Langmuir 1994,10, 381-389
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The Mixing Behavior of Surfactants H. Hoffmann* and G. Possneckert Lehrstuhl fiir Physikalische Chemie I , Universitat Bayreuth, 0 - 8 5 8 0 Bayreuth, FRG Received December 2, 1992. In Final Form: October 6,1993@
The mixing behavior of surfactants was investigated by using the phase separation model. For the nonideal behavior of Surfactantswe took into consideration the most common model of Rubingh. It was shown that this so-called regular solution theory is just an empirical approach and its applicabilityis not a matter of the vanishing excess entropy. The error of the interaction parameter 0 of this theory was calculated and it was shown that the error is higher the more asymmetric the micellar mixing ratio, the more different the cmc values of the used components, and the more positive 0 itself. It was shown that mixtures of surfactants can feign a wrong cmc* value, when the component with the much higher cmc has the highest mole fraction. Mixing an anionic perfluorinated surfactant with a nonionic surfactant with nearly identical cmc values, it was shown that the one-parameter Rubingh model is insufficient. The investigated binary mixtures show S-shaped cmc*(a) curves. These curves can only be described by more parameters. The interactionbetween the different headgroups, that means the decrease of the electrostatic and steric repulsion in a mixture of an ionic and a nonionic surfactant with respect to the pure components, makes AGESnegative for nearly all mixing ratios. The positive contribution of the chain phobicity of the hydrocarbon and perfluorinated chains to AGE=is much smaller and only determines the sign of AGer at very high portions of anionic surfactant and the sign of AV. The asymmetry of the AGExcurve is due to the change of the interaction with the micellar composition x .
Introduction Surface-active compounds play a major role in a lot of applications. For example they are used as coatings to improve wettability of surfaces, as emusifiers to stabilize emulsions, and as detergents to increase the washing activity. The properties of a pure surfactant are mainly determined by its chemical structure and its geometrical arrangement within a micelle. But as known for a long time, surfactant mixtures can show a very different behavior in comparison to their components. The interaction between different surfactants can lead to synergism or to antagonism, depending on the kinds of syrfactants. The most used pairs of surfactants which show synergism are mixtures of a charged and an uncharged compound. The phenomenon of synergism in these cases is due to the interaction between the different headgroups. On the other hand antagonism can be reached by mixing surfactants with the same kinds of headgroups, but different kinds of chains, one hydrocarbon and the other perfluorinated. The phobicity between the different chains causes a demixing tendency, which can result in the formation of two different kinds of micelles (if the headgroups are the same), a phenomenon which is called microscopic demixing. It is very interesting to investigate the behavior of surfactant mixtures, which have different headgroups with synergistic interactions but also different chains with antagonistic interactions. We used pairs of nonionic hydrocarbon surfactants and perfluorinated anionic surfactants. To create special starting conditions, we used a hydrophobic tetraethylammonium counterion of the anionic surfactant. This ion lowers the cmc of the surfactant and it can be lowered further by adding further electrolyte, containing this cation. With this anionic surfactant it was possible to reach nearly identical cmc values of the nonionic and the anionic surfactant.
* To whom correspondenceshould be addressed. t This paper is part of the dissertation of the author, printed in Dec 1991 at the University of Bayreuth, FRG. Abstract published in Advance ACS Abstracte, January 1,1994.
In addition to the experiments, the mixing theory of surfactants and especiallythe phase separation model shall be discussed in detail and new approaches shall be given.
Experimental Methods and Results Materials and Methods. The surfactants tetradecyldimethylamine oxide (CuDMAO) and dodecyldimethylphosphine oxide (C12DMPO) were a gift of the Hoechst-Gendorf Company. The tetraethylammonium perfluorooctanesulfonate NEt4PFOS was a gift of the Bayer AG. The purification of the given surfactants is described elsewhere (ref 1,for the hydrocarbon surfactants; ref 2 for the perfluorinated one). All surfactants showed a constant value of surface tension above the cmc and the measurement of the surface tension did not change with time. The compounds NEtrBr and NEt40H were bought from Fluka in p.a. quality. The water used was twice distilled. All the cmc’s were measured with a ring tensiometer from Lauda which is also described in another paper.3 The measurements of the mixing volume were carried out with a differential refractometer KMX-16 at a wavelength of 632.8 nm. You can read about the theory of these measurements in the literatures4 Surface Tension Measurements. The following systems were chosen with the following cmc values of the components: I, C14DMAO/NEt4PFOS in 10 mM NEt4OH solution, cmc(Cl4DMAO) = 121 f 2 pM,cmc(NEt4PFOS) = 137 f 2 pM; 11, C12DMPO/NEt2FOS in 4.3 mM NEt4Br solution, cmc(Cl2DMPO) = 280 f2 pM, cmc(NEhPFOS)= 266 f 1pM. In the first system, in contrast to the second one, we worked at pH = 12. The reason why we did not w e the bromide but the hydroxide in this system is the basicity of the amine oxide. Amine oxides are weak bases like for example the acetate i0n.5 Mixtures of an anionic surfactant with an amine oxide show a pH maximum as a function of the mixing ratio. This maximum (1) Oetter, G. Dieaertation, Bayreuth, 1989. (2) Reizlein, K.Dissertation, Bayreuth, 1983. (3) Harkins, W.D.; Jordan, H. F. J. Am. Chem. SOC. 1930,62, 1751. (4) Pbsnecker, G.; Hoffman, H. Ber. Bumen-Ges.Phye. Chem. 1990, 94,579. (5) Nylen, P. Z. Anorg. Allgem. Chem. 1941, 246, 227.
0743-7463/94/ 2410-0381$04.50/0 0 1994 American Chemical Society
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302 Langmuir, Vol. 10, No. 2, 1994
/
s
240
.
/ I
I
* u
,.-.-,
/'
/
a(NEt,PFOS) L-L---
log (c/lmd/ll)
Figure 1. Plot of the surface tension vs the logarithm of the totalsurfactantconcentrationof the system (I)ClJlMAO/NEt(PFOS in 10 mM NEtdOH solution at different mixing ratios C14DMAO/NEt,PFOS a, 91; b, 7:3;c, 3.56.5;d, 28.
.1-...1 1--
0.6 0.8 1.0 Figure 3. cmc as a function of the mole fraction a of NEhPFOS for the mixtures (I) Cl&MAO/NEt+PFOS in 10 mM NEhOH
solution and (11) C12DMPO/NEtSFOS in 4.3 mM NEtrBr solution.
phase separation approach. In this model the micelle is treated like a macroscopic phase and a chemical potential can be assigned to a surfactant in the The condition of chemical equilibrium
--
' b,d
20 C -4.6 -4.4 -4.2 -4.0 -3.8 -3.6 -3.4 -3.2 bq ( C / l m d / l l )
Figure 2. Plot of the surface tension vs the logarithm of the totalsurfactantconcentrationof the system (11)ClaMPO/NEh-
PFOS in 4.3 mM NEt&r solution at different mixing ratios C12DMPO/NEt4PFOS: a, 82;b, 55;c, 2:8; d, 1:9.
results in the shift of the equilibrium of protonation of the amine oxide. A cationic N-hydroxyammonium compound is formed to a certain extent. Such systems, as we will describe in a following paper, must be treated as threecomponent mixtures, show quite different cmc values, and can also have other macroscopic properties. On the contrary the protonation of the phosphineoxides can be neglected,because the basicity of these compounds is about 6 to 7 orders of magnitude smaller than that of the amine oxidesn6 Figures 1and 2 show some plots of the surface tension vs the logarithm of the total surfactant concentration for different mixing rations. Every measurement was repeated several times to get some information about the error of our measurements. The plots show a sharp breakpoint which corresponds to the cmc of the mixture. Figure 3 shows the measured cmc values for the two systems as a function of the molar fraction a of NE4PFOS at 25 "C. The cmc values were measured very accurately by the method of surface tension and the error of the cmc values was determined form the upper curves. Both mixtures show nearly an S-shaped curve with two extrema. This behavior, to our knowledge, cannot be compared to any result in the literature.
Theory and Review Ideal and Nonideal Surfactant Mixtures. A very useful tool for the description of micelle formation is the (6) Haake, P.; Cook,R. D.; Hurst, G.H. J. Am. Chem. SOC. 1967,89,
2660.
(1) &,mono 4miz must be satisfied in this approach. Based on this equation the mixing behavior of surfactants in the micelle can be described too. In this case the micelle is treated as a mixed phase with a special a priori unknown composition xi, which means the molar fraction of species i in the micelle. Assuming ideal mixing behavior in the micelle the following equations were calculated for the first time by Lange and Beck.' c
~= xi cmci , ~
~
~
~(2)
At the mixed cmc* the concentration of a species "i" is equal to ita total concentration. Defining the a priori macroscopic molar fraction of the ith component as ai (the molar degree of the surfactant i in the solution with regard to the s u m of the surfactant concentrations) one can write ai cmc* = xi cmci
(3)
By eliminating the unknown micellar molar fractions xi, these equations give a term for the unknown cmc* (4)
The cmc* is not the arithmetic but the harmonic mean of the cmc's of the components weighted by the macroscopic molar ratios. The reason why surfactant mixtures behave this way is the difference between the macroscopic mole fraction ai and the mole fraction in the micelle xi. Mixing surfactants with different cmc values, they will form a t a certain concentration (cmc*) mixed micelles. But the degree in the micelle (x) of the surfactants with the lower cmc's is enhanced at the expense of the other surfactants. On the other hand, elimination of the ai terms from eq 3 shows that the cmc* is a linear function of x i . The same model can be used to describe the behavior of nonideal mixtures of surfactants. In this case, according to the macroscopic model one needs an activity coefficient for each surfactant species which describes the degree of interaction between the different species in the micelle. (7) Lange, H.; Beck,K.-H. Kolloid-2. 2. Polym. 1973,251,424. (8) Shinoda, K.; Nakagawa, T.;Tamamushi, B.; Ieemura, T.Colloidal Surfactants; Academic Press: New York, 1963;p 68 ff.
Mixing Behavior of Surfactants
Langmuir, Vol. 10, No. 2,1994 383
This results in an expansion of the equations (2-4) by these coefficients: Ci,mono
= X i f i cmci
ai cmc* = x i f i cmci
--1
(2a) (3a)
3 '
cmc* -
Apart from these equations one can write an equation which takes care of the constancy of the total amount of a given surfactant in a mixed solution and therefore works in the ideal and nonideal case. x: =
" i Ctot
Ctot -
- Ci,mono
C
cj,mono
J
The introduction of intramicellar activity coefficients means a further unknown variable in comparison to the ideal mixing behavior. Having a system of v surfactants with known cmc values, which are mixed in defined proportions, one has 2v - 1 unknowns in the ideal case (all monomer concentrations and u- 1 micellar mole fractions) and 314unknowns in the nonideal case (in addition to the upper ones: the cmc* and all the activity coefficients). In both cases we neglect the triviality Cxi = 1. The system of equations is completely solvable in the ideal case, because there are the same numbers of independent equations as unknowns (eq 2 v times and eq 5 v - 1 times). In the nonideal case however, one has only one equation more (eq 2a v times, eq 5 v - 1 times, and eq 4a). In other words, to solve the problem of nonideal mixing one needs further assumptions and equations. The first who used activity coefficientsinside the micelle was Rubingh? He solved the upper problem in introducing a further equation for each activity coefficient, where the fi's are connected with an interaction parameter 8. In fi = o(1 - xi)' (6) This parameter, according to its definition was related to the molecular interactions of the surfactants in the mixed micelle. While 8 is unknown too, one has to measure the cmc*, to reduce the number of a priori unknowns and to make the system of equations solvable. The use of this kind of equation was called the regular solution approximation by him. This terminology, as used by Rubingh, goes back to Guggenheim.Io He had developed a lattice model for binary mixtures where he had considered the interaction between the different species by an interaction parameter. In his so-called zeroth approximation he had assumed a complete randomness of the two species, that means an ideal entropy of mixing. This assumption indeed leads to expression 6 for the activity coefficientsand is exactly called the strictly regular solution theory. However, the described model is not the only one, which gives these equations for the activity coefficients. In a purely empirical model of nonideal binary mixtures, which is related with the names of Redlich and Kisterll but also with Guggenheim,I2 the molar excess free enthalpy is written as a power expansion of the difference of the two (9) Rubingh, D.N. Solution Chemistry ojSurfactants; Mittal, K. L., Ed.;Plenum Press: New York, 1979; Vol. 1, p 337 ff. (10)Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952. (11) Redlich, 0.;Kieter, A. T. Znd. Eng. Chem. 1948,40,345. (12) Guggenheim, E.A. Trans. Faraday SOC.1937,33,151. (13) Porter, A. W.Trans. Faraday SOC.1920,16,336.
mole fractions XI - x2. Only using XI as the variable, one gets the following expression which contains the empirical parameters Ai.
The excess chemical potential is related with the activity coefficient and AGex. p i ( ' = R T l n f i = A G e x + ( l - x i ) ( -dAGex ) axi
(8)
Developing eq 7 only to the first power, AGEx is described by the very simple equation (9) AGE' = x1 (1 - x l ) A,
(9)
which was first used by Porter in 1920' and leads to eq 10 for the activity coefficients
RT In fi = A, (1- xi)'
(10)
Because of its structure, eq 9 describes a nonideal mixing behavior which is symmetric to the mole fraction x = 0.5 and therefore the systems which correspond to this equation are called symmetrical mixtures. A comparison between eq 6 and 10 shows that the interaction parameter 0 multiplied with RT is identical with the coefficient AI. This result shows that the same expressions for the activity coefficients are received from this empirical model and Rubingh's model. In Porter's approach there is no assumption made about the entropy of mixing. Using again simple thermodynamics, we get the following expression A 2 P x = - dAGex (~)p=-(%)
x,(l-xJ
(11)
P
Equation 11 shows that the molar excess entropy is vanishing only when the coefficient A1 is independent of temperature. Only in this case is AHex equal to AGex. Porter's approach works quite well for a lot of macroscopic mixtures, in other words AGex is nearly symmetric to the mole fraction 0.5. Obviously AG*' behaves in the same way for a lot of surfactant mixtures. But especially in the case9 of macroscopic mixtures, where AGex can be fitted by a symmetric curve, AHexand ASex are very often S-shaped and far apart from being symmetric with respect to x . The upper statement clearly points out that the strictly regular solution theory is a very special case of Porter's approach. The fitting of the measured cmc* values by one interaction parameter is not an indication to a zero excess entropy. Therefore the terminology "regular solution theory" which was introduced by Rubingh and is used throughout the whole literature of nonideal surfactant mixtures should be replaced by the expression "symmetrical mixture". Besides our original work14 the discrepancy between macroscopic and surfactant mixtures by use of the term "regular solution" has also been published by Christian et al.15
This short review of the history of thermodynamics of macroscopic mixtures also gives the reason for the great differences between the calculated AGEx (from the cmc* ~
(14) PBssnecker, G. Dissertation, Bayreuth, 1991. (15) Christian,S.D.;Tucker,E. E.; Scamehorn, J. F. ACS Symp. Ser. 1992, No. 501, 45 ff.
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384 Langmuir, Vol. 10, No. 2, 1994
values) and the directly measured AHex, which were published for some surfactant systems.16 The symmetric one parameter approach which is used for all nonideal surfactant mixtures is a very useful tool, to get some information about the kind and the magnitude of interaction between different surfactants in the micelle. But as we have seen it does notgive the universal possibility to describe all thinkable mixtures. That means there should be and-as we have seen in this paper-there are surfactant mixtures which cannot be fitted by one parameter. The Solution of the System of Equations. If, as it was shown in the preceding part of thie paper, the behavior of AGex of a binary nonideal surfactant mixture can be described by one parameter 8, one has six unknown quantities and the same number of equations. The unknowns are the two monomer concentrations, the composition of the micelle, the two activity coefficients in the micelle, and the interaction parameter 8. These six unknowns can be calculated unambiguously. But this procedure only works as long as the system can really be described by one constant parameter 8, which means that for every measured cmc* nearly the same value for 8 is calculated. If this is not the case, eq 6 cannot be used to calculate the activity coefficients and has to be replaced by the following equation according to eq 8
The partial differential coefficient in this equation is not known, because of the unknown compositionof the micelle. In general the description of a system by a variable @does not make sense. In these cases higher powers in eq 7 have to be taken into consideration. But the use of higher powers means more than one parameter. This makes the calculation of the unknowns much more complicated, because the number of unknowns is increasing while the number of equations is remaining constant. In other words: if a system cannot be described by one constant parameter 8 and more parameters are hecessary, the system of equations cannot be solved for every single measured cmc*. There are two alternatives to get the needed information about the system. The first is to combine more measured cmc* values and to get a greater system ofequations. When eq 7 is expanded to the uth term (with Y parameters) one has to take u different cmc* values which are measured and one has to solve simultaneously 6 u equations with the same number of unknowns. The second much more elegant method is to calculate the composition of the micelle x in a different way. The knowledge of the micellar composition would give us the possibility to calculate the activity coefficients by eq 3a and further all thermodynamic quantities of the system. Using eq 3a we can differentiate cmc* to a. Replacing the product of CmCi and f i by eq 3a we get the following expression:
The last two terms in the bracket are equal to zero, because of the Gibbs-Duhem equation. Replacing x2 by 1- x1 we get an equation which gives us the composition of the micelle at the cmc* as a function of the a priori mixing ~
~~
(18) Holland, P. M.ACS Symp. Ser. 1981, No. 253,141. Holland, P. M. Adv. Colloid Znterjace Sei. 1986, 26, 111. Osbome-Lee, I. W.;
Schechter, R. S . ACS Symp. Ser. 1986, No. 311, 30.
ratio a, the cmc*, and the differential coefficient dcmc*/ da. 1-ai dcmc* x,(cmc*) = a1 1- (14) cmc* d a l This equation is a special case of the Motomuras equation17 which he calculated for mixtures of two nonionic surfactants. Equation 14 is only valid when eq 3a is correct, that means for mixtures of nonionics and mixtures with ionic components at high ionic strength, when the concentration of the ionic surfactant at the cmc is negligibly small in comparison to the salt concentration. Though a differential coefficient of measured values cannot be determined very accurately, this method of calculating the unknown variables seems to be an applicable tool. For binary mixtures and the one parameter case, the system of equations can partially be separated and one can write two equations with only two unknowns. To manage this one has to put the expression for the activity coefficientfrom eq 3a into eq 6. Doing this for each activity coefficient we get the two equations with the two unknowns 0 and X I .
{
(
cmc* (1- (YJ
)]
(15a)
Though these equations are often used in the literature to calculatethese two unknown variableslLespecidy the interaction parameter 8-there has been, to our knowledge only one empirical attempt to get some information about the possible error of the calculated parameter which could resulk from measuringerrors of the cmcvalues.ls Therefore we calculated an equation which gives us a very good approximation of the error. The way of calculation can be read in Appendix 1. We obtained the following expression, where t is the relative error of the corresponding measured cmc value. 1
(E* + X E ~ +(1- ~ ) < 2 ) (16) (1- x ) x For a constant relative error c this equation can be simplified.
AB=*-
Ab=*-
2e
(Ma) (1- x ) x The absolute value of A8 is plotted in Figure 4 against x for different errors. As you can see, even if the cmc’s can be determined with an accuracy of k l % , the minimum of A@ is nearly 0.1. While a more accurate determination of a cmc in our opinion would be very unrealistic, we think that only the first decimal place of B should be published. The error is strongly increasing when one component in the micelle dominates. It has a minimum at the point x = 0.5 and increases with increasing deviation from the point of equimolar micellar composition. The micellar composition x depends on a lot of other parameters, like for example the macroscopic mixing ratio a,the degree of interaction between the surfactants, and the cmc values (17) Motomurn, K.; Yamanaka, M.;Aratono, M.Colloid Polym. Sci. 1984, 262, 948. (18) Raf9. Holland, P. M.;Rubingh, D. N. J. Phye. Chem. 1983,87, 1984. Roeen, M. J.; Zhu, B. Y. J. Colloid Interface Sei. 1984, 89,427. (19) Kamrath, R. F.; Frances,E. I. 2nd. Eng. ChemFundam. 1983,22,
231.
Langmuir, Vol. 10, No. 2, 1994 385
Mixing Behavior of Surfactants
LLJ / ill
1
E
= 0.05
E
E
=
LI
0.30
1
i 0.20 1 I
0.02
0.10
= 0.01
Figure 4. Error of the interaction parameter 48 as a function of the micellar composition 1c in a binary surfactant system for different relative errore of the measured cmc values.
of the components. The accuracy of 0 therefore is predetermined by intrinsic parameters of the different systems. If, for example, two surfactants have the tendency to demix, the mixing ratio x within the micelle will be very asymmetric over a large region of the ratio a. In this case the error of 0 is very high for most of the measured cmc* values and it should be nearly impossible to decide whether the interaction parameter is smaller or higher than +2, meaning mixing or demixing. For that reason the problem of n)icroscopic mixing or demixing for agiven pair of surfactants cannot be solved very accurately by measuring cmc* values. Besides the magnitude of the interaction for a given pair of surfactants the dependence of the intermicellar mixing ratio x on the cmc values of the two components shall be discussed. The more different the cmc's, the more asymmetric is the x at a given value of a = 0.5. Let us look to two surfactants under the limitation that they should mix ideally. The assumption of ideal mixing bears the big advantage that the behavior can be completely described m d calculated by a given formula. For nonideal mixing it is also possible to calculate a formula, but it will always contain the two unknown and variable activity coefficients. From the value of ideal mixing the qualitative behavior of nonideal systems can be approximated. Equation 14 gives the micellar mixing ratio at the cmc*. In the ideal case the cmc* is given by eq 4. Putting this term and the calculated term of the differential coefficient from eq 4 into eq 14 one gets the following formula x(omc*) =
a?? a(k - 1) 1
+
where, without l i t a t i o n s it is set cmcz 1 cmcl and cmc) = k cmcl
with k 1 1
In this equation the micellar composition at the mixed cmc is only a function of the macroscopic mixing ratio and the quotient of the cmc values of the pure components. While z. is strictly monotonically increasing as a function of a,one is ableto calculate a distinct region of a,id which x will have valued around 0.5, that means, where one can get the highest accuracy. The larger this region, the "ore exact is the calculation by use of the equations (15) Of 0 for any given pair of surfactants, In Figure 5 the region of a,where x lies between 0.2 and 0.8, is plotted against k, that means against the quotient of the two cmc values
*'
'\ \\
c
Figure 5. Extension of the area of a in which the micellar mole fraction x lies between 0.2 and 0.8, as a function of the quotient of the two cmc values of the two Components.
of the pure components. From this plot it can be seen that the region is decreasing, when the cmc's of two surfactants drift apart from each other. With a decreasing difference a(x=0.8) - a(x=0.2), of course the slope dxlda is strongly increasing within this region. This means, in such a case of a mixture of two surfactants with very different cmc values, within this region eq 16 is not valid any more because the error of a must be taken into account. This means: If two surfactants with very different cmc values are mixed, the error of the calculated interaction parameter is very high, because in the first place for most of the mixtures the micellar composition x at the cmc* is very asymmetric and in the second place for some of the mixtures which have micellar compositisnsaround 0.5, the dependenceof the micellar composition on the macroscopic mixing ratio dxlda has such a high value that the error of a must not be neglected, as it is normally done. It should be noted that another mistake in the determination of the cmc* could be made for such systems with highly different cmc values. The region of a,where one gets mean values for x around 0.5 continuously drifts to one side of the whole possible area of the mole fraction a when k is increased. With the foregoing limitation (cmcz > cmcl) this specifically means that only at very high portions of component "2" x reaches values around 0.5 at the cmc*. In ather words a1 must be very high. If the macroacopic mixing ratio is very asymmetric and component "2" with the much higher cmc is predominant, the change of the monomer concentration of component "2" can feign a cmc* at a much too high level. To demonstrate this we again will restrict ourselves to the ideal mixing behavior. The monomer concentralions of the different components of a surfactant mixture, even above the cmc* are not constant, but depend on the total surfactant concentration. This f8d is a direct consequence of the phase separation model and was outlined for some times in the literature.'Sm The following equations can be calculated and are valid above the cmc* in the ideal case of mixing. C1,mono
cmcl = 7j- (1 - x
+ [(l- x)' + 4xa1l1/')
(18a)
x is a positive measure for the total surfactant concen~~
(20) Clint, J. H.J. Chem. Soc., Faraday Trans. 1975, 71,1327,
Hoffmann and PBssnecker
386 Langmuir, Vol. 10, No. 2, 1994 1.2 [ @
0.4 Oe8
0.1
1
i
AG*z/RT
T
-0.41
Figure 7. AGOx as a function of the micellar mole fraction of NEtgFOS for the systems I and 11.
-2.8
Figure 6. Interaction parameter j3 calculated from the cmc* values of the systems I and 11.
tration and is defined in the following way: ctot
= cmcz - cmcl
The monomer concentrations above the cmc* behave strictly monotonically, clpono has a negative slope and C Z , has ~ a~ positive ~ ~ slope and are very "boring" functions in amathematical point of view. Only the third derivative becomes zero at a given value of x. It is 3 %-0
for x = 1 - 2a1 (20) d x3 It can be shown that the zero position of the third derivative means a maximum of the second one. In other words, at a given total concentration which is connected with x the monomer concentration shows a maximum of bending. The more asymmetric the macroscopic mixing ratio, or the smaller the portion of component"1" (because eq 20 only gives positive values of x for the condition a1 < 0.5), the higher is this extremum for the second component. This maximum of bending means a change in the degree of the monomeric surfactant relative to the total amount. Measuring the cmc* with a method, which is sensitive to changes in the monomer concentration, can give in such cases wrong cmc* values or can feign a second cmc*, which could be wrongly interpreted as a second kind of micelle. In reality there does not come into existence a second kind of micelle, but the composition of the given micelle changes dramatically.
Discussion The surfactant pairs used in this study, show nonideal mixing behavior. The cmc values were measured very accurately. The S-shaped cmc curves with two extrema cannot be described by one constant parameter B. The maximum is more pronounced in the amine oxide system than in the mixture of the phosphine oxide. The minimum of the cmc* in both systems lies at a molar fraction of the perfluorinated surfactant of about 0.25 and causes the strong asymmetry. A calculation of B for every point leads to the values which are plotted in Figure 6 and which show a strong dependence on the mixing ratio. The errors in this plot are calculated with the help of eq 16. The plot was only made to show the nonconstancy of B and to give an approximate impression about the magnitude of interaction between the two surfactants. As pointed out earlier in this paper, such an analysis does not make sense in a theoretical point of view. To describe the cmc* values one has to take more parameters.
To calculate the excess free enthalpy of mixing we have made the following procedure: The plots of the cmc* as a function of a were fitted by a polynomial of fourth order (least-squares fitting). This polynomial was then differentiated. By use of this derivative, it was possible to calculate the micellar composition according to eq 14 a t the cmc*. Knowing the micellar mole fraction x it is easy to calculate the activity coefficients according to the general expression3*and furthermore to obtain AGexby simple thermodynamics. The calculated values of AGexwere plotted as a function of x . We tried to use the Redlich-Kister approximation to fit these values by an empirical expression according to eq 7. But the values of AGex could only be fitted in sufficient correspondence by a four-parameter expression. So we kept the advice which was given by MauserZ1to prefer a two-parameter logarithmic expression. We could fit the given values of the amine oxide system by the following function A G e ' = ~ ( l - ~ ) [ Y + Z h (+lx ) ] (21) with the two parameters Y = -7.98 kJ/mol and 2 = 15.3 kJ/mol. The accuracy of this fitting function is nearly the same compared to the four-parameter Redlich-Kister equation. The values of AGexare also plotted in Figure 7. Yand 2 are only empirical parameters and should only be used to calculate the values of AGex as a function of the micellar composition. The accuracy of AGexis not better than 25 J/mol. The values of the parameters of the phosphine oxide system are not published, because the shape of the curve causes only a very inexact calculation at high content of anionic surfactant. To interprete the shape of the curves of cmc*(a) and AGex(x)one has to take into account the different kinds of interaction. The only reason why the minima of AGex and cmc* have the same position is the similarity of the cmc values of the both surfactant components. The higher the difference between the cmc's of used surfactants, the more different are the curves of cmc* and AGex. The mixing of an ionic with a nonionic surfactant causes a decrease of the electric surface charge density. This is one reason why the mixed micelle should be more stable in comparison to the micelle which only contains the ionic surfactant. The process of mixing such kinds of surfactants is further accompanied by a depression of the steric repulsion between the headgroups a t the micellar surface.n The denser packing of the surfactant headgroups in the mixtures can directly be shown by analyzing the surface tension measurements. Using the Gibbs adsorption iso-
(21)Mauser, H.Z. Ekktrochem. Ber. Buneen-Ges.Phys. Chem. 1958, 62, 896. (22) Nagarajan, R.Langmuir 1985, 1, 331.
Langmuir, Vol. 10, No.2,1994 387
Mixing Behavior of Surfactants 3.0 I_
.
2.8
.
b
2.6
E
2.4
CI
/
\
\
w
a
/i'
1.2
0.8 2.0c
t
a (NEt'PFOS)
1.8--I--L-10.0 0.2 0.4 0.6 0.8 1.0 Figure 8. Sum of the excess surface concentrations of the surfactants ClaMAO and NEtaFOS of system I as a function of the molar fraction a of NEQFOS.
therm one gets the following equation for a binary mixture in excess salt.
The right side of eq 22 means the sum of the excess surface concentrations of the surfactants and is a direct measure of the surface area per headgroup at the aidwater interface. This value can be taken for the area of the headgroup at the micellar surface. Figure 8 shows this sum for the amine oxide system. The minimum of the cmc* coincides with the maximum of Cri. These two effects, the reduction of the steric repulsion and the decrease of the electric charge density, both at the surface of the micelle, cause the synergism in these systems, which dominates the cmc* behavior over a large area of composition. Only at high portions of anionic surfactant a positive deviation from Raoult's law can be detected. This behavior must have obviously been caused by the phobicity of the different chains of the surfactants, because the only surfactant pairs which show higher cmc* values than the calculated values of ideal mixing are mixtures of a perfluorinated and a hydrocarbon surfactant with the same headgroup. The phobicity of the chains makes a much smaller contribution to AGQ than the interactions between the headgroups. However, we found the mixing volume being strongly influenced by the chain interactions. AV was determined according to our published method of the measurement of the change of the refractive index An4 with the help of a differential refractometer and employing the formula
AV --=
6n An (23) Vo (n2- l ) ( n 2+ 2) where n is the refractive index of the solution. In Figure 9 AVis plotted for both systems as a function of the composition. The mixing of the different surfactants correlates with a volume expansion which reaches a value of about +2 cms/molsurfactant. It is higher for the mixture of ClDMAO. This could be due to the longer chain of this surfactant in comparison to C&MPO. The different signs of AGax and AV underline the complexity of the surfactant pairs used. We know only about one paper were similar behavior has been published for a surfactant system.* The chain contribution to the energy AGex is only noticeable when the amount of ionic surfactant is high. In (23) F u m d i , N.; Hndn, S.; Neyn, 5.J. Phys. Chem. 1986,90,6469.
:
0.4 7 -
0.0
4:
T4 $
. '- I - A - L _ _ _ L
.L_~I.__~___L__I-
0.0
0.4
0.2
0.6
0.8
1.0
a (NEt,PFOS) Figure 9. Mixing volume of systems I and I1 as a function of the molar fraction a of the surfactant NEQFOS measured at a total surfactant concentration of 20 mM.
these highly charged systems the headgroup interactions are still dominated by the high charge density and therefore the negative contribution of the interaction between the headgroups to AGex is only small. An increasing amount of nonionic surfactant reduces the charge density and stabilizes the micelle. The maximum of synergism is reached when the mixing ratio of the surfactants is 3:l (nonionic:ionic). At a higher content of nonionic surfactant the negative deviation from Raoult's law gets smaller again. A small amount of ionic surfactant in the micelle is energetically favorable. This should be due to the attraction between ions and dipoles at the micellar surface, which should be responsible for the dense packing and the reduction of the area per headgroup. The investigation of the two systems clearly shows that the free energy is dominated by the interactions between the headgroups. A similar result was found by Funasaki and Hada.= The determination of the change of the interaction as a function of the composition of the micelle, in our opinion, was only possible by using the given surfactants with nearly the same cmc values, which was the basis of accurate determinations of the interaction. The asymmetry found in this study should exist also in other systems of a nonionic and an ionic surfactant, even without a perfluorinated component,but should be harder to find, because the cmc of a nonionic surfactant normally is about an order of magnitude lower than that of an ionic surfactant with comparable chain length.
Conclusions Mixed micelle formation was studied by the measurement of the cmc* values for two systems of nonionic hydrocarbon and an anionic perfluorinated surfactant. By use of hydrophobic cation and excess electrolyte the two components of each pair had about the same cmc values. It was found that the cmc's of the mixtures resulted in S-shaped curves with two extrema, when plotted against the macroscopic mole fraction a. In an earlier papers an S-shaped cmc curve was also published, but in a later part of their paper the authors interpreted the behavior of the surfactants and the small 8 values they found as ideal mixing. So, to our knowledge the investigated systems are the first which show this asymmetricbehavior without (24) F u n d , N.; Hndn, S. J. Colloid Interface Sci. 1980, 78, 376. (25) Ravey, J. C.; Cherbi, A.; Stebe, M. J. Prog. Colloid Polym. Sci. 1989, 79, 272.
Hoffmann and Passnecker
388 Langmuir, Vol. 10,No. 2, 1994
doubt. Obviously the data cannot be explained by one nonideality parameter. It could be shown that the one-parameter Rubingh approach, the so-called regular solution theory is only a special case of the empirical Redlich-Kister equation, which is also called Porter's approach. The usefulness of this fundamental equation which connects the activity coefficientwith the interaction parameter 0 does not imply vanishing excess entropy. Therefore surfactant systems which can be fitted by this one parameter model should not be called regular solutions but, according to Porter, symmetrical solutions. According to the Redlich-Kister equation it was shown that more empirical parameters can be used to describe the deviation from ideality and it is not sensible from a physical point of view to describe a system with a nonconstant 8. A formula, which is a special case of an equation, published first by Motomura, was calculated which can be used in such cases, when more than one parameter is needed. For the generally used one-parameter model the error in the interaction parameter due to errors in cmc was calculated for various conditions. It was shown that the error becomes higher, the higher the quotient of the cmc's of the components, the weaker the interaction between the different surfactants (the more positive 81, and the more asymmetric the macroscopic mixing ratio a. It was also shown, that the change of the monomer concentration can feign a cmc* at a much too high level of concentration in special cases, when the concentration of the component with the much higher cmc widely exceeds the concentration of the other component. The experimental results could be fitted by an empirical two-parameter model with a logarithmic term. The cmc* plots showed the synergism of the surfactants at high contents of nonionic surfactant and showed the chaindependent antagonism a t high degrees of the anionic one. Depending on the chain-chain interaction the mixing volume is positive. The change of the degree of interaction as a function of the micellar composition as it was measured in the investigated systems, in our opinion should not be an exception and should not be restricted to mixtures of hydrocarbon andperfluorinated surfactants with different headgroups. All the other surfactant mixtures of a nonionic and an ionic compound should show a similar change of electrostatic and steric interaction and there should be an optimum of the molar ratio of the components which should not be equal to x = 0.5. We do not want to replace the very useful empirical one-parameter model of Rubingh by our fitting equation. We just want to show amore detailed investigation of surfactant mixtures, which in our opinion can only be carried out with special systems, where the possible error is minimized, to allow such detailed interpretations. The main difficulty in the determination of the point of highest synergism and the asymmetry of PGeX is the higher error in ionic-nonionic system, which is due to the great difference of the cmc values of these compounds.
Nomenclature C,,mono = monomer concentration of the ith component cbt = total surfactant concentration cmci = critical micellization concentration of the ith component cmc* = mixed cmc of a system f , = activity coefficient of the ith component in the micelle k = cmc2/cmc1 n = absolute refractive index of a surfactant solution
An = difference between the measured refractive index of a mixture and the calculated value for ideal mixing x , = molar fraction of the ith component in the micelle x = only used in binary mixtures, means x1 A, = empirical parameter of the Redlich-Kister expansion AGex = molar excess free enthalpy A H e x = molar excess enthalpy R = gas constant ASex = molar excess free entropy T = absolute temperature VO= volume of a surfactant solution AV = mising volume of a surfactant mixture Y , 2 = empirical parameters of a logarithmic equation to describe AGeS cy, = macroscopic, a priori, molar fraction of the ith component cy = only used in binary mixtures, means a1 6 = interaction parameter of the symmetrical fit A0 = error of 6 y = surface tension e,, E* = relative error of the corresponding cmc t = error of the cmc's; used when the errors are equal p,ex = excess chemical potential of the ith component in the micelle p,,mono = chemical potential of the monomer of the ith component ~ , , = ~ chemical h potential of the ith component in the micelle v = a whole positive number x = cbt/(cmcz - cmcl) I", = surface excess concentration of the ith component
Appendix I The Geometrical Solution of the Error of the Interaction Parameter 8. The two equations (15a,b) were used as functions in the following way: y1 =P='ln( x12
cmc* (1- cy1) cmcz (1 - xl)
)
(15a)
Both functions have one point of intersection, which is
(1) = Yzo = Bo If the parameters in the functions y1 and yz are not constant, but variable around a constant value, one gets an area of intersection. We define the error of the single cmcs YlO
cmc = cmcO(1f e) (11) The error of the a priori molar fraction a,which could be due to errors of weighting or diluting was neglected. Using this definition, we can calculate the maximum and minimum functions of y1 and yz.
2t
YP,max/min
= Y20 f 7 X
(IV)
The e-containing terms on the right side of the upper equations correspond to the maximum error of the single functions. They can be interpreted as a line, starting from a point (x,yi,o)parallel to they axis with this defined length. The possible error of the interaction parameter is equal to the maximum distance (in the direction of the y axis) of a point of the area of intersection to the central point of intersection Z as it is shown in Figure 10. This error lies between the two upper terms of the equations I11 and
Mixing Behavior of Surfactants
Langmuir, Vol. 10, No. 2, 1994 389
cmc’s are not too high. Let us look to the triangle ACD. With a right angle at the point D you get the expression:
Y
- -
AD=ACcosq (VIII) To calculate the length of AC we take the triangle ABC. In this triangle the distance between A and B is equal to the difference of the two .maximum errors
and it is
0.6
0.65
Figure 10. Example for the regionof intersectionof the functions y1 and yz and their maximum and minimum functions with the pointa A, B, C,D, and Z and the angles cp, w, and y.
IV. The slopes of the two functionsyl and y2 can be written in the following way:
y1 has a negative and yz a positive slope for yi < 2. We must distinguish now between two cases: 1. The parallel line to the y axis from the point Z first crosses the functions y2* and y2This means 2t (1- X ) Z
>2 x2
Putting eq X into eq VI11 we get
- -
sin(o) AD = AB C O S ( ~ )sin(r) The three angles q, w, and y can be expressed with the slopes of the two functionsyl andy2 in the point Z. Putting these values into eq XI, one gets the following term
Into eq XI1 we put the expressions of eqs V and VI and add the values of the distance between Z and A. We get the value of the maminum error of the interaction which we called eq 16a in the main part of parameter @, this paper. 2t (16a) (1- x ) x The other case with x < 0.5 leads to the same equation. The calculation with different errors ti is a little bit more long-windedbut can be carried out the same way and leads to eq 16.
A@=*-
or x
> 0.5
This case is shown in Figure 10. Wh-t we need is the length of the line between Z and D and because we know the distance ZA, we must calculate the distance AD. To get this, we linearizethe functions,that means we generally use the slopes of the functions in the point Z. This approximation gives good values until the errors of the
A@=*-
’
(1 - x ) x
(e*
+ xtl + (1- x)c2)
(16)
