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nonionic mixed micellization
Colloids and Surfaces A: Physicochem. Eng. Aspects 244 (2004) 187–196
Study of the electrostatic and steric contributions to the free energy of ionic/nonionic mixed micellization Hussein Gharibi∗ , B. Sohrabi, S. Javadian, M. Hashemianzadeh Department of Chemistry, Tarbiat Modarres University, PO Box 14155-4838, Tehran, Iran Iranian Information and Documentation Center, PO Box 13185-1371, Tehran, Iran Received 12 January 2004; accepted 3 June 2004
Abstract We have investigated mixed micelle formation in mixtures of ionic and nonionic surfactants. Three types of ionic surfactant were used: CTAB (hexadecyltrimethylammonium bromide), CPC (hexadecylpyridinium chloride) and CPB (hexadecylpyridinium bromide). These surfactants all have a hexadecyl chain but contain different head groups. The use of CPC and CPB, which differ only in counter ion, allowed investigation of counter ion effects. TritonX-100 (p-(1,1,3,3-tetramethylbutyl) polyoxyethylene) was used as the nonionic surfactant in all experiments. PFG-NMR was used to measure the self-diffusion coefficients of the mixed micelles as a function of solution composition and total surfactant concentration. The aggregation number and hydrodynamic radius of each type of mixed micelle were determined by combining viscosity and self-diffusion coefficient measurements. The electrostatic and steric contributions to the free energy of mixed micellization, which are considered to be the most important contributions for mixtures of ionic and nonionic surfactants with different head groups, were determined. The electrostatic free energy was determined by solving an analytical approximation to the Poisson–Boltzmann equation. The electrostatic free energy varied dramatically with increasing mole fraction of ionic surfactant in the CTAB/TritonX-100 system but increased more gradually in the CPC/TritonX-100 and CPB/TritonX-100 systems. The more pronounced change for CTAB/TritonX-100 system can be attributed to the trimethylammonium headgroup of CTAB, which confers a high surface charge density and thus a high electrostatic free energy. © 2004 Published by Elsevier B.V. Keywords: Mixed micelle; PFG-NMR spectroscopy; Micelle size; Aggregation number; CTAB; TritonX-100; CPC; CPB; Electrostatic free energy; Steric free energy
1. Introduction In practical applications, mixtures of surfactants tend to be used rather than a single surfactant. Hence, understanding both the structure and properties of mixed micelles containing ionic and nonionic surfactants is essential for many industrial applications. Aside from their practical applications, mixed surfactant systems are of great theoretical interest in their own right. For example, the aggregation properties of mixtures of surfactants in solution differ substantially from those of pure surfactants [1]. The addition of a nonionic surfactant to an ionic surfactant micelle can reduce the electrostatic repulsions between the charged surfactant head groups, greatly facilitating micelle forma-
∗
Corresponding author. E-mail address: [email protected] (H. Gharibi).
0927-7757/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.colsurfa.2004.06.007
tion. The nonideal behavior of mixtures of an ionic and a nonionic surfactant can also be influenced by the structural characteristics of the two surfactants, such as the relative sizes of their head groups and the lengths of their tails [2]. Several thermodynamic treatments have been developed to describe mixed micellization of surfactants in solution on the basis of macroscopic micellization models. The macroscopic models of mixed micellization include the pseudophase separation model, the mass action model, and the small system thermodynamics and multiple equilibria approaches [3–5]. Molecular-thermodynamic treatments have also been developed to describe mixed micellization [6,7]. The molecular-thermodynamic theory entails a very complex calculation of all the contributions to the free energy of micellization, including the micellar mixing nonidealities resulting from electrostatic and steric interactions between the hydrophilic surfactant head groups and from the packing of hydrophobic surfactant tails of unequal length in the
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Nomenclature Notation a C Dm e g k N R T U V Z
area per molecule concentration self-diffusion coefficient of mixed micelle electronic charge Gibbs free energy Boltzmann constant aggregation number radius of micelle absolute temperature potential energy molecular volume charge of mixed micelle
Greek letters α mole fraction of solution β degree of attachment ε solvent dielectric constant η viscosity κ−1 Debye screening length µ chemical potential ρ density σ surface charge density ψ electrical potential
[11].1 H NMR spectroscopy and 1 H NMR relaxation have been used to study the structure and dynamical properties of mixed micelles of CTAB and TritonX-100 [12]. In the present study, we initially determined the size and aggregation number of mixed micelles by measuring the self-diffusion coefficient and viscosity as a function of solution composition and total surfactant concentration. In addition, we used molecular-thermodynamic theory to analyze the observed experimental behavior in terms of the free energy contributions to mixed micelle formation. This analysis provides a quantitative molecular-level understanding of the contribution of intermicellar surfactant interactions to mixed micelle formation.
2. Experimental 2.1. Materials Hexadecyltrimethylammonium bromide (CTAB), Hexadecyltrimethylpyridinum bromide (CPB) and Hexadecyltrimethylpyridinum chloride (CPC) were obtained from Aldrich. TritonX-100 (p-(1,1,3,3-tetramethylbutyl) polyoxyethylene and D2 O (99.95%)) were obtained from Flucka Co. and Merck Co., respectively. 2.2. Methods
micelle core. A simplified version of the theory assumes that, for binary mixtures of hydrocarbon-based surfactants, the main contributions to the interaction come from the electrostatic interaction between the charged surfactant head groups and differences in head group size. In the present study, we examine mixed surfactant systems in which the ionic surfactant is comprised of a hexadecyl chain and a range of head groups, in combination with different counter-ions. TritonX-100 (p-(1,1,3,3-tetramethylbutyl) polyoxyethylene) was used as the nonionic surfactant. By using ionic surfactants with different head groups and counter ions, we could study the steric contribution to mixed micelle formation as well as the effects of head group type and counter ion on the interactions in mixed surfactant systems. Mixtures of CTAB (hexadecyltrimethylammonium bromide) and TritonX-100 have been previously studied using several experimental techniques and various theoretical approaches. Surface tension and ion selective electrode techniques have been used to measure the critical micelle concentration (CMC), the variation of CMC with composition and the micelle composition of CTAB/TritonX-100 mixed micelles in aqueous solution [8–11]. Fluorescence depolarization measurements of Cn TAB/TritonX-100 indicate that the incorporation of Cn TAB leads to a reduction in the degree of order in the micellar structure of TritonX-100. In addition, the stability of mixed micelles has been shown to increase with increasing alkyl chain length of Cn TAB
2.2.1. NMR measurement NMR self-diffusion studies were performed on a Bruker 500 NMR spectrometer at 25 ◦ C. The pulsed field gradient technique developed by Stejskal and Tanner was used [13]. In this technique, micelle self-diffusion is monitored based on signals from added trace amounts of tetramethylsilane (Me4 Si), which can be assumed to be completely solubilized in the micellar phase. For all samples, a single exponential decay of the echo amplitude was observed. This indicates the existence of only a single self-diffusion mode in the system and represents strong evidence for mixed micelle formation. The Core program was used to calculate the self-diffusion coefficients [14]. 2.2.2. Viscosity The viscosities of the mixed surfactant solutions were measured using semi-micro Cannon Ubbelohde capillary viscometers immersed in a water bath of temperature 25 ± 0.1 ◦ C. The time for a surfactant solution to flow through the capillary was measured with an accuracy of ±0.01 s. The viscosity of each sample was measured at least five times. Standard deviation in these measurements was less than 1.5% and relative standard deviation or coefficient of variation was less than 0.5. The viscosity was measured as a function of total surfactant concentration for mixed surfactant solutions of a given composition by sequentially diluting a concentrated system by adding solvent directly into the
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Table 1 Values of density and viscosity of CTAB/TritonX-100, CPC/TritonX-100, CPB/TritonX-100 mixtures in constant total concentration and different mole fraction of CTAB, CPC and CPB in 25 ◦ C xA
ηCTAB (poise)
ηCPC (poise)
ηCPB (poise)
dCTAB (g/cm3 )
dCPC (g/cm3 )
dCPB (g/cm3 )
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.083 0.084 0.085 0.086 0.087 0.089 0.091 0.091 0.091 0.089 0.092
0.010 0.085 0.091 0.087 0.090 0.092 0.093 0.090 0.089 0.094 0.092
0.093 0.094 0.096 0.093 0.098 0.010 0.091 0.096 0.098 0.091 0.092
1.020 1.014 1.015 1.021 1.016 1.021 1.021 1.021 1.021 1.017 1.019
1.020 1.018 1.023 1.020 1.030 1.020 1.020 1.020 1.021 1.022 1.019
1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.019
viscometer. The results of viscosity of CTAB/TritonX-100, CPC/TritonX-100 and CPB/TritonX-100 mixtures are tabulated in Table 1. 2.2.3. Density The densities of the mixed surfactant solution were measured using a picknometer 25 ml at 25 ◦ C and tabulated in Table 1. To determine density and viscosity data for the mixed surfactant/D2 O system, and to consider the isotopic effect, the density and viscosity data for the mixed surfactant/H2 O system were multiplied by the ratios ρD2 O /ρH2 O or ηD2 O /ηH2 O . The isotopic substitution of the solvent might result in an alternation of the structural properties of the micellar aggregates. In fact, D2 O is thought to be slightly more structured than H2 O [15]. However, Berr showed that these differences are sufficiently small that this effect can be neglected [16].
3. Molecular-thermodynamic theory of mixed micellization 3.1. Free energy of mixed micellization The molecular-thermodynamic theory is based on calculating the size and composition distribution of the mixed micelles, which in turn depend on the free energy of forming a mixed micelle, gmic . The total free energy of mixed micellization is calculated as the sum of the free energy contributions as follows: The first three free energy terms in Eq. (1) involve only the surfactant tails. gmic = gtr + gint + gpack + gelec + gst
(1)
The transfer contribution, gtr , accounts for transforming the surfactant tails of both surfactant types from the aqueous solution to the core of the mixed micelle. The interfacial contribution, gint , accounts for forming the interface between the micelle core and the aqueous solution. And finally, the packing contribution, gpack , accounts for
anchoring one end of the tails of both surfactant types at the micelle core–water interface and packing the surfactant tails in the micelle core. If the two surfactant types have different tail lengths, they may pack better in the micelle core compared to mixtures of surfactants with the same tail length. This improved packing would contribute to the synergistic benefits of mixed micelle formation. The last two free energy terms in Eq. (1) involve only the surfactant heads. The steric contribution, gst , accounts for steric interactions between the surfactant head groups. This free energy depends only on the sizes of the surfactant head groups. If the head groups of the two surfactant types differ in size, the contribution of gst will favor mixed micelle formation due to the synergistic advantages derived from packing head groups of two different sizes. Finally, the electrostatic contribution, gelec , accounts for the electrostatic interactions between the surfactant head groups. If, for example, negatively charged surfactant head groups are intermingled with uncharged surfactant head groups, the uncharged head groups reduce the electrostatic repulsions between the charged head groups, thereby facilitating mixed micelle formation. The electrostatic contributions depend only on the electrostatic characteristics of the surfactant head groups, such as the valance and location of the charge on the head group. In this study, we focus on the electrostatic and steric free energy contributions to the overall free energy of mixed micelles [17–19]. 3.1.1. Determination of the electrostatic free energy, gelec There are two basic approaches for calculating the electrostatic free energy of a charged mixed micelle. The first approach relies on calculating gelec for a pure ionic micelle and then imposing a composition dependence to account for the change in gelec as the ionic composition of the mixed micelle changes. For example, in the simplified version of mix for an ionic–nonionic molecular thermodynamic theory, gelec mixed micelle depends on the electrostatic free energy of pure ionic micelles: mix A gelec = α2 gelec
(2a)
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charge distribution. This equation is valid for charges distributed over a spherical surface of radius R(α) and for fully dissociated 1:1 ionic surfactants and salts.
Fig. 1. The variation of mole fraction of micelle vs. total concentration of mixed surfactant.
The other approximation is made that the electrostatic free energy contribution to the mixed micelle is proportional to the square of the average charge per monomer. Based on this assumption, an approximate analytical solution to the Poisson–Boltzmann equation is used to calculate the electrostatic free energy contributions of the pure ionic surfactant in order to obtain the proportionality constant (for details of the derivation of Eq. (2b), see Appendix A). The following expression will be used to calculate the electrostatic free energy per monomer associated with creating a charged surface for both surfactant and the mixed micelle [20–22]. It has been shown in Fig. 1 at high concentration of mixed surfactants the monomer contribution is negligible, then the mole fraction of micelle is equal to mole fraction of solution (xA = α).
2 1/2 s(α) s(α) gelec = 2kTZ(α) ln + 1+ 2 2 1/2 −1 1 + (s(α)/2)2 4 − − s(α)/2 κ(α)R(α)s(α) 1/2 2 1 + 1 + (s(α)/2) × ln (2b) 2 Here, Z(α) is the magnitude of the average charge per monomer, and is given by the absolute value of Z(α) = αZA + (1 − α)ZB
(3)
where ZA and ZB are the charges on surfactants A and B, respectively, and s(α) is given by: 1 4πeσ(α) (4) s(α) = 2 kTεκ(α) where e is the electronic charge, ε is the solvent dielectric constant, and k−1 (α) is the Debye screening length given by:
1/2 8πC(α)e2 (5) κ(α) = εkT where C(α) is the bulk ionic concentration, σ(α) is the surface charge density, and R(α) is the radius of the spherical
3.1.2. Determination of steric free energy, gst The steric free energy, gst , accounts for the steric interaction between the surfactant head groups and counter ions at the micelle interface. The surfactant head groups and counter ions are treated as a monolayer located at the micelle interface, and their interactions are described by a free energy derived from a two-dimensional equation of state. The free energy associated with the steric repulsion, gst , is calculated by treating the head groups and counter ions present at the interface as a localized monolayer, which reflects the fact that each head group is physically attached to a tail at the interface and counter ions bonded to the micelle with degree of binding, β. Shiloach and Puvvada used same equation for evaluation of Zst for both nonionic–nonionic mixed micelles and ionic–nonionic mixed micelles. But in this work by insertion of the value of degree of dissociation, we improve the proposed equation by Shiloach and Puvvada [7,17]. The steric free energy can be calculated by two approaches. In one approach, the entire micelle interface, consisting of nα1 surfactant A heads, nα2 surfactant B heads and nα3 = nβα1 counter ions of A, is divided into n lattice spaces each occupying an area a, defined as the area per surfactant molecule at the micelle core–water interface. It is further assumed that each space on the lattice can be occupied by a single head group. Denoting the average cross-sectional area of head group A as ahA , the single-particle partition function associated with head group A, ZA , which reflects the fraction of available free lattice area, is given by (a − ahA )/a. Similarly, the single-particle partition functions associated with head group B and the counter ions of surfactant A are given by ZB = (a − ahB )/a and ZCA = (a − aCA )/a, respectively. Assuming that all of these single-particle partition functions are independent of each other, the micellar partition function associated with steric interactions (corresponding to nα1 surfactant A head groups, nα2 surfactant B head groups, and nα3 = nβα1 counter ions of A) can be expressed as the product of the single particle partition func1 nα2 nβα1 tions, that is,Zst = znα A zB zCA . The resulting free energy change (per monomer), gst = −(1/n)kT ln Zst , is given by: ahA ahB gst = −kT α1 ln 1 − + α2 ln 1 − a a aCA +βα1 ln 1 − (6) a From this equation, we see that gst increases as the size of the surfactant head groups increases. Micelles with small areas per surfactant molecule (e.g., cylindrical micelles) will have higher gst values, whereas those with large areas per surfactant molecule (e.g., spherical micelles) will have lower gst values. As noted above, Eq. (6) was derived on the assumption that the single particle partition functions are independent of each other. This assumption is most
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appropriate when the values of ahA and ahB are similar. However, if the values of ahA and ahB differ considerably, the larger head group may occupy more than one lattice space; that is, it may partially occupy some of the lattice spaces occupied by the smaller head groups. In that case, Eq. (6) will over-predict the value of gst . In the second approach, gst is calculated using the test particle approach originally developed to compute chemical potentials by Monte Carlo simulation. In the spirit of this approach, a “test” particle is introduced to an interface having an area nα1 surfactant A head groups, nα2 surfactant B head groups, and nα3 = nβα1 counter ions of A. Note that the test particle is not a real particle in the sense that the real particles in the system are not affected by its presence, although the test particle does sense their presence. The excess chemical potential, µex = µ − µid (where µid is the ideal contribution), has been shown to be equal to −kT lnexp(−βUtest ) , where Utest is the potential energy change associated with interactions between the test particle and the real particles which would result from the addition of the test particle to the interface at random. If the particles interact only through hard steric repulsions, Utest is equal either to infinity (when the test particle occupies a region occupied by a real particle) or zero (when the test particle occupies a region that is not occupied by a real particle). That is, µex is related to the probability of a test particle occupying a region that is not occupied, and this probability is related to the available places, which can be expressed as (na − nα1 ahA − nα2 ahB − nβα1 ah,cont )/na. Thus, µex can be expressed as: aheff ex µ = −kT ln 1 − , a aheff = α1 ahA + α2 ahB + α1 βah,cont
(7)
The excess potential energy is equal to the steric free energy, that is: α1 ahA + α2 ahB + βα1 ah,cont gst = −kT ln 1 − (8) a Thus, using the two approaches, we have derived two expressions for the steric energy, Eqs. (6) and (8). To use Eqs. (2) and (6) to calculate the electrostatic and steric free energies, respectively, it is necessary to know the values of several parameters as a function of micellar composition. These are the average charge per monomer Z(α), the bulk ionic concentration C(α), the surface charge density σ(α), the radius of the spherical surface charge distribution R(α), the cross-sectional areas of the head groups of the ionic and nonionic surfactants, and the area per surfactant at the micelle core–water interface. For mixtures of an ionic and a nonionic surfactant, the average charge per monomer is equal to Z(α) = α for monovalent ionic surfactant head groups. The bulk ionic concentration is given by the monomer concentration: C(α) = Csur (α)
(9)
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and the surface charge is given by: δ(α) =
N(α)Z(α)e A(α)
(10)
Here, N(α) is the total number of surfactant molecules in the micelle and A(α) is the area of the spherical surface charge distribution, which is given by 4πR2 (α), where R(α) is the hydrodynamic radius, which can be calculated by the Stokes–Einstein equation [23,24]: R(α) =
kT 6πηDm
(11)
where k is the Boltzmann constant, T is the absolute temperature, and η is the viscosity of the medium, which is taken to be the ternary solution with the corresponding ionic/nonionic surfactant ratio.The R(α) values obtained using Eq. (11) can be used to compute the aggregation number of the A micelles. Assuming a spherical micelle and neglecting its intrinsic instability, the aggregation number can be computed using the following relation [25]: N(α) =
(4/3)πR(α)3 α[VA + nh,A VD∗2 O + β(Vcon + nh,con VD∗2 O )] +xB (VB + nh,B VD∗2 O )
(12)
where xA and xB are the mole fractions, VA and VB are the molecular volumes, nh,A and nh,B are the hydration numbers of the head groups of surfactants A and B in the mixed micelle, respectively, and VBr− and VD2 O are the molecular volumes of D2 O and Br− that VA(A= CTAB) = 606.9 Å3 [26,27], VB(B= TritonX-100) = 521.48 Å3 [28], VA(A= CPCorCPB) = 595 Å3 [29], VD2 O = 30.2 Å3 [26,27], VCl− = 18 Å3 and VBr− = 39.21 Å3 [26,27]. Finally β is defined as the degree of attachment [9]. ahA(A= CTAB) = 97.98, ahB = 53.46, ahA(A= CPCorCPB) = 65.4 aCl− = 40.69 and aBr− = 47.76 Å2 and a, area per surfactant molecules at the micelle core–water interface is given by (A and B refer to ionic and nonionic surfactants, respectively): a=
A(α) N(α)
(13)
where A(α) is the area of the surface charge of micelle. Then the electrostatic and steric contributions to the free energy of mixed micellization can be calculated for the CTAB/TritonX-100 mixture.
4. Results and discussion 4.1. Self-diffusion coefficient Fig. 2 shows the variation of the self-diffusion coefficient of CTAB/TritonX-100 micelles as a function of the mole fraction of CTAB. The self-diffusion coefficient goes through a minimum as the mole fraction of CTAB is increased from 0 to 1 (except the case of 10 mM concentration). The initial decrease in the self-diffusion coefficient
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Fig. 2. Mixed micelle self-diffusion coefficients vs. the mole fraction of CTAB in mixed micelle: 100 mM ( ), 50 mM (䊐), 10 mM (䉬).
as the nonionic surfactant is replaced by ionic surfactant is attributed primarily to increased micelle–micelle repulsion. This repulsion will inhibit the exchange of monomers between different aggregates. Another factor that would lead to the observed decrease in the self-diffusion coefficient at low fractions of ionic surfactant and high total concentrations of mixed surfactant is micellar growth due to the incorporation of the less bulky ionic head groups into the micellar surface. The small magnitude of the decrease in the self-diffusion coefficient suggests that the reduction in the area per polar group that accompanies the introduction of CTAB has only a small effect on micelle size or that the electrostatic interactions are the dominant factor determining the micelle size at low mixing ratios. The increase in the self-diffusion coefficient at higher fractions of CTAB can be mainly attributed to the smaller size of micelles containing high fractions of CTAB due to the larger amount of electrostatic repulsion between the ionic head groups in these micelles, which increases the effective head group area. Fig. 2 also shows that, as the surfactant concentration is increased, the minimum in the self-diffusion coefficient moves to higher concentration of the ionic surfactant. At low total surfactant concentration, the self-diffusion coefficient increases monotonically, without going through a minimum, because the micelle–micelle interactions are negligible in these systems. Moreover, the second process described above that could potentially cause a decrease in self-diffusion coefficient, namely micellar growth due to the incorporation of less bulky ionic head groups into the micelle surface, is not valid for the CTAB/TritonX-100 system because the head group of CTAB is more bulky than that of TritonX-100 in 50 and 100 mM. CPC/TritonX-100 and CPB/TritonX-100 systems displayed behavior similar to that shown by CTAB/TritonX-100 at 10 mM. As shown in Fig. 3, the self-diffusion coefficients of CPC/TritonX-100 and CPB/TritonX-100 mixed micelles increase monotonically with increasing ionic surfactant content. 4.2. Electrostatic and steric free energy The behavior of the CTAB/TritonX-100, CPC/TritonX100 and CPB/TritonX-100 surfactant mixtures can be understood by analyzing the relevant contributions to the
Fig. 3. Mixed micelle self-diffusion coefficient vs. the mole fraction of CTAB (䊉), CPB ( ), CPC (䊐) at a total surfactant concentration of 10 mM.
overall free energy of forming a mixed micelle (gmic ). Figs. 4 and 5 show the variations in the predicted electrostatic and steric contributions to mixed micelle formation, gelec and gst , respectively, as a function of the mole fraction of ionic surfactant, α, for the CTAB/TritonX-100, CPC/TritonX-100 and CPB/TritonX-100 systems. It may be proposed for micelle systems, gst does not change significantly over the entire solution composition range, indicating that the steric contribution has only a small influence on mixed micelle formation. In contrast, gelec decreases dramatically on adding TritonX-100 to the pure ionic surfactant systems, and therefore the contribution to the total free energy due to electrostatic interactions between the surfactant head groups is expected to make a dominant contribution to any synergistic effects arising from the use of a mixture of surfactants. This is particularly the case for the CTAB/TritonX-100 mixture, which shows a much steeper fall in gelec with decreasing ionic surfactant mole fraction compared to the CPC/TritonX-100 and CPB/TritonX-100 mixtures. The decrease in gelec is due to the decrease in the electrostatic repulsions between charged CTAB, CPC or CPB head groups when TritonX-100 molecules are added to the micelles. In CTAB, the charged trimethylammonium group provides a high charge density, which in turn gives rise to a high electrostatic free energy. In CPC and CPB, however, the charge is distributed over a pyridinium ring, leading to a lower surface charge density and consequently a lower electrostatic free energy (Fig. 4). As has been shown in Fig. 4 the gelec of CPC and CPB are approximately equal, then we do not expect to get a much difference between gelec of CPC and CPB since the electrostatic properties of these two surfactants are near together. At low mole fractions of ionic surfactant, TritonX-100 shields the charged heads sufficiently that provide evidence with a reduction of surface charge density and consequently a lower electrostatic free energy. As has
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Fig. 4. Predicted electrostatic free energy (gelec ) contribution to mixed micelle formation as a function of mixed micelle composition for CTAB/TritonX-100 (䉫), CPC/TritonX-100 (䊊), and CPB/TritonX-100 ( ) at a total surfactant concentration of 10 mM.
been shown in Fig. 4 the obtained results in this work was compared with those results reported by Ref. [17]. It is not a superior agreement between two approaches. Because mix is computationally in the first approach to calculate gelec straight forward, in that only gelec for a pure ionic micelle must be calculated, but depends on the micelle shape, its radius and synergetic effect are not considered [17]. Fig. 5 shows that, when CPC or CPB is introduced into TritonX-100 micelles, the steric free energy remains approximately constant. This is because the CPC and CPB head groups are similar in size to the TritonX-100 head group [24–27]. By contrast, however, the steric free energy increases monotonically with increasing CTAB mole fraction because introduction of large CTAB head groups into the micelles increases the steric repulsions, resulting in a larger area per surfactant molecule. In addition, comparison of the steric free energies for the CPC/TritonX-100 and CPB/TritonX-100 systems calculated using Eqs. (6) and (8)
Fig. 5. Predicted steric free energy (gst ) contribution to mixed micelle formation as a function of mixed micelle composition at a total surfactant concentration of 10 mM for the first approach: CTAB/TritonX-100 (䉫), CPC/TritonX-100 ( ), CPB/TritonX-100 ( ); the second approach: CTAB/TritonX-100 (䊐), CPC/TritonX-100 (×), CPB/TritonX-100 (䊊).
reveals that the nature of the counter ion has little influence on gst . This observation can be attributed to similar size of head groups of CPC, CPB and TritonX-100. However, the values of the steric free energy calculated using Eqs. (6) and (8) for the CTAB/TritonX-100 system show poor agreement. Given the large size of the CTAB head group compared to the TritonX-100 head group, the CTAB head group may have occupied more than one lattice space in the calculation using Eq. (6), and hence the results obtained using Eq. (8) are expected to be closer to the true value. 4.3. Apparent radius and aggregation number The size of the mixed micelles in the studied solutions is mainly determined by the repulsions between head groups, including repulsions of both steric and electrostatic origin. To determine the synergistic effects arising from using a mixture of surfactants, the interactions between micelles are considered above the CMC because with an increase in concentration of surfactants solution, the size and aggregation numbers of micelles above the CMC are changed. At the CMC, the micelles are relatively small and it is difficult to
Fig. 6. Apparent radius of micelle vs. the mole fraction of CTAB (䉬), CPC (䊐), CPB ( ) in mixed micelle at a total surfactant concentration of 10 mM.
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Fig. 7. Aggregation number of mixed micelle vs. the mole fraction of CTAB (䉬), CPC (䊐), CPB ( ) in mixed micelle at a total surfactant concentration of 10 mM.
determine their size experimentally. A surfactant concentration of 10 mM was considered (i.e., 50 times higher than the CMC) [10]. Additional characteristics of solutions of higher surfactant concentration are that spherical micelles cannot form and the Stokes–Einstein equation is not valid. Robson and Dennis [30] proposed that a sharp boundary does not exist between the hydrophobic interior and the oxyethylene chains, in contradiction to the classical picture of micelles. If the first few oxyethylene groups at the octylphenyl end of some TritonX-100 molecules were contained in the hydrophobic core, it would be possible to accommodate a spherical model for the hydrophobic region as well as the total micelle. The core would have to have a radius of about 23 Å to accommodate the volume of the octylphenyl moieties. For the whole micelle, they assumed that the hydrophilic region extends one oxyethylene chain length (17 Å) beyond the hydrophobic core making the radius of the whole micelle about 43 Å. It should be noted that the polydispersity of the oxyethelen chains could result in a nonuniform distribution of oxyethylene groups around the micelle, allowing the overall shape to be closer to spherical. Also, according to Ruiz’ article, the incorporation of CTAB monomers in the TritonX-100 micelles produced a minor distortion in their micelle structure [11]. Figs. 6 and 7 show the apparent radius of the mixed micelles and the weight-average micelle aggregation numbers as a function of the mole fraction of the ionic surfactant for the CTAB/TritonX-100, CPC/TritonX-100 and CPB/TritonX-100 mixtures at a total surfactant concentration of 10 mM. In the CTAB/TritonX-100 system, the average mixed micelle aggregation number decreases monotonically with increasing CTAB content due to the larger head groups of this surfactant compared to TritonX-100, which cause the steric free energy to increase upon introducing increasing numbers of CTAB molecules into the pure TritonX-100 micelle. Another factor contributing to the observed decrease in the average mixed micelle aggre-
gation number is the increase in electrostatic free energy as the ionic surfactant content of the mixed micelles increases, which causes a reduction in the size of the mixed micelles. As a consequence of the smaller micelle size, the optimal average surface area per hydrophilic group increases and the number of surfactant chains per micelle decreases; the smallest R(α) corresponds to the system with the highest surface charge density, i.e., pure CTAB micelles. At low CTAB content, the steric free energy is the main factor determining micelle size, whereas at high mole fractions of CTAB the influence of the electrostatic free energy dominates. In the CPC/TritonX-100 and CPB/TritonX-100 systems, the average mixed micelle aggregation number is approximately independent of CTAB content. This is because, in each of these systems, the head groups of the ionic and nonionic surfactants are of approximately the same size, causing gst to be approximately constant. In addition, the charged CPC or CPB head groups provide lower surface charge density, leading to lower gelec . The results show that the counter ion has no significant effect on the size and average aggregation number of the mixed micelles.
5. Conclusions The micelle self-diffusion coefficient goes through a minimum as the mole fraction of the ionic surfactant is increased from 0 to 1. The initial decrease in the self-diffusion coefficient is due to an increase in micelle–micelle repulsion, while the increase in self-diffusion coefficient at higher ionic surfactant content is due to a decrease in the size of the micelles. As expected, this minimum in the self-diffusion coefficient is not observed at low total surfactant concentration, because the micelle–micelle interactions are negligible in such systems; instead, the self-diffusion coefficient increases monotonically with increasing ionic surfactant content due to the increasing numbers of charged head groups
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in the micelles, which leads to increased repulsion between the head groups and a consequent decrease in the size of the micelles. To study micelle growth at a total concentration above the CMC, PFG-NMR and viscometry were used to measure the average aggregation number and size of the micelles or hydrodynamic radius. The results show that the size of the mixed micelles is determined by interplay between the steric and electrostatic intermicellar interactions. In CTAB/TritonX-100 mixed micelles, the free energy contributions due to the steric and electrostatic interactions (gst and gelec ) caused the aggregation number and hydrodynamic radius to decrease with increasing mole fraction of CTAB. In CPC/TritonX-100 and CPB/TritonX-100 mixed micelles, in contrast, gst remained approximately constant and gelec was low. Due to these factors, the aggregation number and hydrodynamic radius were approximately independent of the mole fraction of ionic surfactant.
which describes the curvature correction to the planar problem drops out. The planar limit can be recovered by writing, X = κr and x = X + ξ, and, taking the limit x → ∞, Eq. (A.4) reduces to
Appendix A
We suppose that the micelle is a spherical micelle of radius R, and that, for simplicity, the solvent contains a 1:1 electrolyte. For this system, the nonlinear Poisson–Boltzmann equation that describes the distribution of ions about the sphere, and the boundary conditions satisfied by the potential ψ = ψ(r), are
d2 ψ 2 dψ 8πn0 e eψ 2 ∇ ψ= 2 + = sinh (A.1) r dr ε kT dr where n0 is the bulk electrolyte concentration, e is the magnitude of the unit charge, ε is the dielectric constant of the solvent, and T is the temperature. The monomers are assumed to be completely dissociated and univalent. The boundary conditions are:
d2 y = sinh y dx2 The first integral is dy y0 s= − = 2 sinh dX X=κR 2 1/2 s s 2 y0 = 2 ln + +1 2 2
where y0 is the scaled surface potential. If we substitute this expression into the curvature term of Eq. (A.4), we obtain
d 1 dy 2 dy 0 dy 2 dx y 0 sinh y dy − 2 =
(A.2)
y0
0
y0 0
1 x
dy dy dx
(A.6)
The main contribution to the second integral of the right side comes from y = y0 , near which x = X = κR. To a first approximation we can remove this factor from the integral to give 1 2
dy dx
2
y=y0
2 y0 dy 2 y0 ∼ − dy = 2 sinh 2 x 0 dx = 2 sinh2
ψ(r) → 0 as r → ∞ dψ 4πσ =− at r = R dr ε
(A.5)
y 0
2
+
y 8 0 cosh −1 κR 2 (A.7)
We used the planar approximation Eq. (A.5) to carry out the last integral. Hence
We introduce dimensionless variables through the substitutions:
(A.4)
y dy 0 = s = 2 sinh dx y=y 2 0 !
" y 4 cosh(y0 /2) − 1 1/2 0 × 1+ = 2 sinh 2 κR 2 sinh (y0 /2) y 4 0 + tanh + ··· (A.8) κR 4
where σ = e/a and a is the area per surfactant molecule head group. In the limit r → ∞, the term in (2/x)(dy/dx)
Comparison of this analytic approximation with the numerical solution of Eq. (A.4) showed that it is quite accurate; the worst error obtained for κR ≥ 0.5 was 5% at κR = 0.5. For larger values of κR the error diminishes rapidly. The electrostatic free energy can now be evaluated explicitly as
y=
eψ kT
x = κr
κ2 =
8πn0 e2 εkT
− (A.3)
where κ−1 is the inverse Debye screening length. Hence Eqs. (A.1) and (A.2) become d2 y 2 dy + = sinh y x dx dx2 dy 4πσe 4πe2 |x=κR = − =− = −s dx εκkT εκakT
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kT 2 εκ # s gel = a 0 ψ0 dσ = a y0 ds ⇒ e 4π 0
s 2 1/2 s gel = 2kT ln + 1+ 2 2 #σ
(1 + (s/2)2 )1/2 − 1 s/2
4 1 + (1 + (s/2)2 )1/2 − ln κRs 2
(A.9)
−
For cylindrical or ellipsoid micelles, the analysis can be carried out in the same way. Thus, for cylindrical micelles the operator ∇ 2 becomes d2 /dr2 + (1/r)(d/dr) instead of d2 /dr2 + 2/r for spheres. Hence, the curvature term will be reduced by a factor of 2. Similarly, it can be evaluated for other shapes. The analysis can also be performed for systems in which the surface of the micelle has dissociated groups, and/or for a mixture of 2:1 and 1:1 electrolyte. At high surfactant concentration ( CMC) the approximations of the Debye–Hückel theory and Poisson–Boltzmann equation break down. In this concentration regime, micelles other than the one under consideration cannot be considered as point ions and intermicellar interactions must be built into the formalism. They could be included either by the Wigner–Seitz method or through a perturbation expansion built by using the above approximate analytic forms.
References [1] M.J. Rosen, Surfactants and Interfacial Phenomena, John Willey & Sons, New York, 1988, p. 395. [2] R.M. Hill, in: K. Ogino, M. Abe (Eds.), Mixed Surfactant Systems, Marcel Dekker, New York, 1993, p. 317.
[3] H. Lange, K.H. Beck, Kolloid-Z. Z. Polym. 251 (1973) 424. [4] R.F. Klamath, E.I. Franses, J. Phys. Chem. 88 (1984) 1642. [5] K. Motomura, M. Yamanaka, M. Aratono, Colloid. Polym. Sci. 262 (1984) 948. [6] S. Puvvada, D. Blankschtein, J. Phys. Chem. 96 (1992) 5567. [7] S. Puvvada, D. Blankschtein, J. Phys. Chem. 96 (1992) 5579. [8] H. Gharibi, S. Javadian, M. Hashemianzadeh, Colloid. Surfaces A: Physicoochem. Eng. Aspects 232 (2004) 77. [9] M. Razavizadeh, M. Mousavi-Khoshdel, H. Gharibi, R. Behjatmanesh-Ardakani, S. Javadian, B. Sohrabi, J. Colloid. Interface Sci. 276 (2004) 197. [10] H. Gharibi, B.M. Razavizadeh, M. Hashemianzadeh, Colloid. Surfaces A: Physicoochem. Eng. Aspects 174 (2000) 375. [11] C.C. Ruiz, J. Aguiar, Langmuir 16 (2000) 7946. [12] H.Z. Yuan, S. Zhao, G.Z. Cheng, L. Zhang, X.J. Miao, S.Z. Mao, J.Y. Yu, L.F. Shen, Y.R. Du, J. Phys. Chem. B 105 (2001) 4611. [13] E.O. Stejskal, J.E. Tanner, J. Chem. Phys. 42 (1965) 288. [14] P. Stilbs, K. Paulsen, P.C. Griffiths, J. Phys. Chem. 100 (1996) 8180. [15] G. Nemethy, H.A. Scheraga, J. Chem. Phys. 41 (1964) 680. [16] S.S. Berr, J. Phys. Chem. 91 (1987) 4760. [17] A. Shiloach, D. Blankschtein, Langmuir 14 (1998) 1618. [18] A. Shiloach, D. Blankschtein, Langmuir 14 (1998) 4105. [19] I. Reif, P. Somasundaran, Langmuir 15 (1999) 3411. [20] A. Shiloach, D. Blankschtein, Langmuir 13 (1997) 3968. [21] C. Sarmoria, S. Puvvada, D. Blankschtein, Langmuir 8 (1992) 2690. [22] D.J. Mitchell, B.W. Ninham, J. Phys. Chem. 87 (1983) 2996. [23] D.F. Evans, B.W. Ninham, J. Phys. Chem. 87 (1983) 5025. [24] A. Eintein, Investigation in the Theory of Brownian Movement, Dover, New York, 1965. [25] P. Stilbs, Prog. NMR Spectr. 19 (1987) 1. [26] D. Ciccarelli, L. Costantino, G. Derrico, L. Paaduano, V. Vitagliani, Langmuir 14 (1998) 7130. [27] S.S. Berr, E. Caponett, J.S. Johnson, R.R.M. Jones, L.G. Magid, J. Phys. Chem. 90 (1986) 5766. [28] T. Image, Shoichi, J. Phys. Chem. 90 (1986) 5216. [29] G. Massiera, L. Ramos, C. Ligoure, Structure Factor of Polymers Interacting via a Short-range Repulsive Potential: Application to Hairywormlike Micelles, 2003, Chemical and Biomolecular Engineering Department, University of Pensylvania, Towne 311A, Philadelphia, PA 19104, USA, http://arxiv.org/abs/cond-mat/0301123 [30] R.J. Robson, E.A. Dennis, J. Phys. Chem. 81 (1977) 1075.
Study of the electrostatic and steric contributions to the free energy of ionic/nonionic mixed micellization Hussein Gharibi∗ , B. Sohrabi, S. Javadian, M. Hashemianzadeh Department of Chemistry, Tarbiat Modarres University, PO Box 14155-4838, Tehran, Iran Iranian Information and Documentation Center, PO Box 13185-1371, Tehran, Iran Received 12 January 2004; accepted 3 June 2004
Abstract We have investigated mixed micelle formation in mixtures of ionic and nonionic surfactants. Three types of ionic surfactant were used: CTAB (hexadecyltrimethylammonium bromide), CPC (hexadecylpyridinium chloride) and CPB (hexadecylpyridinium bromide). These surfactants all have a hexadecyl chain but contain different head groups. The use of CPC and CPB, which differ only in counter ion, allowed investigation of counter ion effects. TritonX-100 (p-(1,1,3,3-tetramethylbutyl) polyoxyethylene) was used as the nonionic surfactant in all experiments. PFG-NMR was used to measure the self-diffusion coefficients of the mixed micelles as a function of solution composition and total surfactant concentration. The aggregation number and hydrodynamic radius of each type of mixed micelle were determined by combining viscosity and self-diffusion coefficient measurements. The electrostatic and steric contributions to the free energy of mixed micellization, which are considered to be the most important contributions for mixtures of ionic and nonionic surfactants with different head groups, were determined. The electrostatic free energy was determined by solving an analytical approximation to the Poisson–Boltzmann equation. The electrostatic free energy varied dramatically with increasing mole fraction of ionic surfactant in the CTAB/TritonX-100 system but increased more gradually in the CPC/TritonX-100 and CPB/TritonX-100 systems. The more pronounced change for CTAB/TritonX-100 system can be attributed to the trimethylammonium headgroup of CTAB, which confers a high surface charge density and thus a high electrostatic free energy. © 2004 Published by Elsevier B.V. Keywords: Mixed micelle; PFG-NMR spectroscopy; Micelle size; Aggregation number; CTAB; TritonX-100; CPC; CPB; Electrostatic free energy; Steric free energy
1. Introduction In practical applications, mixtures of surfactants tend to be used rather than a single surfactant. Hence, understanding both the structure and properties of mixed micelles containing ionic and nonionic surfactants is essential for many industrial applications. Aside from their practical applications, mixed surfactant systems are of great theoretical interest in their own right. For example, the aggregation properties of mixtures of surfactants in solution differ substantially from those of pure surfactants [1]. The addition of a nonionic surfactant to an ionic surfactant micelle can reduce the electrostatic repulsions between the charged surfactant head groups, greatly facilitating micelle forma-
∗
Corresponding author. E-mail address: [email protected] (H. Gharibi).
0927-7757/$ – see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.colsurfa.2004.06.007
tion. The nonideal behavior of mixtures of an ionic and a nonionic surfactant can also be influenced by the structural characteristics of the two surfactants, such as the relative sizes of their head groups and the lengths of their tails [2]. Several thermodynamic treatments have been developed to describe mixed micellization of surfactants in solution on the basis of macroscopic micellization models. The macroscopic models of mixed micellization include the pseudophase separation model, the mass action model, and the small system thermodynamics and multiple equilibria approaches [3–5]. Molecular-thermodynamic treatments have also been developed to describe mixed micellization [6,7]. The molecular-thermodynamic theory entails a very complex calculation of all the contributions to the free energy of micellization, including the micellar mixing nonidealities resulting from electrostatic and steric interactions between the hydrophilic surfactant head groups and from the packing of hydrophobic surfactant tails of unequal length in the
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Nomenclature Notation a C Dm e g k N R T U V Z
area per molecule concentration self-diffusion coefficient of mixed micelle electronic charge Gibbs free energy Boltzmann constant aggregation number radius of micelle absolute temperature potential energy molecular volume charge of mixed micelle
Greek letters α mole fraction of solution β degree of attachment ε solvent dielectric constant η viscosity κ−1 Debye screening length µ chemical potential ρ density σ surface charge density ψ electrical potential
[11].1 H NMR spectroscopy and 1 H NMR relaxation have been used to study the structure and dynamical properties of mixed micelles of CTAB and TritonX-100 [12]. In the present study, we initially determined the size and aggregation number of mixed micelles by measuring the self-diffusion coefficient and viscosity as a function of solution composition and total surfactant concentration. In addition, we used molecular-thermodynamic theory to analyze the observed experimental behavior in terms of the free energy contributions to mixed micelle formation. This analysis provides a quantitative molecular-level understanding of the contribution of intermicellar surfactant interactions to mixed micelle formation.
2. Experimental 2.1. Materials Hexadecyltrimethylammonium bromide (CTAB), Hexadecyltrimethylpyridinum bromide (CPB) and Hexadecyltrimethylpyridinum chloride (CPC) were obtained from Aldrich. TritonX-100 (p-(1,1,3,3-tetramethylbutyl) polyoxyethylene and D2 O (99.95%)) were obtained from Flucka Co. and Merck Co., respectively. 2.2. Methods
micelle core. A simplified version of the theory assumes that, for binary mixtures of hydrocarbon-based surfactants, the main contributions to the interaction come from the electrostatic interaction between the charged surfactant head groups and differences in head group size. In the present study, we examine mixed surfactant systems in which the ionic surfactant is comprised of a hexadecyl chain and a range of head groups, in combination with different counter-ions. TritonX-100 (p-(1,1,3,3-tetramethylbutyl) polyoxyethylene) was used as the nonionic surfactant. By using ionic surfactants with different head groups and counter ions, we could study the steric contribution to mixed micelle formation as well as the effects of head group type and counter ion on the interactions in mixed surfactant systems. Mixtures of CTAB (hexadecyltrimethylammonium bromide) and TritonX-100 have been previously studied using several experimental techniques and various theoretical approaches. Surface tension and ion selective electrode techniques have been used to measure the critical micelle concentration (CMC), the variation of CMC with composition and the micelle composition of CTAB/TritonX-100 mixed micelles in aqueous solution [8–11]. Fluorescence depolarization measurements of Cn TAB/TritonX-100 indicate that the incorporation of Cn TAB leads to a reduction in the degree of order in the micellar structure of TritonX-100. In addition, the stability of mixed micelles has been shown to increase with increasing alkyl chain length of Cn TAB
2.2.1. NMR measurement NMR self-diffusion studies were performed on a Bruker 500 NMR spectrometer at 25 ◦ C. The pulsed field gradient technique developed by Stejskal and Tanner was used [13]. In this technique, micelle self-diffusion is monitored based on signals from added trace amounts of tetramethylsilane (Me4 Si), which can be assumed to be completely solubilized in the micellar phase. For all samples, a single exponential decay of the echo amplitude was observed. This indicates the existence of only a single self-diffusion mode in the system and represents strong evidence for mixed micelle formation. The Core program was used to calculate the self-diffusion coefficients [14]. 2.2.2. Viscosity The viscosities of the mixed surfactant solutions were measured using semi-micro Cannon Ubbelohde capillary viscometers immersed in a water bath of temperature 25 ± 0.1 ◦ C. The time for a surfactant solution to flow through the capillary was measured with an accuracy of ±0.01 s. The viscosity of each sample was measured at least five times. Standard deviation in these measurements was less than 1.5% and relative standard deviation or coefficient of variation was less than 0.5. The viscosity was measured as a function of total surfactant concentration for mixed surfactant solutions of a given composition by sequentially diluting a concentrated system by adding solvent directly into the
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Table 1 Values of density and viscosity of CTAB/TritonX-100, CPC/TritonX-100, CPB/TritonX-100 mixtures in constant total concentration and different mole fraction of CTAB, CPC and CPB in 25 ◦ C xA
ηCTAB (poise)
ηCPC (poise)
ηCPB (poise)
dCTAB (g/cm3 )
dCPC (g/cm3 )
dCPB (g/cm3 )
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
0.083 0.084 0.085 0.086 0.087 0.089 0.091 0.091 0.091 0.089 0.092
0.010 0.085 0.091 0.087 0.090 0.092 0.093 0.090 0.089 0.094 0.092
0.093 0.094 0.096 0.093 0.098 0.010 0.091 0.096 0.098 0.091 0.092
1.020 1.014 1.015 1.021 1.016 1.021 1.021 1.021 1.021 1.017 1.019
1.020 1.018 1.023 1.020 1.030 1.020 1.020 1.020 1.021 1.022 1.019
1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.021 1.019
viscometer. The results of viscosity of CTAB/TritonX-100, CPC/TritonX-100 and CPB/TritonX-100 mixtures are tabulated in Table 1. 2.2.3. Density The densities of the mixed surfactant solution were measured using a picknometer 25 ml at 25 ◦ C and tabulated in Table 1. To determine density and viscosity data for the mixed surfactant/D2 O system, and to consider the isotopic effect, the density and viscosity data for the mixed surfactant/H2 O system were multiplied by the ratios ρD2 O /ρH2 O or ηD2 O /ηH2 O . The isotopic substitution of the solvent might result in an alternation of the structural properties of the micellar aggregates. In fact, D2 O is thought to be slightly more structured than H2 O [15]. However, Berr showed that these differences are sufficiently small that this effect can be neglected [16].
3. Molecular-thermodynamic theory of mixed micellization 3.1. Free energy of mixed micellization The molecular-thermodynamic theory is based on calculating the size and composition distribution of the mixed micelles, which in turn depend on the free energy of forming a mixed micelle, gmic . The total free energy of mixed micellization is calculated as the sum of the free energy contributions as follows: The first three free energy terms in Eq. (1) involve only the surfactant tails. gmic = gtr + gint + gpack + gelec + gst
(1)
The transfer contribution, gtr , accounts for transforming the surfactant tails of both surfactant types from the aqueous solution to the core of the mixed micelle. The interfacial contribution, gint , accounts for forming the interface between the micelle core and the aqueous solution. And finally, the packing contribution, gpack , accounts for
anchoring one end of the tails of both surfactant types at the micelle core–water interface and packing the surfactant tails in the micelle core. If the two surfactant types have different tail lengths, they may pack better in the micelle core compared to mixtures of surfactants with the same tail length. This improved packing would contribute to the synergistic benefits of mixed micelle formation. The last two free energy terms in Eq. (1) involve only the surfactant heads. The steric contribution, gst , accounts for steric interactions between the surfactant head groups. This free energy depends only on the sizes of the surfactant head groups. If the head groups of the two surfactant types differ in size, the contribution of gst will favor mixed micelle formation due to the synergistic advantages derived from packing head groups of two different sizes. Finally, the electrostatic contribution, gelec , accounts for the electrostatic interactions between the surfactant head groups. If, for example, negatively charged surfactant head groups are intermingled with uncharged surfactant head groups, the uncharged head groups reduce the electrostatic repulsions between the charged head groups, thereby facilitating mixed micelle formation. The electrostatic contributions depend only on the electrostatic characteristics of the surfactant head groups, such as the valance and location of the charge on the head group. In this study, we focus on the electrostatic and steric free energy contributions to the overall free energy of mixed micelles [17–19]. 3.1.1. Determination of the electrostatic free energy, gelec There are two basic approaches for calculating the electrostatic free energy of a charged mixed micelle. The first approach relies on calculating gelec for a pure ionic micelle and then imposing a composition dependence to account for the change in gelec as the ionic composition of the mixed micelle changes. For example, in the simplified version of mix for an ionic–nonionic molecular thermodynamic theory, gelec mixed micelle depends on the electrostatic free energy of pure ionic micelles: mix A gelec = α2 gelec
(2a)
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charge distribution. This equation is valid for charges distributed over a spherical surface of radius R(α) and for fully dissociated 1:1 ionic surfactants and salts.
Fig. 1. The variation of mole fraction of micelle vs. total concentration of mixed surfactant.
The other approximation is made that the electrostatic free energy contribution to the mixed micelle is proportional to the square of the average charge per monomer. Based on this assumption, an approximate analytical solution to the Poisson–Boltzmann equation is used to calculate the electrostatic free energy contributions of the pure ionic surfactant in order to obtain the proportionality constant (for details of the derivation of Eq. (2b), see Appendix A). The following expression will be used to calculate the electrostatic free energy per monomer associated with creating a charged surface for both surfactant and the mixed micelle [20–22]. It has been shown in Fig. 1 at high concentration of mixed surfactants the monomer contribution is negligible, then the mole fraction of micelle is equal to mole fraction of solution (xA = α).
2 1/2 s(α) s(α) gelec = 2kTZ(α) ln + 1+ 2 2 1/2 −1 1 + (s(α)/2)2 4 − − s(α)/2 κ(α)R(α)s(α) 1/2 2 1 + 1 + (s(α)/2) × ln (2b) 2 Here, Z(α) is the magnitude of the average charge per monomer, and is given by the absolute value of Z(α) = αZA + (1 − α)ZB
(3)
where ZA and ZB are the charges on surfactants A and B, respectively, and s(α) is given by: 1 4πeσ(α) (4) s(α) = 2 kTεκ(α) where e is the electronic charge, ε is the solvent dielectric constant, and k−1 (α) is the Debye screening length given by:
1/2 8πC(α)e2 (5) κ(α) = εkT where C(α) is the bulk ionic concentration, σ(α) is the surface charge density, and R(α) is the radius of the spherical
3.1.2. Determination of steric free energy, gst The steric free energy, gst , accounts for the steric interaction between the surfactant head groups and counter ions at the micelle interface. The surfactant head groups and counter ions are treated as a monolayer located at the micelle interface, and their interactions are described by a free energy derived from a two-dimensional equation of state. The free energy associated with the steric repulsion, gst , is calculated by treating the head groups and counter ions present at the interface as a localized monolayer, which reflects the fact that each head group is physically attached to a tail at the interface and counter ions bonded to the micelle with degree of binding, β. Shiloach and Puvvada used same equation for evaluation of Zst for both nonionic–nonionic mixed micelles and ionic–nonionic mixed micelles. But in this work by insertion of the value of degree of dissociation, we improve the proposed equation by Shiloach and Puvvada [7,17]. The steric free energy can be calculated by two approaches. In one approach, the entire micelle interface, consisting of nα1 surfactant A heads, nα2 surfactant B heads and nα3 = nβα1 counter ions of A, is divided into n lattice spaces each occupying an area a, defined as the area per surfactant molecule at the micelle core–water interface. It is further assumed that each space on the lattice can be occupied by a single head group. Denoting the average cross-sectional area of head group A as ahA , the single-particle partition function associated with head group A, ZA , which reflects the fraction of available free lattice area, is given by (a − ahA )/a. Similarly, the single-particle partition functions associated with head group B and the counter ions of surfactant A are given by ZB = (a − ahB )/a and ZCA = (a − aCA )/a, respectively. Assuming that all of these single-particle partition functions are independent of each other, the micellar partition function associated with steric interactions (corresponding to nα1 surfactant A head groups, nα2 surfactant B head groups, and nα3 = nβα1 counter ions of A) can be expressed as the product of the single particle partition func1 nα2 nβα1 tions, that is,Zst = znα A zB zCA . The resulting free energy change (per monomer), gst = −(1/n)kT ln Zst , is given by: ahA ahB gst = −kT α1 ln 1 − + α2 ln 1 − a a aCA +βα1 ln 1 − (6) a From this equation, we see that gst increases as the size of the surfactant head groups increases. Micelles with small areas per surfactant molecule (e.g., cylindrical micelles) will have higher gst values, whereas those with large areas per surfactant molecule (e.g., spherical micelles) will have lower gst values. As noted above, Eq. (6) was derived on the assumption that the single particle partition functions are independent of each other. This assumption is most
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appropriate when the values of ahA and ahB are similar. However, if the values of ahA and ahB differ considerably, the larger head group may occupy more than one lattice space; that is, it may partially occupy some of the lattice spaces occupied by the smaller head groups. In that case, Eq. (6) will over-predict the value of gst . In the second approach, gst is calculated using the test particle approach originally developed to compute chemical potentials by Monte Carlo simulation. In the spirit of this approach, a “test” particle is introduced to an interface having an area nα1 surfactant A head groups, nα2 surfactant B head groups, and nα3 = nβα1 counter ions of A. Note that the test particle is not a real particle in the sense that the real particles in the system are not affected by its presence, although the test particle does sense their presence. The excess chemical potential, µex = µ − µid (where µid is the ideal contribution), has been shown to be equal to −kT lnexp(−βUtest ) , where Utest is the potential energy change associated with interactions between the test particle and the real particles which would result from the addition of the test particle to the interface at random. If the particles interact only through hard steric repulsions, Utest is equal either to infinity (when the test particle occupies a region occupied by a real particle) or zero (when the test particle occupies a region that is not occupied by a real particle). That is, µex is related to the probability of a test particle occupying a region that is not occupied, and this probability is related to the available places, which can be expressed as (na − nα1 ahA − nα2 ahB − nβα1 ah,cont )/na. Thus, µex can be expressed as: aheff ex µ = −kT ln 1 − , a aheff = α1 ahA + α2 ahB + α1 βah,cont
(7)
The excess potential energy is equal to the steric free energy, that is: α1 ahA + α2 ahB + βα1 ah,cont gst = −kT ln 1 − (8) a Thus, using the two approaches, we have derived two expressions for the steric energy, Eqs. (6) and (8). To use Eqs. (2) and (6) to calculate the electrostatic and steric free energies, respectively, it is necessary to know the values of several parameters as a function of micellar composition. These are the average charge per monomer Z(α), the bulk ionic concentration C(α), the surface charge density σ(α), the radius of the spherical surface charge distribution R(α), the cross-sectional areas of the head groups of the ionic and nonionic surfactants, and the area per surfactant at the micelle core–water interface. For mixtures of an ionic and a nonionic surfactant, the average charge per monomer is equal to Z(α) = α for monovalent ionic surfactant head groups. The bulk ionic concentration is given by the monomer concentration: C(α) = Csur (α)
(9)
191
and the surface charge is given by: δ(α) =
N(α)Z(α)e A(α)
(10)
Here, N(α) is the total number of surfactant molecules in the micelle and A(α) is the area of the spherical surface charge distribution, which is given by 4πR2 (α), where R(α) is the hydrodynamic radius, which can be calculated by the Stokes–Einstein equation [23,24]: R(α) =
kT 6πηDm
(11)
where k is the Boltzmann constant, T is the absolute temperature, and η is the viscosity of the medium, which is taken to be the ternary solution with the corresponding ionic/nonionic surfactant ratio.The R(α) values obtained using Eq. (11) can be used to compute the aggregation number of the A micelles. Assuming a spherical micelle and neglecting its intrinsic instability, the aggregation number can be computed using the following relation [25]: N(α) =
(4/3)πR(α)3 α[VA + nh,A VD∗2 O + β(Vcon + nh,con VD∗2 O )] +xB (VB + nh,B VD∗2 O )
(12)
where xA and xB are the mole fractions, VA and VB are the molecular volumes, nh,A and nh,B are the hydration numbers of the head groups of surfactants A and B in the mixed micelle, respectively, and VBr− and VD2 O are the molecular volumes of D2 O and Br− that VA(A= CTAB) = 606.9 Å3 [26,27], VB(B= TritonX-100) = 521.48 Å3 [28], VA(A= CPCorCPB) = 595 Å3 [29], VD2 O = 30.2 Å3 [26,27], VCl− = 18 Å3 and VBr− = 39.21 Å3 [26,27]. Finally β is defined as the degree of attachment [9]. ahA(A= CTAB) = 97.98, ahB = 53.46, ahA(A= CPCorCPB) = 65.4 aCl− = 40.69 and aBr− = 47.76 Å2 and a, area per surfactant molecules at the micelle core–water interface is given by (A and B refer to ionic and nonionic surfactants, respectively): a=
A(α) N(α)
(13)
where A(α) is the area of the surface charge of micelle. Then the electrostatic and steric contributions to the free energy of mixed micellization can be calculated for the CTAB/TritonX-100 mixture.
4. Results and discussion 4.1. Self-diffusion coefficient Fig. 2 shows the variation of the self-diffusion coefficient of CTAB/TritonX-100 micelles as a function of the mole fraction of CTAB. The self-diffusion coefficient goes through a minimum as the mole fraction of CTAB is increased from 0 to 1 (except the case of 10 mM concentration). The initial decrease in the self-diffusion coefficient
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Fig. 2. Mixed micelle self-diffusion coefficients vs. the mole fraction of CTAB in mixed micelle: 100 mM ( ), 50 mM (䊐), 10 mM (䉬).
as the nonionic surfactant is replaced by ionic surfactant is attributed primarily to increased micelle–micelle repulsion. This repulsion will inhibit the exchange of monomers between different aggregates. Another factor that would lead to the observed decrease in the self-diffusion coefficient at low fractions of ionic surfactant and high total concentrations of mixed surfactant is micellar growth due to the incorporation of the less bulky ionic head groups into the micellar surface. The small magnitude of the decrease in the self-diffusion coefficient suggests that the reduction in the area per polar group that accompanies the introduction of CTAB has only a small effect on micelle size or that the electrostatic interactions are the dominant factor determining the micelle size at low mixing ratios. The increase in the self-diffusion coefficient at higher fractions of CTAB can be mainly attributed to the smaller size of micelles containing high fractions of CTAB due to the larger amount of electrostatic repulsion between the ionic head groups in these micelles, which increases the effective head group area. Fig. 2 also shows that, as the surfactant concentration is increased, the minimum in the self-diffusion coefficient moves to higher concentration of the ionic surfactant. At low total surfactant concentration, the self-diffusion coefficient increases monotonically, without going through a minimum, because the micelle–micelle interactions are negligible in these systems. Moreover, the second process described above that could potentially cause a decrease in self-diffusion coefficient, namely micellar growth due to the incorporation of less bulky ionic head groups into the micelle surface, is not valid for the CTAB/TritonX-100 system because the head group of CTAB is more bulky than that of TritonX-100 in 50 and 100 mM. CPC/TritonX-100 and CPB/TritonX-100 systems displayed behavior similar to that shown by CTAB/TritonX-100 at 10 mM. As shown in Fig. 3, the self-diffusion coefficients of CPC/TritonX-100 and CPB/TritonX-100 mixed micelles increase monotonically with increasing ionic surfactant content. 4.2. Electrostatic and steric free energy The behavior of the CTAB/TritonX-100, CPC/TritonX100 and CPB/TritonX-100 surfactant mixtures can be understood by analyzing the relevant contributions to the
Fig. 3. Mixed micelle self-diffusion coefficient vs. the mole fraction of CTAB (䊉), CPB ( ), CPC (䊐) at a total surfactant concentration of 10 mM.
overall free energy of forming a mixed micelle (gmic ). Figs. 4 and 5 show the variations in the predicted electrostatic and steric contributions to mixed micelle formation, gelec and gst , respectively, as a function of the mole fraction of ionic surfactant, α, for the CTAB/TritonX-100, CPC/TritonX-100 and CPB/TritonX-100 systems. It may be proposed for micelle systems, gst does not change significantly over the entire solution composition range, indicating that the steric contribution has only a small influence on mixed micelle formation. In contrast, gelec decreases dramatically on adding TritonX-100 to the pure ionic surfactant systems, and therefore the contribution to the total free energy due to electrostatic interactions between the surfactant head groups is expected to make a dominant contribution to any synergistic effects arising from the use of a mixture of surfactants. This is particularly the case for the CTAB/TritonX-100 mixture, which shows a much steeper fall in gelec with decreasing ionic surfactant mole fraction compared to the CPC/TritonX-100 and CPB/TritonX-100 mixtures. The decrease in gelec is due to the decrease in the electrostatic repulsions between charged CTAB, CPC or CPB head groups when TritonX-100 molecules are added to the micelles. In CTAB, the charged trimethylammonium group provides a high charge density, which in turn gives rise to a high electrostatic free energy. In CPC and CPB, however, the charge is distributed over a pyridinium ring, leading to a lower surface charge density and consequently a lower electrostatic free energy (Fig. 4). As has been shown in Fig. 4 the gelec of CPC and CPB are approximately equal, then we do not expect to get a much difference between gelec of CPC and CPB since the electrostatic properties of these two surfactants are near together. At low mole fractions of ionic surfactant, TritonX-100 shields the charged heads sufficiently that provide evidence with a reduction of surface charge density and consequently a lower electrostatic free energy. As has
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Fig. 4. Predicted electrostatic free energy (gelec ) contribution to mixed micelle formation as a function of mixed micelle composition for CTAB/TritonX-100 (䉫), CPC/TritonX-100 (䊊), and CPB/TritonX-100 ( ) at a total surfactant concentration of 10 mM.
been shown in Fig. 4 the obtained results in this work was compared with those results reported by Ref. [17]. It is not a superior agreement between two approaches. Because mix is computationally in the first approach to calculate gelec straight forward, in that only gelec for a pure ionic micelle must be calculated, but depends on the micelle shape, its radius and synergetic effect are not considered [17]. Fig. 5 shows that, when CPC or CPB is introduced into TritonX-100 micelles, the steric free energy remains approximately constant. This is because the CPC and CPB head groups are similar in size to the TritonX-100 head group [24–27]. By contrast, however, the steric free energy increases monotonically with increasing CTAB mole fraction because introduction of large CTAB head groups into the micelles increases the steric repulsions, resulting in a larger area per surfactant molecule. In addition, comparison of the steric free energies for the CPC/TritonX-100 and CPB/TritonX-100 systems calculated using Eqs. (6) and (8)
Fig. 5. Predicted steric free energy (gst ) contribution to mixed micelle formation as a function of mixed micelle composition at a total surfactant concentration of 10 mM for the first approach: CTAB/TritonX-100 (䉫), CPC/TritonX-100 ( ), CPB/TritonX-100 ( ); the second approach: CTAB/TritonX-100 (䊐), CPC/TritonX-100 (×), CPB/TritonX-100 (䊊).
reveals that the nature of the counter ion has little influence on gst . This observation can be attributed to similar size of head groups of CPC, CPB and TritonX-100. However, the values of the steric free energy calculated using Eqs. (6) and (8) for the CTAB/TritonX-100 system show poor agreement. Given the large size of the CTAB head group compared to the TritonX-100 head group, the CTAB head group may have occupied more than one lattice space in the calculation using Eq. (6), and hence the results obtained using Eq. (8) are expected to be closer to the true value. 4.3. Apparent radius and aggregation number The size of the mixed micelles in the studied solutions is mainly determined by the repulsions between head groups, including repulsions of both steric and electrostatic origin. To determine the synergistic effects arising from using a mixture of surfactants, the interactions between micelles are considered above the CMC because with an increase in concentration of surfactants solution, the size and aggregation numbers of micelles above the CMC are changed. At the CMC, the micelles are relatively small and it is difficult to
Fig. 6. Apparent radius of micelle vs. the mole fraction of CTAB (䉬), CPC (䊐), CPB ( ) in mixed micelle at a total surfactant concentration of 10 mM.
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Fig. 7. Aggregation number of mixed micelle vs. the mole fraction of CTAB (䉬), CPC (䊐), CPB ( ) in mixed micelle at a total surfactant concentration of 10 mM.
determine their size experimentally. A surfactant concentration of 10 mM was considered (i.e., 50 times higher than the CMC) [10]. Additional characteristics of solutions of higher surfactant concentration are that spherical micelles cannot form and the Stokes–Einstein equation is not valid. Robson and Dennis [30] proposed that a sharp boundary does not exist between the hydrophobic interior and the oxyethylene chains, in contradiction to the classical picture of micelles. If the first few oxyethylene groups at the octylphenyl end of some TritonX-100 molecules were contained in the hydrophobic core, it would be possible to accommodate a spherical model for the hydrophobic region as well as the total micelle. The core would have to have a radius of about 23 Å to accommodate the volume of the octylphenyl moieties. For the whole micelle, they assumed that the hydrophilic region extends one oxyethylene chain length (17 Å) beyond the hydrophobic core making the radius of the whole micelle about 43 Å. It should be noted that the polydispersity of the oxyethelen chains could result in a nonuniform distribution of oxyethylene groups around the micelle, allowing the overall shape to be closer to spherical. Also, according to Ruiz’ article, the incorporation of CTAB monomers in the TritonX-100 micelles produced a minor distortion in their micelle structure [11]. Figs. 6 and 7 show the apparent radius of the mixed micelles and the weight-average micelle aggregation numbers as a function of the mole fraction of the ionic surfactant for the CTAB/TritonX-100, CPC/TritonX-100 and CPB/TritonX-100 mixtures at a total surfactant concentration of 10 mM. In the CTAB/TritonX-100 system, the average mixed micelle aggregation number decreases monotonically with increasing CTAB content due to the larger head groups of this surfactant compared to TritonX-100, which cause the steric free energy to increase upon introducing increasing numbers of CTAB molecules into the pure TritonX-100 micelle. Another factor contributing to the observed decrease in the average mixed micelle aggre-
gation number is the increase in electrostatic free energy as the ionic surfactant content of the mixed micelles increases, which causes a reduction in the size of the mixed micelles. As a consequence of the smaller micelle size, the optimal average surface area per hydrophilic group increases and the number of surfactant chains per micelle decreases; the smallest R(α) corresponds to the system with the highest surface charge density, i.e., pure CTAB micelles. At low CTAB content, the steric free energy is the main factor determining micelle size, whereas at high mole fractions of CTAB the influence of the electrostatic free energy dominates. In the CPC/TritonX-100 and CPB/TritonX-100 systems, the average mixed micelle aggregation number is approximately independent of CTAB content. This is because, in each of these systems, the head groups of the ionic and nonionic surfactants are of approximately the same size, causing gst to be approximately constant. In addition, the charged CPC or CPB head groups provide lower surface charge density, leading to lower gelec . The results show that the counter ion has no significant effect on the size and average aggregation number of the mixed micelles.
5. Conclusions The micelle self-diffusion coefficient goes through a minimum as the mole fraction of the ionic surfactant is increased from 0 to 1. The initial decrease in the self-diffusion coefficient is due to an increase in micelle–micelle repulsion, while the increase in self-diffusion coefficient at higher ionic surfactant content is due to a decrease in the size of the micelles. As expected, this minimum in the self-diffusion coefficient is not observed at low total surfactant concentration, because the micelle–micelle interactions are negligible in such systems; instead, the self-diffusion coefficient increases monotonically with increasing ionic surfactant content due to the increasing numbers of charged head groups
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in the micelles, which leads to increased repulsion between the head groups and a consequent decrease in the size of the micelles. To study micelle growth at a total concentration above the CMC, PFG-NMR and viscometry were used to measure the average aggregation number and size of the micelles or hydrodynamic radius. The results show that the size of the mixed micelles is determined by interplay between the steric and electrostatic intermicellar interactions. In CTAB/TritonX-100 mixed micelles, the free energy contributions due to the steric and electrostatic interactions (gst and gelec ) caused the aggregation number and hydrodynamic radius to decrease with increasing mole fraction of CTAB. In CPC/TritonX-100 and CPB/TritonX-100 mixed micelles, in contrast, gst remained approximately constant and gelec was low. Due to these factors, the aggregation number and hydrodynamic radius were approximately independent of the mole fraction of ionic surfactant.
which describes the curvature correction to the planar problem drops out. The planar limit can be recovered by writing, X = κr and x = X + ξ, and, taking the limit x → ∞, Eq. (A.4) reduces to
Appendix A
We suppose that the micelle is a spherical micelle of radius R, and that, for simplicity, the solvent contains a 1:1 electrolyte. For this system, the nonlinear Poisson–Boltzmann equation that describes the distribution of ions about the sphere, and the boundary conditions satisfied by the potential ψ = ψ(r), are
d2 ψ 2 dψ 8πn0 e eψ 2 ∇ ψ= 2 + = sinh (A.1) r dr ε kT dr where n0 is the bulk electrolyte concentration, e is the magnitude of the unit charge, ε is the dielectric constant of the solvent, and T is the temperature. The monomers are assumed to be completely dissociated and univalent. The boundary conditions are:
d2 y = sinh y dx2 The first integral is dy y0 s= − = 2 sinh dX X=κR 2 1/2 s s 2 y0 = 2 ln + +1 2 2
where y0 is the scaled surface potential. If we substitute this expression into the curvature term of Eq. (A.4), we obtain
d 1 dy 2 dy 0 dy 2 dx y 0 sinh y dy − 2 =
(A.2)
y0
0
y0 0
1 x
dy dy dx
(A.6)
The main contribution to the second integral of the right side comes from y = y0 , near which x = X = κR. To a first approximation we can remove this factor from the integral to give 1 2
dy dx
2
y=y0
2 y0 dy 2 y0 ∼ − dy = 2 sinh 2 x 0 dx = 2 sinh2
ψ(r) → 0 as r → ∞ dψ 4πσ =− at r = R dr ε
(A.5)
y 0
2
+
y 8 0 cosh −1 κR 2 (A.7)
We used the planar approximation Eq. (A.5) to carry out the last integral. Hence
We introduce dimensionless variables through the substitutions:
(A.4)
y dy 0 = s = 2 sinh dx y=y 2 0 !
" y 4 cosh(y0 /2) − 1 1/2 0 × 1+ = 2 sinh 2 κR 2 sinh (y0 /2) y 4 0 + tanh + ··· (A.8) κR 4
where σ = e/a and a is the area per surfactant molecule head group. In the limit r → ∞, the term in (2/x)(dy/dx)
Comparison of this analytic approximation with the numerical solution of Eq. (A.4) showed that it is quite accurate; the worst error obtained for κR ≥ 0.5 was 5% at κR = 0.5. For larger values of κR the error diminishes rapidly. The electrostatic free energy can now be evaluated explicitly as
y=
eψ kT
x = κr
κ2 =
8πn0 e2 εkT
− (A.3)
where κ−1 is the inverse Debye screening length. Hence Eqs. (A.1) and (A.2) become d2 y 2 dy + = sinh y x dx dx2 dy 4πσe 4πe2 |x=κR = − =− = −s dx εκkT εκakT
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kT 2 εκ # s gel = a 0 ψ0 dσ = a y0 ds ⇒ e 4π 0
s 2 1/2 s gel = 2kT ln + 1+ 2 2 #σ
(1 + (s/2)2 )1/2 − 1 s/2
4 1 + (1 + (s/2)2 )1/2 − ln κRs 2
(A.9)
−
For cylindrical or ellipsoid micelles, the analysis can be carried out in the same way. Thus, for cylindrical micelles the operator ∇ 2 becomes d2 /dr2 + (1/r)(d/dr) instead of d2 /dr2 + 2/r for spheres. Hence, the curvature term will be reduced by a factor of 2. Similarly, it can be evaluated for other shapes. The analysis can also be performed for systems in which the surface of the micelle has dissociated groups, and/or for a mixture of 2:1 and 1:1 electrolyte. At high surfactant concentration ( CMC) the approximations of the Debye–Hückel theory and Poisson–Boltzmann equation break down. In this concentration regime, micelles other than the one under consideration cannot be considered as point ions and intermicellar interactions must be built into the formalism. They could be included either by the Wigner–Seitz method or through a perturbation expansion built by using the above approximate analytic forms.
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