend on the total surfactant concentration, on the micelle composition, and for ionic surfactants on the ionic strength. We have chosen to ignore these effects in order to understand the effect of average size more simply. Moreover, if sizes are narrowly distributed, the effect of distribution of N on wois small. Since in the MAM there is a finite number of micelles at all surfactant concentrations, we have defined the first mixed cmc as the total concentration at which a small fraction t of the surfactant is in micellar form. We have used material balances and equations of chemical equilibria to derive expressions for (i) the cmc(s), (ii) the monomer, micelle(s), and counterion concentrations, and (iii) the micelle composition(s) as a function of the overall surfactant composition, the ratio of
N=30
N=50
N=mC
0.94 1.94 1.74 1.76
0.95 1.97 1.76 1.79
1.01 2.06 1.79 2.00
1.10 1.87 1.80 1.84 2.28 1.92
1.12 1.29 1.90 2.09 1.82 1.91 1.86 1.88 2.44 2.31 1.95 2.06 See calculations for
25 0C.14
the single surfactant cmc's, the total surfactant concentration, the micelle aggregation number, and for ionic surfactants the degree of counterion binding and the concentration of added salt with a common ion. We have used a function w(x) to generally describe the nonideality of mixing in the micelles. If w ( x ) is independent of the micelle composition x (w(x) = wo), then the surfactant chains in the mixed micelles should mix like strictly regular solutions. For this case, we have reported sample calculations which demonstrate the effect of the aggregation number N on the calculated inventories and cmc's. Hall and Pethica have discussed the application of small systems thermodynamics (SST) to micellization.28 Subsequently and Hall and Tiddy3*analyzed the formalism of the multipleequilibrium approach and applied it also to ionic surfactants. They concluded that the two approaches are "essentially equivalent", although they did not demonstrate the equivalence in specific examples. Moreover, although it is unclear to us how the subdivision potential should be estimated from experimental data, the SST/multiple-equilibrium approach is surely more powerful than the MAM. The difficulty of course is in obtaining data on the multitude of equilibrium constants and on how the size distribution depends on concentration of surfactant and salt and on mixed micelle composition. By comparison, our approach, which is more limited in scope, is to formulate specific problems, compare with experiment, and focus on specific cases of mixing nonidealities with emphasis on demixing and azeotrope micellization. Hall's theory can also account for the observed increase of average micelle size with electrolyte c o n c e n t r a t i ~ n .This ~ ~ ~effect ~~ is of course unaccounted for in the single-size MAM. One may also argue that the degree of dissociation (1 - p) may depend on concentration. We feel that specific models of such an effect should await obtaining reliable measurements of p. At present, p can be determined by three methods: (i) light scattering, (ii) ion activity measurements, and (iii) dependence of cmc on electrolyte concentration. The first method would work well if the size distribution were known independently and if intermicellar interference could be accurately calculated. The problems with the second method can be serious and are discussed by Hall and T i d d ~ . ~The * third method is indirect and, for reasons not entirely clear, is consistent with a single value of p.33,34 We have derived conditions for azeotrope micellization and micelle demixing. These conditions are analogous to those derived for the PSM.3,4 Azeotrope micellization occurs when the micelle composition is the same as the monomer surfactant cornpo~ition.~ (28) Hall, D. G.; Pethica, B. A. In "Nonionic Surfactants"; Schick, M. J., Ed.; Marcel Dekker: New York, 1967; p 516. (29) Hall, D. G. Trans. Faraday SOC.1970, 66, 1351, 1359. (30) Hall, D. G. Kolloid-Z. Z . Polym. 1972, 250, 895. (31) Hall, D. G. J . Chem. SOC.,Faraday Trans. 2 1977, 73, 897. (32) Hall, D. G.; Tiddy, G. J. T. In "Anionic Surfactants"; LucassenReynders, E. H., Ed.; Marcel Dekker: New York, 1981; p 5 5 . (33) Elworthy, P. H.; Mysels, K. J. J . Colloid Interface Sei. 1966, 21, 331. (34) Mukerjee, P. Adv. Colloid Interface Sei. 1967, I , 241.
1648 The Journal of Physical Chemistry, Vol. 88, No. 8, 1984
Kamrath and Franses monomer concentration of component 1 cmc of component 1 concentration of micelles concentration of micelles rich in surfactant 1 counterion concentration concentration of added salt with a common ion total surfactant concentration an average ion activity Gibbs free energy per mole of mixed micelles total Gibbs free energy equilibrium constant of formation of micelles of surfactant
Then the ratio of the monomer concentrations is fixed for both the nonionic and the ionic surfactant cases. Only for the mixed nonionic PSM are the monomer concentrations fixed above the cmc when azeotropy occurs. Rosen and Hua have recently presented the condition for optimal synergistic surface tension reduction via binary surfactant m i x t ~ r e s . ~This ~ . ~condition ~ turns out to be the same as the azeotropy condition (with wo < 0) described here. If there are severe positive deviations from ideality of mixing in the micelles, the surfactants can demix into two stable coexisting types of mixed micelles. The compositions of the two coexisting types of micelles are those which minimize the total free energy of the system. Since the MAM micelles are present at all total surfactant concentrations, demixing may occur below what we define as the mixed cmc. However, to make a more direct comparison between the MAM and the PSM, we define the second cmc as the total surfactant concentration which is greater than or equal to the mixed cmc and above which two types of mixed micelles coexist. By plotting the first and second cmc’s for a given system as a function of the overall surfactant composition, we have constructed micellar pseudophase diagrams. In conclusion, the MAM should be preferred over the simpler PSM to model mixed micellization and determine micelle mixing parameters if N is reasonably monodisperse and N is smaller than about 50.
Acknowledgment. This work was supported in part by the Computer Center and the School of Chemical Engineering of Purdue University. It was presented at the 56th Annual Colloid and Surface Science Symposium, Blacksburg, VA, June 13-17, 1982.
Notation A
B C
c*
c*(c,) c** Ci
micelle rich in surfactant 1 micelle rich in surfactant 2 concentration, m critical micellization concentration, cmc cmc of an ionic surfactant in the presence of added salt with a common ion second cmc monomer concentration of component i
(35) Hua, X.Y . ;Rosen, M . J. J . Colloid Interface Sei. 1982, 90, 212. (36) Rosen, M. J.; Hua, X. Y . J . Am. Oil Chem. Soc. 1982, 59, 582.
1
equilibrium constant of formation of mixed micelles with mole fraction x of surfactant 1 surfactant counterion number of surfactant molecules per micelle number of surfactant molecules per pure micelle of surfactant 1 gas constant monomer of ionic surfactant 1 temperature, K excess free energy of mixing excess free energy of mixing for the strictly regular solution model mole fraction of surfactant 1 in the micelle micellar mole fraction at the cmc binodal compositions of coexisting micelles
Greek Letters mole fraction of 1 in a binary surfactant mixture mole fraction for azeotrope micellization counterion binding parameter € fraction of the total surfactant inventory in micellar form at the cmc reference chemical potential chemical potential of surfactant 1 in the micellar pseudoff
phase Pj
Pm
PM+
chemical potential of component j chemical potential of the micelle chemical potential of the counterions
Supplementary Material Available: Figures of sample calculations for single nonionic surfactants, single ionic surfactants, mixed ionic surfactants with wo = -2.0 and 1.8, and the effects of @ and c, on the mixed cmc’s for the MAM and the PSM models (6 pages). Ordering information is given on any current masthead page.
