Langmuir 1996, 12, 3855-3858
3855
Distribution of Surfactants in a Nonideal Mixed Micellar System. Effect of a Surfactant Quencher on the Fluorescence Decay of Solubilized Pyrene Mats Almgren,* Per Hansson, and Ke Wang Department of Physical Chemistry, Uppsala University, Box 532, S-751 21 Uppsala, Sweden Received January 29, 1996. In Final Form: May 3, 1996X We calculate a distribution of the surfactants in nonideal mixed micelles of equal size, which is consistent with the regular solution expressions for the concentrations of the free surfactants in equilibrium with the micellar pseudophase. The distribution gives in particular the fraction of micelles, at given mean composition, which contains only the nonquenching surfactant. This is a quantity that is measured in fluorescence quenching experiments, using a surfactant quencher such as alkylpyridinium chloride, at low concentration, mixed with the main surfactant. In the time-resolved fluorescence quenching method, it is assumed that the distribution is Poissonian, and an apparent aggregation number is determined. The deviation of the apparent aggregation number from the true one is shown to increase linearly with the quencher concentration, at low concentrations, with the slope related to the interaction parameter from the regular solution theory. Experimental results from measurements with a nonionic surfactant and the cationic quencher are reported and compared with the predictions.
Introduction Mixed micelles of two surfactants, 1 and 2, are usually treated within the pseudophase separation model, either as an ideally mixed phase1,2 or within what is usually called a regular solution approach.3,4 The results of such treatments are relations between the composition, x1 and x2 ) 1 - x1, of the mixed micelles and the concentrations c1 and c2 of the surfactants in the aqueous pseudophase at the cmc.
c1 ) c°1f1x1 c2 ) c°2f2x2
(1)
where c°i is the cmc of pure i and the activity factor, fi, is given by
ln fi ) R(1 - xi)2;
i ) 1, 2
(2)
The interaction parameter, R, is zero in the ideal case, giving activity factors equal to unity. Phase separation results from repulsive interactions with a critical point at Rc ) 2, xc ) 1/2, and the coexistence curve symmetric around xc. For micelles, the phase separation amounts to the formation of two coexisting micelle populations, one rich in 1, the other in 2. Such segregation may occur in mixtures of fluorocarbon and hydrocarbon surfactants.5-10 In this first communication we shall only consider R < 2. Apart from attempts to show the coexistence of two types of micelles, few authors have paid attention to how the surfactants distribute among the micelles in mixed, nonideal systems. The molecular theories for mixed X
micelles that have been presented11,12 do not explicitly discuss this aspect. In an ideal case, with micelles of the same total aggregation number, N ) N1 + N2, the surfactants would distribute according to a binomial distribution, that is well approximated by a Poissonian distribution of 2 when N is large and N1 >> N2. In a nonideal mixture, deviations from the binomial or Poissonian distribution will occur. Our particular interest is how these deviations will affect the results of the (timeresolved) fluorescence quenching method for the determination of micelle aggregation numbers.13-16 A basic assumption of the method is a random distribution of quenchers over the micelles, usually taken to imply a Poisson distribution. A broad distribution of micelle sizes may lead to deviations from this assumption, with effects that have been used for estimating the size polydispersity.16-18 More recently, other effective interactions between the quenchers have also been considered.19,20 We have recently used a cationic surfactant, alkylpyridinium chloride, as quencher for the determination of aggregation numbers of micelles formed by the corresponding alkyltrimethylammonium surfactants on interaction with polyelectrolytes.21,22 The two cationic surfactants form close to ideal mixtures, but the interactions are important for the same quencher mixed with nonionic or anionic surfactants. We shall present experimental results for such systems and confront them with the simple theory described next. Theoretical Background The (most probable) size of mixed micelles normally varies with the composition, an exception being an ideal
Abstract published in Advance ACS Abstracts, July 1, 1996.
(1) Lange, H.; Beck, K.-H. Kolloid-Z. Z. Polym. 1973, 251, 424. (2) Clint, J. H. J. Chem. Soc.., Faraday Trans. 1975, 17, 1327. (3) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 1, p 337. (4) Hoffmann, H.; Po¨ssnecker, G. Langmuir 1994, 10, 381. (5) Mukerjee, P.; Yang, A. Y. S. J. Phys. Chem. 1976, 80, 1388. (6) Shinoda, K.; Nomura, T. J. Phys. Chem. 1980, 84, 365. (7) Funasaki, N.; Hada, S. J. Phys. Chem. 1980, 84, 736. (8) Asakawa, T.; Johten, K.; Miyagishi, S.; Nishida, M. Langmuir 1985, 1, 347. (9) Burkitt, S. J.; Ottewill, R. H.; Hayter, J. B.; Ingram, B. T. Colloid Polym. Sci. 1987, 2656, 628. (10) Asakawa, T.; Hisamatsu, H.; Miyagashi, S. Langmuir 1995, 11, 478.
S0743-7463(96)00089-3 CCC: $12.00
(11) Nagarajan, R. Langmuir 1985, 1, 331. (12) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567. (13) Infelta, P. P.; Gra¨tzel, M.; Thomas, J. K. J. Phys. Chem. 1974, 78, 190-195. (14) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (15) Turro, N. J.; Yekta, A. J. Am. Chem. Soc. 1978, 100, 5951. (16) Almgren, M. Adv. Colloid Interface Sci. 1992, 41, 9. (17) Almgren, M.; Lo¨froth, J.-E. J. Chem. Phys. 1982, 76, 2734. (18) Warr, G. G.; Grieser, F. G. J. Chem. Soc., Faraday Trans. 1 1986, 82, 1825. (19) Bales, A. L.; Stenland, C. J. Phys. Chem. 1993, 97, 3418. (20) Bales, B. L.; Almgren, M. J. Phys. Chem. 1995, 99, 15153. (21) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16684. (22) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115.
© 1996 American Chemical Society
3856 Langmuir, Vol. 12, No. 16, 1996
Almgren et al.
mixed system in which the pure components form micelles of similar size. For simplicity, we shall assume that the micelles, at a given mean composition, all have the same total aggregation number (but not necessarily the same at all compositions). We seek a distribution that is consistent with the regular solution assumption in the pseudophase approximation. The Bragg-Williams Lattice Theory. The the Bragg-Williams lattice theory, as applied to solutions (the two components should occupy equal molecular volumes in the mixture; cf. Hill23) reproduces the Rubingh results, eqs 1 and 2. The latter are more general; however, they are restricted neither to only nearest neighbor interactions nor to a well-defined lattice. Since it offers a simple way for the calculation of the distributions, we shall here follow the BW approach. A more general and extended discussion is published as a separate contribution.24 The total interaction energy (or free energy), W, is obtained by adding nearest neighbor pair interactions, uij.
W ) N11u11 + N12u12 + N22u22
(3)
aqueous subphase
µ2 µ°2 ) + ln c2 RT RT For the pure components we find in particular
(
c wi ) uii 2
(4)
N
(5)
where λ1 and λ2 are the absolute activities, defined by
( )
λ1 ) exp
( ) ( ) ( ( )) ( ( )) ( ) ( ) q2 exp -
w2 kt
N2
exp -
(
) ( ( ))
(N - 1)kT
+ ln x2 + (1 - x2)2
2
×
( ) ((
(6)
)
w2 kT
1
))
w N2(N2 - 1) N exp - 2N2xj2 + Nxj22 N2 kT N-1
N N2 ×
N1N2w
xj1N xj2N ∑ N )0 2
N1
-ln q2 exp -
w kT
Introducing the expression (6) for Q into eq 10 and rearranging, we obtain
Z)
where qi is the molecular partition function for surfactant i in a micelle. It is consistent with the pseudophase partitioning model to let N1 and N2 be large so that Stirling’s approximation can be employed, and the chemical potential calculated as
µ2 d ln Q ) RT dN2
(11)
where xj1 is the average composition, and the activity factor given by eq 2
N W N2 exp - kT )
N1
( )
µ1 w1 ) q1-1 exp xj f RT kT 1 1
N
The canonical partition function is
w1 q1 exp kT
λ1N-N2 λ2N2 Q(N - N2,N2) (10)
N2)0
ln f1 ) (1 - xj1)2
N1N2 w + N1w1 + N2w2 N-1
Q(N1,N2) ) q1N1q2N2
∑
Z(N,λ1,λ2) )
eq 3 can be transformed into
W)
(9)
where c°2 is the cmc of 2. Combining eqs 7 and 8, and using eq 9, we retrieve eqs 1 and 2 for the equilibrium, showing the consistency of the regular solution approach with the Bragg-Williams model, and in particular that the interaction free-energy parameter introduced in eq 4 is the same as that of the regular solution approach, R ) w/kT. Let us now consider micelles of fixed size N ) N1 + N2, in a solution with given chemical potentials of 1 and 2. The mean composition of the micelles is then given by eqs 1 and 2. The numbers of micelles of different compositions are determined by the grand partition function
Introducing
c w ) (2u12 - u11 - u22); 2
( ))
µ°2 w2 + ln c°2 ) - ln q2 exp RT kT
where Nij is the number of i,j interactions. We assume that each molecule has c neighbors and that the fraction of 1 among those is the same as the average fraction, e.g., not influenced by the interactions. We then have
cN1(N1 - 1) cN1N2 cN2(N2 - 1) N11 ) ; N12 ) ; N22 ) N-1 N-1 N-1
(8)
w (7) kT
The fraction of micelles with composition x2 ) N2/N is given by
P(N2) )
( ) ( (
N w N2(N2 - 1) xj1N1xj2N2 N exp - 2N2xj2 + Nxj22 kT N-1 2
))
/Z
(13) This equation for the micelle distribution cannot be used when a phase separation occurs. Application to Fluorescence Quenching: Small Values of x2. In fluorescence quenching experiments to determine the average aggregation number, P(0) is determined, and it is usually assumed that the quencher, here surfactant 2, has a Poissonian distribution, Pp(N2), among the micelles, so that
and correspondingly for 1. At pseudophase equilibrium the chemical potential is the same in the micellar and (23) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley Publishing Co.: Reading, MA, 1962. (24) Barzykin, A.; Almgren, M. Submitted for publication in Langmuir.
(12)
h 2) Pp(0) ) exp(-N or
-ln[Pp(0)] ) Nxj2
(14)
Distribution of Surfactants
Langmuir, Vol. 12, No. 16, 1996 3857
In the mixed micelles a binomial distribution results when w ) 0, and from eq 13 we get
P0(0) ) (1 - xj2)N
(15)
For small xj2
(
-ln[P0(0)] ≈ Nxj2 1 +
xj2 2
)
(16)
so that the aggregation number, Napp, determined as if eq 14 were applicable, is given by
(
Napp ) N 1 +
xj2 2
)
(17)
This weak dependence on the quencher concentration would be difficult to observed experimentally. With w * 0, for small xj2, we expand the exponential factor in eq 13 and retain only the first term. After rearrangement
[P(0)]-1 ) N
∑
N2)0
( )( ) [ ( ) ] ∑( )( ) ( ( )) xj2
w N2(N2 - 1)
N2
1 - xj2
1+
N exp N2 kT
N-1
N
xj2
N2)0
1 - xj2
w
N22
N2
N × N2
w
-
kT N - 1
- 2N2xj2 ≈
kT
N2
1
1-N
+ 2xj2
(18)
After some algebra we obtain
(
ln[P(0)] ) -N ln(1 - xj 2) + ln 1 - N
w xj (20) kT 2
)
The apparent aggregation number is then given by
Figure 1. A family of fluorescence decay curves for pyrene in 50 mM C12E8, with the quencher (C12PC) concentration increasing from zero in the uppermost curve to about 4 mol % in the lowest. Measurements were at 25 °C in air-saturated solutions.
A3 ) 〈n〉 ) xj2 Napp
(22b)
A4 ) kq
(22c)
In this form, the model assumes that the probes and quenchers stay in the same micelles during the observation time period and that they are Poisson-distributed over the micelles. F(t) is the fluorescence intensity at time t (after the excitation moment), k0 the decay constant of the excited probe in the absence of quenchers, 〈n〉 the average number of quenchers per micelle, and kq the first-order quenching rate constant in a micelle with one quencher. The uncertainty of the aggregation numbers estimated by this method is influenced by factors such as spurious luminescence from impurities and the precision of the sample preparation. In a series of similar measurements, however, the precision of the relative values within a series is probably limited by the uncertainties of the parameter estimates in the fitting procedure, which are available from the covariance matrix. The relative uncertainty of Napp decreased with increasing quencher concentration, from about 5% at quencher concentration corresponding to 0.5 quenchers per micelle, to about 1.5% at 3 quenchers per micelle.
Results and Discussion
F(t) ) A0 exp{-k0t + A3[exp(-A4t) - 1]}
(22)
A0 ) F(0)
(22a)
TRFQ measurements were performed to determine the variation of the apparent aggregation number of C12E8 micelles, in 50 mM solution, with the concentration of the cationic surfactant quencher. Figure 1 shows a family of decay curves, which could be well fitted with the InfeltaTachiya model, yielding the apparent aggregation numbers presented in Figure 2. The change with quencher concentration may have several causes. It could be a real change of the micelle size on the addition of quencher, or it may be due to deviations from the Poisson distribution, caused by nonideal mixing and size polydispersity. Assuming that only nonideal mixing is important, we may use eq 21 to extract N ) 89.3 for the aggregation number of C12E8, and w ) -5.3 kT for the interaction parameter of C12E8-C12PC. From experimental critical micelle concentration (cmc) data the interaction parameter has been determined for this and many other systems,26 using the simple nonideal mixing model of eqs 1 and 2. Even for strongly nonideal systems, the simple model agreed very well with the experimental results over a wide range of compositions. In the present system a value of -2.7 kT for the interaction parameter gave good agreement; the value was reduced strongly on addition of salt: -1.4 kT in 0.1 M and -1.0 kT in 0.5 M NaCl.27
(25) Almgren, M.; Hansson, P.; Mukhtar, E.; van Stam, J. Langmuir 1992, 8, 2405.
(26) Holland, P. M.; Rubingh, D. N. In Cationic Surfactants. Physical Chemistry; Surfactant Sci Ser. 37, Rubingh, D. N., Holland, P. M., Eds.; Dekker: New York, 1991; p 141.
Napp ) N[1 + (1/2 - w/kT) xj2]
(21)
To test the approximation in eq 21, Napp was calculated from eq 13 for some values of w/kT, with xj2 below 0.05. The approximation was found to be excellent in this range of quencher concentrations, which reaches beyond what is experimentally convenient to use. Experimental Section Chemicals. C12E8 (octaethylene glycol dodecyl ether) was used as received from Nikko Chemicals, Japan. C12TAC (dodecyltrimethylammonium chloride) was prepared by ion exchange from the bromide salt (Serva, analytical grade). The cationic quencher C12PC (N-dodecylpyridinium chloride; from Aldrich) was recrystallized two times from acetone. Pyrene (Aldrich) was recrystallized twice from ethanol. Time-Resolved Fluorescence Quenching. The measurements were performed with methods and equipment described earlier.25 The decay curves were fitted to the Infelta-Tachiya model13,14
3858 Langmuir, Vol. 12, No. 16, 1996
Figure 2. Apparent aggregation numbers determined from the data in Figure 1 by fitting to the model of eq 22. Relative uncertainties vary from about 5% at the lowest to 1.5% at the highest quencher concentration.
Figure 3. Apparent aggregation numbers determined from measurements on solutions as in Figure 1, but with 0.200 M of NaCl, and fitting the data to the model of eq 22.
Qualitatively, the fluorescence quenching results and the cmc data indicate a similar nonideal mixing behavior, although the negative value of the interaction parameter is much larger in the fluorescence measurements. Part of the reason for this discrepancy can be that the fluorescence quenching results refer to mixtures with very little of the ionic component and obtained, therefore, at very low ionic strength. It is possible that the effective interaction obtained in this way is larger than that from the cmc data, which gives the parameter that best fits the results for all compositions. Since the interaction is mainly electrostatic, a series of measurements was performed with 0.200 M NaCl added, and the surfactant quencher concentration varied in the same range as before. The results, Figure 3, give Napp ≈ 91.3 and a weak increase with the quencher concentration, except for a large value at the lowest quencher concentration where the relative uncertainty is largest. Neglecting this value, the weak dependence of Napp on the quencher concentration corresponds to a value of the interaction parameter of only -0.5 kT, which is a stronger screening than indicated by the cmc measurements.27 It is possible that another effect, in addition to the nonideal mixing, changes the apparent aggregation (27) Rosen, M. J. In Phenomena in Mixed Surfactant Systems; ACS Symposium Ser 311, Scamehorn, J. F., Ed.; 1986; p 144.
Almgren et al.
Figure 4. Apparent aggregation numbers determined using a fixed concentration (0.3 mM) of the surfactant quencher and addition of a nonquenching cationic surfactant, using the model of eq 22 to fit the results. The relative uncertainties are about 5%.
number in the opposite direction, and that the two effects partly cancel in 0.2 M salt. A broad size polydispersity is one possibility, another that a real change of size occurs on addition of the cationic surfactant. Figure 4 reports the results from measurements at 50 mM C12E8 using a fixed concentration of the quenching surfactant (0.3 mM of C12PC) and varying amounts of a nonquenching analogue, C12TAC. The changes of the apparent aggregation number are small. If taken as significant, the initial decrease, from 93 with only 0.3 mM of the quencher to a minimum of 87.5 when also 0.6 mM C12TAC is present, can be understood as an offset of the nonideality by the addition of a nonquenching analogue. Further addition of C12TAC to a mole fraction of 0.042 gives a slight increase of the aggregation number (to 90.5). Thus, the size remains practically constant and does certainly not decrease. In conclusion, the experimental results from fluorescence quenching and cmc determinations of the mixed system show qualitative agreement concerning the effect of the nonideality of the mixing, as viewed from the “regular solution” perspective. We could probably not hope for better agreement. The equations for the depression of the free concentration in mixed systems from the cmc values of the pure components have good empirical support and are valid outside the Bragg-Williams theory that we used to derive the equation for the apparent aggregation number from the TRFQ measurements. The BraggWilliams model, assuming nearest-neighbor interactions only, and a random distribution of the nearest neighbors independent of the interactions, does not provide a good description of the system under study. On the other hand, the results obtained are probably more general than this starting point suggests. A separate paper24 will consider some ways to generalize the treatment and explicitly discuss the interesting situation when the interaction is repulsive. Acknowledgment. Support from the Swedish Natural Science Research Council and Knut and Alice Wallenberg’s Foundation are gratefully acknowledged. Ke Wang has been supported by a scholarship from World Laboratory, ICSC, Lausanne. LA960089F
