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Langmuir 1997, 13, 3646-3651
Structure and Thermodynamics of Micellar Solutions in Isotropic and Cell Models N. Rebolj,† J. Kristl,† Yu. V. Kalyuzhnyi,‡ and V. Vlachy*,§ Faculty of Pharmacy, University of Ljubljana, Askerceva 7, 1000 Ljubljana, Slovenia, Institute for Physics of Condensed Matter, 290011 Lviv, Ukraine, and Faculty of Chemistry and Chemical Technology, University of Ljubljana, Askerceva 5, P.O. Box 537, 1000 Ljubljana, Slovenia Received January 21, 1997. In Final Form: April 22, 1997X This paper presents a combined Monte Carlo and integral equation study of micellar solutions. In the first part of the paper, new simulation results for an isotropic model of micellar solutions containing macroions and counterions are compared with the results of the much simpler cell model. The conclusion is that the spherical cell model, in conjunction with the Poisson-Boltzmann equation, yields reliable results for the osmotic pressure over the whole concentration range studied here. The conclusion is valid for solutions with monovalent counterions up to moderate concentrations, which have not been studied before. However, for model solutions containing divalent counterions, the cell model is not an adequate approximation. In the second part of the paper, the results for a three-component model of micellar solutions, containing macroions, counterions, and a free amphiphile, are presented. Again the PoissonBoltzmann cell model results are tested against the results of the isotropic model. The thermodynamics and structure of the isotropic model are obtained via two integral equation theories: (i) the hypernetted chain (HNC) integral equation and (ii) the so-called associative HNC (two-density theory) approximation, developed recently. Overall, the agreement between the isotropic and cell model calculations (note that the latter are based on the Poisson-Boltzmann approximation) for the osmotic pressure is good.
1. Introduction The self-association of ionic surfactant into micelles is strongly influenced by Coulombic forces. In theoretical predictions these interactions are often taken into account using the so-called cell model, with the Poisson-Boltzmann (PB) equation as an additional approximation.1-3 The spherical cell model is based on the asymmetry in size and charge between the macroions and small ions. The basic assumption is that the macroions repel each other strongly and as a consequence are organized mostly at larger distances from each other. Under such conditions, each macroion is assigned its own spherical cell and they are assumed to be completely screened by the available counterions. The volume of a cell defines the concentration of a micellar solution. A more realistic approach to the study of micellar solutions, however, is the istropic model, where the solution is presented as a highly asymmetric electrolyte4-6 and therefore all interacting ionic species, i.e. macroions, counterions, and possibly a free amphiphile, are treated on an equal basis. The validity of the cell model has not been thoroughly tested so far. An important previous investigation was that due to Linse and Jo¨nsson.7 The authors used the Monte Carlo method to assess the validity of the approximation, comparing the thermodynamic properties calculated for the cell model with the results for the isotropic model. The main conclusion of their study was †
Faculty of Pharmacy, University of Ljubljana. Institute for Physics of Condensed Matter. § Faculty of Chemistry and Chemical Technology, University of Ljubljana. X Abstract published in Advance ACS Abstracts, June 15, 1997. ‡
(1) Gunnarsson, G.; Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1980, 84, 3114. (2) Bratko, D.; Lindman, B. J. Phys. Chem. 1985, 89, 1437. (3) Brasher, L. L.; Herrington, K. L.; Kaler, E. W. Langmuir 1995, 11, 4267. (4) Belloni, L. Chem. Phys. 1985, 99, 43. (5) Bratko, D.; Friedman, H. L.; Zhong, E. C. J. Chem. Phys. 1986, 85, 377. (6) Hribar, B.; Kaluyzhnyi, Yu. V.; Vlachy, V. Mol. Phys. 1996, 87, 1317. (7) Linse, P.; Jo¨nsson, B. J. Chem. Phys. 1983, 78, 3167.
S0743-7463(97)00057-7 CCC: $14.00
that the cell model yields good agreement with the isotropic model results for dilute solutions of macroions and counterions. In contrast, for moderate to concentrated solutions the agreement between the two calculations was poor. The seminal calculations of Linse and Jo¨nsson7 have been performed on a relatively small number of macroions, using the so-called ‘minimum image’ boundary conditions, which may not be appropriate for strongly interacting systems.8 The authors themselves indicated7 that the simulation method used in their work may produce erroneous results for more concentrated solutions. It should be emphasized that the cell model, especially if used with a theory based on the solution of the PoissonBoltzmann equation (or with the modified PB equation9), provides a convenient way to analyze the experimental data. Considering this, a more systematic study of the validity of the cell model under conditions valid for typical micellar solutions seems to be warranted and it is presented in this paper. In what follows we present new computer simulations and integral equation results for the isotropic model of micellar solutions. The ions were treated as charged hard spheres in a continuous dielectric. In all simulations the Ewald boundary conditions were used to minimize the effects of a finite sample. The results for the osmotic pressure and the excess internal energy were compared with the results of the much simpler theory, based on the cell model and the solution of the Poisson-Boltzmann theory. In this, the first part of the paper, the model parameters were chosen to facilitate a comparison with previous calculations.7 The results for solutions with mono- and divalent counterions are presented. For highly charged micellar solutions, e.g. for sodium dodecyl sulfate solutions, the computer simulations become very time consuming and therefore impractical. In this case the hypernetted chain (HNC) integral (8) Linse, P. J. Chem. Phys. 1991, 94, 3817. (9) Bhuiyan, L. B.; Outhwaite, C. W.; Bratko, D. Chem. Phys. Lett. 1992, 193, 203.
© 1997 American Chemical Society
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Langmuir, Vol. 13, No. 14, 1997 3647
equation,10 or the newly developed two-density theory,11 was used to compute the properties of the micellar solution in the isotropic model. These calculations were compared with the results of the Poisson-Boltzmann cell model for the same micellar system. We show that the spherical cell model is a very reasonable approximation if monovalent counterions are present in the solution. This conclusion is also valid for more concentrated solutions, which have not been studied extensively before, and for more realistic micellar models where a free amphiphile is present. On the other hand, the cell model may not be an acceptable approximation for solutions containing divalent counterions, with the possible exception of very dilute solutions. 2. Cell Model and Poisson-Boltzmann Theory The Poisson-Boltzmann cell model is now well-known and has been used by numerous authors.1-3,7,9,12 The micelles are modeled as uniformly charged hard spheres of radius rm and of charge -|zm|e, where e is the positive elementary charge and |zm| is the aggregation number. The macroion is located at the center of a spherical cell, and the cell volume, V, is determined by the micellized amphiphile concentration, nb ) |zm|/(NAV), where NA is Avogadro’s number. The counterions are distributed through the cell according to the Poisson-Boltzmann equation, which in spherical symmetry reads (Rc is the radius of a cell)
1 d r dr 2
( ) r2
dψ dr
1 )-
∑i eziFi(r)
r0
Fi(r) ) Fi(Rc)e
(1)
-zieψ/kBT
In eq 1 ψ(r) is the mean electrostatic potential, 0r is the permittivity of the solvent, and Fi(r) is the number density of simple ions of charge zie at a distance r from the center of the cell. The sum in eq 1 is over all mobile ions in the cell. As usual, kB is Boltzmann’s constant and T is the absolute temperature. Equation 1 has to be solved numerically, subject to the boundary conditions given by the Gauss Law. The osmotic pressure, Π, was calculated according to eq 2, where the sum is taken over concentrations ni(Rc) of mobile species at the cell boundary, i.e. at r ) Rc; nm ) nb/|zm| is the concentration of macroions and, as usual, R is the gas constant.
Π ) RT
∑i ni(Rc) + nm
zazbe2 4πr0r uab(r) ) ∞
uab(r) )
for
r g rab
for
r < rab
(3)
where rab is (ra + rb) and the whole solution is treated as a continuous dielectric, with a dielectric constant equal to that of the pure solvent at temperature T. The indices a and b denote either a macroion (m), a counterion (c), or a co-ion (co). The major difference from the cell model is that macroions are allowed to move freely and that they are not fixed in the cell. The asymmetrical electrolyte, as described above, has been studied by various statisticalmechanical techniques.4-8,11 In this paper the canonical Monte Carlo method and two integral equation theories, namely, the HNC approximation4,5 and the recently proposed two-density theory,11 were used to study this model. The Monte Carlo method applied in this paper has been discussed recently.6,11 It is important to mention that the simulations were performed on a cubic box, with 64 macroions (solutions with monovalent counterions) or 128 macroions (solutions with divalent counterions) in the basic cell. Very long simulations are needed to obtain good statistics: in the present simulations averages are taken over 30 million configurations with at least 5 million configurations spent for the equilibration. Ewald boundary conditions13 are used to account for the finite size of the system, and this is an important improvement over the previous study.7 Since the computer simulations of highly asymmetric electrolytes are time consuming, only systems containing macroions and counterions are studied. This is not a realistic representation of micellar solutions, where a free amphiphile is always present. 3.2. Hypernetted Chain Approximation. The HNC equation is an excellent approximation for studying low molecular weight electrolyte solutions.10 For highly asymmetric electrolytes it has been first applied by Belloni4 and subsequently used by many authors.5,6,11 The starting point of the theory is the Ornstein-Zernike equation, which correlates the total correlation function hab(r) and the direct correlation function cab(r):
hab(r12) ) cab(r12) +
∑k Fk∫dr3 cak(r13)hkb(r32)
(4)
where Fk is the number density of component k. The HNC closure condition is given by
hab + 1 ) exp[-Uab(r12)/kBT + hab(r12) - cab(r12)] (5) (2)
The second term in eq 2 gives the (ideal) contribution of a macroion to the osmotic pressure of the solution. 3. Isotropic Model of a Micellar Solution 3.1. Model and Monte Carlo Method. In contrast with the cell model, isotropic models treat all the species on an equal basis. The micellar solution is depicted as an asymmetric electrolyte where the ions differ in size and charge. The solvent averaged potential is given by (10) Rasaiah, J. C. Theories of Electrolyte Solutions. In The Liquid State and its Electrical Properties; Kunhardt, E. E., Christophorou, L. G., Luessen, L. H., Eds.; NATO ASI Series B; Plenum: New York, 1988; Vol. 193. (11) Kalyuzhnyi, Yu. V.; Vlachy, V. Chem. Phys. Lett. 1993, 215, 518. (12) Bratko, D.; Vlachy, V. Colloid Polym. Sci. 1985, 263, 417.
The Monte Carlo simulations indicate that the HNC approximation provides meaningful results for moderately concentrated solutions of macroions and counterions.6,14 However, for dilute solutions and/or highly charged macroions, it becomes more and more difficult to obtain a convergent solution of eqs 4 and 5. Finally, below a certain concentration or above a certain charge on a macroion, the solution of the HNC integral equation disappears.15 In this paper a version of the numerical algorithm developed by Ichiye and Haymet16 is used. The calculations were performed over 4096 integration points. As a result of the numerical calculation on the isotropic model we obtain the pair correlation functions, gab ) hab (13) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; Oxford University Press: Ithaca, New York, 1989. (14) Vlachy, V.; Marshall, C. H.; Haymet, A. D. J. J. Chem. Phys. 1989, 111, 4160. (15) Belloni, L. Phys. Rev. Lett. 1986, 57, 2026. (16) Ichiye, T.; Haymet, A. D. J. J. Chem. Phys. 1988, 89, 4315.
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+ 1, and the thermodynamic parameters. The most important of these for our study is the osmotic pressure, which is calculated via the virial route.17 3.3. Two-Density Theory. In order to avoid the problem of convergency, the two-density theory (the socalled associative HNC approximation) has been proposed.11,18 The theory is based on Wertheim’s formalism,19 extended to solutions of highly asymmetric electrolytes. In this theory, the potential energy between macroions and counterions is divided into an associative (strongly attractive) part Uass(r) and a nonassociative part Un(r):
Uab(r) ) Uab n (r) + (1 - δab)Uass(r)
(6)
where δab is the Kronecker delta. Due to asymmetry in size, we can treat a counterion as bondable to one macroion only, while each macroion can bond an arbitrary number of counterions. The two-density version of the OrnsteinZernike equation is then derived:11
Hab(r12) ) Cab(r12) +
∑k Gk∫Cak(r13) Hkb(r32) dr3
(7)
where H, C, and G are matrices defined as pc Hpp(r) ) hpp(r); Hpc(r) ) (hpc 0 (r),h1 (r));
Hcp(r) )
( )
(
cc hcp h00 (r) 0 (r) cc ; H (r) ) cc hcp h (r) 1 10(r)
Gp ) Fp; Gc )
(
Fc Fc0
Fc0 0
)
cc h01 (r) ; (8) cc h11 (r)
)
4. Results
The subscripts 0 and 1 describe the state (0 for unbonded and 1 for bonded) of the corresponding counterion. The partial correlation functions are related to the regular correlation function, hab(r) ) gab(r) - 1, by the relation FahabFb ) [GaHab(r)Gb]00, where ‘00’ denotes the first element of the matrix. An HNC-like closure (eqs 9 and 10) has been proposed to solve the Ornstein-Zernike equation.11 In these cc equations tpp(r), tpc i (r), and tij (r) are the elements of the matrix T ) H - C and fass ) exp(-βUass) - 1.
hpp(r) ) exp[-βUpp(r) + tpp(r)] - 1 cc cc hijcc(r) ) [δi0δj0 + δi1δj0t10 (r) + δi0δj1t01 (r) + cc cc δi1δj1t11 (r)] × exp[-βUcc(r) + t00 (r)] - δi0δj0 (9) cp pc pc hpc i ) hi ) [δi0 + δi1(ti (r) + fass(r))] exp[-βUn (r) +
tpc 0 (r)] - δi0 (10) The two densities, Fc0 and Fc, are related as Fc0 ) Fc(1 + IFp)-1, where I is given by
∫0∞gcp0 (r) fass(r)r2 dr
I ) 4π
potential is included in the nonassociative part. It is very important for the results to be insensitive to the way the potential is divided, i.e. to the particular choice of Uass(r).18,20 In this calculation that part of the total potential between counterions and macroions, Upc(r), which is less than a fixed value U0, is chosen to be the associative part in eq 6. We assume that the bonded counterions are located within the distance rm, at which the minimum of the function y(r) ) r2gpc(r) is located.6 This means that Uass(r) ) 0, for all rs > rm. The value of U0 can be obtained from the condition that rm is equal to the larger of the distances rs at which Uass(rs) ) 0. Our previous studies indicate that this division yields results which are not sensitive to the choice of U0 in a broad range of U0 values. There were some problems noted with the application of this model of splitting for the highly charged systems studied in section 4.2. At higher concentrations of decyltrimethylammonium bromide and sodium dodecyl sulfate solutions, a secondary minimum in the function y(r) may appear. However, as mentioned before, the results for the osmotic pressure are quite insensitive to the actual choice of U0. The respective values of U0 needed to reproduce the calculation are given with the results. The detailed derivation of the theory and its application to Coulombic systems is presented in ref 18. This new approximate integral equation theory (associative HNC) provides good estimates for both the correlation functions and the thermodynamic properties, even in the region of concentrations where the ordinary HNC does not give convergent results.11,18
(11)
An important step of the analysis is the division of the total pair potential into two parts: an associative and a nonassociative part, as defined by eq 6. The associative potential is of short range; the Coulombic part of the (17) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1986. (18) Kalyuzhnyi, Yu. V.; Vlachy, V.; Holovko, M.; Stell, G. J. Chem. Phys. 1995, 102, 5770. (19) Wertheim, M. S. J. Stat. Phys. 1984, 35, 19.
4.1. Two-Component Systems. First the results for solutions containing only macroions and counterions (no free amphiphile), studied previously in ref 7, are presented. The values of the parameters are rm ) 10 Å, radius of counterion rc ) 1 Å, aggregation number of the micelle |zm| ) 12, and all the calculations apply to LB ) e2/ (4π0rkBT) ) 7.14 Å to mimic aqueous solutions at 298 K. A comparison between the Monte Carlo and PB cell model results for the osmotic coefficient is shown in Figure 1. There is semiquantitative agreement between the two sets of results. This agreement extends to moderate concentrations of macroions which were not extensively studied before. Part of the disagreement between the results for the cell and isotropic model can be ascribed to the approximation of the Poisson-Boltzmann equation used to calculate the ionic distributions within the cell. The deficiencies of the Poisson-Boltzmann equation theory are well documented (for a recent review see ref 21), and there exist improved theories which can be used with the cell model. One of them is the so-called modified PoissonBoltzmann theory.2,9,12 The modified PB theory, when applied to the cell model, yields very good agreement with the Monte Carlo data.9,12 While do we not present these results here, we wish to stress that an increase of |zm| ) 12 to |zm| ) 14 (or an increase of LB from the theoretical value of 7.14 Å to a value of 8.46 Å) significantly improves the agreement of the PB theory with the computer simulations for the isotropic model. This yields the conclusion that the cell model provides a reasonable approximation to a more realistic, but numerically much more complicated, isotropic model. There is some interest in the counterion profile around a macroion, and these results are shown in Figure 2. (20) Kalyuzhnyi, Yu. V.; Holovko, M. F.; Haymet, A. D. J. J. Chem. Phys. 1991, 95, 9151. (21) Kjellander, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 894.
Micellar Solutions in Isotropic and Cell Models
Langmuir, Vol. 13, No. 14, 1997 3649 Table 1. Osmotic Coefficient, Φ, and the Excess Internal Energy E Obtained by the Poisson-Boltzmann Cell Model (PB), Monte Carlo Simulations (MC), and Associative HNC Theory (AHNC) n (M)
ΦPB
0.0249 0.0498 0.0996 0.1993 0.3690
0.627 0.612 0.611 0.633 0.686
0.0249 0.0498 0.0996
0.453 0.447 0.452
ΦAHNC
ΦMC
(a) Monovalent 0.597 0.59 0.574 0.57 0.565 0.56 0.577 0.58 0.634 0.63
EPB/RT
EAHNC/RT
EMC/RT
Counterionsa -2.23 -2.46 -2.69 -2.91 -3.10
(b) Divalent Counterionsb 0.35 -2.98 0.35 -3.08 0.24 -3.18
-2.48 -2.76 -3.04 -3.33 -3.61
-2.49 -2.77 -3.04 -3.32 -3.58
-4.05 -4.18 -4.32
a The values of U /k T in the AHNC calculations are U /k T ) 0 B 0 B -3.35, -3.69, -4.12, -4.68, and -5.31, respectively. The absolute error in Φ obtained by the MC simulations is estimated to be (0.005. b The absolute error in Φ obtained by the MC simulations is estimated to be (0.02.
Figure 1. Osmotic coefficient, φ ) Π/Πid, of a -12:+1 electrolyte as a function of the total amphiphile concentration, n. Solid and broken lines represent Poisson-Boltzmann cell model results for |zm| ) 12 and |zm| ) 14, respectively, and symbols denote the Monte Carlo data.
Figure 3. Same as for Figure 2 but for a -12:+2 model solution.
Figure 2. Macroion-counterion distribution function gmc(r). Solid and broken lines represent Poisson-Boltzmann cell model results for n ) 0.0249 mol/dm3 and n ) 0.0996 mol/dm3, while the symbols apply to Monte Carlo simulations (O for 0.0249 mol/dm3 and 4 for 0.0996 mol/dm3). For the PB cell model the distribution function n(r)/n is plotted.
The associative HNC theory was also applied to this model. The results for the excess internal energy are, together with the osmotic coefficients, given in Table 1. The agreement with the computer simulations is very good, showing the potential of the new theoretical approach. In ref 7 (cf. Table V of ref 7) HNC results are also presented. In spite of considerable effort, we have not been able to reproduce these calculations. Fully convergent HNC results were obtained only for the two highest concentrations of surfactant: i.e., for n ) 0.1993 M and 0.3690 M we calculate the values of the osmotic coefficient to be 0.583 and 0.646, respectively. These values are in good agreement with the associative HNC results and with the Monte Carlo data, but in total disagreement with the calculations reported previously.7
The results for divalent counterions are presented in Table 1 and in Figures 3 and 4. The agreement between the simulations and the PB cell model results for the osmotic pressure is poor. In order to explain these results, we present the macroion-macroion distribution function as obtained from the Monte Carlo simulations for the isotropic model at a total concentration of surfactant n ) 0.0996 M in Figure 4. The pair distribution function indicates that the macroions do not distribute themselves at maximum distances, but they rather stay close to each other. The peak in gmm(r) is located at r ≈ 24 Å, so there are on average two layers of counterions between a pair of macroions. This effect is a consequence of the attraction between the macroions as mediated by divalent counterions, and it is discussed in more detail by Hribar and Vlachy.22 An enhancement in gmm(r) can be noticed at r ≈ 24 Å also at lower concentrations of macroions. An attractive force between the two charged surfaces embedded in a solution containing divalent counterions has been noticed in simulation studies23,24 and also confirmed (22) Hribar, B.; Vlachy, V. J. Phys. Chem. B 1997, 101, 3457. (23) Guldbrand, L.; Jonsso¨n, B.; Wennerstroo¨m, H.; Linse, P. J. Chem. Phys. 1984, 80, 2221. (24) Valleau, J. P.; Ivkov, R.; Torrie, G. M. J. Chem. Phys. 1991, 95, 520.
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Figure 4. Macroion-macroion distribution function, gmm(r), for a -12:+2 electrolyte at n ) 0.0996 mol/dm3.
Figure 6. Same as for Figure 5 but for a model nonyltrimethylammonium bromide (NTAB) solution.
Table 2. Model Parameters for Micellar Solutions Studied in This Paper |zm|a cmca (M) Rm (Å) Rc (Å) Rco (Å) a
SOS
NTAB
DTAB
SDS
26 0.15 15.0b 1.5 3.0
30 0.144 18.5b,c 2.0 3.0
39 0.065 19.5b,c 2.0 3.0
64 0.0081 21.0b 1.5 3.0
Reference 26. b Reference 27. c Reference 28.
Figure 7. Osmotic pressure, Π, divided by 2nRT as a function of n/nf for a model decyltrimethyammonium bromide (DTAB) solution. Here the symbols denote results obtained by the associative HNC approximation. The values of U0/kBT are between -7.5 and -7.8, except for the highest concentration studied here (nm ) 0.002 M), where U0/kBT ) 8.0.
Figure 5. Osmotic pressure, Π, divided by 2nRT as a function of n/nf for a model sodium octanesulfonate (SOS) solution as obtained by the hypernetted chain approximation (symbols) and the Poisson-Boltzmann cell model (broken line). The full line represents the ideal contribution to the osmotic pressure.
experimentally.25 Finally, we have to say that none of the integral equation techniques used here is able to reproduce the simulation results for divalent counterions. 4.2. Three-Component Systems. The Monte Carlo simulations are presently restricted to an asymmetry in (25) Kjellander, R.; Marcˇelja, S.; Pashley, R. M.; Quirke, J. P. J. Chem. Phys. 1990, 92, 4399.
charge of about 20:1, and even these simulations are very time consuming. An alternative approach is to use integral equation theories like the HNC approximation or the recently developed two-density theory. These theories were compared by compute simulations in several papers,6,11,18,22 and in part also in section 4.1. The conclusion is that both of them yield good predictions for the structure and thermodynamics of a highly asymmetric model electrolyte. One advantage of the associative HNC11,18 is that the algorithm is more stable and therefore applicable over a broader range of concentrations and charges. The convergency problem, on the other hand, severely limits the use of the regular (one-density) HNC approximation for these systems. In this section we compare the integral equation (isotropic model) and the Poisson-Boltzmann (cell model) results for solutions containing macroions, counterions, and a free amphiphile. Four different micellar systems
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osmotic pressure Π, divided by 2nRT, is plotted as a function of the ratio of the total versus free surfactant, n/nf. Note that n ) nf + nb ) nf + |zm|nm. The results for SOS and NTAB shown in Figures 5 and 6 were obtained using the HNC integral equation (symbols) and the Poisson-Boltzmann cell model calculation (broken lines). The full lines represent the ideal contribution to the osmotic pressure in these solutions. For more highly charged micelles, i.e. for DTAB and SDS model solutions, the HNC approximation fails to provide convergent results. For these solutions the associative HNC (AHNC) theory is used to obtain the equation of state. These results are presented in Figures 7 and 8. The agreement between the cell model results (broken line) and the integral equation results, based on the isotropic model, is good.
Figure 8. Same as for Figure 7 but for a model sodium dodecyl sufate (SDS) solution. The values of U0/kBT needed to reproduce the associative HNC calculations at micellar concentrations nm equal to 5 × 10-6 M, 1 × 10-5 M, 4 × 10-5 M, 7 × 10-5 M, 1 × 10-4 M, 1.4 × 10-4 M, and 2.1 × 10-4 M are -8.4, -7.9, -6.9, -6.9, -6.8, -7.4, and -8.6, respectively.
were studied, mimicking (i) sodium octanesulfonate (SOS), (ii) nonyltrimethylammonium bromide (NTAB), (iii) decyltrimethylammonium bromide (DTAB), and (iv) sodium dodecyl sulfate (SDS) solutions. In all these calculations a free amphiphile is included (nf ) cmc), but no additional simple electrolyte is present. The model parameters used in the calculations are collected in Table 2.26-28 Again, all calculations apply to water-like solutions at 298 K. The results are presented in Figures 5-8, where the (26) Van Os, N. M.; Haak, J. R.; Rupert, L. A. M. Physico-Chemical Properties of Selected Anionic, Cationic and Nonionic Surfactants; Elsevier: Amsterdam, 1993. (27) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1983, 87, 1264. (28) Berr, S. S.; Caponetti, E.; Johnson, J. S., Jr.; Jones, R. R. M.; Magid, L. J. J. Phys. Chem. 1986, 90, 5766.
5. Conclusions The usefulness of the spherical cell model in interpreting experimental results in micellar solutions has been demonstrated in several papers. In cell model calculations the Poisson-Boltzmann equation is most often used to obtain the measurable properties such as osmotic pressure. In this contribution, inspired by the previous study of Linse and Jo¨nsson,7 we present a critical test of the cell model Poisson-Boltzmann approach, comparing it to a more realistic isotropic model. The Monte Carlo method and two integral equation approximations, the HNC and the associative HNC, were used to calculate the structure and thermodynamics of the isotropic model, where the micellar solution is pictured as a highly asymmetric electrolyte. The conclusion is that the Poisson-Boltzmann theory, based on a spherical cell model, yields a semiquantitative or better agreement with the more complicated numerical techniques applied to micellar solutions with monovalent counterions. The agreement seems to be better at a low concentration of macroions and for higher asymmetries in charge between species. For solutions with divalent counterions the cell model is not an adequate approximation. Acknowledgment. The work was supported in part by the U.S.-Slovene Science and Technology Joint Fund 95/8-06 and by the Ministry of Science of Slovenia. LA9700578
