882
J. Phys. Chem. 1981, 85, 882-885
Mixed Micelle Formation In Two-Phase Systems Fuminori Harusawa’ and Muneo Tanaka Shiseido Laboratories, 1050 Nippa-cho, Kohoku-ku, Yokohania, Japan 223 (Received: August 12, 1980)
An analytical descriptionhas been developed for micellization of multicomponent mixtures of nonionic surfactants partitioned between oil and water. The mixed micelle theory based on the assumptions of ideal mixing in the micelle and of a phase separation model for the micelle allows calculation of both micelle composition and monomer concentration in two-phase systems as a function of total surfactant concentration. The predictions of the theory are in good agreement with the observed results for the binary mixture of hexaoxyethylene nonyl phenyl ether and octaoxyethylene nonyl phenyl ether partitioned between cyclohexane and water. Moreover, it is shown that the theory reproduces the partition isotherm of multicomponent mixtures of polyoxyethylenated octylphenols, having a Poisson distribution of molecular weights, in the isooctane-water system.
Introduction The description of physical behavior of mixed surfactants in equilibrated two-phase systems is very important in relation to the type of emulsion1and interfacial tension m i r ~ i m a . ~There ~ ~ is currently much interest in the ultralow interfacial tension minimum observed in dilute petroleum sulfonate solution/oil systems because of the potential usefulness in tertiary oil I t was found that the minimum in interfacial tension vs. surfactant concentration occurs when petroleum sulfonates having a broad distribution of molecular weights are used, and this was attributed to the partition behavior of surfactant molecules and the changes in monomer concentration and micelle composition with surfactant concentration as well as to the interfacial adsorption mechanism?+ Previous measurements of the partition isotherms of polyoxyethylenated nonylphenols in the cyclohexanewater system’ have shown that the phase in which micelles are formed will be the continuous phase in the formation of emulsions, and the phase inversion temperature of an emulsion stabilized with a mixture of nonionic surfactants varies with composition of mixed micelles. It is also known that very low interfacial tensions are observed near the phase inversion temperature.8 In order to understand above-mentioned phenomena it is essential to have a comprehensive view of the behavior of mixed surfactants in two-phase systems. In the case of the one-phase system, the theoretical treatment for the critical micelle concentration (crnc) of mixed surfactants was developed independently by Langeg and by Shinoda.lo These early works were concerned mainly with the determination of the cmc of mixed systems. Mysels and Otter’l and Tokiwa et a1.12 calculated (1)F. Harusawa, T.Saitb, H. Nakajima, and S. Fukushima, J. Colloid Interface Sci., 74,435 (1980). (2)E. Franses, M. S. Bidner, and L. E. Scriven in “Micellization, Solubilization, and Microemulsion”, Vol. 2, K. L. Mittal, Ed., Plenum Press, New York, 1977,p 855. (3)K. S. Chan and D. 0. Shah, J. Dispersion Sci. Technol., 1, 55
the change in micelle compktion as the concentration increases above the cmc. The analytical description which includes both micelle composition and monomer concentration above the mixed cmc has recently been developed by Clint’3for ideal mixtures and by Rubingh14for nonideal ones. In the case of two-phase systems, only Franses et ale2 appear to have calculated the composition of mixed micelles at the cmc and to have estimated the change in monomer concentration above the cmc for binary surfactant mixtures in connection with interfacial tension minima. However, comparison of the theory with experimental data has not yet been done. We have developed an analytical description for micellization of multicomponent mixtures of nonionics in the oil-water system, assuming ideal mixing in the micelle and a phase separation model for the micelle. The theory can be used to calculate the micelle composition and monomer concentration of multicomponent mixtures of nonionics partitioned between oil and water as a function of surfactant concentration. The experimental and calculated values will be compared for both the binary mixtures of polyoxyethylenated nonylphenols reported in the previous paper1 and the multicomponent mixtures of polyoxyethylenated octylphenols the partition data of which are available in the literature.16J6
Theory List of Symbols. The following symbols will be used for the system consisting of equal volumes of the oil and water phases which are saturated with each other. poi, pwj chemical potentials of monomeric surfactant i in the oil and water phases Po*i, standard chemical potentials of monomeric Pw*i surfactant i in the oil and water phases PMi chemical potential of surfactant i in mixed micelles PMoi chemical potential of surfactant i in pure micelles
(1980). \----.
(4)J. L.Cayias, R. S. Schechter, and W. H. Wade, J.Colloid Interface Sci., 59,31 (1977). (5)P. M. Dunlap Wilson and C. F. Brandner, J. Colloid Interface Sci., 60, 473 (1977). (6)P.H. Doe, M. EL-Emary, W. H. Wade, and R. S. Schechter, J.Am. Oil Chem. Soc., 54,570 (1977). (7)P. R. Antoniewicz and R. Rodriguez, J. Colloid Interface Sci., 64,
mn _ _ _(
~
1 9 7 ~ ~
(8)H. Saito and K. Shinoda, J.Colloid Interface Sci., 32,647(1970). (9) H.Lange, Kolloid-Z., 131, 96 (1953). (10)K. Shinoda, J. Phys. Chem., 58,541 (1954). 0022-3654/81/2085-0882$01.25/0
(11)K. J. Mysels and R. J. Otter, J. Colloid Sci., 16,462,474(1961). (12)F. Tokiwa, K. Ohoki, and I. Kokubo, Bull. Chern. SOC.Jpn., 41, 2845 (1968). (13)J. H.Clint, J. Chern. SOC.,Faraday Trans. I, 71, 1327 (1975). (14)D.N. Rubingh in “Solution Chemistry of Surfactants”, Vol. 1, K. L.Mittal, Ed., Plenum Press, New York, 1979,p 337. (15)H.L.Greenwald, E. B. Kice, M. Kenly, and J. Kelly, Anal. Chern., 33,465 (1961). (16)E.H.Crook, D. B. Fordyce, and G. F. Trebbi, J. Colloid Sci., 20, 191 (1965).
0 1981 American Chemical Society
The Journal of Physical Chemisrry, Vol. 85, No. 7, 198 1 883
Mixed Micelle Formation in Two-Phase Systems
cmc values of pure surfactant i in the oil and water phases cmc in mixed systems1' C* mole fraction of surfactant i in total mixed soai lute mole fraction of surfactant i in mixed micelles Xi total concentration of mixed solute17 C concentrations of monomeric surfactant i in the Comi, CWmi oil and water phases total concentration of mixed monomer^'^ Cm Ki, Kmix partition coefficients of pure surfactant i and mixed solute Theoretical Analysis. Below the cmc (C < C*), at equal volumes of oil and water ai may be expressed as
Coi, Cwi
co,.
P r n i = xicwi
(12)
+ cw,.
= Xi(C"i + Pi)
(13)
Eliminating Co,. and CWmifrom eq 5 and 13 gives
Exi = 1, we have
Kmixcan be defined as
C
From eq 1-3 we have
(11)
which are essentially analogous to the relation between the partial vapor pressure of a component above an ideal solution and the composition of the liquid phases, Le., Raoult's law. From eq 11 and 12 we have
Since
and assuming ideal mixing among the monomers then
corn; = X i C O i
aiC
c + c o i + c w i - c,
=1
Similary from eq 13
+
C(C0,i + CW,i)/(COi Pi) =1 At the cmc since Comi+ CWmi= a$* we have
(16)
(4) which gives partition isotherms for the mixed solute below the cmc. Above the cmc (C > C*) the concentration of component i in mixed micelles is aiC - (Co,. Cwmi)and the total concentration of mixed solute in mixed micelles is C - C,. Therefore, the mole fraction of component i in the mixed micelles is
+
a$ - (C0,i
xi =
+ CW,.)
c - c,
(5)
The chemical potentials of monomeric component i in the oil and water phases can be written as
+ R T In Co,. pwi = pw*i + R T In C",. poi = po*i
(6) (7)
where the activity coefficient of free monomers is assumed to be unity. Assuming the micelles to behave as an ideal mixture, we write for component i in the mixed micelles = pMoi + R T In xi (8) When applying a phase separation model to micellization in equilibrated two-phase (oil-water) systems, one may assume that, when micellization occurs, a saturation concentration for monomers (i.e., cmc) is reached in both phases.l Therefore, considering the equilibrium condition between the oil, water, and micellar phases, we write for component i in the pure micelles pMi
pMoi
= po*i
+ R T In Coi
(9)
pMoi
= pw*i + R T In Cwi
(10)
At equilibrium since pMi = poi for the micellar and oil phases and p M = pwi for the micellar and water phases we obtain from eq 6-10 (17) C, C*, and C, are defined as follows: (1) C = (Co+ C,) where C, and C, are the total concentrations of mixed surfactant in the oil and water phases, respectively. That is, C is the total surfactant content. (2) At the cmc, C = C*. (3) C, = C(C0, + P&). The variable in summation formulas is i throughout this paper.
As the total concentration increases, xi approaches ai because the bulk of surfactants added to the system forms micelles a t very high concentrations where C >> C*. Therefore, above the cmc the monomer concentration (Cd approaches Cai(Coi Cwi)as the total concentration increases (cf. eq 13), starting from the value given by eq 17, namely, C* < C , < Cai(Coi+ Cwi)< C. Assuming a phase separation model for the micelle, in which the cmc is a saturation concentration for monomeric surfactants, then we have an alternative expression18for Ki:
+
Ki = COi/CW, (18) Therefore, in order to calculate the monomeric concentrations of each component and the composition of micelles in the two-phase system as a function of total concentration, we need know only the cmc's of the pure componenta in both phases which are in equilibrium with each other and the mole fraction (ai)of component i in the mixture. The cmc of the mixture is first calculated by using eq 17. Then, for concentrations below the mixed cmc, Comiand CWmiare calculated by using eq 1,3, and 18. For concentrations above the mixed cmc, eq 11,12, 14, and 15 are used. For a given total concentration, the solution of eq 15 is given graphically as the C , coordinate of intersection of the graphs of the functions
cxi = g(C,)
(20) which are schematically shown in Figure 1 for a two-component system. It is apparent from Figure 1 that the line ~~
=1
~
(18)It has been supported experimentally for the two-phase systems containing nonionics that, once the temperature is fixed, micelle formation takes place in only one of the phases (cf. ref l). Therefore, one can write NO*,
+ RT In CO,= pw*,+ RT In P,= OM^, or
p
w
~
~
where poMorand p w ~ , ,are r the chemical potentials of surfactant i in pure micelles formed in the oil and water phases, respectively. This leads to eq 18.
884
Harusawa and Tanaka
The Journal of Physical Chemistry, Vol. 85, No. 7, 1981 b
cx Figure 1. Schematic representationof functions of f(C,) and g(C,) at a total concentration of C for a two-component system: solid line, f(C,); dashed line, g(C,); asymptote (a), C, = C Coi C w l ; asymptote (b), C, = C + Co2 C",; S , solution of eq 15; C " , mixed cmc; C , total concentration of mixed solute.
+
+
+
g(C,) intersects the graph of f(C,) at two points. This means that for the two-component system eq 15 has two solutions mathematically. However, one of the solutions obtained between the C, asymptotes (a and b) should be omitted because it does not satisfy the requirement, C* C C, C C. For an n-component system, eq 15 may have n roots, that is, the line g(C,) intersects the graph of f(C,) at n points. However, one will find only one solution under the condition, C* C C , C C,as supposed from Figure 1.
E
=
C02 + Cw2+ (1/2)[(C - A) - {(C - A)2 + 4alACJ1/2](21)
where A = (CO, + Cw2)- (Col + Cwl). From eq 14 and 21, we have XI
= (1/2A)[-(C - A)
+ ((C
-
A)'
+ 4~xlAC)'/~]
(22)
Partition isotherms of hexaoxyethylene nonyl phenyl ether (NPE,), octaoxyethylene nonyl phenyl ether (NPE8), and their mixutres in the water-cyclohexane system were determined previous1y.l I t was found from the partition isotherms of homogeneous NPE6 and NPEs that the concentration of monomeric surfactants remains constant at concentrations above the cmc. Hence, the phase separation model is applicable to these systems. Partition data have shown that the cmc values of NPE6 and NPE8 in the cyclohexane phase and those in the water phase are 1.30 X 2.85 X 2.70 X and 4.05 X 10" M, respectively. Then, according to eq 18, we obtain K1 = 481 and K2 = 70 where subscripts 1 and 2 refer to W E 6 and W E 8 , respectively. For the binary mixtures of NPE6 and NPE8 partitioned between cyclohexane and water, one can calculate the monomer concentrations of each component for both phases throughout the whole concentration range from eq 1, 3, 11, 12, and 22 by using the above partition coefficient and cmc values. The monomer concentrations for each component in the cyclohexane phase together with their sum were calculated as a function of total concentration for the NPE,-NPE, (a1= 0.542) mixture where a1 is the mole fraction of NPE6 in the mixture. The calculated lines together with the observed values1Qare plotted in Figure 2, and indicate that the predictions of the theory are in good agreement with the experimental data. The (19) The observed values were obtained from the partition data reported in ref 1.
in one-phase (water)
................................................
Comparison of Theory with Experiment Binary Mixtures. For binary mixtures in two-phase systems, eq 15 gives a quadratic expression in C, which can be solve to yield
c,
i o 3 (M)
Figure 2. Plot of monomer concentrations(Corn) In the cyclohexane phase against total concentration ( C )for the NPE,-NPE, (a,= 0.542) mixture partitioned between cyclohexane and water at 25 O C : curve 1, monomer concentratlon of NPE6;curve 2, monomer concentration of NPE,; curve 3, total monomer concentration; (-), calculated lines; (0, 0,O), observed values.
.-
ti
0.2
g o o
2
4 6 8 c x 103 (M)
10
Flgure 3. Plot of the mole fraction (xi) of NPE, in mixed micelles as a function of total concentration ( C ) for the NPE6-NPE8(a,= 0.542) mixture in the one-phase (water) or two-phase (cyclohexane-water) system at 25 O C : (-), calculated lines: (0),observed values; (---),
a,.
total concentration of monomers continues to increase even after the cmc is passed, although, in the one-component system containing NPE6 or NPEs, the monomer concentration remains constant at concentrations above the cmc as mentioned above. It is clear that this increase in monomer concentration above the cmc is due to only the NPE6, since the concentration of the monomeric NPE8 actually decreases after reaching a maximum at the cmc as seen from Figure 2. For the NPE6-NPE8 (al = 0.542) mixture partitioned between cyclohexane and water, the mole fraction of N P G in the mixed micelles (xl) against C was also calculated from eq 22. The calculated line is shown together with the observed valueslg in Figure 3 where x1 for the one-phase (water) system, which was calculated according to Rubingh's equation14by assuming ideal mixing in the micelle, is also plotted against C. For a one-phase system x1 is greater than a1at the cmc, suggesting that mixed micelles, the composition of which is more lipophilic compared with al,are formed at first, and x1 approaches to alwith increase in total concentration. This reflects the fact that the Cw2value for NPE8 is greater than the Cwl value for NPEG. On the contrary, for a two-phase system more hydrophilic micelles are formed at first and then x1 approaches to a1with increasing total concentration. This reflects the fact that the sum of Col and Cwl for NPEe is greater than that of Co2and Cw2for NPE& The fact that the composition of mixed micelles varies with total concentration can explain the surface13or interfacial2tension of mixed surfactant systems as well as the behavior of the
The Journal of Physical Chemistty, Vol. 85, No. 7, 1981 885
Mlxed Micelle Formation in Two-Phase Systems
TABLE I: Cmc Values of Homogeneous OPE’s in the Isooctane and Water Phases (C”i and Cwi) at 25 “C and Mole Fraction (ai) of Each Component in Normal Distribution OPE,, Used in the Calculation
EO chain length 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
c o i , ~. 6.85 X 3.33 x 1.51 X 7.27 x 3.82 x 1.48 x 5.65 x 2.19 x 8.42 x 3.20 x 1.23 x 4.70 X 1.80 x 7.00 x 2.00 x 1.00 x 3.00 X 1.00 x 5.00 x
lo-’
10-3 10-3 10-3 10-4 10-4 10-5 10-5 10-5
lo-$ 10-7 10-7 10-7 10-9
104cwi, M 0.765 1.03 1.29 1.72 2.50 2.68 2.83 3.04 3.23 3.40 3.60 3.80 4.00 4.15 4.30 4.45 4.60 4.75 4.90
ffi
0.001 0.005 0.015 0.034 0.061 0.091 0.117 0.132 0.132 0.119 0.097 0.073 0.050 0.032 0.019 0.011 0.006 0.003 0.001
phase inversion temperature for emulsion.’ Multicomponent Mixtures. I t is very important to understand the behavior of multicomponent mixtures of nonionics in two-phase systems, since commercial nonionics having a broad distribution of ethylene oxide (EO) chain lengths are widely used as emulsifying agents in various fields of industry. Therefore, the mixed micelle theory was applied to the multicomponent mixture of nonionics partitioned between oil and water. In order to calculate the micelle composition and monomer concentration of multicomponent mixtures as a function of total concentration, it is necessary to know the cmc values of each component in both the oil and water phases which are in equilibrium with each other. Hydrocarbons such as cyclohexane, heptane, and decane are solubilized in the interior of the micelle. The addition of such nonpolar hydrocarbons tends to decrease the cmc of surfactants in aqueous solution. A decrease of about 5-30% in the cmc is observed.20i21 However, the effect is far smaller compared to that of polar compounds such as long-chain alcohols which penetrate into the palisade layer of the micelle. Therefore, the application of the cmc values in pure water, which are available in the literature, to the nonpolar hydrocarbon-water system will not lead to large errors. On the other hand, one can calculate the cmc values in the oil phase from eq 18 using the cmc values in the water phase if one knows the partition coefficients of each component. Crook et a1.16 studied the partition of normal (Poisson) distribution poly(oxyethy1ene) octyl phenyl ethers and homogeneous ones (OPE’s) between isooctane and water, and found that the logarithm of the partition coefficient changes linearly with the EO chain length, giving the relation log Ki = 3.836 - 0.442n (23) where n is the EO chain length. The cmc values of homogeneous OPEl-lo and normal distribution OPE1-10,16,40 in pure water were measured by Crook et a1.,22and the (20) K. Shinoda, T. Nakagawa, B. Tamamushi, and T. Isemura, “Colloidal Surfactants”, Academic Press, New York, 1963, Chapter 1. (21) T. Nakagawa in “Nonionic Surfactants”, M. J. Schick, Ed., Marcel Dekker, New York, 1967, Chapter 17.
log Cw(M)
Flgure 4. Partition isotherms of normal distribution OPE’s in the isooctane-water system; C, and C, are equilibrium concentrations in the organic and aqueous phases, respectively: (A), experlmentalline for OPEe,, at 26 OC obtained by Greenwald et al.; (B), experimental line for OPEe,eat 25 O C obtained by Crook et ai.; (C), calculated line for OPElo at 25 OC.
numerical data are summarized in ref 23. The differences in cmc between the normal distribution OPE’S and homogeneous ones seems to be small for higher EO chain homologues. For the isooctane-water system, Greenwald et al.15 and Crook et determined the partition isotherm of normal distribution OPE9.7over the whole concentration range below and above the cmc and the partition coefficient of normal distribution OPE9.9below the cmc, respectively. Thus, in order to compare the theory with the experimental data, we have obtained the partition isotherm of normal distribution OPElo in the isooctane-water system based on the mixed micelle theory presented. Table I lists the cmc values of homogeneous OPE’S for both the isooctane and water phases, together with the mole fraction of each component in the normal distribution OPElo, used in our calculation. Unavailable cmc values for the water phase were estimated graphically, and the distribution of molecular weights in the inhomogeneousOPElois assumed to be Poisson, which must slightly be modified for the reaction of ethylene oxide with alkylated phenol^.^^^^^ The cmc of normal distribution OPElo was first calculated from eq 17. Then, for concentrations below the cmc, eq 4 was used. For concentrations above the cmc, eq 14 and 15 were used to calculate x , and then eq 11was used to calculate C.o, Since the experimental data indicate that the micellization of OPE9.7occurs in the water phase,lSthe concentrationsof the surfactant in the oil and water phases are equal to CComiand C - CComi, respectively. A comparison of the theory with the experimental data is shown in Figure 4. The theory is seen to reproduce the experimental data well in spite of an inexactness in the cmc values of higher EO chain homologues. Thus, the mixed micelle theory for two-phase systems based on the assumptions of ideal mixing in the micelle and of a phase separation model for the micelle appears to be able to predict the monomer concentration and micelle composition of the multicomponent mixtures of nonionics partitioned between the two phases as a function of total concentration. (22) E.H.Crook, D. B. Fordyce, and G. F. Trebbi, J. Phys. Chem., 67, 1987 (1963). (23) P. Becher in “Nonionic Surfactants”, M. J. Schick, Ed., Marcel Dekker, New York, 1967, Chapter 15. (24) S.A. Miller, B. Bann, and R. D. Thrower, J. Chem. SOC., 3623 (1950).
