J. Phys. Chem. B 2000, 104, 1137-1140
1137
COMMENTS Reliable Aggregation Numbers Are Obtained for Polyelectrolyte Bound Cationic Micelles Using Fluorescence Quenching with a Cationic Surfactant Quencher Per Hansson* and Mats Almgren Department of Physical Chemistry, Uppsala UniVersity, Box 532, S-75121 Uppsala, Sweden ReceiVed: July 12, 1999; In Final Form: October 27, 1999 Introduction In a series of papers, we have studied the aggregation of a cationic surfactant, dodecyltrimethylammonium bromide (C12TAB), in dilute solutions of various anionic polyelectrolytes,1-5 determining aggregation numbers from time-resolved fluorescence quenching (TRFQ). The results represent a significant contribution to the current knowledge of polyelectrolyte/ surfactant complexes. Central to this work was the application of dodecylpyridiniumions (C12P+) as quenchers. This quenching surfactant mixes ideally with C12TA+ in the micelles, a fact that was used in calculating the mole fraction, XQ, of quencher in the micelles from the bulk mole fraction RQ. For this purpose, we adapted the ideal mixing model for binary surfactant mixtures6 to the case of mixed micelles in complexes with polyelectrolytes.3 In a recent paper,7 Rodenhiser and Kwak (R&K) investigated how the surfactants we used as quenchers affect the surfactant binding isotherm and what implications that will have on the TRFQ-method. Unfortunately, it is easy to get the impression that their paper corrects mistakes made by us. We show in this communication that the problems pointed out by R&K were, in fact, considered by us and that the method we used gives reliable results. The major point made by R&K is that XQ may be substantially larger than RQ which is demonstrated from calculations of the distribution of the surfactant (S) and the quencher (Q) between the polyelectrolyte and the aqueous subphase. This was shown in one of our papers3 and was taken into account in the analysis of TRFQ data in that and in all our subsequent papers.8 In a preliminary study of C12TAB/sodium polyacrylate (NaPA), where we first applied C12P+ as a quencher,2 we assumed that RQ ≈ XQ. The rational for this was that, even if C12P+ is slightly more hydrophobic than C12TA+, RQ should not differ too much from XQ since the major part of the surfactant was in the micelles. In a subsequent paper,3 we noted that this criterion was not completely fulfilled and corrected the previously reported value. The second point made by R&K is that the effect of the quencher on the critical aggregation concentration (cac) and the binding isotherms of the mixture needs to be taken into account. The idea of this derives from a paper by Liu et al.9 who * Corresponding author. E-mail:
[email protected]. Fax: +46-18508542.
combined the Satake-Yang binding model10 with our ideal mixing treatment3 to explain the binding observed in a binary surfactant mixture. In this study, RQ is on the order of 0.5, thus in a range where both components strongly influence the micelle properties and the interaction with the polyelectrolyte. This is quite different from the situation in TRFQ experiments where Q is present only in small amounts. In fact, the concentration of Q is always adjusted so that XQ is typically in the range 0.010.02. As expected, the calculations presented by R&K indicate that the effect of Q on the binding of S is very small for most situations relevant to TRFQ, and in particular, in the systems used by us. As will be demonstrated below, our data treatment, which actually takes into account information from the binding isotherms of both the pure S and Q systems, gives negligible errors. Theory Quite generally, the chemical potential of a surfactant i (S or Q) in a mixed micelle can be described as a sum of free energy contributions (per mole monomer):11
µmic ) µ0,mic + µsurf + µeli + RT ln Xi + i i i
Gmix mic N
(1)
where the first three terms on the right-hand side represent the standard contribution, the free energy due to the hydrocarbon/ water contact at the micelle surface, and the electrostatic free energy, respectively. The remaining terms are the (ideal) entropy of mixing the components in the micelles and the free energy of mixing the micelles in the system. N is the surfactant aggregation number. In the general case, µeli is a function of surfactant concentration. With the polyelectrolyte wrapped around the micelle as a counterion12 this dependence is expected to be small. There is also a contribution from the change in free energy of the polyelectrolyte monomers as they bind to the micelles. In the presence of excess salt, which is the case considered here, this contribution can be treated as constant. For the free surfactant monomers in the water, we have
+ RT ln Cf,i µwi ) µ0,w i
(2)
Thus, at chemical equilibrium,
RT ln Cf,i ) -RT ln Ki + RT ln Xi +
Gmix mic N
(3)
where
- µ0,w + µsurf + µeli -RT ln Ki ) µ0,mic i i i
(4)
Next, we relate this to the binding isotherms in Figure 1. The most simple way to model the binding is to consider the binding of uniform micelles to the polyelectrolyte coils. The concentration of bound monomers Cb can then be expressed as5,13
10.1021/jp9923217 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/19/2000
1138 J. Phys. Chem. B, Vol. 104, No. 5, 2000
Comments
Figure 1. Binding isotherms for C12PB21 (open circles) and C12TAB3 (filled circles) in solutions containing 0.5 mM NaPA and 10 mM NaBr: (solid lines) eq 5 with Cb,max ) 0.3 mM, K ) 4.5 mM-1, N ) 18 (C12PB); Cb,max ) 0.3 mM, K ) 2.7 mM-1, and N ) 23 (C12TAB); (dotted line) isotherm from eq 6 with the additional term RTθ/N(1 - θ) (see text), Cb,max ) 0.32 mM, K ) 2.75 mM-1, N ) 60.
Cb ) Cb,max
(KCf)N
(5)
1 + (KCf)N
where Cb,max is the maximum concentration of surfactant associated with the polymer coils. Cb,max is not well defined, but as shown in Figure 1, eq 5 gives a good mathematical description of the relevant part of the binding isotherms in the single surfactant systems. Here, we put Cb,max equal to 0.3 mM and fitted the other two parameters; see legend of Figure 1. By writing eq 5 in the form
RT ln Cf ) -RT ln K +
RT θ ln N 1-θ
(
)
(6)
where θ ) Cb/Cb,max, K (≈ 1/cac) can be identified with Ki in eqs 3 and 4 (Xi ) 1 in the single surfactant system). Furthermore, the second term on the right-hand side of eq 6 corresponds to the entropy (per monomer) of mixing the micelles. Since θ can be considered as the fraction of “sites” occupied by micelles, the micelles bound to the polyelectrolyte are described simply as an ideal lattice gas (as in the Langmuir adsorption isotherm14); see below. It is straightforward to extend the binding model to binary surfactant mixtures by adding RT ln Xi to the right-hand side of eq 6. The free concentrations of S and Q are then given by
Cf,i )
Xi
(1 -θ θ) Ki
1/N
(i ) S,Q)
(7)
where KS and KQ can be obtained from fits of eq 5 to experimental data for the single surfactant systems. It is important to note that the entropy of mixing the micelles corresponds to mixing of the mixed micelles. In principle, we need to know the properties of the mixed micelles to evaluate this contribution. However, when XQ is small (one or two Q per micelle), it will be almost the same as for the pure S system. (For C12P+/C12TA+ this assumption is particularly good since the pure micelles of both components have similar aggregation numbers.15,16) We emphasize this because the treatment by Liu et al.,9 where the binding is described as a nearest neighbor interaction between bound surfactant monomers,10 gives a different result. In their model, the free concentration at a given
binding ratio Cb/Cp in the mixed system (Cp is the concentration of polyelectrolyte) is related to the composition of the micelles using the relationship Cf,i ) XiCf,i*, where Cf,i* is the free concentration in the single surfactant system at the same Cb/ Cp. This relation, also used by R&K, is replaced by eq 7 in the above treatment. This means that the reference state in their treatment, i.e., the pure S and Q micelles in complexes with the polyelectrolyte at a particular binding ratio, changes with the binding ratio. As long as the binding isotherms in the pure systems have the same shape (as for C12P+/C12TA+), the result will be the same as with eq 7. It is impossible, however, to analyze for instance a mixture of ionic and nonionic surfactant (or nonionic quencher) in this way, where the latter do not interact with the polyion in its pure state. Results and Discussion In Figure 2, we have used eq 7 to calculate XQ for the mixed system of C12PB/C12TAB/NaPA. KS, KQ, and N are equal to 2.7 mM-1, 4.5 mM-1, and 23, respectively; see legend to Figure 1. RQ ) 0.02 is typical for the TRFQ experiments in ref 3. The solid line represents XQ as a function of Ctot, the total surfactant concentration. Shown in the figure is also the result from the approximate procedure used to calculate XQ in our previous work3 (squares). This involves the following approximations. The concentration of free surfactant Cf,S is taken from the binding isotherm for the pure C12TAB/NaPA system at a total concentration, Ctot, equal to the concentration of S + Q in the mixed system. Cf,Q was calculated from Cf,S using the empirical relation3
log(Cf,Q/molar) ) 1.08 log(Cf,S/molar) + 0.0159
(8)
Finally, XQ was calculated using the ideal mixing model,3,6
(
XQ2 - 1 -
)
Ctot RQCtot )0 XQ ∆ ∆
(9)
where ∆ ) Cf,S - Cf,Q. (Note that we did not take ∆ as the difference in the cac of C12TAB and C12PB, as may be inferred from the paper by R&K.) As evident from Figure 2, the agreement is very good. In fact, the error is never larger than 3% in the entire range of concentrations.
Comments
J. Phys. Chem. B, Vol. 104, No. 5, 2000 1139
Figure 2. Mole fraction of quencher in micelles XQ as a function of total surfactant concentration Ctot. Upper curves: C12PB/C12TAB/NaPA, RQ ) 0.02; (solid line) XQ calculated from eqs 7 with KS, KQ, and N equal to 2.7 mM-1, 4.5 mM-1, and 23, respectively; (squares) the result from the approximate procedure used in ref 3, see text. Lower curves: C16PB/C12TAB/NaPA; (dotted line) XQ ) 0.01 (the theoretical value); (circles) XQ obtained assuming no free Q, see text. The double arrow covers the range of concentrations investigated in ref 3.
We have also applied C16P+ as quencher in the C12TAB/ NaPA system.3 In this case, it is not appropriate to calculate XQ from eq 7 for a fixed RQ. Instead, XQ is put equal to 0.01 in agreement with the experimental conditions and then used to calculate RQ and Ctot corresponding to equilibrium between micelles and free monomers. The calculation gives RQ in the range from 0.0002 to 0.004. Here, KQ ) 291 (mM)-1 (calculated from the ratio between the cac values for C16PC and C12TAC) and N ) 23. In the paper,3 it was assumed that 100% of Q was in the micelles. Furthermore, the concentration of micellized surfactant (S + Q) was taken from the C12TAB binding isotherm corresponding to a concentration equal to Ctot. To test these approximations we calculate XQ in this way from the values of RQ and Ctot that theoretically (eq 7) give XQ ) 0.01. As shown in Figure 2, the approximations (circles) are in perfect agreement with the theoretical value (dotted line), except close to the cac. For the range of concentrations used in our TRFQ experiments (see Figure 2) the difference is again less than 3%.8 In the above calculations, it is justified to use eqs 5-7 since we are only interested in the mutual distribution of S and Q between micelle and aqueous subphases. In contrast, eq 5 should not be used to analyze binding data to obtain aggregation numbers. For instance, N ) 23 gives a good fit to the NaPA/ C12TAB data in Figure 1, but the aggregation number obtained from TRFQ (Cb ) 0.25 mM) is 60.3 A better description of the entropy of mixing the micelles is expected to improve the binding model. Excluded volume effects must be of importance since, during the course of binding, the number of micelles at each polyion coil increases considerably. As an illustration to this we treat the micelles in a polyion/surfactant coil (comprising several micelles) as a van der Waals gas with zero potential energy of interaction between micelles. This means that each micelle excludes a volume b, not available to the other micelles. If b is chosen so that θ ) 1 corresponds to close-packing of the micelles, the expression for the chemical potential of the surfactant in the micelles is obtained by adding RTθ/N(1 - θ) to the right-hand side of eq 6; see, for example, Hill.14
Interestingly, with this simple modification, the slope of the isotherm calculated using N ) 60 is in reasonable agreement with the experimental data (except at low degrees of binding); see Figure 1. Our conclusion from these and other TRFQ studies is that surfactant quenchers with properties similar to the surfactant studied offers important advantages over the more conventional quenchers. Originally, we had hoped that ideal mixing of the quencher and the studied surfactant would allow a very simple approach, but as demonstrated in our papers, and now also in R&K’s study, it is not so. But still, the partitioning of both surfactants can be controlled from measurements and an appropriate analysis. Furthermore, it is a strength, e.g., for the assumption of random distribution of quenchers over micelles, that the quencher and the surfactant are similar in their interactions with each other and with the polyelectrolyte. Pyrene and other nonpolar, aromatic molecules usually employed as probes and quenchers have serious drawbacks. They are qualitatively different than the surfactants, and much less is known of their interactions in each specific case. It is more difficult to assess and control the distribution among the micelles and between micelle and water, and interactions with the polyelectrolyte cannot always be excluded. Much is known of pyrene in micelles. Even for this strongly hydrophobic molecule, the fraction in water cannot be neglected when the amount of micellized surfactant is low. From distribution data determined in saturation measurements,17 one can estimate that with 1 × 10-4 M surfactant in micelles (corresponding to β ) 0.2 in R&K Table 2), about 10% of pyrene would be in the aqueous subphase, if the micelles where similar to C12TAB micelles in water, and 25% for SDS micelles. If the micelles are smaller, however, there can be appreciable differences in the partitioning; tributylaniline favors free SDS micelles over PEO-bound SDS by a factor of about 1.8.18 The presence of appreciable amounts of pyrene in the aqueous subphase affects of course all results using pyrene as a probe, but it is more serious when excimer formation is involved. Other molecules
1140 J. Phys. Chem. B, Vol. 104, No. 5, 2000 employed as quenchers, such as dimethylbenzophenone, have much less affinity for micelles; under conditions such as those above from R&K Table 2, only about 25% would be found in the micelles. It is also clear that pyrene interacts strongly with cationic surfactants; it was for instance clearly shown that the static quenching of pyrene occurs below the cmc with CPC as the quencher.19 Similar interactions would probably be found if looked for also with other compounds used as quenchers; if nothing else, the effects from the mere size of the hydrophobic additive would certainly be important in small micelles.20 Few investigations have addressed such effects, which are specific and difficult to assess. We know at least something regarding surfactant quenchers. References and Notes (1) Almgren, M.; Hansson, P.; Mukhtar, E.; van Stam, J. Langmuir 1992, 8, 2405. (2) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115. (3) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16684. (4) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 16694. (5) Hansson, P.; Almgren, M. J. Phys. Chem. 1996, 100, 9038. (6) Clint, J. H. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1327. (7) Rodenhiser, A. P.; Kwak, J. C. T. J. Phys. Chem. B 1999, 103, 2970.
Comments (8) In ref 7, R&K wrongly quoted ref 3, whereas in that work we had allready taken into account that XQ > RQ. We also showed that the aggregation numbers, obtained using C12PC as quencher with XQ calculated from the ideal mixing model, should have been the same as those obtained with C16PC as quencher assuming 100% binding. (9) Liu, J.; Takisawa, N.; Shirahama, K. J. Phys. Chem. B 1998, 102, 6696. (10) Satake, I.; Yang, J. T. Biopolymers 1976, 15, 2263. (11) Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1987, 91, 338. (12) Wallin, T.; Linse, P. Langmuir 1996, 12, 305. (13) Linse, P.; Piculell, L.; Hansson, P. Models of Polymer-Surfactant Complexation. In Polymer-surfactant systems; Kwak, J. C. T., Ed.; Marcel Dekker: New York, 1998; Vol. 77. (14) Hill, T. L. An Introduction to Statistical Thermodynamics, 2nd ed.; Dover Publications: New York, 1986. (15) Ozeki, S.; Ikeda, S. J. Colloid Interface Sci. 1982, 87, 424. (16) Fujio, K.; Ikeda, S. Langmuir 1991, 7, 2899. (17) Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. Soc. 1979, 101, 279. (18) van Stam, J.; Brown, W.; Fundin, J.; Almgren, M.; Lindblad, C. In Colloid-Polymer Interactions; Dubin, P., Ed.; ACS Symp. Ser.; American Chemical Society: Washington, DC, 1993; p 195. (19) Almgren, M.; Wang, K.; Asakawa, T. Langmuir 1997, 13, 4535. (20) Almgren, M.; Swarup, S. J. Phys. Chem. 1983, 87, 876. (21) Shimizu, T. Colloids Surf. A: Physicochemical and Engineering Aspects 1995, 94, 115.
