5756
Langmuir 1997, 13, 5756-5758
Notes Critical Micelle Concentration and Average Aggregation Number Estimate of Zwitterionic Amphiphiles: Salt Effect F. H. Florenzano† and L. G. Dias*,‡ Departamento de Bioquı´mica, Departamento de Quı´mica Fundamental, Laborato´ rio Interdepartamental de Cine´ tica Ra´ pida, Instituto de Quı´mica, Universidade de Sa˜ o Paulo, Caixa Postal 26077, Sa˜ o Paulo, S.P., 05599-970 Brazil Received February 19, 1997
Introduction Zwitterionic aqueous micelles are electrically neutral charged aggregates composed of a hydrophobic core and a solvated dipolar region. In principle, these macromolecular aggregates should not be influenced by the addition of electrolytes and therefore should be modeled as simple spherical capacitors. Accordingly, salt condensation at the interfacial region should not occur and micellar parameters such as critical micelle concentration (cmc) and average aggregation number (〈g〉) should be invariant with salt.1,2 However, ionic conductance,3 static light scattering,3 cmc, and 〈g〉4,5 data indicate that these micelles are affected by the bulk ionic strength. This apparent contradiction was solved by allowing the entrance of ions into the dipolar region. Using the Debye-Huckel approximation, the mean field potential of the double charged layer (spherical capacitor with outer shell permeable and inner shell impermeable to ions) was obtained.3 The model fitted qualitatively second virial coefficients and conductivity data. In order to complement previous work by Baptista et al.3 and Nagarajan and Ruckenstein,1,2 an extension of the thermomolecular model which takes into account explicitly the influence of the ionic strength in the micellar parameters is presented. The model predicts qualitatively or semiquantitatively salt effects on the cmc and 〈g〉 of zwitterionic micelles. Theory
∆µ°g,dip(sphere) ) g|e|2d/[2(Rs + δ)(Rs + d + δ)]
(2)
where g ) aggregation number, |e| ) proton charge, ) dielectric constant, d ) distance between the charges of the capacitor, δ ) separation between the hydrophobic surface and the location of charge on the head group, and Rs ) radius of the hydrocarbon core (4πRs3/3 ) gvs, with vs ) volume of the hydrophobic tail of the surfactant molecule). Equation 2 is also employed for globular aggregates and endcaps of spherocylinders. The cylindrical portion of the spherocylinders is1,2
∆µ°g,dip(cylinder) ) (g - gs)|e|2 ln[(ls + d + δ)/(ls + δ)]/Lc (3) where gs ) aggregation number of the largest spherical micelle,1,2,6 ls ) extended length of the amphiphile tail, and Lc ) length of the cylindrical part of the micelle. The total contribution for the spherocylindrical micelle is simply
g∆µ°g,dip ) gs∆µ°g,dip(sphere) + (g - gs)∆µ°g,dip(cylinder) (4)
Nagarajan and Ruckenstein define ∆µg (micellization free energy) as the sum of independent factors:1,2,6
∆µ°g ) ∆µ°g,tr + ∆µ°g,def + ∆µ°g,int + ∆µ°g,st + ∆µ°g,dip
water to aggregate, ∆µ°g,def ) deformation free energy of the surfactant tail, ∆µ°g,int ) aggregate core-water interfacial free energy, ∆µ°g,st ) head group steric interactions, and ∆µ°g,dip ) head group dipole interactions. The factors to be considered in eq 1 can be therefore specified, and their contribution can be calculated separately; thus, the term ∆µ°g,dip is extended to include the salt effect. Although all terms on the right side of eq 1 are salt dependent, only ∆µ°g,dip is significantly affected. Thus corrections for the other terms in eq 1 were neglected. In the following, the calculations of the geometric parameters and the various energy contributions to eq 1 employ Nagarajan and Ruckenstein’s expressions. Globular aggregates are treated as equivalent spheres. Dipole-dipole interactions can be estimated by visualizing the arrangement of the poles of the head groups as constituting an electrical capacitor. The distance between the surfaces of the capacitor is equated with the average distance of charge separation of the zwitterionic head group. For spherical aggregates, Nagarajan and Ruckenstein’s expression for the contribution of the dipole interactions is1,2
(1)
where ∆µ°g,tr ) transfer free energy of the surfactant tail from * Author for correspondence. E-mail:
[email protected]. Fax: (55) (011) 815-5579. Permanent address: Faculdade de Engenharia & Cieˆncias Tecnolo´gicas, Universidade Metropolitana de Santos, Rua da Constituic¸ a˜o 374, Santos, S.P., CEP 11015-470. Phone: (55) (013) 233-3400. Fax: (55) (013) 235-2990. † Departamento de Bioquı´mica. ‡ Departamento de Quı´mica Fundamental. (1) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580. (2) Nagarajan, R.; Ruckenstein, E. Langmuir 1991, 7, 2934. (3) Baptista, M. S.; Cuccovia, I. M.; Chaimovich, H.; Politi, M. J.; Reed, W. F. J. Phys. Chem. 1992, 96, 6442. (4) Herman, K. W. J. Colloid Interface Sci. 1966, 22, 352. (5) Essadam, H.; Pichot, C.; Guyot, A. Colloid Polym. Sci. 1988, 266, 462. (6) Tanford, C. The Hydrophobic Effect, 2nd ed.; John Wiley: New York, 1980. (7) Hartley, G. S. Aqueous Solutions of Paraffin Chain Salts; Hermann: Paris, 1936.
S0743-7463(97)00176-5 CCC: $14.00
Expressions 2 and 3 are valid for an electrical capacitor impermeable to ions. Thus to model the effect of salt on the properties of the zwitterionic aggregates, it is necessary to generalize them to the case of a permeable electrical capacitor. In the Debye-Huckel limit, the potential of mean field in the region between the surfaces of a permeable spherical capacitor can be expressed as3
φ h (r) ) {-g|e| exp[-κ(Rs + d + δ)]/[2κ(Rs + d + δ)]} × exp(-κr)/r + {{g|e| exp[κ(Rs + δ)]/[(1 + κRs + κδ)]} + {g|e| exp[κ(Rs + δ - d)][1 - κ(Rs + δ)]/[2κ(Rs + d + δ)(1 + κRs + κδ)]}} exp(κr)/r (5) where κ ) reciprocal Debye length, given by
κ2 ) 4π|e|2(
∑F z )/(k T) 2 i i
B
(6)
i
where Fi ) number density of the ion of species i (particles/cm3), zi ) valence of the ion i, kB ) Boltzmann constant, and T ) absolute temperature.
© 1997 American Chemical Society
Notes
Langmuir, Vol. 13, No. 21, 1997 5757 Chart 1. Zwitterionic Surfactant Structures
For a permeable cylindrical capacitor, expression 7 is obtained:
φ h (r) ) -2(g - gs)|e|K0(κRout)I0(κr)/(Lc) + 2(g - gs)|e|I0 × (κRin)K0(κr)/(Lc) + 2(g - gs)|e|I1(κRin)[K0(κRout) - K0 × (κRin)]K0(κr)/(LcK1(κRin)) (7) where Rout ) ls + d + δ ) outer radius of the cylinder, Rin ) ls + δ ) inner radius, and I0, I1, K0, and K1 are the modified Bessel functions of the first and second kind of zero and first order, respectively. Having the potentials, the electrical work (Wel) to charge the capacitors (via Debye process) can be calculated:
Wel )
∫
Q
0
dQ φ h (Q;Rin)
(8)
The electrical work is related to the dipole-dipole interaction contribution:
∆µ°g,dip ) Wel/g For spherical and cylindrical symmetries, one obtains
∆µ°g,dip(sphere) ) g|e|2d/{2(Rs + δ)(Rs + d + δ)[1 + κ(Rs + δ)]} (9)
Table 1. Calculated Molecular Constants for Zwitterionic Surfactants DDPS and OXC18 surfactant head group
vpa (cm3/mol)
apa,b (Å2)
a0a (Å2)
db (Å)
δ (Å)
DDPS OXC18
170.00 191.01
47.05 82.60
21 21
6.00 3.84
1.5 1.5
a Symbols are defined as follows: v ) effective volume of the p polar head group, ap ) effective cross-sectional area of the polar head group near the micellar surface, and a0 ) area per molecule of the core surface shielded from contact with water by the polar head group. b Parameters ap and d were optimized by the best agreement between cmc and 〈g〉 in the absence of salt, keeping vp ) apd.
and
v(CH3) ) 54.6 + 0.124(T - 298) (in Å3)
(16)
2
∆µ°g,dip(cylinder) ) (g - gs)|e| [K0(κRout) - K0(κRin)] × [(1/κRin) - 2I0(κRin)K1(κRin)]/[LcK1(κRin)] (10) ∆µ°g denotes the difference in the standard chemical potential of a surfactant molecule present in an aggregate with g monomers from that in the singly dispersed state expressed by the sum of various factors (eq 1). In the case of spherical or globular micelles, the average aggregation number (〈g〉) is considered equal to the maximum value of g in the aggregate distribution curve (g). An appropriate expression for this is
(∂Xg/∂g)|g)g ) 0 ∴ Xg ) X1g exp(-g∆µ°g/kBT)
(11)
0 ) -kBT ln X1 + ∆µ°g + g∂∆µ°g/∂g
(12)
where Xn ) molar fraction of the aggregates with n monomers in solution. The critical micelle concentration can be defined as the value where the total amount of the surfactant in the micellized form at which g is a maximum is equated to that in the dispersed form:1,2,7
Xcmc ) X1 ) gXg
∑
gXg
(14)
g
Comparison between Theoretical Prediction and Experimental Results Predictions for the salt effect of the zwitterionic surfactants DDPS and OXC18 (Chart 1) are compared with experimental data (Tables 2 and 3). Calculated molecular constants for DDPS and OXC18 are summarized in Table 1. The molecular volume (vs) and the extended length (ls) of the surfactant tail are calculated from the additive contributions of methylene and methyl groups using eq 15:1,2,6
vs ) v(CH3) + (nc - 1)v(CH2) where
and
ls ) 1.50 + 1.265nc (in Å)
(15)
(17)
where nc ) number of carbon atoms in the hydrocarbon tail. Calculated and experimental cmc data as a function of the ionic strength are collected in Table 2. For OXC18 Table 2. Influence of NaCl Concentration on the cmc of DDPS and OXC18 surfactant DDPSa OXC18b
(13)
When micelles are spherocylinders, the cmc can be calculated by1,2
Xcmc ) X1 )
v(CH2) ) 26.9 + 0.0146(T - 298) (in Å3)
a
Csalt (mol/L)
cmcexp (mol/L × 103)
cmctheor (mol/L × 103)
0 0.15 1.0 0 0.02 0.1 1.0
3.6 3.0 2.1 0.0061 0.0060 0.0034 0.0017
3.9 0.61 0.41 0.0061 0.0033 0.0025 0.0019
Data from ref 4, t ) 30 °C. b Data from ref 5, t ) 25 °C.
theoretical values are in good agreement with experimental ones; for DDPS, however, theoretical values decrease faster. In Table 3, calculated 〈g〉 values are shown. When these values are compared with the calculated aggregation numbers for the largest spherical geometry (gs) possible for DDPS and OXC18 (55 and 116, respectively), it can be deduced that DDPS micelles are within spherical to globular geometries whereas OXC18 micelles are well below the largest spherical geometry (Table 3). Modeling shows that although cmc estimates for DDPS are not so good, 〈g〉 values are reasonable. The divergence between theoretical and experimental cmc values can be assigned exclusively to the approximation used for the dipolar term. The Debye-Huckel approximation leads to a large ion density and exaggerates the mean field potential shielding between the surfaces of the permeable capacitor. In this manner, the calculated dipolar term decreases sharply with salt as evidenced in the DDPS cmc estimate (Table 2). On the other hand,
5758 Langmuir, Vol. 13, No. 21, 1997
Notes
Table 3. Influence of NaCl Concentration on the Average Aggregation Number (〈g〉) of DDPS and OXC18 surfactant
Csalt (mol/L)
〈g〉exp
〈g〉theor
DDPSa
0 0.15 1.0 0 0.02 0.1 1.0
55 60 61
39 58 62 35 36 36 37
OXC18
a
Data from ref 4, t ) 30 °C.
OXC18 shows good agreement between calculated and experimental data. For this surfactant, the sterical term (∆µ°g,st) hinders the micellar growth (〈g〉 is practically invariant with added salt), avoiding abrupt changes in X1 and therefore in Xcmc. Thus this quantitative agreement seems to be specific to systems having high ap values. To improve the fit, the effect of the ionic strength on ap
and d, that is on the head group conformation parameters, should be included. The use of formalisms which explicitly account for the ionic size would also lead to a better fit. These effects are currently being studied. In summary, the salt effect on zwitterionic aggregates is similar to the effect on ionic ones. That is, head group repulsive interactions are shielded by electrolyte, leading to a decrease in the cmc and an increase in 〈g〉. On this theoretical level, qualitative or semiquantitative concordance must be expected. Acknowledgment. The authors are thankful for valuable discussions with Dr. M. J. Politi. They also acknowledge the Ph.D. sponsorship program of the Brazilian agency CNPq. This work is part of L.G.D.’s Ph.D. thesis. LA970176N
