Molecular Thermodynamic Modeling of Specific Ion Effects on

pubs.acs.org/Langmuir © 2010 American Chemical Society

Molecular Thermodynamic Modeling of Specific Ion Effects on Micellization of Ionic Surfactants Livia Moreira and Abbas Firoozabadi* Department of Chemical and Environmental Engineering, Mason Laboratory, Yale University, New Haven, Connecticut 06520-8286 Received June 22, 2010. Revised Manuscript Received August 13, 2010 Specific ion effects are ubiquitous in biological and colloidal systems. The addition of electrolytes to ionic surfactant solutions has pronounced effects on micellar properties, such as critical micelle concentration (cmc), micellar size, and shape. Ions play an important role in colloid stability and aggregation behavior of ionic surfactant solutions. Despite extensive experimental data, there is no well established molecular theory on specific ion effects. Published molecular thermodynamic theories for ionic surfactants do not properly account for ion-specific effects such as the inversion of the lyotropic series for the cmc of alkyl sulfates and carboxylates. In this work, we present a molecular thermodynamic theory for ionic surfactant solutions to take into account the headgroup-counterion specificity and address ion-specific effects on the cmc and aggregation number. We assume that the charged headgroup and the counterion at the Stern layer form solvent-shared ion pair with different degrees of cosphere overlap. The thickness of the Stern layer is estimated from molecular structures of hydrated surfactant heads and hydrated counterions, and from the knowledge of the qualitative strength of headgroup-counterion interaction in line with Collins’ concept of matching water affinities. Our proposed thermodynamic model properly predicts the cmc of both anionic and cationic surfactants of various counterions, and the effect of different inorganic salts on micellization of ionic surfactants.

Introduction Surfactants find application in many industrial products and processes including pharmaceuticals, agrochemicals, detergents, food, paints, cosmetics, and the emerging field of biotechnology and nanomedicine.1 In the oil industry, surface active agents play a key role, for example in enhanced oil recovery.2 As a result, surfactants enter the environment in effluents from industrial and household wastes. As environmental considerations increasingly shape the developmentof many industrial products and processes, studies on the physical chemistry of surface active agents is essential in order to learn how to design surfactants with minimal environmental impact. Thus, investigations on structureperformance relationship of surface active agents engender a unique ability to actively improve efficiency of surfactantbased processes as well as to seek novel applications. Despite the considerable progress that has been made in the study of ions in aqueous micellar solutions and extensive experimental data, there is no established molecular thermodynamic theory, which can adequately describe ion specific effects in micellization of ionic surfactants. To the best of our knowledge there are two molecular thermodynamic models that assess the counterion specificity on micellization of ionic surfactants: the works of Nagarajan and Ruckenstein3 and Srinivasan and Blankschtein.4,5 Nagarajan and Ruckenstein3 use an approximate analytical solution to the Poisson-Boltzmann equation derived by Evans *To whom correspondence should be addressed. E-mail: abbas.firoozabadi@ yale.edu. (1) Minnes, R.; Ytzhak, S.; Weitman, H.; Ehrenberg, B. Chem. Phys. Lipids 2008, 155, 38–42. (2) Surfactants: Fundamentals and Applications in the Petroleum Industry, 1st ed.; Schramm, L. L., Ed.; Cambridge University Press: London, 2000. (3) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934–2969. (4) Srinivasan, V.; Blankschtein, D. Langmuir 2003, 19, 9932–9945. (5) Srinivasan, V.; Blankschtein, D. Langmuir 2003, 19, 9946–9961.

Langmuir 2010, 26(19), 15177–15191

and Ninham.6 In their approach, ionic surfactants are assumed to be fully dissociated and the released counterions are assumed to be distributed only in the electrical diffuse layer surrounding the micelle. This formulation has a parameter δ that is the distance from the hydrophobic core surface to the surface where the center of the counterion is located. Nagarajan7 points out that δ depends on the sizes of the ionic headgroup and hydrated counterion, and also on the proximity of the counterion to the charge of surfactant ion. However he does not investigate these dependencies further. The molecular thermodynamic theory developed by Srinivasan and Blankschtein4,5 is an effort aimed at modeling the counterion specificity on micellization of ionic surfactants. In their formulation, the counterions are allowed to bind onto the micelle surface. They analyze the influence of the bound counterions on various contributions to the free energy of micellization of an ionic surfactant. Even though their model takes into account important aspects of aggregation of ionic surfactants, they do not take into account the ion specific interactions between headgroup and counterion, which is an important aspect as shown in this work. Experimental and theoretical investigations have pointed out that counterion binding to micellar aggregates depend on the surfactant headgroup. Vlachy and co-workers8,9 have investigated the effect of various cations on salt-induced micelle-tovesicle transition. They observe that in the case of an excess of dodecyl sulfate, Liþ is the less efficient cation whereas Csþ is the most efficient one. By contrast, the cation series inverts for an (6) Evans, D. F.; Ninham, B. W. J. Phys. Chem. 1983, 87, 5025–5032. (7) Nagarajan, R. In Theory of Micelle Formation; Esumi, K., Ueno, M., Eds.; CRC Press: Boca Raton, FL, 2003; Vol. 112, Chapter 1. (8) Renoncourt, A.; Vlachy, N.; Bauduin, P.; Drechsler, M.; Touraud, D.; Verbavatz, J.-M.; Dubois, M.; Kunz, W.; Ninham, B. W. Langmuir 2007, 23, 2376– 2381. (9) Vlachy, N.; Drechsler, M.; Verbavatz, J.-M.; Touraud, D.; Kunz, W. J. Colloid Interface Sci. 2008, 319, 542–548.

Published on Web 09/01/2010

DOI: 10.1021/la102536y

15177

Article

Moreira and Firoozabadi

Table 1. Cmc (mM) of Various Cations of Dodecyl Sulfate and Dodecanoate dodecyl sulfate10 Liþ Naþ Kþ Rbþ Csþ

8.92 8.32 7.17 6.09

dodecanoate11 26.7 27.2 28.3 28.5

Figure 2. Schematic representation of the micelle interfacial region. The dashed line marks the radius of the hydrocarbon core. The adsorbed counterions and surfactant headgroups are located on the micelle surface of charge represented by the dotted line. The Stern surface is represented by the dash-dot line and is located at a distance dst from the micelle surface of charge.

Figure 1. Alkali metal and halogen ions arranged according to their effect on the structure of water.15 The ions are drawn to scale of their bare radii.16,17

excess of dodecanoate in the aggregates, and Csþ is the less efficient cation whereas Liþ is the most efficient one. An inversion in the cation series also is experienced in the simplest surfactant system composed of one type of surfactant and water as shown in Table 1. The series of alkali metals for the critical micelle concentration (cmc) of dodecyl sulfate is cmcLiþ > cmcNaþ >cmcKþ >cmcRbþ, while the series for the cmc of dodecanoate is cmcCsþ >cmcRbþ >cmcKþ >cmcNaþ. Vlachy and co-workers12 explain the opposite trends of the two surfactant headgroups by applying the concept of matching water affinities proposed by Collins.13,14 Ions have long been classified as being either kosmotropes (structure makers) or chaotropes (structure breakers) according to their relative abilities to alter the structure of surrounding water molecules.15 Just as anions and cations have been arranged according to their effect on the structure of water (Figure 1), Vlachy and co-workers12 propose that surfactant headgroups can also be arranged according to their effect on the structure of water. In line with the concept of matching water affinities, the headgroup sulfate can be classified as a chaotrope, whereas the headgroup carboxylate can be considered as kosmotrope.12 Following Collins’ concept, chaotropes can form direct ion pairs with other chaotropes, much as kosmotropes with other kosmotropes (although for different reasons), but chaotropes do not come into close contact with kosmotropes. This implies that Csþ will form the closest ion pairs with dodecyl sulfate and Liþ with dodecanoate. Gustavsson and Lindman’s NMR studies of alkali ion binding to micellar aggregates present direct evidence for specific interactions between alkali ions and surfactant headgroups.18 Their (10) Mukerjee, P. Adv. Colloid Interface Sci. 1967, 1, 241–275. (11) Kale, K. M.; Zana, R. J. Colloid Interface Sci. 1977, 61, 312–322. (12) Vlachy, N.; Jagoda-Cwiklik, B.; Vacha, R.; Touraud, D.; Jungwirth, P.; Kunz, W. Adv. Colloid Interface Sci. 2009, 146, 42–47. (13) Collins, K. D. Methods 2004, 34, 300–311. (14) Collins, K. D.; Neilson, G. W.; Enderby, J. E. Biophys. Chem. 2007, 128, 95–104. (15) Marcus, Y. Chem. Rev. 2009, 109, 1346–1370. (16) Israelachvili, J. N. Intermolecular and surface forces, 2nd ed.; Academic Press: San Diego, CA, 1991. (17) Nightingale, E. R., Jr. J. Phys. Chem. 1959, 63, 1381–1387. (18) Gustavsson, H.; Lindman, B. J. Am. Chem. Soc. 1975, 97, 3923–3930.

15178 DOI: 10.1021/la102536y

experiments indicate that Naþ binds to premicellar complexes in the case of octanoate, but not in the case of octyl sulfate. Further, micellar shape may vary drastically with the counterion. Reiss-Husson and Luzzati19 have studied the structure of micellar solutions of cetyltrimethylammonium chloride (C16TACl) and cetyltrimethylammonium bromide (C16TABr) by small-angle X-ray scattering methods over a large concentration range at 27 C. They observe that (C16TABr) forms spherical micelles at low concentrations and forms rodlike micelles at high concentrations. However, C16TACl forms spherical micelles at concentrations up to 0.38 g of surfactant per gram of solution. The explanation based on Collins’ concept also applies to cationic surfactants.20,21 The headgroups trimethylammonium15 and pyridinium22 can be considered as chaotropes. Applying the concept of matching water affinities, the headgroup trimethylammonium will form closer ion pairs with Br- (more chaotropic) than with Cl- (more kosmotropic). This has been experimentally confirmed by Hedin and co-workers.23 They perform NMR studies on hydration dynamics of Cl- and Br- in cetyltrimethylammonium micelles, and observe that the shell of the Br- is more distorted than the hydration shell of the Cl- by the micellar surface. We present a molecular thermodynamic theory to model the micellization of ionic surfactants in aqueous electrolyte solutions. In our model, as in Srinivasan and Blankschtein’s model,4,5 we account for counterions binding.The counterions, released by the surfactant heads and any added electrolytes, are allowed either to bind onto the micelle surface or to be distributed in the diffuse region (Figure 2). The fraction of the counterions bound is modeled as being intercalated among the surfactant heads on the micelle surface of charge (dotted line in Figure 2). The remaining counterions are distributed according to Boltzmann equation in the diffuse region, which lies beyond the Stern layer of steric exclusion (dot-dash line in Figure 2). The accumulation of the surfactant coion in the Stern layer is negligible due to the strong electrostatic repulsion by the charge of surfactant ions. For spherical micelles, Rc is the radius of the spherical hydrocarbon core. The center of charge of the ionic surfactant heads lies (19) Reiss-Husson, F.; Luzzati, V. J. Phys. Chem. 1964, 68, 3504–3511. (20) Vlachya, N.; Drechslerb, M.; Tourauda, D.; Kunz, W. C. R. Chim. 2009, 12, 30–37. (21) Specific Ion Effects, 1st ed.; Kunz, W., Ed.; World Scientific: Singapore, 2010. (22) Abezgauz, L.; Kuperkar, K.; Hassan, P. A.; Ramon, O.; Bahadur, P.; Danino, D. J. Colloid Interface Sci. 2010, 342, 83–92. (23) Hedin, N.; Furo, I.; Eriksson, P. O. J. Phys. Chem. B 2000, 104, 8544–8547.

Langmuir 2010, 26(19), 15177–15191


Figure 3. Schematic representation of the different types of ion pair. The dashed circles represent water molecules. (a) contact ion pairs, (b) solvent-shared ion pair, (c) solvent-separated ion pair. and (d) unpaired solvated ions.

on a surface of radius Rch referred to as the micelle surface of charge (where Rch =Rc þ dch), and dch can be obtained from the molecular structure of the surfactant. The Stern surface, defined by radius Rs, marks the distance of closest approach of the ions in the diffuse layer to the micelle surface of charge due to their finite size (where Rs =Rch þ dst=Rc þ dch þ dst). The thickness of the Stern layer, dst, can be estimated from the molecular structure of the hydrated surfactant headgroups and the hydrated ions, and the headgroup-counterion interaction. In practice, there is no unambiguous distinction between bound and free counterions. The counterion concentration as a function of the distance from the micelles shows a gradual decrease in going outward with no well-defined transition point.24 Furthermore, different methods make different distinctions between free and bound counterions, and it is not surprising that data on counterion binding from different types of studies may be very different. The adsorption of counterions at the micellar surface is modeled in this work as suggested by Srinivasan and Blankschtein.4 The bound counterions penetrate into the Stern layer all the way to adsorb at the same Helmholtz plane as the ionic surfactant heads, namely, at the micelle surface of charge. In addition, the model assumes that the counterions adsorbed onto the micelle surface of charge preserve their freedom, which implies that a counterion can be shared by several surfactant heads, and that no ion-pairing of a surfactant head and a counterion occurs. The surfactant ions and the bound counterions are considered to be arranged randomly on the micelle surface of charge, and possible ordering effects resulting from the interactions between the surfactant ions and the bound counterions are ignored. In the diffuse ion cloud, all the ions are treated as point charges having no physical excluded volume except for a minimum distance of closest approach to the charged micelle surface in the context of the Stern layer model. In the Stern layer, we assume that the charged headgroup and the counterion will form ion pairs. The definition of the ion pair is based on the mutual geometry of the ions and the solvent molecules. The literature on ion-pairing points to three types of ion pairs:25,26 (a) contact ion pairs (Figure 3a) which are formed immediately after ionization. There are no solvent molecules between the two ions that are in contact. The contact ion pair constitutes an electric dipole having only one common primary solvation shell. (b) The ion pair separated by the thickness of only (24) Wennerstr€om, H.; Lindman, B. Phys. Rep. 1979, 52, 1–86. (25) Conway, B. E. Ionic hydration in chemistry and biophysics, 1st ed.; Elsevier Scientific Publishing Company: Netherlands, 1981; Vol. 12. (26) Marcus, Y. Ion solvation, 1st ed.; John Wiley and Sons Ltd.: London, 1985.

Langmuir 2010, 26(19), 15177–15191

Article

Figure 4. Schematic representation of headgroup-counterion pairing at the interface. The dashed area represents the hydration shell around the ions. The dotted area represents the cosphere overlap of the hydration shells.

one layer of solvent molecule is called solvent-shared ion pair (Figure 3b). In solvent-shared ion pairs, the two ions have their own primary solvation shells. These, however, interpenetrate each other. Further dissociation leads to (c) solvent-separated ion pair (Figure 3c). Here, the primary solvation shells of the anion and cation are in contact, so that some overlap of secondary and further solvation shells takes place. The designations solventshared and solvent-separated ion pairs are sometimes interchangeable, since there is no clear experimental distinction between them. Eventually, further dissociation of the two ions leads to (d) free or unpaired solvated ions with independent primary and secondary solvation shells (Figure 3d). Strong electrolytes in water such as aqueous solutions of ionic surfactants are highly dissociated even at high concentrations. However, investigations on electrolytic dissociation have shown that even strong electrolytes present a significant degree of ion pairing among the solvated ions at sufficiently high concentrations. Among different types of ion pairing, the solvent-shared ion pairs (Figure 3b) are probably the most common, especially in high polar solutions.25 Furthermore, Conway27 has shown that two-dimensional ionic concentrations commonly accumulated in double layers at charged interfaces (about 0.3 C m-2) are comparable to relatively high equivalent concentrations in a three-dimensional solution (4.25 M). Under these conditions, interactions due to overlap of solvation shells may give important contribution to the free-energy of electrolytes in solution, as originally recognized by Gurney.27,28 In this work, we assume that the charged headgroups and counterions in the Stern layer form solvent-shared ion pairs of different degrees of cosphere overlap (Figure 4). The thickness of the Stern layer is estimated from the molecular structures of the hydrated surfactant headgroup and hydrated counterions, and from knowledge of the qualitative strength of headgroupcounterion interaction in line with the matching water affinities concept. If the headgroup and counterion have one kosmotropic and the other chaotropic characteristics, the ion pair interacts loosely, and we assume a small cosphere overlap of their hydration shells (Figure 4a). If the headgroup and counterion have both chaotropic or kosmotropic characteristics, the ion pair interacts strongly, and we assume a large cosphere overlap of their hydration shells (Figure 4c). We also include ion specificity between added coion and counterion. Since counterions are adsorbed at the micellar surface of charge, coions may also form ion pairs with the adsorbed counterions, and the degree of cosphere overlap of their hydration shell will depend on their classification with respect to their ability (27) Conway, B. E. J. Electroanal. Chem. 1981, 123, 81–94. (28) Gurney, R. W. Ionic processes in solution, 1st ed.; Dover: New York, 1953.

DOI: 10.1021/la102536y

15179

Article


to influence the water molecules in their vicinities. The thickness of the Stern layer is estimated from the molecular structures of the hydrated coion and counterion, and from knowledge of the qualitative strength of counterion-coion interaction in line with the matching water affinities concept. The proximity of the counterion to the headgroup is also assumed to affect the cross-sectional area of the headgroup, aA (=πrA2). The closer the counterion is to the headgroup, the larger is the cross-sectional area experienced by the ion pair (Figure 4c). As the counterions become farther apart from the headgroup, due to a change of the alignment of ions at the interface, there is a decrease of the cross-sectional area experienced by the ion pair (Figure 4a). It is worth noticing that with this model we do not intend to give a well founded general description of counterion binding phenomena. The reason is that there exists in the field of electrostatics no theories which can adequately deal with interactions in a medium which cannot be approximated as a continuum.24 We aim to develop a model that keeps the simplicity of the molecular thermodynamic modeling approach introduced by Nagarajan and Ruckenstein,3 allowing a priori quantitative prediction of aggregation behavior of surfactants, starting from their molecular structures and solution conditions. Besides the ionic specificity due to the cosphere overlap between headgroups and counterions, our model includes ionic specificity from different contributions. We take into account the decrease in solubility of hydrocarbons in electrolyte solutions. On the micelle surface of charge, we account for the finite size of ions adsorbed and for the entropic mixing due to the adsorption of different types of ions. In addition to that, the model incorporates the effect of salts on the interfacial tension, dielectric constant, and density of the electrolyte solution. These contributions are described in the following section.

Thermodynamics of Micellization Gibbs Free Energy Expression. Consider a system composed of NW water molecules, NsA surfactant A molecules, and Nadd ionic pairs of an inorganic salt at temperature, T, and pressure, p. The total Gibbs free energy of the solution, G, is modeled as the sum of two contributions: the free energy of formation, Gf, and the free energy of mixing, Gm: G ¼ Gf þ Gm

ð1Þ

Let us assume that the micelles are monodisperse based on the recognition that for spherical and globular micelles, the size distribution is usually narrow. Thus, the average properties of the solution are strongly influenced by the species present in the largest amount.7 Then, we may write the free energy of formation as: Gf ¼ NW μoW þ N1A μo1A þ

X

Nifree μoi þ Ng μog

ð2Þ

i

where μol is the standard state chemical potential of the species l. The subscript W refers to water, 1A to the singly dispersed surfactant, i to inorganic ions from the dissociation of ionic surfactants and added salts, and g to the aggregate containing g surfactant molecules. N1A stands for the number of singly dispersed is the number of ionic surfactant molecules in solution. Nfree i species i free in solution. Ng denotes the number of micelles composed of g surfactant monomers. The standard state chemical potential of water is defined here as pure water. The standard states of all species other than water are taken as those corresponding to infinitely dilute solution conditions. 15180 DOI: 10.1021/la102536y

The free energy of mixing under the maximum term approximation is: X free Ni ln Xifree þ Ng ln Xg Gm ¼ kT½NW ln XW þ N1A ln X1A þ i

ð3Þ where k is the Boltzmann constant, and Xl is the mole fraction of the species l: Xl ¼

Nl for l ¼ W, 1A, i, g P NW þ N1A þ Nifree þ Ng

ð4Þ

i

The mass balance equations are NsA ¼ N1A þ gNg Ni ¼ Nifree þ gβi Ng ,

for i ¼ 1, 2, 3, 4, :::

ð5aÞ ð5bÞ

We assume that the counterions may adsorb at the monolayer formed by surfactant headgroups. The number of counterions of species j adsorbed to the micelle, per surfactant molecule in the micelle, is referred to as the degree of binding of counterion j and is denoted by βj.4 Let zA be the valence of the ionic polar head of the surfactant and zi is the valence the counterion, where zi = z1 if the ionic surfactant is anionic and zi = z2 if the ionic surfactant is cationic. If the surfactant A is anionic, zA < 0, then N1 = |zA/z1|NsA and N2=0. And, if the surfactant A is cationic, zA>0, then N2=|zA/z2| NsA and N1 = 0. This work accounts for the presence of one inorganic salt in the mixture. However the same methodology can be readily extended to the presence of two or more inorganic salts in the mixture. Let Cadd be the molar concentration of inorganic salt added,where z3 and z4 are the valences of the cation and anion of the added salt, respectively. Assuming that the volume of the solution is primarily determined by the volume of water, one may use the following relation to calculate the number of ion pairs in the mixture, Nadd Nadd ¼

Cadd NW MW Fsol

ð6Þ

where MW is the molecular mass of water and Fsol is the density of the electrolyte solution in kg/m3. By adding a z3:z4 salt, the number of ions in the system can be calculated as N3 ¼ jz4 jNadd

ð7aÞ

N4 ¼ jz3 jNadd

ð7bÞ

Using the mass balance equations, eq 5a and eq 5b, into eq 2: X Gf ¼ NW μoW þ ðNsA - gNg Þμo1A þ ðNi - gβi Ng Þμoi þ Ng μog i

ð8Þ Substituting eq 8 and eq 3 into eq 1: G ¼ NW μoW þ NsA μo1A - gNg μo1A X X þ Ni μoi gβi Ng μoi þ Ng μog i

i

þ kT½NW ln XW þ N1A ln X1A þ

X

Nifree ln Xifree þ Ng ln Xg

i

ð9Þ Langmuir 2010, 26(19), 15177–15191


Article

Since some of the terms of the expression above only depend on the fixed variables, we can define G0 as 0

G ¼

G - NW μoW X

¼ - gNg μo1A -

i

þ N1A ln X1A þ

- NsA μo1A -

X

Table 2. Geometrical Properties of Surfactant Aggregates29 Spherical Micelles (Rc e lsA)

Ni μoi

gβi Ng μoi þ Ng μog þ kT½NW ln XW

X


4πRc 3 ¼ gvsA 3

Vg ¼

i

Ag ¼ 4πRc 2 ¼ ga

ð10Þ

Req ¼ Rc ,

i

Vg vsA 1 ¼ ¼ 3 Ag Rc aRc

P ¼

Reorganizing the expression above and dividing by kT: 0

G ¼ Ng kT

ðμog - gμo1A þ

kT X

P i

gβi μoi Þ

þ NW ln XW þ N1A ln X1A


Globular Micelles (Semiminor Axis Rc=lsA, Semimajor Axis b e 3lsA, Eccentricity E) Vg ¼

ð11Þ

4πRc 2 b ¼ gvsA 3

i

Δμog (= (μog - gμo1A - Σigβiμoi )/g) from the expression above is the difference in the standard chemical potentials between a surfactant molecule present in an aggregate that contains g surfactant molecules and βj adsorbed counterions, and singly dispersed surfactants and dissolved ions in water. Δμog is referred to as the free energy of micellization which not only depends on the type of the surfactant, but also on the aggregation number, aggregate shape, mole fraction of singly dispersed surfactant, degree of binding of counterions for ionic surfactants, and type and concentration of salts. By using the definition of the free energy of micellization into eq 11, we have: Δμog G0 ¼ gNg þ NW ln XW þ N1A ln X1A kT kT X free þ Ni ln Xifree þ Ng ln Xg

ð12Þ

i

If the surfactant concentration is lower than the cmc, then G0 is: X G0 ¼ NW ln XW þ N1A ln X1A þ Ni ln Xi kT i

ð13Þ

The addition of salts may result in micellar change from spherical to rodlike shape above a certain concentration. The free energy of micellization is defined differently for rodlike micelles. Rodlike micelles are not included in this study. Geometrical Relations. Even though the surfactant molecules may form aggregate of different shapes, in this work we consider spherical and globular aggregates. For an aggregate containing g molecules, the following geometrical parameters characterize the aggregate: Vg, the volume of the hydrophobic domain of the aggregate, Ag, the surface area of contact between the aggregate and water, and P, the packing factor. The geometrical relations are listed in Table 2. In these relations, a is the surface area of the hydrophobic core per surfactant molecule, and Rc is the radius of the hydrophobic core for spherical micelles or the semiminor axis for globular micelles. The molecular volume vsA of the surfactant tail is calculated from the methylene and methyl group contributions. These group molecular volumes are estimated from the density versus temperature data available for aliphatic hydrocarbons3 and are given by the expressions for T in Kelvin: vCH3 ¼ 0:0546 þ 1:24 10 - 4 ðT - 298Þ nm3 Langmuir 2010, 26(19), 15177–15191

ð14Þ

" Ag ¼ 2πRc 2 1 þ

Req ¼

#

sin - 1 E Eð1 - E 2 Þ1=2

3Vg 1=3 , 4π

P ¼

" ¼ ga,

E ¼

Vg vsA ¼ , Ag Rc aRc

1-

2 #1=2 Rc b

1=3 e P e 0:406

vCH2 ¼ 0:0269 þ 1:46 10 - 5 ðT - 298Þ nm3

ð15Þ

The extended length of the surfactant tail lsA at 298 K is calculated using a group contribution of 0.1265 nm for the methylene group and 0.2765 nm for the methyl group.3 Free Energy of Micellization. In the molecular thermodynamic modeling approach, the free energy change associated with the formation of the surfactant aggregate is expressed as the sum of several free-energy contributions, all of which can be computed given the chemical structure of the various micellar components and the solution conditions:3 Δμog ðΔμog Þtrans ðΔμog Þdef ðΔμog Þint ¼ þ þ kT kT kT kT þ

ðΔμog Þsteric ðΔμog Þent ðΔμog Þionic þ þ kT kT kT

ð16Þ

Explicit expressions are presented here for each of the free energy contributions in terms of the molecular characteristics of the surfactant. Detailed discussion of the origin of the following expressions are provided in refs 3 and 4. Transfer of the Surfactant Tail. The transfer free energy accounts for the energy to transfer the surfactant tail from its contact with the solvent to the hydrophobic core of the aggregate. The contribution to the free energy of this transfer process is estimated by considering the aggregate core to be like a liquid hydrocarbon. The fact that the aggregate core differs somewhat from a liquid hydrocarbon gives rise to an additional free energy contribution that is evaluated immediately below. The addition of salts to the aqueous solution decreases the solubility of hydrocarbons. The low solubility of hydrocarbons in aqueous salt solutions makes solubility measurements difficult, hence the absence of available experimental solubility data. (29) Nagarajan, R. In Theory of micelle formation; Esumi, K., Ueno, M., Eds.; Structure-Performance Relationships in Surfactants; Marcel Dekker: New York, 1997; Chapter 1, pp 1-81.

DOI: 10.1021/la102536y

15181

Article


Table 3. Experimental Salting-Out Constants and Estimated Liquid Salt Volumes salt

Vso33 (cm3/mol)

ks31 (L/mol)

Vs (cm3/mol)

LiCl 0.141 16.95 21.02 NaCl 0.195 16.62 22.25 KCl 0.166 26.85 31.64 CsCl 0.088 39.17 41.71 KBr 0.119 33.73 37.16 45.24 47.14 KI 0.066a -2.37 4.96 NaF 0.254b NaBr 0.155 23.48 27.95 NaI 0.095 35.04 37.78 a No data for benzene solubility in KI solutions are available. An estimate of the ks for KI is made by using an additive relation (ksKI ≈ ksKCI - ksNaCl þ ksNaI). b From ref 34.

In view of this, we will use the strategy used by Carale and co-workers.30 They decompose the transfer of the surfactant tails from the aqueous salt solution to bulk hydrocarbon liquid into two transfer steps: (1) the transfer of the tail from the aqueous salt solution to water, and (2) the subsequent transfer of the tail from water to the micelle’s interior. ðΔμog Þs=w ðΔμog Þw=hc ðΔμog Þtrans ¼ þ kT kT kT

kT

¼ - ks Cadd

CH2

ð17Þ

The free-energy change in the first transfer step is estimated using the McDevit-Long theory,31 and the second transfer step is estimated from solubility data of hydrocarbons in water. McDevit and Long have measured the effect of electrolytes on the solubility of benzene in aqueous solutions and have observed that the free-energy of transferring nonpolar molecules from aqueous salt solutions to water varies linearly with salt concentration over a wide range of salt concentrations. ðΔμog Þs=w

and is reported in Table 3. Values of Vs cannot be determined directly from experiments (except at temperatures above the melting point of the salt) and, therefore, need to be estimated. The estimation of Vs is made from McDevit-Long theory (eq 19), using the experimentally determined variation of ks from solubility experiments for benzene, as well as data such as Vio, Vso, and βo. Considerable experimental data on the solubility of benzene in various salt solutions is available31,34,35 (Table 3). These data are used to estimate Vs given in Table 3. In these calculations, Vio o of benzene (Vbenzene = 89.48 cm3/mol) is estimated from density 36 data. The transfer free energy of the surfactant tail from water to a liquid hydrocarbon state (second term of right-hand side of eq 17) can be estimated from independent experimental data regarding the solubility of hydrocarbons in water. The expressions for the methylene and methyl group contribution to the free energy of transfer of an aliphatic tail as a function of temperature (in Kelvin) from pure water is estimated by Nagarajan and Ruckenstein:3 ! ðΔμog Þw=hc 896 - 36:15 - 0:0056T ð20Þ ¼ 5:85 ln T þ T kT

ð18Þ

where ks is the salting-out constant which depends on a particular salt. In order to relate the observed variations in the effects of different salts to other observed properties of a particular salt solution, McDevit and Long assume that the only role of the nonpolar solute (the hydrocarbon tails, in this case) is to modify the ion-water interactions by occupying volume. Building on this simple concept, they show that the excess work done against the ion-solvent forces upon the introduction of the nonpolar solute volume, Vio, into an aqueous salt solution having low salt molarity, is proportional to the volume change upon mixing (liquid) salt and water. The salting-out constant based on the work of McDevit and Long is given by31

ðΔμog Þw=hc kT

! ¼ 3:38 ln T þ CH3

4064 - 44:13 - 0:02595T T ð21Þ

Packing and Deformation of the Surfactant Tail. The surfactant tails inside the hydrophobic core of the aggregate are not in a state identical to that in liquid hydrocarbons. This is because one end of the surfactant tail in the aggregate is constrained to remain at the aggregate-water interface, while the entire tail has to assume a conformation consistent with the maintenance of a uniform density equal to that of liquid hydrocarbons in the aggregate core. Consequently, the formation of aggregates is associated with a positive free energy contribution stemming from the conformational constraints on the surfactant tail.3 The packing and deformation free energy expression for spherical and globular micelles is given by: " # ðΔμog Þdef 9Pπ2 R2c ð22Þ ¼ kT 80 NA L2

where Vs is the volume occupied by the salt as a liquid, Vso is the partial molar volume of salt at infinite dilution, βo is the compressibility of water (βo = 4.524 10-5 bar-1 at 25 C and atmospheric pressure32), and R is the gas constant. The volume of the hydrocarbon tail, Voi , can be estimated by eqs 14 and 15. A compilation of Vos data has been made by Millero33

where L is the characteristic segment length for the tail (L = 0.46 nm), P is the packing factor defined in Table 2, and NA is the number of segments in the tail of surfactant A, (NA = lsA/L, where lsA is the extended length of the tail). Headgroup Steric Interactions. The steric free energy contribution accounts for steric interactions between surfactants heads and adsorbed counterions at the micelle-water interface. The molecules at the micelle surface are treated as components of an ideal localized monolayer:4 0 1 P aA þ βj ah, j o X ðΔμg Þsteric j B C ¼ - ð1 þ βj Þ ln@1 A ð23Þ kT a j

(30) Carale, T. R.; Pham, Q. T.; Blankschtein, D. Langmuir 1994, 10, 109–121. (31) McDevit, W. F.; Long, F. A. J. Am. Chem. Soc. 1952, 74, 1773–1777. (32) D. R Lide, E. CRC Handbook of Chemistry and Physics, version 84th; CRC Press: Boca Raton, FL, 2004. (33) Millero, F. J. In Water and aqueous solutions: Structure, thermodynamics, and transport processes; Horne, R. A., Ed.; Wiley Interscience: New York, 1972; Chapter 13, The partial molal volumes of electrolytes in aqueous solutions.

(34) Saylor, J. H.; Whitten, A. I.; Claiborne, I.; Gross, P. M. J. Am. Chem. Soc. 1952, 74, 1778–1781. (35) Wen-Hui, X.; Jing-Zhe, S.; Xi-Ming, X. Thermochim. Acta 1990, 169, 271– 286. (36) Perry’s chemical engineers’ handbook, 7th ed.; Perry, R. H., Green, D. W., Maloney, J. O., Eds.; McGraw-Hill Companies: New York, 1997.

ks ¼

Voi ðVs - Vso Þ βo RT

15182 DOI: 10.1021/la102536y

ð19Þ

Langmuir 2010, 26(19), 15177–15191


Article Table 5. Hard Sphere Radii of Hydrated Ions17

Table 4. Effective Radius of Headgroup, rA,j (nm)

Liþ Naþ Kþ Rbþ Csþ

-

F ClBrI-

alkyl sulfate

alkyl carboxylate

ion

rh,j (nm)

0.270 0.273 0.277 0.281 0.284

0.254 0.250 0.244 0.242 0.238

alkyl trimethylammonium

alkyl pyridinium

0.17 0.20 0.23 0.26

0.21 0.24 0.26 0.3

Liþ Naþ Kþ Rbþ Csþ FClBrI-

0.238 0.184 0.125 0.118 0.119 0.166 0.121 0.118 0.119

where aA and ah,j are the effective cross-sectional areas of the hydrated headgroup of surfactant A and of the hydrated counterions j, respectively, and a is the surface area of the hydrophobic core per surfactant molecule (Table 2). In this contribution, headgroup and counterions are modeled as hard spheres occupying areas aA and ah,j at the micelle core-water interface. Equation 23 reflects the fact that the presence of the bound counterions along with the surfactant heads at the micelle surface leads to an increase in the steric repulsions. Besides the well established dependence of the effective crosssectional area of the headgroup on the molecular structures of the surfactant headgroup, in this work we assume that the type of the counterion at the interfacial region also affects the cross-sectional area of the headgroup. The closer the counterion is to the headgroup, the larger is the cross-sectional area experienced by the ion pair (Figure 4c). When the counterion is farther from the headgroup, due to a change of the alignment of ions at the interface, there is a decrease of the cross-sectional area experienced by the ion pair (Figure 4a). The degree of proximity between the ion and the surfactant headgroup is affected by the strength of the headgroup-counterion interaction as discussed in the Introduction. For the case that there is more than one type of counterion in solution, the effective cross-sectional area of the headgroup is estimated as aA ¼

X Nj P aA, j Nk j

ð24Þ

k

where the sub indices j and k stand for the counterions present in solution. aA,j is the effective cross-sectional area of the headgroup for the pair composed of surfactant A and counterion j. We assume that the headgroup of the surfactant is spherical. So we can use the radius of the surfactant headgroup, rA,j, to calculate the cross-sectional area, aA,j. The estimated effective headgroup radii are listed in Table 4. The cross-sectional area of the counterions is obtained from knowledge of the hydrated ionic radii17 and the assumption that they are spherical. The counterion radii are listed in Table 5. Formation of Aggregate Core-Solvent Interface. The effect of salts on interfacial tension may be used to capture the corresponding effects at the micellar level, that is, on the micellar core-aqueous salt solution interfacial free energy as done by Carale and co-workers.30 The free energy associated with the formation of the hydrophobic core-aqueous solution interface is given by:3 ðΔμog Þint Ragg ¼ ða - aA Þ kT kT

ð25Þ

where Ragg is the macroscopic interfacial tension between bulk hydrocarbon and the aqueous salt solution. aA can be obtained Langmuir 2010, 26(19), 15177–15191

from eq 24 and the radii listed in Table 4 as discussed in the previous subsection. In our work, unlike the past work,3,5,7 we assume that the area per molecule of the core surface shielded from contact with water by the polar headgroup of the surfactant is equal to the effective cross-sectional area of the headgroup. The aggregate core-water interfacial tension Ragg is taken to be equal to the interfacial tension between the aliphatic hydrocarbon of the same molecular mass as the surfactant tail and the surrounding electrolyte solution. Because of the limited experimental data on the effect of salt on hydrocarbon/electrolyte solution interfacial tension,37-39 we use experimental data on the effect of salt on surface tension of aqueous salt solutions instead.40-42 We assume that the relation between the surface tensions of the aqueous salt solution and hydrocarbon with the aqueous salt solution/hydrocarbon interfacial tension holds the same as the relation between the surface tensions of water and hydrocarbon with the water/hydrocarbon interfacial tension. The interfacial tension σagg can be calculated in terms of the surface tensions σA of the aliphatic surfactant tail and σsalt of the electrolyte solution via the relation interpolated from experimental data of water/hydrocarbon interfacial tension43,44 σagg ¼ 0:7562ðσA þ σsalt Þ - 0:4906ðσA σ salt Þ0:5

ð26Þ

where the surface tension is in mN/m. We assume that the linearity in the variation of the hydrocarbon-aqueous salt solution interfacial tension with salt concentration observed in experimental studies37 extends to higher salt concentrations, similar to the observed linearity in the variation of the aqueous salt solution surface tension with salt concentration. With this in mind, the following relation between σsalt and Cadd is obtained: dσ Cadd ð27Þ σ salt ¼ σ W þ dCadd The values for dσ/dCadd are listed in Table 6. The surface tension of pure water is given by45 σW

¼ 235:8 1 -

" # T 1:256 T ð28Þ 1 - 0:625 1 647:15 647:15

(37) Aveyard, R.; Saleem, S. M. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1609– 1617. (38) Ikeda, N.; Aratono, M.; Motomura, K. J. Colloid Interface Sci. 1992, 149, 208–215. (39) Cai, B.-Y.; Yang, J.-T.; Guo, T.-M. J. Chem. Eng. Data 1996, 41, 493–496. (40) Weissenborn, P. K.; Pugh, R. J. J. Colloid Interface Sci. 1996, 184, 550–563. (41) Tuckermann, R. Atmos. Environ. 2007, 41, 6265–6275. (42) Ali, K.; Shaha, A. A.; Bilal, S.; Shahb, A. A. Colloids Surf. A: Physicochem. Eng. Aspects 2009, 337, 194–199. (43) Aveyard, R.; Haydon, D. A. Trans. Faraday Soc. 1965, 61, 2255–2261. (44) Zeppieri, S.; Rodrı´ guez, J.; de Ramos, A. L. L. J. Chem. Eng. Data 2001, 46, 1086–1088. (45) Vargaftik, N. B.; Volkov, B. N.; Voljak, L. D. J. Phys. Chem. Ref. Data 1983, 12, 817–820.

DOI: 10.1021/la102536y

15183

Article


Table 6. Variation of Surface Tension with Salt Molarity

Table 7. Distance between the Hydrocarbon Core and the Center of Charge of the Headgroup

salt

dσ/dCadd mN/(m M)

ref

LiCl NaCl KCl CsCl KBr KI NaF NaBr NaI

2.20 2.10 1.84, 1.943 1.60 1.425 1.104 2.34 1.89 1.44

40 40 40, 42 40 42 42 40 40 40

where σW is expressed in mN/m and the temperature in Kelvin. The surface tension of normal alkanes is fitted from experimental data:46 σA ¼ 29:7003½1 - expð- 0:1532nc Þ - 0:0896ðT - 298:15Þ ð29Þ where nc corresponds to the number of carbons of the normal alkane tail. Headgroup-Counterion Mixing Entropy. This contribution accounts for the entropic gain associated with mixing of the surfactant heads and the bound counterions at the micelle surface. The surfactant ionic heads and bound counterions are considered to be arranged randomly on the micelle surface.4 0 1 0 1 o X ðΔμg Þent 1 B B βP C j P C ¼ ln@ βj ln@ ð30Þ Aþ A 1 þ βj kT 1 þ β j j j

j

Headgroup Ionic Interactions. The free energy of the double layer is equal to the amount of work performed in building up the double layer around the colloidal particle by a reversible and isothermal process. The ionic free energy contribution is accounted for by the double layer free energy of an isolated charged particle:47,48 ðΔμog Þionic ach ¼ kT kT

Z

σ

φo ðσ 0 Þ dσ 0

ð31Þ

o

where σ is the surface charge density [charge/area] at the surface of charge, φo is the electrical potential at the micellar surface of charge, and ach is the surface area per surfactant molecule at the micelle surface of charge: ach ¼

4πRch 2 g

ð32Þ

The radius of the micelle surface of charge, Rch, is calculated using the equivalent radius, Req, since the equivalent radius is equal to the radius of the spherical hydrocarbon core for spherical micelles (see Table 2): Rch ¼ Req þ dch

ð33Þ

The distance between the hydrocarbon core and the center of charge of the ionic surfactant head, dch, is estimated from molecular structure of the surfactant and is given in Table 7. The integral in eq 31 is the free energy per unit area of the particle surface. Trapezoidal rule is used for the numerical (46) Janczuk, B.; Bialopiotrowicz, T.; Wojcik, W. Colloids Surf. 1989, 36, 391– 403. (47) Derjaguin, B. V. Trans. Faraday Soc. 1940, 36, 203–215. (48) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the stability of lyophobic colloids, First reprint ed.; Elsevier Publishing Company: New York, 1948.

15184 DOI: 10.1021/la102536y

surfactant headgroup

dch (nm)

alkyl sulfate alkyl carboxylate alkyl trimethylammonium alkyl pyridinium

0.4 0.2 0.1 0.1

Table 8. Thickness of the Stern Layer, dst (nm)

þ

Li Naþ Kþ Rbþ Csþ alkyl sulfate alkyl carboxylate

alkyl sulfate

alkyl carboxylate

0.4 0.35 0.3 0.25 0.2 0.5

0.2 0.25 0.3 0.35 0.4 0.4

alkyl trimethylammonium alkyl pyridinium -

F ClBrIalkyl trimethylammonium alkyl pyridinium

0.5 0.4 0.3 0.2 0.3

0.5 0.4 0.3 0.2 0.4

integration. The domain is discretized in 20 equally distributed nodes. The charge density is P eðzA þ zj βj Þ j ð34Þ σ ¼ ach where e is elementary charge, zA is the valence of the surfactant headgroup, and zj is the valence of the counterion j. The electrical potential at the surface of charge φo is determined by solving the Poisson-Boltzmann equation which in spherical coordinates is given by: d2 φ 2 dφ -e X ¥ ½zi eφðxÞ þ Ui ðxÞ ¼ ð35Þ þ z n exp i i dx2 x dx εo εsol i kT In the above equation φ is the self-consistent electrical potential. The electrical potential depends on the spatial distance from the micelle particle, x. n¥ i is the ion concentration infinitely far from the charged interface, and εo and εsol are vacuum permittivity and dielectric constant of the solvent, respectively. The total ¥ ionic concentration n¥ i is related to the molar concentration ci by ¥ ¥ 3 the relation ni = 10 Navci where Nav is the Avogadro number. Ui is the hard sphere repulsion which is infinite when the ion is located closer to the micellar surface of charge than the thickness of the Stern layer, dst,i. charge. ð36Þ Ui ðxÞ ¼ ¥, x < Rch þ dst, i 0, x G Rch þ dst, i As discussed in the Introduction, we assume that the charged headgroups and the counterions at the Stern layer form solventshared ion pair of different degrees of overlap (Figure 4). The thickness of the Stern layer is thus estimated from the molecular structure of the hydrated surfactant heads and hydrated counterions, and from the knowledge of the qualitative strength of headgroup-counterion interaction based on the concept of matching water affinities. For the surfactant headgroups, the closest distance that the headgroup can approach the micelle surface of charge is estimated from molecular structure of the Langmuir 2010, 26(19), 15177–15191


Article

Table 9. Dielectric Decrements, δsalt50 -

F

Cl

-

Br

-

Table 10. Constants of Equation 44 -

I

Liþ -11.77a -13.07 -13.57 -14.57 -9.17 -11.27 -11.87 -13.07 Naþ þ -8.67 -9.67 -10.37 -10.77 K -7.77 -8.87 -9.07 -10.07 Rbþ -7.37 -7.87 -8.27 -8.67 Csþ a Estimated using additive relation as proposed by Hasted.49

surfactant headgroup. The thickness of the Stern layer for the counterions and surfactant headgroups are given in Table 8. We also account for coion specificity when salt is added to the mixture. We assume that if one ion is chaotropic and the other is kosmotropic, the thickness of the Stern layer is the sum of the hydrated radius of the coion and the counterion; and if both ions of the salt are chaotropic or both kosmotropic, the thickness of the Stern layer is assumed to be the sum of the hydrated radius of the coion and counterion minus half of the radius of a water molecule (dH2O= 0.276 nm).15 The dielectric constant εsol is the static dielectric constant of the saline solution. In the limit of low salt concentration the dependence of εsol upon salt concentration can be approximated by a linear relationship:49 εsol ¼ εW þ δsalt Cadd

ð37Þ

where δsalt is a negative quantity known as the dielectric decrement given in Table 9. εW is available from experimental measurements. For pure water at temperatures ranging from 273.15 to 373.15 K:32 εW ¼ - 1:0677 þ 306:4670 expð - 4:52 10 - 3 TÞ

ð38Þ

The Poisson-Boltzmann equation is a second-order differential equation with two boundary conditions. The first boundary condition is that the potential vanishes at infinity, i.e., far away from the charged interface: lim φ ¼ 0

xf¥

ð39Þ

At infinity the derivative of the potential also vanishes: lim rφ ¼ 0

xf¥

ð40Þ

The second boundary condition gives information about the charged surface. The electrical field or the electrical potential at the charged interface can be fixed. Here, by knowing the charge density at the interface, we calculate the electrical field at the interface: rφjx ¼ Rch ¼ -

σ εo εsol

ð41Þ

The Poisson-Boltzmann equation is solved by finite differences. The polar discretization is done after Strikwerda.51 Details on the finite difference method are given in the Appendix. The domain is discretized in 100 equally distributed nodes. The (49) Hasted, J. B. Aqueous dielectrics, 1st ed.; Chapman and Hall Ltd.: London, 1973. (50) Giese, K.; Kaatze, U.; Pottel, R. J. Phys. Chem. 1970, 74, 3718–3725. (51) Strikwerda, J. C. Finite Difference Schemes and Partial Differential Equations, 2nd ed.; SIAM Society for Industrial and Applied Mathematics: Philadelphia, PA, 2004.

Langmuir 2010, 26(19), 15177–15191

a

salt

-2

A 10

-B 10

C 103

-D

E 102

-F 104

LiCl 0.2446 0.5505 0.8671 0.7927 1.169 NaCl 0.4485 0.9634 0.6136 2.712 1.009 KCl 0.4971 0.7150 0.6506 2.376 0 CsCl 1.327 1.511 1.251 3.113 4.181 KBr 0.9057 1.876 1.425 4.019 5.985 KI 1.256 2.125 1.515 3.022 5.980 NaF 0.4940 2.985 3.365 4.752 16.22 NaBr 0.8362 1.872 1.353 2.847 4.791 NaI 1.196 2.120 1.396 2.502 5.095 a See ref 54 for ranges of temperature and concentration.

1.761 0 0 3.319 4.092 4.090 18.72 3.413 3.346

domain starts at the distance from the surface of charge and goes to up 4 times the Debye length. For an electrolyte solution, P K ¼ 2

i

n¥i ðezi Þ2

εo εsol kT

ð42Þ

1/κ is the Debye length which is the characteristic length of the electrical double layer. Gibbs Free Energy Minimization. In order to find the minimum of the total Gibbs free energy of a system consisting of NW water molecules, NsA surfactant A molecules, and Nadd ionic pairs of an inorganic salt at temperature T and pressure p, we proceed with the following algorithm. First, eq 12 is minimized with respect to the independent variables, g, Ng, and βj, subject to the material balance constraint under the maximum term approximation (eqs 5a and 5b). The minimization is executed using the FSQP algorithm.52 The result of the minimization of eq 12 is compared with the result of eq 13 and the minimum of the total Gibbs free energy is determined as well as the aggregation number, g, the number of aggregates in solution, Ng and the degree of binding of the counterion j, βj. Calculating the Cmc. In order to compare the model predictions to experimental data, we calculate the cmc by constructing a plot of X1A versus the total concentration Xtot (=X1A þ gXg). The Gibbs free energy minimization is performed for different values of NsA for fixed values of NW.53 The plot is marked by a sharp change in slope as the concentration reaches the cmc, and an extrapolation procedure is used to determine the total concentration at the cmc (Xtot =Xcmc). Then, the cmc is calculated by the following expression: 1000 3 cmcðmMÞ ¼ 2 ! 1 X M cmc W 4 þ vsA Nav 10 - 24 5 Xcmc Fsol

ð43Þ

where vsA is in nm3 (eqs 14 and 15) and Fsol in kg/m3. The density of the electrolyte solution Fsol in kg/m3 for the temperature T in Kelvin is given by54 Fsol ¼ FW þ ACadd þ BCadd ðT - 273:15Þ þ CCadd ðT - 273:15Þ þ DCadd 3=2 þ ECadd 3=2 ðT - 273:15Þ þ FCadd 3=2 ðT - 273:15Þ2 ð44Þ (52) Zhou, J. L.; Tits, A. L.; Lawrence, C. T. User’s Guide for FFSQP Version 3.7: A FORTRAN Code for Solving Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality and Linear Constraints; Electrical Engineering Dept. and Institute for Systems Research: University of Maryland, College Park, MD, 1997. (53) Moreira, L. A.; Firoozabadi, A. Langmuir 2009, 25, 12101–12113. (54) Novotny, P.; Sohnel, O. J. Chem. Eng. Data 1988, 33, 49–55.

DOI: 10.1021/la102536y

15185

Article


where A through F are specific constants for each salt and the values are provided in Table 10, and FW is the water density in kg/m3 for the temperature T in Kelvin:36 FW ¼

5:459 ð1 þ ð1 - ðT=647:13ÞÞ

0:081

0:30542

Þ

ð45Þ

Results and Discussions First we apply the model for single surfactants in water. Four surfactants are considered: alkyl sulfates, alkyl carboxylates, alkyl trimethylammoniums, and alkyl pyridiniums. The calculated cmc and aggregation number are compared with experimental data. For alkyl sulfates, alkyl trimethylammoniums (CnTAþ), and alkyl pyridiniums (CnPþ), we also present results of micellar properties for different types of salts as a function of concentration. The calculated cmc is at 25 C. Alkyl Sulfates. Cmc. The cmcs of decyl and dodecyl sulfates of various counterions are presented in Figures 5a and 5b. The calculated cmc is plotted versus the experimental cmc from different authors and techniques. Ideally all points in these figures would fall on the diagonal line. However, due to the fact that the cmc is not a specific concentration, rather a narrow range of concentrations and its value depends on the experimental techniques, not all points on these plots fall on the diagonal line. For example, as can be seen in Figure 5b, the experimental cmc of sodium dodecyl sulfate (SDS) may vary from 7.4 to 8.4 mM at 25 C. In Figure 6, we plot the cmc of sodium alkyl sulfates of various tail lengths to show that our model predicts the generality established by Nagarajan and Ruckenstein3 with respect to tail length. To evaluate the effect of salt concentration on the cmc, we apply the model to calculate the variation of the cmc of SDS due to addition of NaCl, and present the comparison between our model and experimental data in Figure 7. The agreement at low electrolyte concentrations is good, but as the salt concentration increases, the agreement weakens. Possible explanations to the decrease in agreement at higher NaCl concentrations in Figure 7 are: change in micellar shape due to high salt concentrations, limitation of Poisson-Boltzmann equation to lower electrolyte concentrations; and neglected nonelectrostatic potentials. The salt specificity on the cmc is studied by calculating the variation of the cmc of SDS due to addition of different salts. (55) Tartar, H. V. J. Colloid Sci. 1959, 14, 115–122. (56) Mysels, K. J.; Otter, R. J. J. Colloid Sci. 1961, 16, 462–473. (57) Shedlowsky, L.; Jakob, C. W.; Epstein, M. B. J. Phys. Chem. 1963, 67, 2075–2078. (58) Elworthy, P. H.; Mysels, K. J. J. Colloid Interface Sci. 1966, 21, 331–347. (59) Benton, D. P.; Sparks, B. D. Trans. Faraday Soc. 1966, 62, 3244–3252. (60) Moroi, Y.; Nishikido, N.; Uehar, H.; Matuura, A. R. J. Colloid Interface Sci. 1975, 50, 254–264. (61) Paredes, S.; Tribout, M.; Ferreira, J.; Leonis, J. Colloid Polym. Sci. 1976, 254, 637–642. (62) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; Zana, R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976, 80, 905–922. (63) de la Ossa, E. M.; Flores, V. Tenside Surfact. Det. 1987, 24, 38–41. (64) Lu, J. R.; Marrocco, A.; Su, T. J.; Thomas, R. K.; Penfold, J. J. Colloid Interface Sci. 1993, 158, 303–316. (65) Kim, D.-H.; Oh, S.-G.; Cho, C.-G. Colloid Polym. Sci. 2001, 279, 39–45. (66) Dutkiewicz, E.; Jakubowska, A. Colloid Polym. Sci. 2002, 280, 1009–1014. (67) Varga, I.; Gilanyi, T.; Meszaros, R. Prog. Colloid Polym. Sci. 2004, 125, 151–154. (68) Das, C.; Das, B. J. Mol. Liq. 2008, 137, 152–158. (69) Huisman, H. F. Koninkl. Ned. Akad. Wet. Proc. Ser. B 1964, 67, 388–406. (70) Saito, S. Colloid Polym. Sci. 1967, 215, 16–21. (71) Williams, R. J.; Phillips, J. N.; Mysels, K. J. Trans. Faraday Soc. 1955, 51, 728–737. (72) Mysels, K. J.; Princen, L. H. J. Phys. Chem. 1959, 63, 1696–1700.

15186 DOI: 10.1021/la102536y

Figure 5. Calculated vs experimental10,55-68 cmc of (a) decyl sulfates and (b) dodecyl sulfates at 25 C.

Figure 6. Variation of cmc of sodium alkyl sulfates with surfactant tail length. Markers are experimental data10,55-70 at 21-30 C and solid line is model prediction at 25 C.

Figure 7. Cmc of SDS as a function of NaCl concentration at 25 C. Markers are experimental data63,66,71,72 and solid line is model prediction. Langmuir 2010, 26(19), 15177–15191


Article

Table 11. Influence of Counterion of Chloride Salts on Cmc (mM) of SDS salt 0.01 M

exp. data73 (29.85 C)

our model (25 C)

LiCl NaCl KCl CsCl

7.71 6.32 5.86 4.32 3.77

7.6 5.6 5.1 4.3 3.4

Figure 9. Calculated cmc of SDS as a function of the concentration of (a) alkali metal chlorides and (b) sodium halides in the aqueous solution.

Figure 8. Cmc of SDS as a function of the concentration of alkali metal chlorides in the aqueous solution. Markers are experimental data and lines are model predictions. Solid line and diamonds63 for LiCl, dashed line and circles63 for NaCl, and dash-dot line, upwardpointing triangles,63 and downward-pointing triangles66 for KCl. Key: (a) with headgroup-counterion specificity; (b) without headgroup-counterion specificity.

First, we investigate the counterion effect on cmc, and then, we investigate the coion effect. In Table 11, we compare experimental data73 with calculated cmc for SDS in the presence of various chloride salts. We can predict the lyotropic series with reasonable quantitative agreement. The counterion effect on the cmc of SDS also can be verified in Figure 8a, where we plot predictions from the model versus experimental data63,66 for cmc of alkali metal chlorides in SDS solutions. In order to show that the size effects are not responsible for the ionic specificity in the cmc, we calculate the cmc of alkali metal chlorides in SDS solutions without taking into account the headgroup-counterion specificity, and we show the results in Figure 8b for comparison. As we can see in parts a and b of Figure 8, if the headgroupcounterion interactions are not taken into account, the experimentally verified change in the cmc is not predicted by the model. In Figure 9a, we present predictions for the effect of different chloride salts on the cmc of SDS at various salt concentrations, the counterion effect. In Figure 9b, we investigate the coion effect. (73) Maiti, K.; Mitra, D.; Guha, S.; Moulik, S. P. J. Mol. Liq. 2009, 146, 44–51.

Langmuir 2010, 26(19), 15177–15191

Figure 10. Cmc of SDS as a function of the concentration of sodium halides in the aqueous solution. Markers are experimental data63,69 and lines are model predictions. Solid line for NaF, dashed line for NaCl, dash-dot line for NaBr, and dotted line for NaI.

We present predictions for the cmc of SDS as a function of the concentration of sodium halides in the aqueous solution. We observe that the coion effect is small at low electrolyte concentrations, but it grows as the electrolyte concentration increases. For the coion effect, we compare calculated cmc with experimental data63,69 in Figure 10. The results indicate very little effect of different sodium halides on cmc. In both experimental investigations,63,69,74 the data analysis concluded that the deviation in the experimental data reported is of the same order of magnitude (74) Huisman, H. F. Koninkl. Ned. Akad. Wet. Proc. Ser. B 1964, 67, 407–424.

DOI: 10.1021/la102536y

15187

Article


Figure 11. Degree of counterion binding onto SDS micelles as a function of KCl concentration at 25 C.

Figure 13. Calculated vs experimental11,87,90-92,94,96-98,100 cmc of decanoate at 25 C. Key: (a) with headgroup-counterion specificity; (b) without headgroup-counterion specificity. Figure 12. Influence of added NaCl concentration on the average aggregation number of SDS. Markers are experimental data66,72,76-78 and solid line is model prediction.

as the error in the measurements and that the difference in the experimental cmc is in the error margin. For completeness, in Figure 11, we plot the degree of counterion binding onto SDS as a function of KCl concentration. We observe that the Naþ is promptly replaced by Kþ as KCl concentration increases. The calculated degree of binding is less than experimental data,75 because the degree of binding in our model only accounts for the lateral adsorption at the micelle surface of charge; we neglect other types of adsorption such as the ion pairing. Aggregation Number. The error in experimental data for the aggregation number may be much higher than the error in cmc measurements. However a comparison between data and model prediction for aggregation number will be useful for the examination of the trends. For sodium alkyl sulfates, we calculate the average aggregation number at the cmc as a function of the tail length. Our model has the same limitations as the original model by Nagaranjan and Ruckenstein3,7 in predicting large aggregation numbers for tail length with more than 12 carbons. Experimental data shows that the aggregation number of the micelles increases due to the addition of monovalent inorganic salt.62,72,76-78 Since different techniques measure different aggregation numbers for SDS,55,62,72,77-80 we compare in Figure 12 the increase of the aggregation number of the micelles as a function of (75) (76) (77) (78) (79) (80)

Stigter, D. J. Colloid Interface Sci. 1967, 23, 379–388. Suzuki, H. Bull. Chem. Soc. Jpn. 1976, 49, 1470–1474. Kushner, L. M.; Hubbard, W. D. J. Colloid Sci. 1955, 10, 428–435. Phillips, J. N. Trans. Faraday Soc. 1955, 51, 561–569. Hutchinson, Z. Phys. Chem. 1954, 2, 363–374. Tartar, H. V.; Lelong, A. L. M. J. Phys. Chem. 1955, 59, 1185–1190.

15188 DOI: 10.1021/la102536y

NaCl concentration. The agreement at low electrolyte concentrations is reasonably good, but as the salt concentration increases, the agreement weakens. This is possibly due to the same reasons discussed for the case of cmc predictions. Alkyl Carboxylates. For alkyl carboxylates of various counterions we present comparisons between experimental and calculated cmc in Figures 13a and 14a. And once more, not all calculated cmc values coincide with the experimental data values since cmc is a narrow range of concentrations. We have also calculated the cmc of sodium and potassium alkyl carboxylates of various tail lengths,11,81-100 and verified that the model gives results in agreement with data (plots omitted for the sake of brevity). We calculate the cmc of alkyl carboxylates without taking into account the headgroup-counterion specificity, and we show the (81) Davies, D. G.; Bury, C. R. J. Chem. Soc. 1930, 133, 2263–2267. (82) Hess, K.; Philippoff, W.; Kiessig, H. Kolloid Z. 1939, 88, 40–51. (83) Ekwall, P. Kolloid Z. 1942, 101, 135–149. (84) Klevens, H. B. J. Colloid Sci. 1947, 2, 301–303. (85) Merrill, R. C.; Getty, R. J. Phys. Chem. 1948, 52, 774–787. (86) Herzfeld, S. H. J. Phys. Chem. 1952, 56, 953–959. (87) Herzfeld, S. H. J. Phys. Chem. 1952, 56, 959–963. (88) Klevens, H. B. J. Am. Oil Chem. Soc. 1953, 30, 74–80. (89) Maron, S. H.; Elder, M. E.; Ulevitch, I. N. J. Colloid Sci. 1954, 9, 382–384. (90) Klevens, H. B. J. Phys. Colloid Chem. 1950, 54, 1012–1016. (91) Shinoda, K. J. Phys. Chem. 1954, 58, 541–544. (92) Klevens, H. B. Kolloid Z. 1958, 158, 53–58. (93) Ekwall, P.; Eikrem, H.; Mandell, L. Acta Chem. Scand. 1963, 17, 111–122. (94) Campbell, A. N.; Lakshminarayanan, G. R. Can. J. Chem. 1965, 43, 1729– 1737. (95) Lindman, B.; Brun, B. J. Colloid Interface Sci. 1973, 42, 388–399. (96) Brun, T.; Høiland, H.; Vikingstad, E. J. Colloid Interface Sci. 1978, 63, 590– 592. (97) Vikingstad, E.; Skauge, A.; Hoiland, H. J. Colloid Interface Sci. 1978, 66, 240–246. (98) Vikingstad, E. J. Colloid Interface Sci. 1979, 72, 68–74. (99) Zemb, T.; Drlfford, M.; Hayoun, M.; Jehanno, A. J. Phys. Chem. 1983, 87, 4524–4528. (100) Maa, Y. F.; Chen, S. H. J. Colloid Interface Sci. 1987, 115, 437–442.

Langmuir 2010, 26(19), 15177–15191


Article

Figure 15. Calculated vs experimental61,101-111 cmc of alkyl trimethylammoniums at 25 C. Table 12. Influence of Counterion of Sodium Salts on Cmc (mM) of C14TABr salt 0.01 M

exp. data73 (29.85 C)

our model (25 C)

NaF NaCl NaBr

3.61 3.15 2.53 1.94

3.5 2.9 2.2 1.5

Figure 14. Calculated vs experimental11,84-86,90,92,94,96,97 cmc of

dodecanoate at 25 C. (a) with headgroup-counterion specificity; (b) without headgroup-counterion specificity.

results in Figures 13b and 14b. As we can see, if the headgroup-counterion interactions are not taken into account, the experimentally verified change in the cmc is not predicted by the model. Furthermore, the model without the headgroupcounterion specificity predicts a decrease in cmc as the bare ion radius increases, while measurements show an increase in cmc as the bare ion radius increases. Therefore, without headgroupcounterion interactions the correct lyotropic series of counterions for the cmc of alkyl carboxylates (see Table 1) cannot be predicted. Alkyl Trimethylammonium. Cmc. The experimental cmc data of alkyl trimethylammonium bromide and chloride are compared with the calculated cmc in Figure 15. Also for this surfactant headgroup, the counterion effect is examined in Table 12. The model prediction for the lyotropic series for C14TABr is in quantitative agreement with data for various sodium salts. The results for the addition of NaBr to alkyl trimethylammonium surfactants, C14TABr and C16TABr, presented in (101) Corrin, M. L.; Harkins, W. D. J. Am. Chem. Soc. 1947, 69, 683–688. (102) Norman, D.; Weiner, G. Z. J. Pharm. Sci. 1965, 54, 436–442. (103) Evans, D. F.; Allen, M.; Ninham, B. W.; Fouda, A. J. Solution Chem. 1984, 13, 87–101. (104) Barry, B. W.; Russell, G. F. J. J. Colloid Interface Sci. 1972, 40, 174–194. (105) Sepulveda, L.; Cortes, J. J. Phys. Chem. 1985, 89, 5322–5324. (106) Berr, S. S.; Caponetti, E.; J. S. J. Jr.; Jones, R. R. M.; Magid, L. J. J. Phys. Chem. 1986, 90, 5766–5770. (107) Roelants, E.; Schryver, F. C. D. Langmuir 1987, 3, 209–214. (108) Cipiciani, A.; Onori, G.; Savelli, G. Chem. Phys. Lett. 1988, 143, 505–509. (109) Kim, H.-U.; Lim, K.-H. Bull. Korean Chem. Soc. 2004, 25, 382–388. (110) Para, G.; Jarek, E.; Warszynski, P. Adv. Colloid Interface Sci. 2006, 122, 39–55. (111) Lopez-Diaz, D.; Velazquez, M. M. Chem. Educator 2007, 12, 327–330. (112) Barry, B. W.; Morrison, J. C.; Russell, G. F. J. J. Colloid Interface Sci. 1970, 33, 554–561. (113) Per Ekwall, P. S.; Leo Mandell J. Colloid Interface Sci. 1971, 35, 519–528. (114) Debye, P. J. Phys. Colloid Chem. 1949, 53, 1–8. (115) Trap, H. J. L.; Hermans, J. J. Koninkl. Ned. Akad. Wet. Proc. Ser. B 1955, 58, 97–108.

Langmuir 2010, 26(19), 15177–15191

Figure 16. Cmc of (a) C14TABr and (b) C16TABr as a function of NaBr concentration at 25 C. Squares103-108,110,111 and circles112 are experimental data and solid line is model prediction.

Figures 16a and 16b, show quantitative agreement for range of salt concentration tested. We also investigated the coion effect for cetyltrimethylammoniums. In Table 13, we compare experimental versus calculated cmc for cetyltrimethylammonium bromide and chloride with potassium bromide and chloride. We observe that ion bromide is more effective to decrease the cmc of cetyltrimethylammonium than the ion chloride both as coion and as counterion. DOI: 10.1021/la102536y

15189

Article


Table 13. Influence of Counterion and Co-Ion on Cmc (mM) of Cetyl Trimethylammoniums surfactant

salt 0.1 M

exp. data110

our model

C16TABr C16TACl C16TABr C16TACl

KBr KBr KCl KCl

0.034 0.035 0.06 0.065

0.025 0.027 0.043 0.047

Figure 17. Dependence of the average aggregation number of micelles at the cmc, on the tail length of the surfactant for alkyl trimethylammonium bromides. Markers are experimental data55,80,113-115 and solid line is model prediction. Figure 19. Calculated vs experimental115,123,126,128,131 cmc of (a) decyl and undecyl pyridinium and (b) dodecyl pyridinium at 20-30 C.

Figure 18. Influence of added KBr concentration on the average aggregation number of C12TABr. Markers are experimental data114,115 and solid line is model prediction.

Aggregation Number. We compare experimental data of aggregation number with model prediction (Figure 17). For alkyl trimethylammonium, we predict the trend of the aggregation number for various surfactant tail lengths; however, the values predicted are higher than the experimental data. For the effect of salt on the aggregation number of alkyl trimethylammonium, various authors have verified growth of micelles with addition of salt,114-117 however, not many systematic studies have been conducted on the growth with salt concentration. In Figure 18, we show that the aggregation number of C12TABr increases with addition of KBr. The model prediction for C12TABr-KBr shows an initial fast growth and a subsequent slow growth as salt concentration increases, as it has been observed in experimental studies77,78 and model prediction for SDS-NaCl. (116) Fujio, K. Bull. Chem. Soc. Jpn. 1998, 71, 83–89. (117) Aswal, V. K.; Goyal, P. S. Chem. Phys. Lett. 2002, 364, 44–50.

15190 DOI: 10.1021/la102536y

Alkyl pyridiniums. Cmc. The experimental data and model predictions are compared for alkyl pyridinium surfactants. In parts a and b of Figure 19, we observe the dependence of the cmc on the type of counterion. The model predicts reasonably well the experimental data. It is interesting to notice in Figure 19a that the cmc of decyl pyridinium iodine C10PI is closer to undecyl pyridinium bromide C11PBr than to decylpyridinium choride C10PCl. We also calculate the cmc of alkyl pyridinium chloride and bromide of various tail lengths22,59,88,102,115,118-130 and verify that the model gives good agreement. The plots are omitted for the sake of brevity. We have also calculated the effect of NaCl on the cmc of dodecyl pyridinium chloride C12PCl and hexadecyl pyridinium chloride C16PCl (Figures 20a and 20b). We have verified that the cmc dependence on salt concentration is predicted by the model. Aggregation Number. We compare experimental data of aggregation number with model predictions for alkyl pyridinium (118) Evers, E. C.; Kraus, C. A. J. Am. Chem. Soc. 1948, 70, 3049–3054. (119) Cushman, A.; Brady, A. P.; McBain, J. W. J. Colloid Sci. 1948, 3, 425–436. (120) Adderson, J. E.; Taylor, H. J. Colloid Sci. 1964, 19, 495–500. (121) Mukerjee, P.; Ray, A. J. Phys. Chem. 1966, 70, 2150–2157. (122) Ford, W. P. J.; Ottewill, R. H.; Parreira, H. C. J. Colloid Interface Sci. 1966, 21, 522–533. (123) Mandru, I. J. Colloid Interface Sci. 1972, 41, 430–436. (124) Paluch, M. J. Colloid Interface Sci. 1978, 66, 582–583. (125) Rosen, M. J.; Dahanayake, M.; Cohen, A. W. Colloids Surf. 1982, 5, 159– 172. (126) Malovikova, A.; Hayakawa, K.; Kwak, J. C. T. J. Phys. Chem. 1984, 88, 1930–1933. (127) Skerjanc, J.; Kogej, K.; Vesnaver, G. J. Phys. Chem. 1988, 92, 6382–6385. (128) Mehrian, T.; de Keizer, A.; Korteweg, A.; Lyklema, J. Colloids Surf. A: Physicochem. Eng. Aspects 1993, 71, 255–267. (129) Semmler, A.; Kohler, H.-H. J. Colloid Interface Sci. 1999, 218, 137–144. (130) Varade, D.; Joshi, T.; Aswal, V. K.; Goyal, P. S.; Hassan, P. A.; Bahadur, P. Colloids Surf. A: Physicochem. Eng. Aspects 2005, 259, 95–101. (131) Harkins, W. D.; Krizek, H.; Corrin, M. L. J. Colloid Sci. 1951, 6, 576–583.

Langmuir 2010, 26(19), 15177–15191


Article

described by only ion properties. Experimental and theoretical investigations have pointed out that the specificity behavior of the ions is dependent on the interacting part of the interface, macromolecule, or counterion. We present a molecular thermodynamic theory for ionic surfactant solutions that takes into account the headgroupcounterion specificity, and addresses ion-specific effects on the cmc and aggregation number. We assume that the charged headgroup and the counterion at the Stern layer form a solventshared ion pair with different degrees of cosphere overlap. The thickness of the Stern layer is estimated from molecular structures of hydrated surfactant heads and hydrated counterions, and from knowledge of the qualitative strength of headgroup-counterion interaction in line with Collins’ concept of matching water affinities. The thermodynamic model properly predicts the cmc of both anionic and cationic surfactants of various counterions, and the effect of different inorganic salts on micellization of ionic surfactants. It also successfully predicts the inversion of the cmc of alkyl carboxylates and sulfates in the lyotropic series of counterions. Acknowledgment. The authors thank the Electrical Engineering Dept. and the Institute Systems Research (ISR) at University of Maryland for kindly providing the code of the FSQP algorithm and the member companies of the Reservoir Engineering Research Institute (RERI) in Palo Alto, CA, for financial support. Figure 20. Cmc of (a) C12PCl and (b) C16PCl as a function of the concentration of NaCl in the aqueous solution. Markers are experimental data22,127,127,128 and solid line is model prediction.

bromides. The model predicts well the trend of the aggregation number for alkyl pyridinium. The plot is omitted for the sake of brevity. Ionic Dispersion Interactions. As an alternative to counterion-binding models, the Poisson-Boltzmann theory has been generalized to account for counterion-specific factors such as the counterion polarizability.132 At the beginning of our work, we have implemented the theory developed by Ninham and coworkers132-136 in the context of the molecular model of micellization, by accounting for the dispersion interactions between the micelle and counterions in the evaluation of Gibbs free energy of micellization. We take into account the ionic specificity via the modified Poisson-Boltzmann equation (MPBE). The MPBE incorporates the ionic dispersion contribution to the potential of mean force132 and ion size effects using hydrated ion size.134 The results (not shown) reveal that the dispersion interactions between the micelle and counterions do not adequately describe the specific ion effect on the cmc. The ion specific effect on cmc from dispersion interactions are small unlike the results we show in Figure 5.

Conclusions In this work, we have developed a molecular thermodynamic model of specific ion effects on micellization of ionic surfactants. Our work is in line with general layout by Kunz and co-workers21 which expose the fact that the ion specific behavior cannot be (132) Bostr€om, M.; Williams, D. R. M.; Ninham, B. W. Langmuir 2002, 18, 6010–6014. (133) Parsons, D. F.; Ninham, B. W. J. Phys. Chem. A 2009, 113, 1141–1150. (134) Parsons, D. F.; Bostrom, M.; Maceina, T. J.; Salis, A.; Ninham, B. W. Langmuir 2010, 26, 3323–3328. (135) Parsons, D. F.; Ninham, B. W. Langmuir 2010, 26, 1816–1823. (136) Parsons, D. F.; Ninham, B. W. Langmuir 2010, 26, 6430–6436.

Langmuir 2010, 26(19), 15177–15191

Appendix The Finite Difference Iterative Method. The PoissonBoltzmann equation is solved by finite differences. The Cartesian and polar discretizations are carried out after Strikwerda.51 With the discretization of the Laplacian, we get a tridiagonal matrix that is then solved by the Thomas algorithm.137 The iterative process is established in the source term f(x,φ). r2 φ ¼ f ðx, φÞ

ð46Þ

The source term is linearized by taking the two first terms of Taylor series expansion around the value of potential from the previous iteration, φ*. r2 φ ¼ f ðx, φÞjφ þ f 0 ðx, φÞjφ ðφ - φÞ

ð47Þ

r2 φ - f 0 ðx, φÞjφ φ ¼ f ðx, φÞjφ - f 0 ðx, φÞjφ φ

ð48Þ

For a salt zþ:z-

" e - z þ eφ - U þ f ðx, φÞ ¼ z þ n¥þ exp - z - n¥εo εsol kT # z - eφ - U ð49Þ exp kT "

- z þ eφ - U þ f ðx, φÞ ¼ exp εo εsol kT kT # z - eφ - U exp kT 0

e2

z2þ n¥þ

þ z2- n¥ð50Þ

(137) Faires, J. D.; Burden, R. L. Numerical Methods, 3rd ed.; Brooks Cole: Pacific Grove, CA, 2002.

DOI: 10.1021/la102536y

15191

Molecular Thermodynamic Modeling of Specific Ion Effects on

Recommend Documents