Prediction of Conformational Characteristics and Micellar Solution

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Prediction of Conformational Characteristics and Micellar Solution Properties of Fluorocarbon Surfactants Vibha Srinivasan† and Daniel Blankschtein*,‡ Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Department of Chemistry and Biochemistry, University of Texas, Austin, Texas 78712 Received July 7, 2004. In Final Form: November 17, 2004 A molecular-thermodynamic theory is developed to model the micellization of fluorocarbon surfactants in aqueous solutions, by combining a molecular model that evaluates the free energy of micellization of fluorocarbon surfactant micelles with a previously developed thermodynamic framework describing the free energy of the micellar solution. In the molecular model of micellization developed, a single-chain mean-field theory is combined with an appropriate rotational isomeric state model of fluorocarbon chains to describe the packing of the fluorocarbon surfactant tails inside the micelle core. Utilizing this singlechain mean-field theory, the packing free energies of fluorocarbon surfactants are evaluated and compared with those of their hydrocarbon analogues. We find that the greater rigidity of the fluorocarbon chain promotes its packing in micellar aggregates of low curvatures, such as bilayers. In addition, the mean-field approach is utilized to predict the average conformational characteristics (specifically, the bond order parameters) of fluorocarbon and hydrocarbon surfactant tails within the micelle core, and the predictions are found to agree well with the available experimental results. The electrostatic effects in fluorocarbon ionic surfactant micelles are modeled by allowing for counterion binding onto the charged micelle surface, which accounts explicitly for the effect of the counterion type on the micellar solution properties. In addition, a theoretical formulation is developed to evaluate the free energy of micellization and the size distribution of finite disklike micelles, which often form in the case of fluorocarbon surfactants. We find that, compared to their hydrocarbon analogues, fluorocarbon surfactants exhibit a greater tendency to form cylindrical or disklike micelles, as a result of their larger molecular volume as well as due to the greater conformational rigidity of the fluorocarbon tails. The molecular-thermodynamic theory developed is then applied to several ionic fluorocarbon surfactant-electrolyte systems, including perfluoroalkanoates and perfluorosulfonates with added LiCl or NH4Cl, and various micellar solution properties, including critical micelle concentrations (cmc’s), optimal micelle shapes, and average micelle aggregation numbers, are predicted. The predicted micellar solution properties agree reasonably well with the available experimental results.

1. Introduction Fluorocarbon surfactants are surface-active materials in which some, or all, of the hydrogen atoms in the hydrophobic moiety, referred to hereafter as the surfactant tail, are replaced by fluorine atoms. The resulting structural and chemical properties of fluorocarbon surfactants lead to differences in their micellar solution properties as compared to those of their hydrocarbon analogues (for an overview of the bulk solution behavior of fluorocarbon surfactants, see Chapters 6 and 7 in refs 1 and 2). Fluorocarbon surfactants exhibit lower critical micelle concentrations (cmc’s) as compared to hydrocarbon surfactants having the same tail length.3-5 In addition, experimental investigations of fluorocarbon surfactants, including neutron scattering, light scattering, and X-ray scattering measurements,6-9 cryogenic transmission elec* To whom correspondence should be addressed. Mail: Department of Chemical Engineering, Room 66-444, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139. Tel: (617) 253-4594. Fax: (617) 252-1651. E-mail: dblank@ mit.edu. † University of Texas. ‡ Massachusetts Institute of Technology. (1) Kissa, E. In Fluorinated Surfactants; Surfactant Science Series 50; Marcel-Dekker: New York, 1994. (2) Matsuoka, K.; Moroi, Y. Curr. Opin. Colloid Interface Sci. 2003, 8, 227-235. (3) Shinoda, K.; Hato, M.; Hayashi, T. J. Phys. Chem. 1972, 76, 909914. (4) Kunieda, H.; Shinoda, K. J. Phys. Chem. 1976, 80, 2468-2470. (5) Ravey, J. C.; Gherbi, A.; Stebe, M. J. Prog. Colloid Polym. Sci. 1988, 76, 234-241.

tron microscopy studies,9,10 and rheological studies,11,12 reveal that these surfactants display a greater tendency to form aggregates having low curvatures, including cylindrical micelles and bilayer structures (which include planar disklike micelles and closed bilayer vesicles), as compared to their hydrocarbon counterparts. Investigations of the phase behavior of fluorocarbon surfactants reveal that, in many cases, these surfactants tend to form lamellar liquid crystalline phases, instead of the hexagonal liquid crystalline phases typically formed by hydrocarbon surfactants.9,13,14 Fluorocarbon surfactants are also very important commercially. They are powerful wetting agents and are indispensable as emulsifiers in many industrial applications, including emulsion polymerization of chlorocarbons and fluorocarbons, and in a variety of biomedical applications, including the development of oxygen-carrying (6) Hoffmann, H.; Kalus, J.; Thurn, H. Colloid Polym. Sci. 1983, 261, 1043-1049. (7) Iijima, H.; Kato, T.; Yoshida, N.; Imai, M. J. Phys. Chem. B 1998, 102, 990-995. (8) Burkitt, S. J.; Ottewill, R. H.; Hayter, J. B.; Ingram, B. T. Colloid Polym. Sci. 1987, 265, 619-627. (9) Ravey, J. C.; Stebe, M. J. Colloids Surf., A 1994, 84, 11-31. (10) Wang, K.; Karlsson, G.; Almgren, M.; Asakawa, T. J. Phys. Chem. B 1999, 103, 9327-9246. (11) Oelschlaeger, C.; Waton, G.; Buhler, E.; Candau, S. J.; Cates, M. E. Langmuir 2002, 18, 3076-3085. (12) Tamori, K.; Kihara, K.; Esumi, K.; Meguro, K. Colloid Polym. Sci. 1992, 270, 927-934. (13) Fontell, K.; Lindman, B. J. Phys. Chem. 1983, 87, 3289-3297. (14) Boden, N.; Jolley, K. W.; Smith, M. H. J. Phys. Chem. 1993, 97, 7678-7690.

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fluorocarbon emulsions (see Chapter 8 in ref 1). In view of the commercial relevance of fluorocarbon surfactants, the development of a theoretical formulation capable of quantitatively predicting their micellar solution properties, including their cmc’s, optimal micelle shapes, and average micelle aggregation numbers, would be very useful in understanding and optimizing their bulk micellar solution behavior and practical performance characteristics. Moreover, since fluorocarbon surfactants are nonbiodegradable and generally more toxic than their hydrocarbon counterparts (see Chapter 10 in ref 1), the fundamental understanding gained from such a theoretical development should also be useful for the design of novel substitute hydrocarbon-based surfactant structures, or hybrid hydrocarbon-fluorocarbon surfactants, that would exhibit similar micellization behavior as the fluorocarbon surfactants and can therefore replace them in applications where increased biodegradability and reduced toxicity are desired. On the theoretical front, fluorocarbon surfactants have not been studied as extensively as their hydrocarbon counterparts. The packing parameters, P,15,16 of selected fluorocarbon surfactants have been evaluated by Guilieri and Krafft,17 and their calculations reproduced qualitatively the experimental observations that the surfactants examined formed closed bilayer vesicles. The main theoretical advance in this area was made by Nagarajan,18,19 who developed a quantitative molecular-thermodynamic description of the micellization of fluorocarbon surfactants. The focus of these studies was mainly to predict the micellar solution properties of fluorocarbonhydrocarbon surfactant mixtures, and the only fluorocarbon surfactant considered was sodium perfluorooctanoate. The predicted cmc’s and micelle compositions of the fluorocarbon-hydrocarbon surfactant mixtures examined were found to be in good agreement with the experimental results.18,19 More recently, molecular dynamics simulations of micelles formed by model fluorocarbon surfactants and model hydrocarbon surfactants have been carried out,20 which illustrate the conformational characteristics of the fluorocarbon (or the hydrocarbon) surfactant tails in the micelle core. With the existing theoretical work in mind, in this paper, we undertake the development of a molecular-thermodynamic theory of micellization of single fluorocarbon surfactants, with the aim of predicting quantitatively their micellar solution properties, as well as the conformational characteristics of the fluorocarbon chains within the micelle core. We have essentially implemented a thermodynamic framework of micellization developed recently for ionic hydrocarbon surfactants,21,22 where binding of counterions, released by the surfactant heads and any electrolytes added to the solution, onto the charged micelle surface is accounted for explicitly, and have evaluated the various free-energy contributions to the free energy (15) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1991. (16) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525-1568. (17) Guilieri, F.; Krafft, M. P. Colloids Surf., A 1994, 84, 121-127. (18) Nagarajan, R. In Mixed Surfactant Systems; Holland, P. M., Rubingh, D. N., Eds.; ACS Symposium Series 501; American Chemical Society: Washington, DC, 1992; Chapter 4, pp 54-95. (19) Nagarajan, R. In Structure-Property Relationships in Surfactants; Esumi, K., Ueno, M., Eds.; Surfactant Science Series 70; Marcel Dekker: New York, 1997; Chapter 1, pp 1-81. (20) Mei, D.; O’Connell, J. P. Langmuir 2002, 18, 9067-9079. (21) Srinivasan, V.; Blankschtein, D. Langmuir 2003, 19, 99329945. (22) Srinivasan, V.; Blankschtein, D. Langmuir 2003, 19, 99469961.

Srinivasan and Blankschtein

of micellization, g˜ mic, for these fluorocarbon surfactants. All the fluorocarbon surfactants examined here, including the anionic perfluoroalkanoates (CncF2nc+1COO-) and perfluorosulfonates (CncF2nc+1SO3-), where nc denotes the number of carbon atoms in the surfactant tail, consist of linear perfluorocarbon tails. The main new aspects of this paper include the following: (i) the implementation of a model of counterion binding in the case of fluorocarbon ionic surfactant micelles, which accounts explicitly for the effect of the counterion type on the micellar solution properties; (ii) the implementation of a single-chain mean-field theory23-25 to model the packing of fluorocarbon chains within the micelle core, which yields detailed information about the conformational characteristics of the fluorocarbon surfactant chains packed within the micelle core, in addition to the free-energy penalty associated with confining the fluorocarbon chains within the micelle core; and (iii) the modeling of finite disklike micelles, which often form in the case of fluorocarbon surfactants, and their inclusion in the micelle size distribution. The resulting conformational characteristics and micellar solution properties of the fluorocarbon surfactants examined are then contrasted with those corresponding to their hydrocarbon analogues. The observed differences in the aggregation properties of these two classes of surfactants are rationalized by examining the various contributions to the free energy of micellization corresponding to the two surfactant classes. The remainder of this paper is organized as follows. The salient features of the thermodynamic framework and the molecular model utilized to describe the micellization phenomenon are presented in section 2. In section 3, we apply the molecular-thermodynamic theory developed to predict the average conformational characteristics of the fluorocarbon chains within the micelle core, as well as to predict various micellar solution properties, including cmc’s, optimal micelle shapes, and average micelle aggregation numbers. Finally, in section 4, we present our concluding remarks. 2. Molecular-Thermodynamic Theory of Micellization 2.1. Thermodynamic Framework. To model ionic surfactant systems, we developed recently a new thermodynamic framework, which allowed for a fraction of the counterions released by the charged surfactant heads and any electrolytes added to the solution to bind onto the charged micelle surface and, thereby, to influence the surfactant monomer-micelle equilibrium (for details, see refs 21 and 22). In our thermodynamic formulation for ionic surfactant-electrolyte systems, when there is a single counterion species, c, released by the surfactant heads and any added electrolytes, the micelle is characterized by its aggregation number, n, and by the degree of counterion binding, denoted by β (which represents the number of counterions, c, bound per surfactant ion, s, in the micelle). The population density of the ionic surfactant micelles (nβ-mers) composed of the surfactants, s, and the bound counterions, c, is given by21 (23) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985, 83, 3597-3611. (24) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985; Chapter 4, p 404. (25) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1986, 85, 5345-5358.

Properties of Fluorocarbon Surfactants

Xnβ )

()

[

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]

g˜ mic(β, Xc) 1 n Xs exp -n e kT

(1)

where Xi is the mole fraction of component i (i ) nβ-mer, s, or c), k is the Boltzmann constant, and T is the solution absolute temperature. In eq 1, the pre-exponential factor represents the translational entropy lost by the surfactant molecules upon aggregation, while g˜ mic, referred to as the free energy of micellization, represents the net free-energy gain (on a per surfactant molecule basis) associated with transferring the n surfactant monomers and the nβ counterions from the bulk aqueous solution to form the micelle (nβ-mer). The free energy of micellization, g˜ mic, is given by21

g˜ mic ) (gtr + gint + gpack + gster + gelec + gent) βkT ln(Xce) - kT (2) In eq 2, gtr, gint, and gpack represent the free-energy contributions associated with the formation of the hydrophobic micelle core (see section 2.2), and gster, gelec, and gent represent the free-energy contributions associated with the formation of the hydrated micelle interfacial shell containing the surfactant heads and the bound counterions (see section 2.3). The -βkT ln(Xce) term reflects the translational entropy lost by the nβ counterions upon binding onto the charged micelle surface. The last term in eq 2 (-kT) arises from our definition for the entropy of mixing of the various species in solution (s, c, and nβmers), which is based on the number densities of these species rather than on their mole fractions.26 Expressions for the various free-energy contributions shown in eq 2 have been developed previously for the case of nonionic and ionic hydrocarbon surfactants.21,26-29 To utilize this thermodynamic framework to model fluorocarbon surfactant systems, we need to develop expressions for the various free-energy contributions associated with the formation of the micelle core, consisting of the fluorocarbon tails, which differ from the previously considered hydrocarbon tails in both their molecular structure and chemical properties (see section 2.2). The free-energy contributions associated with the formation of the micelle interfacial shell can be evaluated using the same expressions derived in the case of hydrocarbon surfactants21,26 (see section 2.3). 2.2. Formation of the Micelle Core. In this section, we describe the evaluation of the various free-energy changes associated with the formation of the micelle core composed of the fluorocarbon surfactant tails. We should stress that for the ionic surfactant-electrolyte systems considered in this paper, the associated counterions are inorganic and, therefore, do not penetrate into the hydrophobic micelle core. Consequently, the micelle core consists solely of the fluorocarbon surfactant tails. Note also that for a fluorocarbon surfactant in a micelle, we assume that the -CF2 group attached to the polar head of the surfactant lies within the hydration sphere of the head and, therefore, does not possess any hydrophobic character. This assumption parallels that made in the case of hydrocarbon surfactants21,26 and was made based on an examination of the hydrophobic and the hydrophilic groups in the fluorocarbon surfactant molecule using a (26) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 37103724. (27) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 55675579. (28) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 55795592. (29) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 1618-1636.

chemistry software, Molecular Modeling Pro.30 Therefore, a fluorocarbon surfactant having nc carbon atoms in the tail is modeled as consisting of an effective tail of (nc - 1) carbon atoms (which include (nc - 2) -CF2 groups and a terminal -CF3 group) and of an effective head, which includes the polar moiety of the surfactant and the adjacent -CF2 group. The following three free-energy contributions are involved in the formation of the micelle core: 1. The transfer free energy, gtr, has been evaluated using the experimental aqueous solubility data for fluoroalkanes at 25 °C, measured by Kabalnov.31 Based on the experimental solubility data, gtr for fluoroalkanes at 25 °C is given by

gtr/kT ) -2.30(nc - 1) - 2.55

(3)

Equation 3 shows that gtr is more negative for fluorocarbon tails than for hydrocarbon tails (for a hydrocarbon chain at 25 °C, gtr/kT ) -1.50(nc - 1) - 2.01,26 as estimated from the experimental aqueous solubility data for alkanes32). This greater hydrophobicity of the fluorocarbon tails results in lower cmc’s in the case of a fluorocarbon surfactant having the same polar head and nc value as the hydrocarbon surfactant, consistent with the experimental observations.3-5 Due to the lack of experimental aqueous solubility data for fluoroalkanes as a function of temperature, we have not been able to obtain a temperature-dependent expression for gtr. Since the temperature range that we have considered in our predictions is 25-30 °C, using the expression for gtr at 25 °C given in eq 3 is appropriate. In addition, since gtr is independent of micelle shape, its value should only affect the predicted cmc’s and should have no effect on the predicted optimal micelle shapes and average micelle aggregation numbers. 2. The interfacial free energy, gint, is evaluated as follows:21,26

gint ) σs(a - a0,s)

(4)

where σs is the micelle core (composed of the fluorocarbon tails)-water interfacial tension, a is the interfacial area available per surfactant molecule, and a0,s is the interfacial area per surfactant molecule screened from contact with the aqueous solvent due to the physical connection of the surfactant tail to the surfactant head (see below). In eq 4, the curvature dependence of the micelle corewater interfacial tension, σs, is accounted for explicitly, similar to the case of hydrocarbon surfactants.21,26 For curved interfaces, σs < σ∞s , where σ∞s denotes the interfacial tension of a planar fluorocarbon-water interface, and the decrease of σs relative to σ∞s is described using the Gibbs-Tolman-Koenig-Buff equation33 (details are given in ref 26). Based on the interfacial tension data for fluoroalkanes reported by Kabalnov,31 which indicate a weak dependence of σ∞s on the fluorocarbon chain length, nc (σ∞s ) 54-56.1 dyn/cm), we have selected an average value of σ∞s ) 55 dyn/cm for all the fluorocarbon tails examined. The interfacial area, a (on a per surfactant molecule basis), depends on the micelle geometry and is given by (30) Molecular Modeling Pro, Version 3.2; ChemSW: Fairfield, CA. (31) Kabalnov, A. S.; Makarov, K. N.; Shcherbakova, O. V.; Nesmeyanov, A. N. J. Fluorine Chem. 1990, 50, 271-284. (32) Abraham, M. H. J. Chem. Soc., Faraday Trans. 1 1984, 80, 153181. (33) Tolman, R. C. J. Chem. Phys. 1949, 17, 333-337.

1650


a)

( ) Svt lc


(5)

where S is a shape factor (3 for spheres, 2 for infinitesized cylinders, and 1 for infinite-sized disks or bilayers), vt is the molecular volume of the surfactant tail, and lc is the minor radius of the micelle core (or the half-thickness of the core, in the case of bilayers). Based on experimental data for the liquid densities of perfluorocarbons,31,34 their molecular volume, vt, is given by vt ) 41.6(nc - 1) + 42.4 Å3 (that is, v(-CF2) ) 41.6 Å3 and v(-CF3) ) 84.0 Å3). In other words, fluorocarbon tails are bulkier than their hydrocarbon analogues, for which vt ) 26.9(nc - 1) + 27.4 Å3 (that is, v(-CH2) ) 26.9 Å3 and v(-CH3) ) 54.3 Å3).35 On the other hand, the maximum extended length of a fluorocarbon tail, lmax ) 1.30(nc - 1) + 2.04 Å, as computed from knowledge of the C-C bond length in a fluorocarbon chain and the van der Waals radius of the terminal -CF3 group, is comparable to that of hydrocarbon tails, lmax ) 1.27(nc - 1) + 1.54 Å.35 The area screened at the fluorocarbon core-water interface, a0,s, is given by the cross-sectional area of the fluorocarbon tail as a0,s ) 29.80 Å2 (which is larger than the cross-sectional area of a hydrocarbon tail, a0,s ) 21.26 Å2,26 since the fluorocarbon tail is bulkier than the hydrocarbon tail). Equation 5 shows that due to the larger volume of the fluorocarbon tail, as compared to a hydrocarbon tail with the same nc value, the interfacial area, a, occupied by a fluorocarbon surfactant is much larger, which implies higher values of gint in the case of the fluorocarbon surfactant (see eq 4). As a result, the drive to reduce the interfacial free energy, gint, is one of the main factors driving fluorocarbon surfactants to assemble into micellar structures possessing low curvatures (characterized by smaller values of a). 3. The packing free energy, gpack, represents the freeenergy change per surfactant molecule associated with the constrained arrangement of the surfactant tails inside the micelle core, reflecting the fact that the tails are pinned to the micelle core-water interface at one end and must pack in a manner that minimizes the intersegmental excluded-volume interactions within the micelle core. In our previous theoretical treatments of micellization of hydrocarbon surfactants, the single-chain mean-field theory developed by Ben-Shaul, Szleifer, and Gelbart24,25 was implemented to evaluate gpack. In this paper, we have generalized this single-chain mean-field theory to model chain packing in fluorocarbon surfactant micelles, as described below. In the single-chain mean-field theory of Ben-Shaul, Szleifer, and Gelbart,24,25 various conformations of a single central (arbitrary) surfactant tail in the micelle core are generated, including its internal conformations (associated with the torsional states of each of the bonds in the tail) and its external conformations (associated with the orientations of the surfactant tail and the fluctuations of the surfactant head with respect to the micelle core-water interface). The fluorocarbon surfactant tail is modeled using the united-atom approach, where each -CF2 or -CF3 segment in the tail is treated as a single unit, and the molecular volumes of these units are given by v(-CF2) ) 41.6 Å3 and v(-CF3) ) 84.0 Å3, as discussed earlier in this section. The other structural parameters of the fluorocarbon tail, including the length of the C-C bond, lCC, and the C-C-C bond angle, θ, are given by lCC ) 1.53 Å and θ ) 116°.36 (34) Nostro, P. L.; Chen, S. H. J. Phys. Chem. 1993, 97, 6535-6540. (35) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Wiley: New York, 1980.

To generate the various internal conformations of a linear fluorocarbon chain (namely, the sequence of torsional states in the chain), we make use of a three-state rotational isomeric state (RIS) model developed by Bates and Stockmayer.37,38 According to this RIS model, each bond in the chain can exist in three possible torsional states, a low-energy trans (t) state (torsional angle φ ) 0°) and two high-energy gauche (g() states (torsional angles φ ) (115°), with the gauche-trans energy difference, g, taken to be 1400 cal/mol (Bates and Stockmayer report a value of 1200 ( 200 cal/mol for g38). Note that this is analogous to the three-state RIS model for hydrocarbon chains, where g ) 500 cal/mol,36 which was utilized in our previous packing calculations for hydrocarbon surfactants.21,26 The higher value of g for fluoroalkanes is due to the larger van der Waals radius of the fluorine atom as compared to the hydrogen atom, as well as due to the electronegativity of the fluorine atoms. This, in turn, leads to larger steric and Coulombic repulsions between the fluorine atoms attached to second-neighbor carbon atoms in the fluoroalkyl chain, thus making gauche conformations of the fluoroalkyl chain less likely.36-38 In other words, the fluorocarbon chain is stiffer than the hydrocarbon chain. The effect of the greater stiffness of the fluorocarbon chain on its packing free energy, gpack, its implications on the conformational characteristics of the fluorocarbon surfactant, and its implications on the micellization properties of the fluorocarbon surfactant will be discussed in sections 3.1, 3.2, and 3.3, respectively. Utilizing this three-state RIS model, we then generate the 3(nc-3) possible internal conformations of the fluorocarbon surfactant tail (note that the surfactant tail contains (nc - 1) carbon atoms and the H-C1 bond is counted as well, where H refers to the effective surfactant head). Of these, we eliminate conformations for which g+g- or g-g+ bond sequences occur, since these bond sequences are highly energetic (g+g- > 3000 cal/mol) and, therefore, highly improbable. The fluorocarbon chain three-state RIS model described above is then incorporated into the single-chain meanfield theory, which has been discussed in detail in refs 23-25. In Appendix A, we briefly discuss the implementation of the mean-field theory to model chain packing in fluorocarbon surfactant micelles. The packing free energy, gpack, is then given by

gpack ) Ac - Ac,free

(6)

where Ac denotes the conformational free energy of the tails in the micelle core (on a per surfactant molecule basis), and Ac,free denotes the conformational free energy of the free tail in the bulk fluorocarbon phase. The quantities Ac and Ac,free are given by eqs A2 and A3, respectively, in Appendix A. In addition, in Appendix A, we discuss the evaluation of the chain conformational characteristics from the mean-field theory, specifically, the order parameters of the C-F (or the C-H) bonds in the surfactant tail, denoted by SCk-F (or SCk-H). The bond order parameters are quantitative indicators of the extent of ordering of the surfactant tail in the micelle core with respect to the micelle core-water interface.39,40 In section 3.2, we will (36) Flory, P. J. Statistical Mechanics of Chain Molecules; Interscience: New York, 1969. (37) Bates, T. W. Trans. Faraday Soc. 1967, 63, 1825-1834. (38) Bates, T. W.; Stockmayer, W. H. Trans. Faraday Soc. 1968, 1, 12-17. (39) Wennerstrom, H.; Lindman, B.; Soderman, O.; Drakenberg, T.; Rosenholm, J. B. J. Am. Chem. Soc. 1979, 101, 6860-6864.


Langmuir, Vol. 21, No. 4, 2005 1651

(

Table 1. Hydrated Radii of the Counterions, rh,c, Examined in This Papera

gdis,i ) -

counterion c rh,c (Å)

Cs+

K+

NH4+

Na+

Li+

1.69

1.72

2.13

2.18

2.43

a The values of r h,c were deduced from the experimental data for ion hydration numbers compiled in ref 41.

utilize the procedure discussed in Appendix A to predict the bond order parameters corresponding to fluorocarbon surfactant tails and contrast these predictions with those for hydrocarbon surfactant tails. 2.3. Formation of the Micelle Interfacial Shell. In our model of the interfacial shell of ionic surfactant micelles,21,22 the bound counterions are assumed to be located on the same plane as the charged surfactant heads, referred to as the micelle surface of charge. The micelle surface of charge is located at a distance r ) Rch ) lc + dch from the center of the micelle, where dch depends on the geometry of the surfactant head and on the hydrated radius of the counterion, rh,c. The hydrated radii of the counterions examined in this paper, rh,c, are listed in Table 1, where these values were computed utilizing the hydration numbers reported from ionic activity coefficient measurements.41 Adjacent to the micelle surface of charge is the Stern layer, located at r ) Rst ) Rch + dst, where dst is the Stern layer thickness. Note that the Stern layer marks the distance of closest approach of the diffuse counterion cloud to the micelle surface of charge. Expressions quantifying the free-energy contributions associated with the formation of the micelle interfacial shell have been derived in our recent theoretical formulation to model ionic surfactant-electrolyte systems21 and include the following: 1. The steric free energy, gst, which accounts for the steric repulsions between the surfactant heads and the bound counterions at the micelle surface of charge and is given by21

(

gst ) -kT(1 + β) ln 1 -

)

ah,s + βah,c a

(7)

where ah,s and ah,c are the effective cross-sectional areas of the surfactant head, s, and of the counterion, c, respectively. 2. The electrostatic free energy, gelec, associated with the electrostatic repulsions between the ionic surfactant heads at the micelle surface of charge, which is given (on a per surfactant molecule basis) by21

gelec ) gdis +

∫0q ψ0(q) dq f

(8)

In eq 8, gdis represents the electrostatic self-atmosphere energy released by the surfactants and the counterions when they are discharged in the bulk aqueous solution and assembled to form an initially neutral micelle surface. The Debye-Hu¨ckel expression is utilized to evaluate gdis,i, the electrostatic self-atmosphere energy released by each ionic species i (surfactant s or counterion c), and is given by21 (40) Ahlnas, T.; Soderman, O.; Walderhaug, H.; Lindman, B. In Surfactants in Solution, Vol. 1; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; p 107. (41) Marcus, Y. Ion Solvation; Wiley: Chichester, U.K., 1985; p 79.

zi2e02 1 bkT rh,i(1 + κrh,i)

)

(9)

where e0 is the electronic charge, b ) 4πηb0, where 0 is the dielectric permittivity of vacuum and ηb is the dielectric constant of the bulk aqueous solvent, and zi and rh,i are the valence and the hydrated radius of ionic species i, respectively. The Debye screening length of this ionic solution, κ-1, represents a quantitative measure of the length scale over which electrostatic effects decay. The second term in eq 8 represents the work done in recharging the discharged ions (the surfactant heads and the bound counterions) on the micelle surface of charge, which leads to the formation of a charged micelle surface and the accompanying formation of the diffuse counterion cloud which balances the charge of the bare micelle. In eq 8, the quantity q denotes the instantaneous micelle charge on a per surfactant molecule basis, which varies from zero to qf as the micelle evolves from being uncharged to being fully charged, and ψ0(q) denotes the instantaneous electrostatic potential at the micelle surface of charge. Note that for a monovalent surfactant ion-counterion pair, |qf| ) (1 - β)e0. The work of charging the micelle is obtained by solving the Laplace equation for the Stern region of the micelles and by utilizing analytical approximations to the Poisson-Boltzmann equation (specifically, the analytical approximations developed by Ohshima, Healy, and White42) to describe the diffuse region of the micelles (for details, see ref 21). 3. The entropic gain associated with mixing the surfactant heads and the bound counterions at the micelle surface of charge, gent, is evaluated using a random mixing model and is given by21

gent ) -kT ln

(1 +1 β) - βkT ln(1 +β β)

(10)

2.4. Evaluation of Various Micellar Solution Properties. In this section, we briefly review the evaluation of various micellar solution properties, including cmc’s, optimal micelle shapes, and average micelle aggregation numbers, when spherical or cylindrical micelles form in solution (note that the evaluation of these micellar solution properties has been discussed in detail in our previous treatments of micellization21,26). In addition, since disklike micelles often form in the case of fluorocarbon surfactants, in section 2.4.1, we present a model to estimate g˜ mic and the average micelle aggregation numbers of finite-sized disklike micelles. For each of the three regular micelle shapes, spheres (S ) 3), infinite cylinders (S ) 2), and infinite bilayers (S ) 1), the free energy of micellization, g˜ mic(S), is evaluated and minimized with respect to lc and β to determine the optimal core minor radius, l/c (S), and optimal degree of counterion binding, β*(S), for each micelle shape. The cmc of the surfactant is then estimated using the expression cmc ) exp(g˜ /mic/kT), where g˜ /micis the minimum value of g˜ mic, obtained by minimizing g˜ mic(S) with respect to the micelle shape, S.21,26 The corresponding optimal micelle shape, S*, indicates the shape of the micelles that form predominantly in the solution. If S* corresponds to a sphere (that is, if S* ) 3), then spherical micelles having aggregation number nsph ) 4π(l/c (S* ) 3))3/3vt form in solution. If S* corresponds to an infinite-sized cylinder (that is, if S* ) 2), then a wide (42) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 90, 17-26.

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number n, denoted by g˜ mic(n), can be represented as a linear interpolation between the g˜ mic values corresponding ˜ rim to the planar bilayer, g˜ bil mic, and to the toroidal rim, g mic. Specifically, disk g˜ mic(n) ) g˜ bil mic + (nrim/n)∆µ

Figure 1. Schematic representation of a disklike micelle of radius Rb, showing the toroidal semicylindrical rim (of core minor radius l/c (cyl)) and the bilayer body (of half-thickness l/c (bil)).

distribution of polydisperse finite cylindrical micelles will form. The free energy of micellization of a finite spherocylinder of aggregation number n, denoted by g˜ mic(n), is obtained as a linear interpolation between the g˜ mic values for the infinite cylinder, g˜ cyl mic, and for the spherical (known as the ladder model of micelle endcaps, g˜ sph mic growth43), as shown below:21,26 cyl g˜ mic(n) ) g˜ cyl mic + (nsph/n)∆µ

(11)

In eq 11, the micelle growth parameter, ∆µcyl ) g˜ sph mic cyl g˜ mic, reflects the tendency of a surfactant molecule to enter into the cylindrical body of the micelle rather than into the spherical endcaps, and the average micelle aggregation numbers, including the number-average micelle aggregation number, 〈n〉n, and the weight-average micelle aggregation number, 〈n〉w, depend exponentially on ∆µcyl, through the quantity K ) exp(nsph∆µcyl).43 2.4.1. Modeling of Disklike Micelles. If S* corresponds to an infinite-sized bilayer (that is, if S* ) 1), then finite polydisperse disklike micelles, whose size distribution is given by eq 1, will form in the solution. The modeling of finite disklike micelles and the evaluation of their size distribution has been carried out in previous studies focusing on understanding the conditions governing the optimal micelle shape.16,44,45 In our approach to model a finite disklike micelle, we proceed as follows. We assume that a finite disklike micelle is composed of a planar bilayer body, surrounded by a closed toroidal rim (where the closed toroidal rim is formed by bending a semicylinder uniformly). In Figure 1, we present a schematic representation of a disklike micelle, where we show the planar bilayer body and a half-section of the toroidal rim. Note that a finite disklike micelle is characterized by the following geometrical parameters: (i) the optimal half-thickness of the bilayer body, denoted by l/c (bil) ()l/c (S ) 1)), (ii) the optimal core minor radius of the toroidal rim, denoted by l/c (cyl) ()l/c (S ) 2)), and (iii) the radius of the planar bilayer body, Rb. Similar to the expression given in eq 11 for the free energy of micellization of a finite spherocylinder, the free energy of micellization of a finite disk of aggregation (43) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044-1057. (44) McMullen, W. E.; Ben-Shaul, A.; Gelbart, W. M. J. Colloid Interface Sci. 1984, 98, 523-536. (45) Eriksson, J. C.; Ljunggren, S. Langmuir 1990, 6, 895-904.

(12)

In eq 12, the micelle growth parameter, ∆µdisk ) ˜ bil (g˜ rim mic - g mic), reflects the tendency of a surfactant molecule to enter into the bilayer body of the disklike micelle rather than into the toroidal rim. The quantity nrim refers to the total number of surfactant molecules in the toroidal rim. Note that nrim can be related to the aggregation number of the disklike micelle, n, by making use of simple geometrical considerations (see Appendix B) and is given by

nrim )

4πl/c (cyl)3 + 3vt

π3l/c (cyl)4 4vtl/c (bil)

[x (

1+ n-

]

)

4πl/c (cyl)3 8vtl/c (bil) - 1 (13) 3vt π3l/c (cyl)4

Next, we need to develop a model for the free energy of micellization of the toroidal rim, g˜ rim mic. The interfacial area per surfactant molecule in the toroidal rim, arim, can be obtained from geometrical considerations (see Appendix B) and is given by

arim )

( )[ 2vt / lc (cyl)

1+

]

1 2[1 + 3πRb/4l/c (cyl)]

(14)

From eq 14, it follows that arim is larger than the value corresponding to a straight cylinder ()2vt/l/c (cyl)), due to the positive curvature of the toroidal rim, reflected in 1/Rb. When the curvature of the toroidal rim, 1/Rb, is low, we can treat the toroidal rim as a straight semicylinder, as expected intuitively (note that when Rb f ∞, the curvature of the toroidal rim 1/Rb f 0, and one recovers arim ) 2vt/ l/c (cyl) in eq 14). We then evaluate g˜ rim mic by utilizing the expressions for the various free-energy contributions corresponding to cylindrical micelles presented in sections 2.2 and 2.3, with the interfacial area per surfactant molecule, a, now given by arim (from eq 14) instead of by 2vt/l/c (cyl) corresponding to a straight cylinder. In other words, to evaluate g˜ rim mic, we utilize eqs 3, 4, 6, and 7-10, with the geometrical inputs S ) 2, lc ) l/c (cyl), a ) arim, ˜ cyl and β ) β*(cyl). The difference g˜ rim mic - g mic represents the bending free energy involved in creating the toroidal rim from a straight cylindrical micelle.46 In general, for any given micelle shape, the free energy of micellization, g˜ mic, is a function of the local micelle interfacial curvatures (that is, of the two local principal curvatures at any point on the micelle core-water interface, denoted by c1 and c2), as well as of the interfacial area per surfactant molecule, a. For nonregular micelle geometries, including the toroidal rim under consideration, the interfacial curvatures vary locally, and therefore, strictly, one needs to integrate the local free energy of micellization, g˜ mic(c1, c2, a), over the entire micelle corewater interface in order to calculate the total free energy of micellization, g˜ mic. However, since an analytical expression for g˜ mic(c1, c2, a) is not available in the toroidal rim (46) May, S.; Bohbot, Y.; Ben-Shaul, A. J. Phys. Chem. B 1997, 101, 8648-8657.


Langmuir, Vol. 21, No. 4, 2005 1653

case, implementing the integration procedure to evaluate g˜ rim mic is computationally very tedious. Consequently, to simplify our analysis, we have used instead the approximate procedure described above to evaluate g˜ rim mic, where we account primarily for the increased interfacial area in the toroidal rim, arim, as compared to that in a straight cylinder ()2vt/l/c (cyl)), and ignore the effect of the two locally varying interfacial curvatures on g˜ rim mic. Clearly, our is most appropriate when the approximation for g˜ rim mic curvature of the toroidal rim is low, that is, when Rb . l/c (cyl), arim f 2vt/l/c (cyl), and g˜ rim ˜ cyl mic f g mic. Efforts have been made to incorporate the effect of the varying local interfacial curvatures on the packing free-energy contribution in toroidal micelles by Ben-Shaul et al.,46 in the context of predicting the bending elasticity of cylindrical micelles. Finally, our approximation for g˜ rim mic is not suitable for small disklike micelles, say, disks having Rb < l/c (cyl), where the variation of the local interfacial curvatures cannot be ignored. The formation of such small disklike micelles would be extremely unfavorable due to the high bending energies involved in forming the toroidal rim in that case. Consequently, we assume that when Rb < l/c (cyl), the bending energy involved in creating the toroidal rim is infinitely high, and therefore, these disklike micelles do not form. In other words, the lower cutoff value for the aggregation number in disklike micelles, denoted by nd, corresponds to the case of a disklike micelle having Rb ) l/c (cyl). Note that, in this respect, in the theoretical approach of Eriksson and Ljunggren,45 nd was chosen by the authors to be 100. 2.4.2. Size Distribution of Disklike Micelles, {Xnβ}. Once g˜ bil mic is obtained utilizing the expressions presented in sections 2.2 and 2.3 and g˜ rim mic is estimated using the approximations discussed in section 2.4.1, g˜ mic(n) can be calculated using eq 12 and then substituted in eq 1 to evaluate the size distribution of disklike micelles present in the solution. The mass balance on the total surfactant concentration, Xsurf, is given by ∞

Xsurf ) Xs +

∑ n)n

sph

∞

nXnβ,cyl +

∑ n)n

nXnβ,disk

(15)

d

where the second and third terms in eq 15 account for the concentrations of surfactant molecules in the cylindrical and disklike micellar forms, respectively. Note that due to the large aggregation numbers typically encountered in disklike micelles, the translational entropy loss of the surfactant molecules opposing the formation of these aggregates, reflected in the Xns factor in eq 1, can be significant. Therefore, even when bilayers are the optimal micelle shape, the fraction of surfactant molecules aggregating into short cylindrical or spherical micelles (included in nXnβ,cyl) cannot be ignored, due to the translational entropic advantage associated with forming these smaller aggregates. Using eqs 11, 12, and 1, we can solve eq 15 for Xs and evaluate numerically the size distributions of cylindrical and disklike micelles. The various average aggregation numbers, including 〈n〉n and 〈n〉w, can then be evaluated from the moments of the micelle size distribution.26,43 In Figure 2, we illustrate our ability to model the coexistence of disklike and cylindrical micelles formed by a model surfactant system, where a bilayer is the preferred micelle shape. For this purpose, we consider the following disk two illustrative conditions: (i) g˜ bil mic ) -8.0 kT, ∆µ

Figure 2. Predicted size distribution of cylindrical micelles and disklike micelles formed by a model surfactant system, for disk the following two illustrative cases: (i) g˜ bil ) mic ) -8.0 kT, ∆µ 0.05 kT, ∆µcyl ) 0.5 kT, and Csurf ) 250 mM (dashed lines); (ii) disk ) 0.1 kT, ∆µcyl ) 0.5 kT, and C g˜ bil surf ) 250 mic ) -8.0 kT, ∆µ mM (solid lines). cyl ) 0.5 kT, and (defined here as g˜ cyl ˜ bil mic - g mic) ) 0.05 kT, ∆µ ) -8.0 kT, ∆µdisk Csurf ) 250 mM (dashed lines); (ii) g˜ bil mic cyl bil cyl ()g˜ mic - g˜ mic) ) 0.1 kT, ∆µ ) 0.5 kT, and Csurf ) 250 mM (solid lines). Note that in cases i and ii, the bilayer is the optimal micelle shape, as follows from the positive values of ∆µdisk in both cases. Figure 2 reveals that, in case i, most of the surfactant molecules aggregate into cylindrical micelles, and the concentration of surfactant molecules in disklike micelles is low. Our calculations indicate that the total concentration of surfactant molecules assembled ∞ nXnβ,disk, in case i is about 35 into disklike micelles, ∑n)n d mM. On the other hand, in case ii, Figure 2 reveals a significant increase in the concentration of surfactant molecules assembled into disklike micelles, and our ∞ nXnβ,disk is about 155 mM. calculations indicate that ∑n)n d In other words, an increase in ∆µdisk implies an increased tendency of the surfactant molecules to assemble into bilayers as opposed to cylinders, leading to an increase in the concentration of finite disklike micelles in the solution. Several qualitative aspects concerning the coexistence of cylindrical and disklike micelles in the solution have been discussed in the theoretical work of Gelbart et al.44 Note also that finite disklike micelles form only over a very narrow range of ∆µdisk values. Indeed, in our calculations, we typically find that if ∆µdisk > 0.15 kT, infinite bilayers form (corresponding to surfactant monomer-bilayer phase separation), as proposed by Israelachvili.16 This is because in disklike micelles, the number of surfactant molecules needed to form the toroidal rim is large (nrim is proportional to n1/2, see eq 13). Therefore, the free-energy disadvantage associated with forming the toroidal rim, nrim∆µdisk, can be very high, and this drives the formation of infinitely sized bilayer aggregates.

3. Results and Discussion This section is organized as follows. In section 3.1, we present our predictions of the packing free energies, gpack, of representative fluorocarbon and hydrocarbon surfactant chains packed within micellar aggregates, to investigate the effect of the chain conformational rigidity on gpack. In section 3.2, we compare the predicted bond order parameter profiles of fluorocarbon and hydrocarbon surfactant chains with available experimental results. In section 3.3, we present our predictions of the free energy of micelli-

1654


Figure 3. Predicted packing free energies, gpack, of a C10fluorocarbon chain (solid lines) and of a C10-hydrocarbon chain (dashed lines) packed in spherical micelles and in bilayers, as functions of the micelle core minor radius (or bilayer halfthickness), lc.

zation, g˜ mic, and of the micelle growth parameter, ∆µcyl, of representative fluorocarbon and hydrocarbon surfactants, to investigate the increased tendency of fluorocarbon surfactants to assemble into micellar structures of low curvatures, as compared to hydrocarbon surfactants. In section 3.4, we compare our predicted optimal micelle shapes and average micelle sizes of selected fluorocarbon and hydrocarbon surfactant-electrolyte systems with available experimental results. In section 3.5, we compare our predicted cmc’s of selected fluorocarbon and hydrocarbon surfactants with available experimental results. 3.1. Prediction of Packing Free Energies, gpack. In Figure 3, we compare the predicted packing free energies, gpack, of a C10-fluorocarbon chain (solid lines) and of a C10hydrocarbon chain (dashed lines) packed in spherical micelles and in bilayers, as functions of the micelle core minor radius (or bilayer half-thickness), lc. As seen in Figure 3, for both types of surfactant tails, gpack in a bilayer increases with the bilayer half-thickness, lc, while gpack in spherical micelles decreases with the spherical micelle core minor radius, lc. The penalty involved in packing a surfactant tail (chain) in the micelle core results from two main effects: (i) the confinement of the chain within the micelle core, since it cannot cross over to the aqueous side of the micelle core-water interface, and (ii) the stretching of the chain to fill the available volume in the inner regions of the micelle core, as required by imposing the condition of uniform and liquidlike density within the micelle core. The confinement effect is especially severe in highly curved geometries, such as those encountered in small spherical micelles, since the chain can easily slip out of the micelle core in that case. Therefore, for spherical micelles, as lc increases, the chain is in a less confined state, and gpack decreases. For bilayers, the stretching of the chain is the main source of the free-energy penalty. Accordingly, as lc increases in bilayers, the chain needs to stretch to a greater extent, leading to an increase in gpack. In addition, Figure 3 reveals that, for bilayers, gpack of the fluorocarbon chain (solid line) is lower than that of the hydrocarbon chain (dashed line) and that this difference increases with lc. On the other hand, in spheres, gpack is higher for the fluorocarbon chain (solid line) than for the hydrocarbon chain (dashed line) for low values of lc, and the trend is reversed for high values of lc. As discussed in section 2.2 (see item 3), the fluorocarbon chain is stiffer than the hydrocarbon chain, and gauche con-


Figure 4. Comparison between the predicted (lines) and the experimental (various symbols) results for (i) the SCk-F order parameter profile of the fluorocarbon octanoate chain in cesium perfluorooctanoate micelles (solid line, squares [refs 47 and 48]) and (ii) the SCk-H order parameter profile of the protonated octanoate chain in sodium octanoate micelles (dashed line, circles [refs 39 and 40]). Note that the two lines connect the predicted SCk-F (or SCk-H) values for each Ck group (k ) 3-8) in the surfactant tail and are shown to help guide the eye.

formations of the fluorocarbon chain are therefore less likely. Therefore, due to the natural tendency of the free fluorocarbon chain to adopt an extended conformation, the chain can stretch more easily to pack in bilayers, and the penalty incurred is lower, as reflected in the lower gpack values. On the other hand, in spherical micelles having low lc values, the chain needs to adopt a bent conformation in order to stay inside the hydrophobic micelle core, and the penalty incurred is higher than that for a hydrocarbon chain. As lc increases in spherical micelles, the entropic confinement is reduced, and the chain stretching effect begins to dominate, leading to the reverse trend observed in Figure 3. Based on the predictions shown in Figure 3, we can expect that the rigidity of the fluorocarbon chain, including its impact on gpack, will promote its assembly into micelles of low curvatures, such as cylinders or bilayers (in addition to the drive to reduce the interfacial free energy, gint). 3.2. Comparison between Predicted and Experimental Chain Conformational Characteristics. In Figure 4, we present (i) the predicted (solid line) and the experimental (squares47,48) SCk-F order parameter profile of the fluorocarbon octanoate chain in cesium perfluorooctanoate (CsPFO, C7F15COOCs) micelles and (ii) the predicted (dashed line) and the experimental (circles39,40) SCk-H order parameter profile of the hydrocarbon octanoate chain in sodium octanoate (C7H15COONa) micelles. Recall that, as discussed in section 2.2, in our model of the surfactant chain, the -CF2 (or -CH2) group connected to the polar head is assumed to be part of the surfactant hydrophilic head which lies in the aqueous side of the micelle core-water interface. In other words, the surfactants under consideration are modeled as H-(CF2)5CF3 (or H-(CH2)5CH3), where H denotes the -CF2COO(or the -CH2COO-) unit, whose conformations in the micelle are not modeled explicitly. Consequently, we can only predict the order parameters for the carbon groups (Ck) numbered k ) 3-8 as shown in Figure 4 (where k ) 1 refers to the carbon atom in the -COO- unit, k ) 2 refers to the carbon atom in the -CF2 (or the -CH2) group attached to the -COO- unit, and k ) 3-8 refers to the (47) Furo, I.; Sitnikov, R. Langmuir 1999, 15, 2669-2673. (48) Dvinskikh, S. V.; Furo, I. Langmuir 2000, 16, 2962-2967.


carbon atoms comprising the effective surfactant tail). The theoretical predictions of the bond order parameters have been made for the micelle of optimal structural characteristics, characterized by S* and l/c , and were obtained by averaging over all the chain conformations in the context of the mean-field theory, as discussed in Appendix A. According to our calculations, the sodium octanoate micelles are predicted to be spheres with l/c equal to 9.13 Å, and the CsPFO surfactant is predicted to form disklike micelles with l/c equal to 8 Å (as will be shown later in Figure 7 and discussed in section 3.4). As seen in Figure 4, both the predicted and the experimental SCk-F and SCk-H values decay from a certain value corresponding to the bonds close to the surfactant head to a nearly negligible value corresponding to the terminal -CF3 or -CH3 tail segments. This is due to the fact that the presence of the interface hinders random rotational motion of the tail, which is felt most significantly by the tail segments closer to the interface. In addition, the volume-filling constraint leads to a stretching of the bonds in the surfactant tail normal to the micelle corewater interface, so that the terminal segments can reach the interior of the micelle core, resulting in a high degree of ordering of the bonds which are close to the interface. As seen in Figure 4, the order parameter profile is significantly higher for the fluorocarbon chain (solid line) than for the hydrocarbon chain (dashed line), indicating a higher degree of order in the fluorocarbon chain. Moreover, the decay of the SCk-F order parameter profile from the k ) 3 group to the terminal -CF3 group (k ) 8) is much slower than the decay of the SCk-H order parameter profile, indicating that most of the fluorocarbon tail is in a highly ordered state. The differences in the conformational characteristics discussed above reflect the different micellar geometries adopted by the fluorocarbon surfactant (bilayer) and by the hydrocarbon surfactant (spherical micelle). As already stressed in our discussion of Figure 3 in section 3.1, the extent of chain stretching and alignment is higher for chains packed inside bilayers, as compared to spherical micelles. In other words, in spherical micelles, the terminal chain segments need to reside in the regions close to the micelle center only in a small fraction of the total conformations in order to fill the available volume, and therefore, the chain is relatively less ordered than a chain packed in a bilayer. The predicted order parameter profiles are in reasonable agreement with the experimental results, which indicates that the meanfield theory of Ben-Shaul, Szleifer, and Gelbart implemented here, along with a suitable RIS model of the fluorocarbon chains, captures reasonably well the conformational characteristics of the fluorocarbon chains inside micelles. In addition, we have used our predicted micelle shapes as inputs to predict the chain conformational characteristics, which indicates that the molecularthermodynamic theory developed here is successful in predicting quantitatively the micelle shapes of fluorocarbon surfactants (as will be illustrated further in our average micelle size predictions presented in section 3.4). 3.3. Prediction of the Free Energy of Micellization, g˜ mic, and of the Growth Parameter, ∆µcyl. In Table 2, we compare the shape-dependent free-energy contributions to g˜ mic (that is, excluding the gtr term and the kT term in eq 2) for two surfactant systems: (1) C11H21COONa + 0.1 M NaCl at 25 °C and (2) C11F21COONa + 0.1 M NaCl at 25 °C, in both spherical and cylindrical micelle geometries. Our aim here is to identify the main driving forces responsible for the differences in the optimal shapes of the micelles formed by fluorocarbon surfactants and

Langmuir, Vol. 21, No. 4, 2005 1655 Table 2. Theoretically Predicted Values (in kT Units) of (i) gint, (ii) gpack, (iii) the Remaining Shape-Dependent Free-Energy Contributions,a gster + gelec + gent - βNa ln(XNae), and (iv) g˜ mic, for Spherical and Cylindrical Micelles in (1) the C11H23COONa + 0.1 M NaCl Surfactant System and (2) the C11F23COONa + 0.1 M NaCl Surfactant System, Both at 25 °Cb C11H23COONa + 0.1 M NaCl (25 °C) gint cylinder sphere g(sph) - g(cyl) C11F23COONa + 0.1 M NaCl (25 °C) cylinder sphere g(sph) - g(cyl)

gpack

2.31 2.93 4.18 2.44 1.87 -0.49

gint

gpack

3.38 2.18 6.09 2.05 2.71 -0.13

gster + gelec + gent - βNa ln(XNae)

g˜ mic

3.96 2.17 -1.79

-7.87 -8.28 ∆µcyl ) -0.41

gster + gelec + gent - βNa ln(XNae)

g˜ mic

3.30 1.79 -1.51

-16.72 -15.65 ∆µcyl ) 1.07

a That is, those associated with the formation of the micelle interfacial shell and with the translational entropy loss of the bound counterions. b In addition, for each of the free-energy contributions i-iii, we have indicated the differences in the values for the spherical and for the cylindrical micelle shapes, g(sph) - g(cyl), as well as sph cyl the micelle growth parameter, ∆µcyl ) g˜ mic - g˜ mic .

hydrocarbon surfactants. For this purpose, we have included the various free-energy contributions to the micelle growth parameter, ∆µcyl ) g˜ sph ˜ cyl mic - g mic (in other words, we have indicated the differences gint (sph) gint(cyl), gpack(sph) - gpack(cyl), and so on). In Table 2, we present our predictions of the following shape-dependent free-energy contributions to g˜ mic: (i) the interfacial free energy, gint; (ii) the packing free energy, gpack; and (iii) the “net free-energy contribution associated with the micelle interfacial shell”, given by gster + gelec + gent, along with the translational entropy-loss contribution, βNa ln(XNae); as well as (iv) the free energy of micellization, g˜ mic. Note that we have lumped several free-energy contributions in term iii, because these free-energy contributions are associated with the formation of the micelle interfacial shell and the entropic penalty associated with binding the Na+ counterions and, therefore, depend only indirectly on the nature of the surfactant tail (hydrocarbon or fluorocarbon) through the interfacial area, a (see section 2.3). On the other hand, the two shape-dependent contributions associated with the formation of the micelle core, gint and gpack, depend explicitly on the nature of the surfactant tail, through the interfacial area, a, as well as through other molecular characteristics of the surfactant tail, including σs, a0,s, and the torsional states of the surfactant tail (see section 2.2). Note also that for the sake of this comparison, the core minor radius has been fixed at lc ) lmax in both the spherical and the cylindrical micelle geometries, for the fluorocarbon as well as for the hydrocarbon surfactants. These qualitative trends are well-understood based on the “opposing forces” model35,15 and other molecular-thermodynamic theories of micellization,49 and our main aim here is to quantify the various free-energy contributions to the micelle growth parameter, ∆µcyl. We first examine the free-energy contributions in the hydrocarbon surfactant-electrolyte system, C11H21COONa + 0.1 M NaCl at 25 °C. As expected,15,35 the interfacial free-energy contribution promotes the sphere-to-rod transition in micelle shape, as reflected in its positive contribution (1.87 kT) to ∆µcyl. On the other hand, the (49) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934-2969.

1656


repulsive steric and electrostatic free-energy contributions are larger in cylinders than in spheres,15,35 and this gives rise to the main opposing force to the sphere-to-rod micelle shape transition, as reflected in its negative contribution (-1.79 kT) to ∆µcyl. Finally, the packing free energy, gpack, is larger in cylinders (2.93 kT) than in spheres (2.44 kT), since the chain is in the highly stretched regime (lc ) lmax in our analysis) and, therefore, prefers to pack in micelles possessing high curvatures, as already stressed in our discussion of Figure 3. The packing contribution to ∆µcyl is negative (-0.49 kT) and, therefore, opposes micelle growth. The overall micelle growth parameter, ∆µcyl, is equal to -0.41 kT, which indicates the formation of spherical micelles in this hydrocarbon surfactantelectrolyte system. We next consider the free-energy contributions in the fluorocarbon surfactant-electrolyte system, C11F21COONa + 0.1 M NaCl at 25 °C. The qualitative trends observed in this system, namely, the contributions of the interfacial, packing, and electrostatic and steric effects to ∆µcyl, are the same as those observed in the hydrocarbon surfactantelectrolyte system. Comparing the fluorocarbon surfactant with the hydrocarbon surfactant, Table 2 reveals that the contribution of the interfacial effects to ∆µcyl is much higher for the bulkier fluorocarbon surfactant (2.71 kT) than for the hydrocarbon surfactant (1.87 kT). This is because the fluorocarbon surfactant has a larger tail volume and corresponding larger interfacial area, a (see eq 5), which increases its drive to assemble into cylinders instead of into spheres. In addition, our predictions indicate that the contribution of the packing effects to ∆µcyl is higher for the fluorocarbon surfactant (-0.13 kT) as compared to the hydrocarbon surfactant (-0.49 kT). In other words, the increased preference of the fluorocarbon chain to remain in an extended conformation enables its assembly into cylindrical micelles, as compared to its hydrocarbon counterpart. The contribution of the rigidity of the fluorocarbon chain in promoting the formation of micelles of low curvatures has been suggested qualitatively by several researchers in the past,10,17 and our quantitative predictions corroborate the fact that the rigidity of the fluorocarbon chain, in addition to its increased hydrophobic volume, does indeed promote the formation of micelles with low curvatures. When cylindrical micelles form, the average micelle size depends exponentially on the quantity nsph∆µcyl, where nsph is of the order of 50-100, and therefore, even small contributions to ∆µcyl (of the order of 0.01 kT) will have a significant effect on the predicted micelle sizes. Therefore, it is essential to estimate all the free-energy contributions to g˜ mic with a high degree of accuracy, to achieve quantitative accuracy in the predicted micelle sizes. Overall, our predictions in Table 2 indicate that the micelle growth parameter, ∆µcyl, is significantly higher for the fluorocarbon surfactant (1.07 kT), indicating the formation of cylindrical micelles, as compared to the formation of spherical micelles by the hydrocarbon surfactant (for which, ∆µcyl ) -0.41 kT). This observation will be verified quantitatively for related surfactant systems in Figure 5, as discussed in section 3.4. 3.4. Comparison between Predicted and Experimental Micelle Shapes and Average Micelle Aggregation Numbers. 3.4.1. Prediction of Average Micelle Sizes: Comparison of Fluorocarbon and Hydrocarbon Surfactants. In Figure 5, we present our predicted average micelle hydrodynamic radii, R h h, for two fluorocarbon and one hydrocarbon surfactant-electrolyte systems: (i) lithium perfluorononaoate, C8F17COOLi (LiPFN, dashed line), (ii) lithium fluorooctyl sulfonate, C8F17SO3Li (LiFOS, dotted line), and (iii) lithium dodecyl


Figure 5. Comparison between the predicted (lines) and the experimental (symbols) average micelle hydrodynamic radii, R h h (Å) [ref 50], of (i) lithium perfluorononaoate, C8F17COOLi (dashed line and circles), (ii) lithium fluorooctyl sulfonate, C8F17SO3Li (dotted line and squares), and (iii) lithium dodecyl sulfate, C12H25OSO3Li (solid line and triangles), as functions of the added LiCl concentration, CLiCl, at 25 °C and a surfactant concentration Csurf ) 100 mM.

sulfate, C12H25OSO3Li (LiDS, solid line), as functions of the added LiCl concentration, CLiCl, at 25 °C and a surfactant concentration Csurf ) 100 mM. When spherical micelles form, the average micelle hydrodynamic radius was evaluated as R h h ) l/c (sph) + lhead, where lhead is the length of the surfactant head ()4.5 Å for the carboxylate head in LiPFN, 5.1 Å for the sulfonate head in LiFOS, and 6.2 Å for the sulfate head in LiDS). The average micelle hydrodynamic radii, R h h, for cylindrical micelles were evaluated from the predicted weight-average micelle aggregation numbers, 〈n〉w, by utilizing expressions that relate the hydrodynamic radius of a rodlike micelle, Rh, to its aggregation number and to its predicted core minor h h values for all three radius, l/c (cyl).43 Our predicted R surfactant systems examined compare reasonably well with the experimental values deduced from measurements of the micelle diffusion coefficients made using micellesolubilized probes.50 In the absence of added LiCl, the aggregation numbers of fluorocarbon surfactant micelles (LiFOS and LiPFN), which are spherical, are rather small due to the bulkiness of the fluorocarbon chains (that is, nsph ) 4πl/c (sph)3/3vt is small due to the large vt values). The addition of LiCl to the micellar solution results in screening of the electrostatic repulsions between the charged surfactant heads, leading to a transition from spherical to cylindrical micelles, as reflected in the increasing R h h values in Figure 5. Our predictions indicate that the salt-induced micelle growth is more pronounced for the fluorocarbon surfactant micelles (LiPFN and LiFOS), which show a steady growth with increasing LiCl concentration (see dashed and dotted lines), as opposed to the LiDS micelles, which grow only upon addition of over 1 M LiCl (see solid line). As illustrated in Table 2, due to the larger hydrophobic volume (or equivalently, larger cross-sectional area) and the conformational rigidity of the fluorocarbon chains, fluorocarbon surfactants exhibit an increased tendency to form micelles of lower curvatures, such as cylinders or bilayers, as compared to hydrocarbon surfactants. Therefore, upon screening of the electrostatic repulsions by the added LiCl, the interfacial effects and the rigidity of the fluorocarbon chains promote (50) Asakawa, T.; Sunagawa, H.; Miyagishi, S. Langmuir 1998, 14, 7091-7094.


Langmuir, Vol. 21, No. 4, 2005 1657

Table 3. Comparison between the Predicted and the Experimental Micelle Shapes and Micelle Structural Characteristics for Three Surfactant Systemsa surfactant system

predicted

experimental

C7F15COONa (SPFO) at the cmc, T ) 25 °C

sphere nsph ) 18

sphere nsph ) 23

C7F15COONH4 (APFO) at CAPFO ) 120 mM, T ) 25 °C, and 0.1 M NH4Cl

cylinder l/c ) 9.0 Å L ) 45.3 Å

cylinder l/c ) 10.0 Å L ) 48.0 Å

C7F15COONH4 (APFO) at CAPFO ) 115-575 mM, T ) 50 °C, and 0.5 M NH4Cl

cylinder l/c ) 9.0 Å L ) 132.1 Å disk l/c ) 5.8 Å Rb ) 21.0 Å

cylinder l/c ) 12.2 Å L ) 300.0 Å disk l/c ) 12.0 Å Rb ) 24.0 Å

a (i) C F COONa (SPFO) at its cmc at T ) 25 °C [ref 51], (ii) 7 15 C7F15COONH4 (APFO) + 0.1 M NH4Cl at T ) 25 °C and CAPFO ) 120 mM [ref 8], and (iii) C7F15COONH4 + 0.5 M NH4Cl at 50 °C and CAPFO ) 115-575 mM [ref 52]. The micelle structural characteristics include (1) the aggregation number, nsph, of spherical micelles; (2) the core minor radius, l/c , and the length, L, of cylindrical micelles; and (3) the half-thickness, l/c , and the bilayer radius, Rb, of disklike micelles. The calculation of these quantities is explained in the text.

the formation of rodlike micelles in the two fluorocarbon surfactant systems examined. This trend, namely, the transition from spherical micelles to rodlike or threadlike micelles, disklike micelles, and closed bilayer vesicles upon addition of very small quantities of added electrolyte, has also been observed experimentally in other fluorocarbon surfactants.10 3.4.2. Prediction of Average Micelle Sizes in Fluorocarbon Surfactants: Effect of Counterion Size. In Table 3, we compare our predicted micelle structural characteristics for selected anionic fluorocarbon surfactants, C7F15COONa (sodium perfluorooctanoate, SPFO) and C7F15COONH4 (ammonium perfluorooctanoate, APFO), with the available experimental results. As indicated in Table 3, we predict the formation of spherical micelles by SPFO surfactants, and the predicted aggregation number, nsph, compares well with the experimental value.51 In the APFO + 0.1 M NH4Cl system, our predictions indicate that polydisperse cylindrical micelles are formed, and we can deduce an average value of the cylindrical micelle length, L, from the weight-average aggregation number, 〈n〉w. Our predicted structural characteristics of the cylindrical micelles formed, core minor radius l/c ) 9 Å and length L ) 45.3 Å, compare favorably with the small-angle neutron scattering (SANS) measurement of Burkitt et al.,8 namely, l/c ) 10 Å and L ) 48 Å (see Table 3). Finally, in Table 3, we also present our predicted micelle structural characteristics for the APFO + 0.5 M NH4Cl system at 50 °C and compare these with the experimental results of Kunze et al.52 In our theory, by minimizing the free energy of micellization, g˜ mic, we predict that the optimal micelle shape is a bilayer for this system. In other words, due to the higher ionic strength (0.5 M NH4Cl), the electrostatic repulsions between the surfactant heads are reduced sufficiently to promote the formation of planar disklike micelles in solution. Finite disklike micelles are modeled as described in section 2.4.1. We can then evaluate the complete population distribution of spherical, polydisperse cylindrical, and polydisperse disklike micelles (51) Berr, S. S.; Jones, R. R. M. J. Phys. Chem. 1989, 93, 2555-2558. (52) Kunze, B.; Kalus, J.; Boden, N.; Brandao, M. S. B. Physica B 1997, 234-236, 351-352.

Figure 6. Comparison between the predicted (solid line) and the experimental (circles [ref 52]) fraction of surfactant molecules that assemble into disklike micelles for the ammonium perfluorooctanoate (APFO) + 0.5 M NH4Cl system, as a function of the APFO concentration, CAPFO, at 50 °C. The dashed line represents the predictions obtained by increasing nd arbitrarily by a value of 50 (see section 2.4.1).

in the solution, as described in section 2.4.2, to obtain the geometrical characteristics of the rodlike micelles (core minor radius, l/c , and length, L) and of the disklike micelles (bilayer half-thickness, l/c , and bilayer radius, Rb). The geometrical characteristics, L and Rb, are deduced disk from the values of 〈n〉cyl w and 〈n〉w , the weight-average aggregation numbers of the cylindrical and disklike micelles, respectively, and vary with the APFO concentration, CAPFO. In Table 3, we present our predicted average values for L and Rb over the range CAPFO ) 115-575 mM, which corresponds to the range studied experimentally by Kunze et al. (APFO weight fraction of 0.05 ≈ 115 mM to a weight fraction of 0.25 ≈ 575 mM). Our predictions of the geometrical characteristics are in good agreement with the values reported by Kunze et al.52 Note that Kunze et al. analyzed their scattering data by assuming that the rodlike and disklike micelles that formed in the solution were monodisperse and that their relative proportion varied as the surfactant concentration increased from 115 to 575 mM. In Figure 6, we present our predictions (solid line) of the fraction of surfactant molecules that assemble into disklike micelles for the same system, APFO + 0.5 M NH4Cl at 50 °C, as a function of the APFO concentration, CAPFO, and compare these predictions with the experimental results of Kunze et al.52 The experimental results indicate that as CAPFO increases from about 115 to 575 mM, there is a gradual change from rodlike micelles to disklike micelles. Our predictions reproduce qualitatively the experimentally observed trend, and the observed behavior reflects the fact that, as CAPFO increases, the entropic penalty associated with forming larger micellar aggregates, such as disklike micelles (which have larger aggregation numbers than cylindrical micelles) decreases, leading to an increase in the population of disklike micelles in the solution. This is similar to the case of onedimensional micelle growth, where an increase in the surfactant concentration promotes the formation of longer cylindrical micelles. However, as seen in Figure 6, we tend to overpredict the concentration of disklike micelles at low values of CAPFO and underpredict that concentration at high values of CAPFO. At low CAPFO values, small disklike micelles form, and as discussed in sections 2.4.1 and 2.4.2, the bending free energy involved in forming the toroidal rim in these micelles could be higher than that predicted

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Figure 7. Comparison between the predicted (line) and the experimental (circles [ref 7]) weight-average micelle aggregation numbers, 〈n〉w, of cesium perfluorooctanoate (CsPFO) micelles, as a function of the CsPFO concentration, CCsPFO, at 30 °C.


Figure 8. Comparison between the predicted (open bars) and the experimental (shaded bars [refs 56 and 57]) cmc’s of the alkali metal perfluorooctanoates, C7F15COO-M+, where M+ ) Cs+, K+, Na+, and Li+, at 30 °C. Note that the hydrated counterion radii increase in the order Cs+ < K+ < Na+ < Li+ (see Table 1).

by our model, in which case we would tend to overpredict the population of finite disklike micelles formed. The dashed line in Figure 6 corresponds to the predictions obtained by increasing nd arbitrarily by a value of 50, and it can be seen that the discrepancy between our predictions and the experimental results at low CAPFO values could be improved by choosing a higher value for nd (see section 2.4.1). On the other hand, the discrepancy between our predictions and the experimental results at high CAPFO values is related to the fact that intermicellar excludedvolume interactions, which were neglected altogether in our analysis, become important at high surfactant concentrations. These interactions would tend to further promote the formation of disklike micelles over cylindrical micelles, since the average intermicellar distance can be maximized by forming disklike micelles. Theoretical models for the excluded-volume interactions between micelles have shown the importance of the excludedvolume effects in promoting the formation of larger micelles in the solution.53 In Figure 7, we present our theoretical predictions of the weight-average micelle aggregation numbers, 〈n〉w, of cesium perfluorooctanoate (CsPFO) micelles at 30 °C, as a function of the surfactant concentration, CCsPFO, in the range 65-500 mM. Our predictions are compared with the experimental results of Iijima, Kato, and co-workers,7 who deduced the micelle shape to be an oblate ellipsoid from their SANS and small-angle X-ray scattering measurements, indicating bidimensional micelle growth in this system.7 In addition, Boden et al. found that a discotic nematic phase is formed in this system at about 1 M CsPFO concentration and that there is no significant discontinuity in the micelle aggregation number from the isotropic to the nematic phase, suggesting the presence of disklike micelles even in the isotropic micellar solution.14,54,55 Our predictions indicate that the optimal micelle shape is a bilayer, which is consistent with these experimental findings regarding the shapes of the micelles formed. The Cs+ ion is smaller than the NH4+ ion (see Table 1), which leads to lower steric and electrostatic repulsions at the charged micelle surface. This promotes the formation of

disklike micelles in the CsPFO case, even in the absence of any added electrolyte, as opposed to the formation of disklike micelles in the APFO case only with added electrolyte (as discussed earlier in this section). Figure 7 shows that our predicted 〈n〉w values are in reasonable agreement with the experimental values of Iijima, Kato, and co-workers, which they deduced using the oblate ellipsoid model of micelle growth.7 As stressed in section 2.4.1, when disklike micelles form in the solution, the micelle size predictions are extremely sensitive to the micelle growth parameters, ∆µdisk and ∆µcyl (even more so than in the case of cylindrical micelles). Keeping this in mind, the agreement between our predictions and the experimental results in Figure 7 can be considered to be reasonably good. Our underprediction of the micelle sizes is again related to the fact that the intermicellar excludedvolume interactions, which were neglected altogether in our analysis, would promote the aggregation of the surfactant molecules into disklike micelles. 3.5. Comparison between Predicted and Experimental Critical Micelle Concentrations. In Figure 8, we present our theoretical cmc predictions for several alkali metal perfluorooctanoates, C7F15COO-M+, where M ) Cs, K, Na, and Li, at 30 °C, and compare these with the experimental cmc’s.56,57 Figure 8 shows that both the experimental and the predicted cmc’s increase as the radius of the hydrated counterion, rh,M, increases (see Table 1). The observed cmc behavior parallels that observed in the case of hydrocarbon surfactants.58 As discussed in section 3.4, an increased counterion radius, rh,M, leads to higher gst and gelec values, which contribute to an increase in the cmc. Figure 8 also shows that the predicted cmc’s agree reasonably well with the experimental cmc’s.56,57 In Figure 9, we present our theoretical predictions of the cmc’s of two surfactant series: (i) sodium alkyl sulfates, CncH2nc+1OSO3Na, at 30 °C (dashed line), and (ii) lithium perfluorocarboxylates, CncF2nc+1COOLi, at 30 °C (solid line), both as a function of nc. As seen in Figure 9, for both surfactant series, ln cmc varies linearly with the carbon number of the surfactant tail, nc. This linearity of ln cmc versus nc arises primarily from the transfer free-energy contribution, gtr.35 As discussed in section 2.2, fluorocarbon

(53) Zoeller, N.; Lue, L.; Blankschtein, D. Langmuir 1997, 13, 52585275. (54) Boden, N.; Jackson, P. H.; McMullen, K.; Holmes, M. C. Chem. Phys. Lett. 1979, 65, 476-479. (55) Holmes, M. C.; Reynolds, D. J.; Boden, N. J. Phys. Chem. 1987, 91, 5257-5262.

(56) Muzzalupo, R.; Ranieri, G. A.; La Mesa, C. Colloids Surf., A 1995, 104, 327-336. (57) Iijima, H.; Koyama, S.; Fujio, K.; Uzu, Y. Bull. Chem. Soc. Jpn. 1999, 72, 171-177. (58) Rosen, M. J. Surfactants and Interfacial Phenomena; Wiley: New York, 1989.


Figure 9. Comparison between the predicted (lines) and the experimental cmc’s (symbols) of two surfactant series: (i) sodium alkyl sulfates, CncH2nc+1OSO3Na at 30 °C (dashed line and triangles [ref 59]), and (ii) lithium perfluorocarboxylates, CncF2nc+1COOLi at 30 °C (solid line and circles [ref 59]), both as a function of nc.

surfactants are more hydrophobic than hydrocarbon surfactants having the same nc value, and the transfer free energy of a -CF2 group is -2.3 kT, while that of a -CH2 group is -1.5 kT. Due to the greater hydrophobicity of the fluorocarbon tail, the cmc’s of the fluorocarbon surfactants are lower than those of the corresponding hydrocarbon surfactants, as seen in Figure 9. The predicted cmc’s are in good agreement with the experimental cmc’s.59 4. Conclusions In this paper, we have developed a molecularthermodynamic theory to model the micellization behavior of fluorocarbon surfactants by combining a molecular model that evaluates the free energy of micellization of fluorocarbon surfactant micelles with a previously developed thermodynamic formulation of the size distribution of micelles. The main new aspects of this paper include the following: (i) the incorporation of a recently developed model of counterion binding21 for the description of the electrostatic effects in ionic fluorocarbon surfactant micelles, (ii) the implementation of a single-chain meanfield theory of chain packing in fluorocarbon surfactant micelles, and (iii) the modeling of finite disklike micelles and their inclusion in the micelle size distribution. Utilizing the single-chain mean-field theory of chain packing in micelles, the average conformational characteristics (specifically, the bond order parameter profile) of a fluorocarbon surfactant tail in the micelle core were predicted and compared with those of the hydrocarbon analogue. The predicted bond order parameter profiles of the fluorocarbon and hydrocarbon surfactant tails were found to agree well with the available experimental results. In addition, the molecular-thermodynamic theory developed was applied to several ionic fluorocarbon surfactant-electrolyte systems, and micellar solution properties, including cmc’s, optimal micelle shapes, and average micelle aggregation numbers, were predicted. The predicted micellar solution properties were found to agree reasonably well with the available experimental results for these properties. We find that compared to their hydrocarbon analogues, fluorocarbon surfactants exhibit (59) Moroi, Y.; Take’uchi, M.; Yoshida, N.; Yamauchi, A. J. Colloid Interface Sci. 1998, 197, 221-229.

Langmuir, Vol. 21, No. 4, 2005 1659

a greater tendency to undergo one-dimensional micelle growth (for example, in the lithium perfluorononanoateLiCl case) or two-dimensional micelle growth (for example, in the cesium perfluorooctanoate case). These observations were rationalized by examining the various predicted freeenergy contributions to the free energy of micellization of both surfactant systems. We found that the larger hydrophobic volume (and consequently, the larger crosssectional area) of the fluorocarbon surfactant chain and its greater conformational rigidity promote its assembly into micelles of low curvatures, such as cylinders or bilayers. In the future, it would be interesting to extend the molecular model of micellization presented here, which includes a rigorous description of the electrostatic and packing free-energy contributions, to other interesting systems, such as hybrid fluorocarbon-hydrocarbon surfactants,60,61 fluorocarbon surfactants with lipophilic counterions,62 and fluorocarbon-hydrocarbon surfactant mixtures.63,64 Acknowledgment. We are grateful to Professor Igal Szleifer for very useful discussions, which provided insight into the computation of the packing free energy in surfactant micelles. V.S. and D.B. also acknowledge Unilever Research USA and Dow Chemicals for partial funding of this work. Appendix A: Implementation of the Mean-Field Theory to Model Fluorocarbon Chain Packing in Micelles and Evaluation of the Chain Conformational Characteristics Briefly, in the mean-field theory of chain packing, the micelle core of radius (or half-thickness, in the case of bilayers) lc is divided into L layers of equal widths which are parallel to the micelle surface (see Figure 1 of ref 25). For each internal conformation of the central fluorocarbon chain, we randomly generate 50 orientations of the chain within the micelle core. The surfactant head is assumed to be located within a region of thickness 1.5 Å from the micelle core-water interface, and we sample five positions of the surfactant head in this region.23-25 Conformations for which any segment of the surfactant tail crosses over to the aqueous side of the micelle core-water interface are discarded, under the assumption of a sharp fluorocarbon core-water interface.23-25 The influence of the neighboring surfactant tails on this central tail (or chain) is accounted for in a mean-field manner. Specifically, the mean field is described by a series of L lateral pressures, {πi}, where πi represents the pressure exerted by the neighboring tails to stretch, or squeeze, the central tail in layer i of the micelle core, so as to minimize the excludedvolume repulsions between the tails. The distribution {πi}, where 1 e i e L, is evaluated by imposing the constraint that the density within the micelle core be uniform and equal to that of the bulk liquid fluoroalkane (implying no water penetration into the micelle core). Once the distribution {πi}, where 1 e i e L, has been obtained, the probability of the central tail adopting a (60) Abe, M. Curr. Opin. Colloid Interface Sci. 1999, 4, 354-356. (61) Guo, W.; Li, Z.; Fung, B. M.; O’Rear, E. A.; Harwell, J. H. J. Phys. Chem. 1992, 96, 6738-6742. (62) Regev, O.; Leaver, M. S.; Zhou, R.; Puntambekar, S. Langmuir 2001, 17, 5141-5149. (63) Almgren, M.; Wang, K.; Asakawa, T. Langmuir 1997, 13, 45354544. (64) Muto, Y.; Esumi, K.; Meguro, K.; Zana, R. J. Colloid Interface Sci. 1987, 120, 162-171.

1660



given conformation, Ω, denoted by P(Ω), can be obtained using the following expression:24,25

[

P(Ω) ) exp

1

L

∑

]/

(-(Ω) - πiφi(Ω)) y kT i)1

(A1)

where (Ω) is the internal (conformational) energy of the central tail in conformation Ω, and φi(Ω) is the volume occupied by the central tail in layer i of the micelle core, when it adopts conformation Ω. The conformational energy, (Ω), is evaluated as (Ω) ) ng(Ω)g, where ng(Ω) is the number of gauche bonds in the fluorocarbon tail in conformation Ω. The term y in eq A1 is the normalization factor of the probability distribution function, obtained by demanding that ∑ΩP(Ω) ) 1. The conformational free energy of the surfactant tails in the micelle core, on a per surfactant molecule basis, Ac, is given by23,25

Ac ) Ec - TSc )

(Ω)P(Ω) + kT∑P(Ω) ln P(Ω) ∑ Ω Ω

(A2)

The conformational free energy of the free tail in the bulk fluorocarbon phase, Ac,free, is obtained by substituting {πi} ) 0 in eqs A1 and A2 and is given by

exp[-(Ω)/kT] ∑ Ω

Ac,free ) Ec,free - TSc,free ) -kT ln

(A3)

Next, the probability distribution function, P(Ω), in eq A1 can be utilized to compute any average conformational property of the surfactant tails, 〈X〉, as follows:

〈X〉 )

X(Ω)P(Ω) ∑ Ω

c

c

Appendix B: Geometrical Considerations for Disklike Micelles The total number of surfactant molecules in the toroidal rim, nrim, can be related to the aggregation number of the disklike micelle, n, by making use of simple geometrical considerations. Specifically, the volume of the toroidal rim needed to wrap around the bilayer body, Vrim, is given by (see Figure 1)

4πl/c (cyl)3 1 Vrim ) nrimvt ) πl/c (cyl)2(2πRb) + 2 3

n ) nrim +

(A5)

where θk is the angle between bond k and the normal to the micelle core-water interface passing through the surfactant head. In eq A5, the averaging is carried out over all conformations of the surfactant tail in the micelle core, as shown in eq A4. Utilizing the mean-field theory, we can compute the skeletal bond order parameters, SCk-1-Ck+1, which are defined as in eq A5, where, in this case, θk represents the angle between the vector joining the Ck-1 and the Ck+1 carbon atoms in the surfactant chain and the normal to the micelle core-water interface passing through the surfactant head.25 The maximum value of the skeletal bond order parameter, SCk-1-Ck+1, equals unity, and for a chain in an all-trans conformation aligned always normal to the micelle core-water interface, all the SCk-1-Ck+1 values equal unity (since θk ) 0°, 〈cos2 θk〉 ) 1,

(B1)

The aggregation number, n, which includes nrim and the surfactant molecules residing in the bilayer body, is given by (see Figure 1)

(A4)

where X(Ω) is any conformational property of the tail (chain) in conformation Ω. In particular, below, we discuss the evaluation of the bond order parameters of the surfactant tail, Sk, which are quantitative indicators of the extent of ordering of the surfactant tail in the micelle core with respect to the micelle core-water interface. The order parameter of bond k along the surfactant tail is defined as follows:39,40

3 1 Sk ) 〈cos2 θk〉 2 2

and SCk-1-Ck+1 ) 1, see eq A5). When the surfactant chain adopts a completely random orientation (corresponding to a free chain), 〈cos2 θk〉 ) 1/3, and SCk-1-Ck+1 ) 0 (see eq A5). Typically, experimental techniques, such as 13C NMR spin relaxation, measure the SCk-F (or the SCk-H) order parameter profile, where SCk-F (or SCk-H) is defined as in eq A5, with θk now representing the angle between the Ck-F bond (or the Ck-H bond) and the normal to the micelle core-water interface passing through the surfactant head.39,40,47,48 Note that the SCk-F (or the SCk-H) order parameter is related to the SCk-1-Ck+1 order parameter. Specifically, for chains with tetrahedral geometry, which include fluorocarbon and hydrocarbon chains, SCk-F (or SCk-H) ) -SCk-1-Ck+1/2. For the terminal -CF3 (or -CH3) group, the SCk-F (or SCk-H) order parameter is related to the terminal skeletal bond order parameter as SCk-F (or SCk-H) ) -SCn -1-Cn /3.25

2πRb2l/c (bil) vt

(B2)

Equation B1 can be rearranged to yield an expression for Rb in terms of nrim, and by substituting the resulting expression for Rb in eq B2, it is possible to relate the aggregation numbers, nrim and n, as follows:

nrim )

4πl/c (cyl)3 + 3vt

π3l/c (cyl)4 4vtl/c (bil)

[x (

1+ n-

]

)

4πl/c (cyl)3 8vtl/c (bil) - 1 (B3) 3vt π3l/c (cyl)4

Next, the total surface area, Arim, of the toroidal rim is given by

1 Arim ) 4πl/c (cyl)2 + [2πl/c (cyl)(2πRb)] 2

(B4)

The interfacial area per surfactant molecule in the toroidal rim, arim () Arim/nrim), can be obtained by combining eqs B4 and B3. Specifically,

arim ) LA048304C

( )[ 2vt / lc (cyl)

1+

]

1 2[1 + 3πRb/4l/c (cyl)]

(B5)

Prediction of Conformational Characteristics and Micellar Solution

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