Role of Condensed Counterions in the Thermodynamics of Surfactant

occur in ionic surfactant micelles.1,2 However, the role of the condensed counterions in the thermodynamics of micellization has often been ignored. T...
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Langmuir 1999, 15, 58-65

Role of Condensed Counterions in the Thermodynamics of Surfactant Micelle Formation with and without Oppositely Charged Polyelectrolytes Andrew J. Konop and Ralph H. Colby* Department of Materials Science and Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802 Received June 29, 1998. In Final Form: September 28, 1998 We present a simple model of micelle formation that includes any effects of condensed counterions. The critical micelle concentration (cmc) of ionic surfactants has a much weaker dependence on surfactant tail length than nonionic surfactants because condensed counterions on the ionic surfactant micelle lose their translational entropy, as predicted by Murray and Hartley in 1935. The critical aggregation concentration (cac) of ionic surfactants in the presence of oppositely charged polyelectrolytes is much smaller than the cmc, and has a strong dependence on surfactant tail length, similar to the cmc of nonionic surfactants. The oppositely charged polyelectrolyte has a strong electrostatic interaction with the micelle surface, partially accounting for the low cac. The binding of micelle and polyelectrolyte also means that no condensed counterions are necessary on the micelle surface, which also lowers the cac and changes the dependence of the cac on surfactant tail length. For strongly charged polyelectrolytes, the cac is lowered further still because counterions are released from the polyelectrolyte when the surfactant binds to it. Our model correctly predicts that the cac increases as salt is added while the cmc decreases with added salt, owing to the different roles of counterion condensation. The experimental increases in the cac as either the charge density or concentration of the polyelectrolyte increases are also explained by the model. We experimentally find that the number of counterions released from the polyelectrolyte per bound surfactant is roughly equal to the fraction of neutralized monomers with bound counterions.

I. Introduction Counterion condensation has long been recognized to occur in ionic surfactant micelles.1,2 However, the role of the condensed counterions in the thermodynamics of micellization has often been ignored. Therefore, we first would remind the reader of the importance of counterion condensation in ionic surfactant micelles and review the calculation of the critical micelle concentration (cmc) of both nonionic and ionic surfactants. Like the surfactant ions, any condensed counterions must relinquish their translational entropy when the micelle is formed. For ionic surfactants with no added salt this greatly weakens the dependence of the cmc on surfactant tail length,1,3-5 compared to either nonionic surfactants or ionic surfactants with an excess of salt. Complexes between surfactants and oppositely charged polyelectrolytes have many important uses.6 In the presence of oppositely charged polyelectrolytes, there is no need for any counterions to condense on the micelle surface, because the polyelectrolyte can stabilize the surface charge of the micelle. For this reason, ionic surfactant aggregation in the presence of an oppositely charged polyelectrolyte has a critical aggregation concentration (cac) that has a dependence on surfactant tail length similar to that of nonionic surfactants. Further(1) Murray, R. C.; Hartley, G. S. Trans. Faraday Soc. 1935, 31, 183189. (2) Botre, C.; Crescenzi, V. L.; Mele, A. J. Phys. Chem. 1959, 63, 650-653. (3) Mukerjee, P. J. Phys. Chem. 1962, 66, 1375-1376. (4) Nakagaki, M.; Handa, T. In Structure/Performance Relationships in Surfactants; Rosen, M. J., Ed.; ACS Symposium Series 253; American Chemical Society: Washington, DC, 1984; pp 73-86. (5) Zana, R. Langmuir 1996, 12, 1208-11. (6) Hayakawa, K.; Kwak, J. C. T. In Cationic Surfactants; Rubingh, D. N., Holland, P. M., Eds.; Marcel Dekker: New York, 1991, pp 189248.

more, strongly-charged polyelectrolytes release their counterions when bound to the surfactant micelle, causing the cac to increase as salt is added. Thermodynamic treatments that ignore counterion condensation effects7 incorrectly predict the cac to decrease as salt is added (like the cmc does). We first discuss the essential physics of counterion condensation on the surface of a spherical micelle at the level of a two-state model in section II. Sections III and IV review the minimal model for surfactant micelle formation, that dates back to the 1935 work of Murray and Hartley,1 in the cases of nonionic and ionic surfactants, respectively. We then extend this minimal model to the case of micelle formation by ionic surfactants in the presence of oppositely charged polyelectrolyte in section V. The minimal model focuses on the dominant terms in the free energy. One dominant term is the energetic gain in bringing hydrocarbon tails out of water into the interior of the micelle. For ionic surfactants there are also electrostatic energy changes when the micelle is formed. The other dominant terms arise from the loss in translational entropy for the surfactant and any change in counterion condensation that results. II. Counterion Condensation We estimate the extent of counterion condensation on a spherical micelle formed from ionic surfactant by considering a simple balance between electrostatic energy and counterion entropy, as is common in polyelectrolytes.8,9 We do not solve the Poisson-Boltzmann equation but instead derive an approximate result, relying on the twostate model for counterions. Either each counterion is (7) Wallin, T.; Linse, P. Langmuir 1998, 14, 2940-49. (8) Manning, G. S. J. Chem. Phys. 1969, 51, 924-933. (9) Manning, G. S. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 909922.

10.1021/la980782y CCC: $18.00 © 1999 American Chemical Society Published on Web 12/04/1998

Role of Condensed Counterions

Langmuir, Vol. 15, No. 1, 1999 59

condensed on the micelle (thereby having no entropy and reducing the micelle’s effective charge), or the counterion is completely free (with considerable translational entropy but leaving the micelle with an effective charge). The real counterion distribution is a smoothly decreasing concentration of counterions with distance from the micelle, but the two-state model is simpler and captures the essential physics for our purposes. We define β to be the ratio of condensed counterions to surfactant ions on a micelle. For monovalent surfactants with monovalent counterions, the surface charge on the micelle is (1 - β)Ne, where N is the aggregation number (number of surfactants per micelle) and e is the elementary charge. The Coulomb potential10 surrounding the micelle is (1 - β)Ne/(r), where  is the dielectric constant of the surrounding medium (water) and r is the radial coordinate from the micelle center. The surface potential (1 - β)Ne/ (R) is simply the value of the Coulomb potential at the surface of the micelle (r ) R, the micelle radius). If another counterion of charge e condenses onto the micelle, it lowers the Coulomb repulsion energy but also loses all of its translational entropy. The equilibrium value of β is determined from the net free energy change being zero.

()

cco (1 - β)Ne2 + kT ln )0 R cw

(1)

Here k is Boltzmann’s constant, T is temperature, cco is the number density of free counterions and cw is the number density of water molecules. The first term in eq 1 is the reduction in Coulomb repulsion energy of condensing an additional counterion and the second term arises from the loss of translational entropy. Solving eq 1 for β provides an estimate for the condensed fraction of ions on a micelle.

β)1+

()

cco R ln NlB cw

(2)

The Bjerrum length lB ≡ e2/(kT) is the scale where the Coulomb energy of two charges equals the thermal energy kT (lB ) 7 Å in water). Equation 2 is identical to the result of Alexander et al.11 for spherical charged colloids. Using sodium dodecyl sulfate (SDS) as an example, R ) 20 Å and N ) 58, making R/(NlB) ) 0.049. The fact that this number is small compared to unity means that the fraction of condensed counterions on the micelle surface is nearly independent of surfactant concentration, as observed in experiments.2,12-14 We treat the case of an aqueous solution of surfactant in the presence of a monovalent salt that shares a counterion with the surfactant (i.e., sodium dodecyl sulfate with sodium chloride). The free counterions comprise noncondensed counterions from the surfactant and counterions from added salt. (10) The correct form of the electrostatic potential is the Yukawa potential, which incorporates electrostatic screening. This reduces to the Coulomb potential for length scales small compared to the Debye length. Our treatment of counterion condensation is only approximately valid for sufficiently small salt concentrations where the Debye length is larger than the micelle size. (11) Alexander, S.; Chaikin, P. M.; Grant, P.; Morales, G. J.; Pincus, P.; Hone, D. J. Chem. Phys. 1984, 80, 5776-5781. (12) Lindman, B.; Puyal, M. C.; Kamenka, N.; Brun, B.; Gunnarsson, G. J. Phys. Chem. 1982, 86, 1702-1711. (13) Jansson, M.; Stilbs, P. J. Phys. Chem. 1985, 89, 4868-4873. (14) Li, P.; Jansson, M.; Bahadur, P.; Stilbs, P. J. Phys. Chem. 1989, 93, 6458-6463.

cco ) (1 - β)Ncmic + csurf + csalt The first term counts the free counterions from surfactants in micelles (where cmic is the number density of micelles). The second term counts the counterions from free surfactant (csurf is the number density of free surfactant). The last term counts the counterions from added salt (csalt is the number density of added salt). In general, therefore, care must be taken in using eq 2 since the concentration of free counterions also depends on β. Since we are concerned here with the critical micelle concentration, we take cmic ) 0 and csurf ) ccmc, the critical concentration of surfactant where micelles first form. This allows a simple estimate of the counterion condensation on a micelle at the cmc.

β)1+

(

)

ccmc + csalt R ln NlB cw

(3)

With no added salt, the cmc of SDS is ccmc ) 8.3 mM. Using R/(NlB) ) 0.049 (calculated above) and cw ) 55 M, eq 3 predicts β ) 0.57, in excellent agreement with the experimental value15 of 0.60. These numbers are typical for spherical monovalent ionic short-chain surfactant micelles, and the extent of counterion binding increases somewhat as surfactant tail length increases, as discussed at the end of section IV. Equation 3 also predicts the experimentally observed weak increase in condensed counterions as salt is added.2 As more counterions are added to the solution (by adding salt) the entropic penalty for condensing ions is diminished, thereby favoring more condensation. Another interesting consequence of R/(NlB) , 1 is that micelles effectively choose a level of condensation that allows them to have a certain surface potential. When mixed micelles are formed from nonionic and ionic surfactants of the same hydrocarbon chain length, counterion condensation makes the effective (net) charge on the micelle independent of mixture composition over a wide range.16,17 A final cautionary note is in order regarding counterion condensation on ionic micelles. While the agreement between predicted and measured levels of counterion condensation is impressive, the calculations presented here are really only crude estimates of counterion condensation. A genuine theory for condensation requires a much more elaborate calculation that includes a full solution of the Poisson-Boltzmann equation.18-20 III. Cmc of Nonionic Surfactants Here we review the minimal model for the free energy of micelle formation of nonionic surfactants that focuses on the two largest terms of the free energy. We restrict our attention to the critical concentration at which micelles first form (the cmc) and write the following expression for the free energy change on forming one micelle of N surfactants, each having a hydrocarbon tail with n carbons. (15) Barchini, R.; Pottel, R. J. Phys. Chem. 1994, 98, 7899-7905. (16) Treiner, C.; Khodja, A. A.; Fromon, M. J. Colloid Interface Sci. 1989, 128, 416-421. (17) Clint, J. H. Surfactant Aggregation; Blackie: Glasgow, U.K., 1992. (18) Gunnarsson, G.; Jonsson, B.; Wennerstrom, H. J. Phys. Chem. 1980, 84, 3114-3121. (19) Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1983, 87, 5025-32. (20) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971.

60 Langmuir, Vol. 15, No. 1, 1999

Konop and Colby

( )

∆G ) -kTN ln

ccmc - ψNn + ∆Gcorr cw

(4)

The first term arises from the translational entropy loss in forming a micelle from N free surfactant molecules. The second term is the energetic gain in bringing the Nn carbons of the hydrocarbon chains out of the water and into the hydrophobic interior of the micelle (with ψ being the energy per carbon atom in the tail). The third term is a correction that includes a number of rather subtle effects that are not essential to the present discussion. The terms in this correction include surfactant headgroup interactions, surface terms arising from hydrophobic interactions and curvature, and tail entropy losses due to packing into the micelle.21,22 Proper account of these terms allows the calculation of the size and shape of micelles, but this is not our concern here. We instead assume the micelles are spherical at the cmc, with aggregation number N. At the cmc, ∆G ) 0 and we can solve eq 4 for ccmc.

∆Gcorr ψn + ln cw ln ccmc ) kTN kT

(5)

The dependence of the cmc of nonionic surfactants on surfactant tail length is known to follow this prediction.17,23,24 The data for alkylpoly(ethyleneoxides),25 for instance, obey the form of eq 5, with ψ/(kT) ) 1.1 and ∆Gcorr/(kTN) ) 0.4. The terms comprising ∆Gcorr all have smaller magnitudes than the terms we keep, and the dominant parts of ∆Gcorr roughly scale with aggregation number, making ∆Gcorr/(kTN) a meaningful constant for a given family of surfactants, even though the aggregation number itself is a function of n. IV. Cmc of Ionic Surfactants

( )

(

)

ccmc ccmc + csalt - kTβN ln cw cw 2

ψNn +

2

2

( )

∆G ) -kTN ln

ln ccmc )

ccmc e2(1 - β)2N2 + ∆Gcorr - ψNn + cw R csalt . ccmc (7)

∆Gcorr (1 - β)2NlB ψn + + ln cw kTN R kT csalt . ccmc (8)

Comparing eqs 5 and 8, we see that ionic surfactants with excess salt should have a similar tail length dependence of their cmc as nonionic surfactants. Plots of ln ccmc vs n should be parallel for the two systems, with the cmc of the ionic surfactant considerably larger due to electrostatic repulsion of head groups. This electrostatic repulsion is contained in the second term of eq 8, and using the values previously discussed for SDS (R/(NlB) ) 0.049 and β ) 0.60) we estimate its magnitude (1 - β)2NlB/R ) 3.3. Thus we expect the cmc of ionic surfactants with added salt to be roughly a factor of exp(3.3) = 30 larger than the cmc of nonionic surfactants. With no added salt (csalt , ccmc) we can write eq 6 in a simpler form, and determine the cmc.

∆G ) -kT(1 + β) N ln

Here we present the minimal model, based on the model of Murray and Hartley,1 for micelle formation by ionic surfactants and compare with the literature data. The free energy of formation of a micelle from ionic surfactants is in fact quite similar to the nonionic case. There are two important differences. When the ionic surfactant forms a micelle, there is a large electrostatic repulsion between head groups. This large repulsion is partially reduced by the condensed counterions on the micelle, but these condensed counterions also lose their translational entropy. The following expression estimates the free energy change in forming one ionic surfactant micelle.

∆G ) -kTN ln

surface of the spherical micelle. The fifth term ∆Gcorr again contains the terms of the free energy that we ignore, including surface and tail packing entropy effects.26 There are two limits of salt concentration that are of interest. With a large amount of added salt, such that csalt . ccmc, the second term of eq 6 can be ignored relative to the first term. This leads to the following equations in the high salt limit, for the free energy to form one micelle and the critical micelle concentration.

e (1 - β) N + ∆Gcorr (6) R

The first and third terms are identical to the nonionic case and thus also appear in eq 4. The second term arises from the loss of translational entropy from the βN condensed counterions on the micelle. The fourth term is the Coulombic repulsion of the (1 - β)N charges on the (21) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 37103724. (22) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934-2969. (23) Shinoda, K.; Nakagawa, T.; Tamamushi, B. I.; Isemura, T. Colloidal Surfactants; Academic Press: New York, 1963. (24) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1973. (25) Meguro, K.; Takasawa, Y.; Kawahashi, N.; Tabata, Y.; Ueno, M. J. Colloid Interface Sci. 1981, 83, 50-56.

( )

ccmc - ψNn + cw

e2(1 - β)2N2 + ∆Gcorr csalt , ccmc (9) R ln ccmc )

∆Gcorr

(1 - β)2NlB

+ (1 + β)R ψn c , ccmc (10) ln cw kT(1 + β) salt

kT(1 + β)N

+

Without added salt, the cmc of nonionic surfactants has a much weaker dependence on surfactant tail length n than either nonionic surfactants (eq 5) or ionic surfactants in the presence of added salt (eq 8). This fact is reflected in the (1 + β) in the denominator of the last term of eq 10, which is recognized in the literature as a direct manifestation of counterion condensation.1,3,4,27 The general form of eqs 5, 8, and 10, that ln ccmc ) A - Bn, is well established, with values of A and B listed for many surfactant systems.23,28-30 In Figure 1 we present literature data for the tail length dependence of the cmc for a nonionic homologous surfactant family (alkyl poly(26) Evans, D. F.; Wennerstrom, H. The colloidal domain: where physics, chemistry, biology, and technology meet; VCH: Weinheim, Germany, 1994. (27) Attwood, D.; Florence, A. T. Surfactant Systems: Their chemistry, pharmacy and biology; Chapman and Hall: London, 1983. (28) Rosen, M. J. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1989. (29) Ananthapadmanabhan, K. P. In Interactions of Surfactants with Polymers and Proteins, Goddard, E. D., Ananthapadmanabhan, K. P., Eds.; CRC Press: Boca Raton, FL, 1993; pp 5-58. (30) Klevens, H. B. J. Am. Oil Chem. Soc. 1953, 30, 74-80.

Role of Condensed Counterions

Figure 1. Dependence of the cmc on surfactant tail length for the nonionic family of alkyl poly(oxyethylenes) CnE8 at 25 °C25 (triangles), the ionic family of sodium alkyl sulfates31 (open circles, with no salt; filled circles with 0.1 M NaCl), and the ionic family of alkyltrimethylammonium bromides without salt32 (open squares).

(oxyethylenes)),25 an anionic family of surfactants with and without added salt (sodium alkyl sulfates),31 and a cationic family of surfactants without added salt (alkyltrimethylammonium bromides).32 The ionic surfactants with excess salt have the same slope as the nonionic surfactant, with ψ ) 1.1kT. Using this value for ψ allows us to estimate β from the n dependence of ln ccmc using eq 10 for the ionic surfactants without salt, assuming for the moment that β is independent of surfactant tail length. From the smaller slopes of the two data sets of ionic surfactants with no salt in Figure 1, we get β ) 0.65 for the sodium alkyl sulfates and β ) 0.64 for the alkyl trimethylammonium bromides. These numbers are in reasonable agreement with the levels of counterion condensation expected from section II. Shinoda’s model23 of micellization results in the form of eq 10 with β an empirical constant. A value of β ) 0.65 is known to fit the data of sodium alkyl sulfate cmcs as functions of salt and surfactant tail length.33 While we will consider the prediction that β actually has a weak n dependence next, we can already conclude that the difference in slopes of ln ccmc vs n between nonionic and ionic surfactants is a simple counterion condensation effect. The prevalent view in the literature23,24 is that this difference in slopes is a salt effect, arising from the free surfactant concentration at the cmc. As tail length is decreased, the cmc rises, and the higher free surfactant concentration should reduce the electrostatic repulsion between head groups on the micelle surface. This view stems from the incorrect idea that the cmc of ionic surfactants is determined by a balance of the energy gain in bringing the tails together and the electrostatic repulsion of the head groups on the micelle surface. A simple comparison of the terms in eq 9 allows us to largely ignore the electrostatic repulsion of the head groups. The magnitude of the first two terms are each always in the range 10-20kTN, whereas we previously estimated the electrostatic repulsion (third term) for SDS to be 3.3kTN, and expect ∆Gcorr = 0.4kTN, as observed in the nonionic surfactants. Thus the electrostatic repulsion plays a minor role in the micellization free energy. The cmc is mainly determined by the balance of entropy loss from surfactants and counterions and the energy gain from the tails.1,3,4,27 (31) Mukerjee, P.; Mysels, K. J. Critical Micelle Concentrations of Aqueous Surfactant Systems; NSRDS-NBS 36; NBS: Washington, DC, 1971. (32) Evans, D. F.; Allen, M.; Ninham, B. W.; Fouda, A. J. Solution Chem. 1984, 13, 87-101. (33) Funasaki, N. J. Colloid Interface Sci. 1978, 67, 384-386.

Langmuir, Vol. 15, No. 1, 1999 61

As mentioned in section II, the extent of counterion condensation β at the cmc is predicted to increase weakly as the surfactant tail length increases because the cmc decreases (see eq 3). Using eq 10, for monovalent surfactants we plot (1 + β) ln(ccmc/cw) vs. n to determine ψ. Vikingstad, et al.34 measured β and ccmc for a series of sodium alkanecarboxylates with n ) 6, 7, 8, 9, 10, and 11. Their data obey eq 10, yielding ψ ) 1.3kT. Zana35 determined β and ccmc for a series of alkyltrimethylammonium bromides with n ) 8, 10, 12, 14, and 16, from which we conclude ψ ) 1.4kT. Vikingstad and Saetersdal36 reported β and ccmc for a series of divalent sodium alkylmalonates with n ) 11, 12, 13, 14, and 15. For divalent surfactants, the relevant plot of (1 + 2β) ln(ccmc/cw) vs. n determines ψ ) 1.2kT. The higher cmc for divalent surfactants has been attributed to the “increase in electrical work required to form a micelle”.27 In reality, ionic micelles always seek a constant surface potential on the micelle surface, as a direct consequence of the counterion condensation discussed in section II. Thus the electrostatic term in eq 10 should be roughly independent of surfactant valency. The higher cmc for divalent surfactants is merely caused by the increased amount of condensed monovalent counterions required to achieve that constant surface potential. Ionic surfactants have similar energies of tail interactions as nonionic surfactants (similar ψ) when counterion condensation effects are included in the free energy.5 The difference in slope of plots of ln ccmc vs n between divalent, monovalent, and nonionic surfactants arise from counterion condensation, as originally predicted in 1935 by Murray and Hartley.1 V. Cac of Ionic Surfactants in the Presence of Oppositely Charged Polyelectrolytes In the presence of polymers, surfactants aggregate to form micelles at lower concentrations than the cmc. The concentration at which micelles form in the presence of a polymer is called the critical aggregation concentration (cac). In the case of both nonionic surfactants and ionic surfactants forming micelles in the presence of a nonionic polymer, the cac is only slightly below the cmc.37,38 In the case of ionic surfactants with oppositely charged polyelectrolytes, where a relatively short section of polyelectrolyte is adsorbed to the micelle surface,39 the cac is orders of magnitude lower than the cmc for three important reasons: (1) There is an electrostatic interaction between the polyelectrolyte and the micelle surface. (2) There is no need for any condensed counterions on the surfactant micelle. (3) A highly charged polyelectrolyte can release some of its condensed counterions when it binds to the micelle. We consider the free energy difference between two states. The initial state has long strongly charged polyelectrolyte chains (monomer number density cp) with many condensed counterions (fraction f of the monomers have free counterions) in solution with free surfactant (with number density ccac). The final state has a single spherical micelle of N surfactants with part of a single polyelectrolyte chain wrapped around it. We assume that the net charge (34) Vikingstad, E.; Skauge, A.; Hoiland, H. J. Colloid Interface Sci. 1978, 66, 240-246. (35) Zana, R. J. Colloid Interface Sci. 1980, 78, 330-337 (36) Vikingstad, E.; Saetersdal, H. J. Colloid Interface Sci. 1980, 77, 407-412. (37) Cabane, B. J. Phys. Chem. 1977, 81, 1639-1645. (38) Nikas, Y. J.; Blankschtein, D. Langmuir 1994, 10, 3512-3528. (39) Dubin, P. L.; The, S. S.; McQuigg, D. W.; Chew, C. H.; Gan, L. M. Langmuir 1989, 5, 89-95.

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on the surface of the micelle is very nearly zerosroughly N oppositely charged polyelectrolyte monomers are also on the micelle surface. Thus, there is no electrostatic repulsion term in the free energy change. In fact, part of the electrostatic repulsion of the polyelectrolyte chain is reduced by the micelle formation. We estimate this reduction in electrostatic repulsion to be of order fNe2/(lB) ) fkTN, since the Bjerrum length lB sets the spacing between effective charges on the polyelectrolyte. We therefore expect the net electrostatic energy difference between the micellization in the presence of oppositely charged polyelectrolyte compared with micellization of the same ionic surfactant without polyelectrolyte to be slightly larger than the 3.3kTN estimated for SDS in the electrostatic terms of eqs 6, 7, and 9. Recent microcalorimetry experiments40 indicate that the energy of micellization is 9 kJ/mol ()3.6kTN) smaller for alkylpyridinium cations (with n ) 11, 13, 14, and 15) binding to poly(styrene sulfonate) than for the surfactant alone, independent of tail length and salt concentration. Since this energetic difference is very close to our estimate for the electrostatic energy of micelle formation of ionic surfactants without polyelectrolytes, we can safely ignore the electrostatic energy terms in the case of micellization of ionic surfactants with oppositely charged polyelectrolytes. The free energy to form a micelle is quite similar to the two cases previously discussed. Without the surfactant, a strongly charged polyelectrolyte has many of its counterions condensed on the chain. When the surfactant binds to the polyelectrolyte, some of the condensed counterions on the polyelectrolyte will be released. We define γ as the ratio of released polyelectrolyte counterions to bound surfactant molecules (γN is the number of polyelectrolyte counterions released when a micelle of N surfactants is formed). Motivated by experimental results discussed below, we assume γ to be a constant for a given polyelectrolyte/surfactant system. Such considerations lead to the following expression for the free energy change in forming one micelle bound to an oppositely charged polyelectrolyte.

∆G ) -kTN ln

( )

(

)

ccac fcp + csalt + kTγN ln cw cw ψNn + ∆Gcorr (11)

The first and third terms are identical to the previous two cases, except that we now take the cac to be the concentration of free surfactant instead of the cmc. The second term is a lowering of free energy when γN of the condensed counterions on the polyelectrolyte are replaced by surfactant (from the gain in translational entropy of released counterions). We treat the case where the polyelectrolyte and any added salt have a common a counterion, so that fcp + csalt is the concentration of free counterions (cp is the number density of polyelectrolyte monomers and f is the fraction of those monomers that have released a counterion). The last term now contains a small effect of the loss in configurational entropy of the polyelectrolyte when it coils around the micelle, in addition to the usual surface and tail packing entropy effects. Although the polyelectrolyte monomers on the micelle surface are highly localized, they did not have much configurational entropy in the locally rodlike state of the polyelectrolyte without surfactant, thus making the configurational entropy loss small. (40) Bezan, M.; Malavasic, M.; Vesnaver, G. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 1054-1058.

Figure 2. Dependence of the cac on (a) the surfactant tail length for the indicated salt concentrations and (b) the salt concentration for the indicated surfactant tail lengths. Data are from Malovikova, Hayakawa, and Kwak41 for the binding of alkylpyridinium cations to sodium dextran sulfate. The cac is taken to be the point where half of the added surfactant is bound to the polyelectrolyte. The slopes in part b give the value of γ, the number of condensed counterions on the polyelectrolyte released per bound surfactant. For n ) 11, γ ) 0.78; for n ) 12, γ ) 0.71; for n ) 13, γ ) 0.67; for n ) 14, γ ) 0.64.

Using eq 11 with ∆G ) 0 gives the following expression for the cac.

ln ccac )

(

)

∆Gcorr fcp + csalt ψn + ln cw + γ ln kTN cw kT

(12)

The effect of added salt on the cac of polyelectrolytestabilized ionic micelles is opposite to the normal effect of salt on the cmc of ionic micelles, because of the opposite role of condensed ions. The ionic micelle cmc decreases as salt is added, primarily because the entropic penalty for condensing counterions on the micelle surface diminishes. However, the cac of polyelectrolyte-stabilized ionic micelles increases as salt is added because the entropic gain from releasing condensed counterions from the polyelectrolyte is diminished. The cac increasing with added salt has been seen experimentally.41,42 The slope of ln ccac vs n, the surfactant tail length, should be independent of salt concentration, and identical to that of nonionic surfactants (see eq 5). Experimentally it is well-known that the cac is both much smaller than the cmc of ionic surfactants, and has a stronger dependence on surfactant tail length.43 In Figure 2a, we plot the cac data of Malovikova, Hayakawa, and Kwak41 for the system of sodium dextran sulfate (cp ) 5 × 10-4 equiv/kg) with alkylpyridinium bromides of varying n, for different salt concentrations. Clearly ψ is independent of salt concen(41) Malovikova, A.; Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1984, 88, 1930-1933. (42) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 1668416693. (43) Thalberg, K.; Lindman, B. In Surfactants in Solution; Mittal, K. L., Shah, D. O., Eds.; Plenum: London, 1991; Vol. 11, pp 243-260.

Role of Condensed Counterions

tration, and ψ ) 1.3kT for this system. In Figure 2b we plot the same data as a function of salt concentration. The slopes of Figure 2b directly determine γ, and we find that it varies from γ ) 0.78 for n ) 11 to γ ) 0.64 for n ) 14. Hayakawa and Kwak44 have studied the effect of salt on the cac of two other systems and found γ ) 0.68 for sodium dextran sulfate/dodecyltrimethylammonium bromide (n ) 12) with NaCl and γ ) 0.75 for sodium polystyrene sulfonate/dodecyltrimethylammonium bromide (n ) 12) with NaCl. A second way to determine γ is to study the dependence of the cac on polyelectrolyte concentration. Equation 12 predicts the cac will increase as the polyelectrolyte concentration is increased, as has recently been reported by Shimizu45 for the system sodium polyacrylate and dodecylpyridinium chloride. While Shimizu’s data only cover a factor of 4 in polyelectrolyte concentration, Hansson and Almgren46 cover a factor of 50 in polyelectrolyte concentration, for the system sodium polystyrene sulfonate/dodecyltrimethylammonium bromide. Plotting their data as ln ccac vs ln cp yields γ as the slope, with γ ) 0.76, in excellent agreement with the γ ) 0.75 obtained by Hayakawa and Kwak44 on the same system as salt is added (discussed above). Recently, Hansson47 has combined data on sodium polyacrylate solutions and gels with dodecyltrimethylammonium bromide to cover more than three decades of polymer concentration and found γ ) 0.69. A third way to determine γ is to directly measure the concentration of the free polyelectrolyte counterions as surfactant is added. This approach has been used by Shimizu45 in a study of sodium polyacrylate/dodecylpyridinium chloride. Shimizu used a sodium selective electrode and a surfactant selective electrode to make a plot of released sodium counterions as a function of bound surfactant. Roughly linear relationships were observed, with slopes γ of 0.6-0.8. Slight variations in γ were seen for different polyelectrolyte concentrations, but no systematic trend was observed. We have experimentally investigated the effect of charge density of the polyelectrolyte on γ, for poly(acrylic acid) that is partially neutralized with NaOH. We mixed various amounts of dodecyl trimethylammonium bromide (C12TAB) with 10 mM polyelectrolyte, which had previously been neutralized to 30%, 50%, or 90%. The number of counterions released from the polyelectrolyte during binding of the surfactant was monitored by a measurement of sodium ion activity with a sodium selective electrode (Orion). The cac was determined in the usual way with a surfactant selective electrode (Orion).6 We assumed that concentration of free surfactant equals the cac (for added surfactant levels above the cac) to estimate the amount of surfactant bound to the polyelectrolyte. We assume for simplicity that all activity coefficients are unity. In Figure 3, we plot concentration of free sodium ions vs concentration of bound surfactant. The slopes of the lines are γ. For the 30% neutralized sample, γ ) 0.46, for the 50% neutralized polymer γ ) 0.58, and for the 90% neutralized sample γ ) 0.79. At low degrees of neutralization, a smaller fraction of sodium ions are released as surfactant binds to the polyelectrolyte. The intercept in Figure 3 gives f, the fraction of monomers on the chain with dissociated counterions (and hence an effective charge). Increasing the neutralization of the polyelectro(44) Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1982, 86, 38663870. (45) Shimizu, T. Colloids Surf. A 1994, 84, 239-248. (46) Hansson, P.; Almgren, M. Langmuir 1994, 10, 2115-2124. (47) Hansson, P. Langmuir 1998, 14, 2269-77.

Langmuir, Vol. 15, No. 1, 1999 63

Figure 3. Concentration of free sodium ions as a function of bound surfactant for dodecyl trimethylammonium bromide and poly(acrylic acid) neutralized 30% (triangles), 50% (open circles), and 90% (filled circles). The slope is γ and the intercept is f.

Figure 4. Correlation of the released polyelectrolyte counterions per bound surfactant with the fraction of neutralized monomers having condensed counterions. Filled symbols denote γ determined by the variation of the cac with salt concentration, the open symbol denotes γ determined by the variation of the cac with polyelectrolyte concentration, × and + symbols denote γ determined by ion selective electrodes. Circles are data for sodium dextran sulfate (R ) 1) with various cationic surfactants41,44 (see Figure 2b), for which we estimated f ) 0.3. The hexagon is for sodium polyacrylate (R ) 1) with C12TAB,47 for which f ) 0.29. Triangles are for sodium polystyrene sulfonate (R ) 1) with C12TAB,44,46 for which f ) 0.25. The + is for sodium polyacrylate (R ) 1) with dodecylpyridinium chloride45 and the x symbols are for the data of Figure 3, partially neutralized poly(acrylic acid) (R ) 0.3, 0.5, 0.9) with C12TAB. The solid line is eq 13.

lyte releases more counterions into solution but also causes more counterions to bind to the polyelectrolyte. The γ-condensed counterions released per bound surfactant apparently depend on the level of counterion condensation on the polyelectrolyte. We assume that only the neutralized polyelectrolyte monomers bind to the micelle surface and that they bind randomly, with no distinction between monomers with and without condensed counterions. With these assumptions, we expect the fraction of counterions released from the polyelectrolyte per bound surfactant to equal the fraction of neutralized monomers with bound counterions.

γ)

R-f R

(13)

The fraction of neutralized monomers is R. We test eq 13 in Figure 4, using our data and the literature data discussed above. Since we estimate the uncertainty in all determinations of γ to be (0.1, we conclude that the data are not inconsistent with eq 13. However, it is clear that more data, particularly at lower neutralization extents, are needed before any firm conclusions can be drawn about

64 Langmuir, Vol. 15, No. 1, 1999

the physics behind γ. Some firm conclusions are justified by Figure 4. The three techniques for determining γ seem to agree, and the observed γ values are all significantly less than unity. The fraction of neutralized monomers with condensed counterions, (R - f)/R, has a stronger effect on γ than the surfactant tail length has (see Figure 2b; those data are filled circles in Figure 4). The tail length dependence may in fact be within the bounds of experimental error on γ. The fact that the many different polyelectrolyte/surfactant systems all roughly fall on the line provides some hope of the universality promised by eq 13. The cac is also expected to increase as the charge density of the polyelectrolyte increases (larger f in eq 12). Such behavior has indeed been reported for many polyelectrolyte-surfactant systems.45,48-50 The effect of charge density of the polyelectrolyte is at first glance counterintuitive, because the electrostatic interaction might increase with polyelectrolyte charge density. We assume that there are always more than enough charges on a single polyelectrolyte chain to satisfy all charges on the micelle. This is the case for most experiments, but some computer simulations51 treat such short chains that they see the cac decrease when charge density of the polyelectrolyte increases: an electrostatic effect caused by an insufficient number of charges on the polymer that could conceivably be seen in experiments with either very short chains or chains of extremely low charge density. The cac has been reported to decrease as the fraction of potentially charged monomers on the polyelectrolyte is increased in one system.52 In this case, the charge density on the polyelectrolyte was sufficiently high that there is considerable counterion condensation. Thus f does not change appreciably (only the number of condensed ions changes) as the nominal charge density of the polyelectrolyte is increased. Increasing the condensed ions on the polyelectrolyte increases γ, which in turn decreases the cac. It is important to note that our model treats micelle formation in the presence of long oppositely charged polyelectrolytes. As long as there are more charges on each chain than on a micelle, we expect to be in the long chain limit, where the cac is independent of chain length. Experimentally, it is known that the cac increases as the chain length of the polyelectrolyte is lowered below this threshold53 (where a single polymer has fewer charges than the micelle). VI. Discussion An important point that was circumvented in the above discussion is the role of the surfactant aggregation number N. For ionic micelles without polyelectrolytes, the aggregation number generally increases with surfactant tail length,54,55 and increases as salt is added.27,54 Aggregation number of polyelectrolyte-stabilized ionic micelles has been found to increase as either the polyelectrolyte charge density is increased,52 or the ratio of surfactant to (48) Shimizu, T.; Kwak, J. C. T. Colloids Surf. A 1994, 82, 163-171. (49) Chandar, P.; Somasundaran, P.; Turro, N. J. Macromolecules 1988, 21, 950-953. (50) Kiefer, J. J.; Somasundaran, P.; Ananthapadmanabhan, K. Langmuir 1993, 9, 1187-1192. (51) Wallin, T.; Linse, P. J. Phys. Chem. 1996, 100, 17873-80. (52) Hansson, P.; Almgren, M. J. Phys. Chem. 1996, 100, 90389046. (53) Liu, J.; Takisawa, N.; Shirahama, K.; Abe, H.; Sakamoto, K. J. Phys. Chem. 1997, 101, 7520-23. (54) Birdi, K. S. In Micellization, Solubilization, and Microemulsions; Mittal, K. L., Ed.; Plenum: London, 1977; Vol. 1, pp 151-169. (55) Van Os, N. M.; Haak, J. R.; Rupert, L. A. M. Physico-chemical Properties of Selected Anioic, Cationic and Nonionic Surfactants; Elsevier: Amsterdam, 1993.

Konop and Colby

polyelectrolyte is increased.56 Furthermore, the aggregation number of polyelectrolyte-stabilized ionic micelles has been found to be either slightly larger than,57,58 slightly smaller than,56,59,60 or similar to42,46,52 the aggregation numbers of the same ionic surfactants without the polyelectrolyte. The key fact that allows us to ignore changes in the aggregation number is that N only enters into eq 5 for the cmc of nonionic surfactants and eq 12 for the cac of ionic surfactants with oppositely-charged polyelectrolytes, through a term ∆Gcorr/(kTN). The largest terms in ∆Gcorr are proportional to N, making ∆Gcorr/(kTN) and hence the cmc and cac essentially independent of N. The fact that all measured values of ψ, the energetic gain in bringing the hydrocarbon tails out of water (per carbon in the tail), are nearly identical, for the many different systems considered, provides some assurance that this is reasonable. The 10-20% differences in ψ observed for different systems probably reflect small differences in terms of ∆Gcorr that are not proportional to N. However, the electrostatic terms of eqs 8 and 10 assure that the cmc of ionic micelles depends on the surfactant aggregation number N. Since the micelles are dense spheres, R ∼ N1/3, the electrostatic term (1 - β)2NlB/R ∼ N2/3. In our analysis of the cmc, we effectively assumed that N does not change with either surfactant tail length or salt concentration. What justifies our analysis is the fact that the magnitude of the electrostatic term is small ((1 - β)2NlB/R = 3.3). This term is simply dwarfed by the main term driving micellization, the tail energy term, ψn/ (kT) ) 10-20. This electrostatic energy term does not appear for either nonionic surfactants or ionic surfactants in the presence of oppositely charged polyelectrolytes, so these systems should have cmc and cac insensitive to aggregation number. VII. Conclusion We have considered the role of counterion condensation and release in the formation of spherical micelles from ionic surfactants, in the presence and absence of oppositely-charged polyelectrolytes. Without polyelectrolytes, a fraction β of the surfactant’s counterions condense on the surface of the micelle and lose their translational entropy, as originally predicted by Murray and Hartley in 1935.1 This loss of translational entropy, coupled with the energy of charge repulsion on the surface of the micelle, makes ionic surfactants have a higher cmc than nonionic surfactants. The loss of translational entropy from condensed counterions also makes ionic surfactants have a weaker dependence of the cmc on surfactant tail length than nonionic surfactants, when there is no added salt. When salt is added (with an ion in common with the surfactant’s counterion), the cmc decreases and has a similar dependence on tail length as nonionic surfactants because the entropy loss for counterion condensation on the micelle surface is greatly reduced. These results are summarized in Figure 1. An oppositely charged polyelectrolyte can stabilize the surface of the ionic micelle, virtually eliminating electrostatic repulsion and allowing the micelle to form without (56) Fundin, J.; Hansson, P.; Brown, W.; Lidegran, I. Macromolecules 1997, 30, 1118-1126. (57) Chu, D.; Thomas, J. K. J. Am. Chem. Soc. 1986, 108, 62706276. (58) Hansson, P.; Almgren, M. J. Phys. Chem. 1995, 99, 1669416703. (59) Abuin, E. B.; Scaiano, J. C. J. Am. Chem. Soc. 1984, 106, 62746283. (60) Almgren, M.; Hansson, P.; Mukhtar, E.; Van Stam, J. Langmuir 1992, 8, 2405-2412.

Role of Condensed Counterions

condensing appreciable numbers of counterions. This stabilization makes the cac much lower than the cmc, and the lack of importance of counterion condensation makes the cac have a dependence on surfactant tail length that is similar to that of nonionic surfactants (Figure 2a). Additionally, strongly charged polyelectrolytes have condensed counterions associated with them that can be released when the surfactant binds to the polyelectrolyte, lowering the cac further still. The effect of added salt on the cac is opposite to its effect on the cmc. Added salt (with an ion in common with the polyelectrolyte’s counterion) increases the cac because the entropy gain on releasing a counterion from the polyelectrolyte is reduced (Figure 2b). For the same reason, the cac also increases as either polyelectrolyte concentration or charge density is increased, in agreement with experimental results in the literature. We have studied the counterion release from polyelectrolytes with poly(acrylic acid) as a function of its neutralization with NaOH. These results are presented in Figure 3. The fraction of counterions released from the polyelectrolyte per bound surfactant, γ, has been determined in the literature by three independent means. One

Langmuir, Vol. 15, No. 1, 1999 65

method involves determining the dependence of the cac on concentration of added salt.41,44 The second method uses the variation of the cac with polyelectrolyte concentration.46 The other method directly measures the polyelectrolyte counterions released as a function of bound surfactant, using ion selective electrodes.45 The literature data for γ suggest that it is determined by the fraction of neutralized monomers with bound counterions (see Figure 4). This implies that neutralized monomers bind to the micelle without regard for whether the monomer has a condensed counterion or not. A systematic study of γ covering wider ranges of salt concentration, polyelectrolyte concentration and polyelectrolyte charge is needed to better test these ideas. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. We thank Andrey Dobrynin, Phil Pincus, and Michael Rubinstein for helpful discussions. LA980782Y