314
Langmuir 2001, 17, 314-322
An Isotropic Model for Micellar Systems: Application to Sodium Dodecyl Sulfate Solutions Miguel Molero,* Rafael Andreu, David Gonza´lez, Juan Jose´ Calvente, and Germa´n Lo´pez-Pe´rez Department of Physical Chemistry, University of Seville, 41071-Seville, Spain Received July 20, 2000. In Final Form: September 29, 2000 A theoretical description of ionic micellar solutions, which are considered as a mixture of two fully dissociated electrolytes with a common ion, is proposed. The solution is modeled within the framework of an improved primitive model, in which the solution permittivity as well as the charge and size of the solutes are allowed to vary with surfactant concentration. The MSA theory is used to estimate the electrostatic interactions, and the Carnahan-Starling approximation is employed to evaluate the hard-sphere contributions to the osmotic and activity coefficients. The experimental osmotic coefficients of sodium dodecyl sulfate (SDS) solutions are reproduced by using a value of 1.9 + 4CT nm for the micellar radius, CT being the total surfactant molar concentration. This is the only adjustable parameter in the theory, and the agreement with published estimations of the size of SDS micelles is excellent.
1. Introduction A variety of models have been proposed to describe the properties of the micellar solutions.1-3 The pseudophase model,4-7 the mass action law model,8-13 and mixed models14 have been developed to analyze thermodynamic properties of micellar solutions. Some micellization parameters can also be obtained from these models, although a detailed description of the molecular interactions is lost. The pseudophase model seems to be preferable when dealing with long chain surfactants, whereas models based on the mass action law are more convenient for short chain length surfactants.3 On the other hand, the cell model represents a fruitful microscopic approach, where the micellar solution is divided into spherical cells around each micelle.15 Then the osmotic coefficient, surface potential, and other properties are calculated from this cell, isolated from the rest of the solution, using several theoretical approximations.15-21 Although micelle-micelle interactions are clearly neglected, some experimental * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Hunter, R. J. Introduction to Modern Colloid Science; Oxford University Press: Oxford, 1993. (2) Hiemenz, P. C. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1986. (3) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press Limited: San Diego, CA, 1992. (4) Kale, K. M.; Zana, R. J. Colloid Interface Sci. 1977, 61, 312. (5) Vikingstad, E.; Skange, A.; Hoiland, H. J. Colloid Interface Sci. 1978, 66, 240. (6) Desnoyers, J. E.; De Lisi, R.; Perron, G. Pure Appl. Chem. 1980, 52, 433. (7) Rosenholm, J. B. Colloid Polym. Sci. 1981, 259, 1116. (8) Shinoda, K.; Hutchinson, E. J. Phys. Chem. 1962, 66, 577. (9) Hall, D. G.; Pethica, B. A. Nonionic Surfactants; Schick, M. J., Ed.; Marcel Dekker: New York, 1967; Chapter 16. (10) Mukerjee, P. Physical Chemistry: Enriching Topics from Colloid and Surface Science; Van Olfen, H., Mysels, K. J., Eds.; Theorex: La Jolla, CA, 1975; Chapter 9. (11) Rosenholm, J. B.; Burchfield, T. E.; Hepler, L. G. J. Colloid Interface Sci. 1979, 71, 147. (12) Desnoyers, J. E.; De Lisi, R.; Perron, G. Pure Appl. Chem. 1983, 87, 1397. (13) Moroi, Y.; Yoshida, N. Langmuir 1997, 13, 3909. (14) Douheret, G.; Viallard, P. Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 2. (15) Bell, G. M.; Dunning, A. J. Trans. Faraday Soc. 1970, 66, 500. (16) Westra, S. W. T.; Leyte, J. C. Ber. Bunsen-Ges. Phys. Chem. 1979, 83, 672.
results have been successfully explained with this model. However, as the concentration, size, or charge of the micelles grows, isotropic models are preferable.22,23 In these models, the solution is treated as a whole, and the micellemicelle interactions are then taken into account explicitly. The micellar solution is thus considered as an asymmetric electrolyte solution, where the micelle is a large and highly charged macroion.24 We will adopt this last theoretical approach in this paper. One of the first difficulties in the analysis of a micellar solution is to determine its composition. The system may present a high degree of polydispersity, with aggregates having different number of monomers. The starting point is then to estimate the concentration, size, and shape of the different aggregates. In some cases, it is possible from experimental measurements to calculate several of these parameters of the system.25 In other cases, theoretical treatments are developed to predict their values, such as association constants of different aggregates or critical micellar concentration (cmc).13 In this work, we will assume a previous knowledge of the composition of the micellar solution, and we will analyze some of its thermodynamic properties from a theoretical point of view. The primitive model of electrolytes has been applied previously to model micellar solutions under several theoretical approximations such as HNC,26-28 MSA,28-30 and SPB.31 On the other hand, a recent extension of the (17) Wennerstrom, H.; Jonsson, B.; Linse, P. J. Chem. Phys. 1982, 76, 4665. (18) Belloni, L.; Drifford, M.; Turp, P. Chem. Phys. 1984, 83, 147. (19) Mukherjee, A. K.; Bhuiyan, L. B.; Outhwaite, C. W.; Chan, D. Y. C. Langmuir 1999, 15, 4940. (20) Lauza-Amaro, M.; Bhuiyan, L. B.; Outhwaite, C. W. Mol. Phys. 1995, 86, 725. (21) Bhuiyan, L. B.; Outhwaite, C. W.; Bratko, D. Chem. Phys. Lett. 1992, 193, 203. (22) Linse, P.; Jo¨nsson, B. Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 1. (23) Lobaskin, V.; Linse, P. J. Chem. Phys. 1999, 111, 4300. (24) Burchfield, T. E.; Woolley, E. M. J. Phys. Chem. 1984, 88, 2149. (25) Van Os, N. M.; Haak, J. R.; Rupert, L. A. M. Physicochemical properties of selected anionic, cationic and nonionic surfactants; Elsevier: Amsterdam, 1993. (26) Bratko, D.; Friedman, H. L.; Zhong, E. C. J. Chem. Phys. 1986, 85, 377. (27) Derian, P.-J.; Belloni, L.; Drifford, M. J. Chem. Phys. 1987, 86, 5708.
10.1021/la0010267 CCC: $20.00 © 2001 American Chemical Society Published on Web 12/28/2000
Isotropic Model for Micellar Systems
Langmuir, Vol. 17, No. 2, 2001 315
primitive model has been developed, and it has been successfully applied to analyze the properties of bulk electrolyte solutions.32-34 Within this improved primitive model, both the ionic sizes and the solution permittivity are allowed to vary with the solute concentration. The goal of this paper is to apply this extended primitive model to the study of micellar solutions. The main difference of our treatment compared with previous micellar theories based on isotropic models24,26-31 is that our theory allows for a variation of the micelle aggregation number (and then the micelle size and charge) with the solution composition. Besides, we also include, for the first time in a micellar model, a solute dependent dielectric permittivity of the solutions. The main advantage of the proposed model is that only one microscopic property of the system is used as a fitting parameter: the micellar radius. The rest of the required information is taken from the literature. In next section, we will develop the theoretical aspects of the model, and we will derive the working equations. In section 3, we will show how to apply the theory to a given experimental system, and we will analyze the theoretical contributions to the observed values of the osmotic coefficients of sodium dodecyl sulfate (SDS) solutions. 2. Theory In this section, we will consider successively the physical approximations, which define the model, and the mathematical approximations, which define the theory. (a) Physical Approximations. As a first simplification, it will be assumed that an ionic surfactant behaves as a strong electrolyte, and it is completely dissociated in solution according to BA + solvent a B+ solv + Asolv
(1)
This is a common assumption,1-3 and it really does not introduce a strong limitation in the model. As a second approximation, it will be accepted that for surfactant concentrations higher than the cmc, only one type of spherical aggregate is formed:
nAsolv
a
Mnsolv
(2)
where Mnsolv stands for the micelle in solution, which is characterized by its aggregation, or charge, number (n) and its diameter (σM). Here, negatively charged micelles (Mn-) and positive counterions (B+) will be considered, without loss of generality. In the literature, the mass action equilibrium usually includes a number of counterions bound to the micelle.1 In our model, counterions can only be electrostatically bound to the surface of the micelle. The amount of such ions can be estimated through the micelle-counterion pair distribution function, but it will be left for future work. Thus, in this treatment the counterions are not subjected to extra chemical forces or ionic association. (28) Sheu, E. Y.; Wu, C.-F.; Chen, S.-H.; Blum, L. Phys. Rev. A 1985, 32, 3807. (29) Caccamo, C.; Malescio, G. J. Phys. Chem. 1990, 94, 909. (30) Amos, D. A.; Markels, J. H.; Lynn, S.; Radke, C. J. J. Phys. Chem. B 1998, 102, 273. (31) Outhwaite, C. W.; Molero, M. Chem. Phys. Lett. 1992, 197, 643. (32) Simonin, J. P.; Blum, L.; Turp, P. J. Phys. Chem. 1996, 100, 7704. (33) Simonin, J. P. J. Chem. Phys. B 1997, 101, 4313. (34) Lo´pez-Pe´rez, G.; Gonza´lez-Arjona, D.; Molero, M. J. Electroanal. Chem. 2000, 480, 9.
Third, the monomer concentration (CA-) will be assumed to remain constant, independent of the total surfactant concentration (CT), whenever CT g Ccmc, and equal to Ccmc. This is a common assumption,1-3 although there is some evidence indicating that CA- varies to some extent with the surfactant concentration.35 However, such a variation is small, and for the sake of simplicity we will neglect it. Anyway, changes in the free monomer concentration can be easily incorporated into the model. Thus, the micellar solution will be treated as a mixture of two strong electrolytes, BA and BnM, with a common cation B+. From the electroneutrality condition of the solution and the mass balances, the molar concentrations of both electrolytes can be expressed in terms of known variables as follows:
CBA ) Ccmc CBnM )
CT - Ccmc n
(3) (4)
The electrolyte mixture is treated within the framework of the primitive model; ions are considered as charged hard spheres with diameter σi, charge zi, and molar concentration Ci, immersed in a continuum dielectric medium with a relative permittivity . Micelles are then modeled as big spherical ions whose size and charge depend on the total surfactant concentration CT, according to experimental evidence.3 The solution permittivity will be allowed to vary with CT too. (b) Mathematical Approximations. The Helmholtz free energy per unit volume (A), the molar individual activity coefficient (yi), and the osmotic coefficient (φ) computed for the previous model can be split in two contributions originating in electrostatic (denoted by the el superscript) and hard-sphere (denoted by the HS superscript) interactions:
∆A ) ∆Ael + ∆AHS
(5)
HS ln yi ) ∆ ln yel i + ∆ ln yi
(6)
φ ) 1 + ∆φel + ∆φHS
(7)
where the symbol ∆ refers to excess thermodynamic quantities. To solve our model, two mathematical approximations will be adopted: the electrostatic interactions will be obtained from the MSA theory,36-38 and the hard-sphere contributions will be calculated by using the CarnahanStarling (CS) approach.39 Therefore,
∆Ael ) ∆AMSA
(8)
∆AHS ) ∆ACS
(9)
and the analogous equations for the activity and osmotic coefficients. The MSA activity and osmotic coefficients are calculated using the thermodynamic route of the energy,40 through (35) Frahm, J.; Diekmann, S.; Hoase, A. Ber. Bunsen-Ges. Phys. Chem. 1980, 84, 566. (36) Waisman, E.; Lebowitz, J. L. J. Chem. Phys. 1970, 52, 4307; 1972, 56, 3086. (37) Blum, L. Mol. Phys. 1975, 30, 1529. (38) Blum, L.; Hoye, J. S. J. Phys. Chem. 1977, 81, 1311. (39) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. (40) Outhwaite, C. W. Statistical Mechanics; The Chemical Society: London, 1975; Vol. 2.
316
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Molero et al.
in our notation for the sake of completeness:
the equations
(
)
(10)
∂(β ∆AMSA/F) ∂F
(11)
∂β ∆AMSA ∂Fi
∆ ln yiMSA )
(
∆φMSA ) F
Γ
)
Γ
where i refers to any individual ionic species, Fi is the particle number density of i, and F represents the total particle density of solute species in the solution. β ) 1/kT where k is the Boltzmann constant and T the absolute temperature. Last, Γ is the MSA screening parameter,36-38 defined in eq 15. The MSA expressions for the free and internal energies in the primitive model are36-38
β ∆A β ∆EMSA )
MSA
) β ∆E
) ∑k β ∆EMSA k -
βe2
∑k
[
Fk
Γ + 3π
MSA
Q) 1+
π 6
zk2Γ
1 + Γσk
∑k Fk 1 + Γσ
+
zkσk 1 + Γσk
]
k
(14)
∑k Fk1 + Γσ (2 - Γσk)
∑k
Fk
(
1 + Γσk
)
()
(
∂σj ∂Fi
)
∂β ∆AMSA
Γ,Fk(k*i), σk(k*j),zk
∂Fi +
∑j
(
+
Γ,Fk(k*i) σk,zk,
∑j
(
2
(15)
∂σj
) () ( ) ()
Γ,Fk, σk(k*j),zk
∂zj
Γ,Fk, σk,zk(k*j)
∂β ∆AMSA ∂
)
)
(
(
)
∂β ∆AMSA ∂zj
∂Fi
+
Γ,Fk(k*i), σk,zk(k*j)
∂
Γ,Fk σk,zk
]
Γ,Fk σk,zk
∂Fi Γ,Fk(k*i) σk,zk
1 ) - β ∆EMSA
(19)
ΓFk, σk,zk(k*j)
(16)
where summations extend over all j species in solution. Partial derivatives of ∆AMSA with respect to Fi, σj, and obey analytical expressions, which have been derived in the literature.32,33,37,38 Here, we reproduce the equations
2 ) β ∆EMSA j zj
(20)
Equation 16 relates individual activity coefficients to the dσj/dFi, dzj/dFi, and d/dFi derivatives, which cannot be obtained experimentally. From now on, we will concentrate on magnitudes that are experimentally accessible, such as the mean ionic activity coefficient (ym), which is defined by
∑i xi ∆ ln yi
(21)
where xi is the molar fraction of the i species in the mixture. Then, from eqs 16 and 21 is obtained MSA MSA ) ∆ ln yMSA ∆ ln yMSA m m,0 + ∆ ln ym,σ + ∆ ln ym,z + MSA (22) ∆ ln ym,
where
∆ ln yMSA m,0 )
β ∆EMSA βe2 2 2 Q F πF
1
MSA ) ∆ ln ym,σ
F
∂β ∆AMSA
∂β ∆AMSA ∂zj
)
]
ΓFk, σk(k*j),zk
∆ ln ym )
To solve the MSA equations, Γ is first computed from eqs 14 and 15 by an iterative procedure, starting with σk ) 0 in eq 15. Implicit differentiation of eq 10 is required to account for the dependence of ionic diameters, micellar charge, and solution permittivity with concentration, and the mathematical chain rule gives32
) ∆ ln yMSA i
)
∂β ∆AMSA ∂
k
zk - σk2Q
)
Γ,Fk(k*i) σk,zk,
zj2Γ2 2zj 2 - Γ2 σj2 2 2 βe2 F Q+ σj Q (18) j (1 + Γσ )2 (1 + Γσ )2 (1 + Γσj)2 j j
Q (13)
σk3
βe2
Γ )π
[
)
∂β ∆AMSA ∂Fi
The third term in eq 16, related to the charge variation, has not been described previously in the literature and it may be expressed as
which is usually denoted in the literature32,33 as Q ≡ πPn/ 2∆, and Γ satisfies the following relationship: 2
(
( [
2 2ziσi 2 - Γσi σi3 2 βe2 zi Γ Q (17) + Q 1 + Γσi 1 + Γσi 1 + Γσi 3
∂β ∆AMSA ∂σj
(12)
zkσk
2
-
3
where e is the electron charge, and
π
∆ ln yMSA ) i,0
MSA ) ∆ ln ym,z
1 F
MSA ) ∆ ln ym,
( ∑(
∑k
) )
∂β ∆AMSA ∂σk
∂β ∆AMSA ∂zk
k
(
(23)
D(σk)
(24)
D(zk)
(25)
)
1 ∂β ∆AMSA D() F ∂
(26)
where D(Ps) represents an operator of the property Ps, defined by
D(Ps) )
( ){ ∂Ps
∑k Fk ∂F
k
Γ,Fi(i*k) σi,zi,*Ps
}
(27)
and the summation runs over all species in solution. The
Isotropic Model for Micellar Systems
Langmuir, Vol. 17, No. 2, 2001 317
values of D(σi), D(zi), and D() depend on each particular experimental system and in the present approach have to be estimated independently. The osmotic coefficient is also experimentally accessible and may be obtained from the thermodynamic relationship:
∆φMSA ) ∆ ln yMSA m
β ∆AMSA F
(28)
which is equivalent to eq 11. The alternative virial equation to calculate the MSA osmotic coefficient gives poorer results, because of the low values of the pair distribution function at contact.40 Thus, the electrostatic part of the osmotic coefficient can also be expressed like eq 22 as a sum of four terms:
+ ∆φMSA + ∆φMSA + ∆φMSA (29) ∆φMSA ) ∆φMSA 0 σ z represents a zeroth order electrostatic where ∆φMSA 0 contribution,
∆φMSA )0
βe2 2 2 Γ3 Q 3πF πF
(30)
MSA MSA ) ∆ ln ym,σ , ∆φMSA ) ∆ ln ym,z , and whereas ∆φMSA σ z MSA MSA ∆φ ) ∆ ln ym, are the contributions due to the dependence of the size, charge, and permittivity on the solution concentration, respectively. Analogously, hard-sphere contributions to the activity and osmotic coefficients are calculated from the Carnahan-Starling closure:
CS CS ∆ ln yCS m ) ∆ ln ym,0 + ∆ ln ym,σ
(31)
CS ∆φCS ) ∆φCS 0 + ∆φσ
(32)
∆φCS σ
) ∆ ln
CS ym,σ
)
1 F
∑k
(
)
∂β ∆ACS ∂σk
D(σk)
(33)
In the literature, the expressions derived from the CS closure for the properties of a multicomponent mixture of hard spheres are messy,39,41-47 so we have rewritten them, in a more compact form, in Appendix B. For the calculation of osmotic and activity coefficients, experimental information is necessary to estimate the operators appearing in eqs 24-26 (D(σk), D(zk), and D()). As indicated in eq 27, these operators involve derivatives with respect to individual ionic concentrations keeping constant all other variables. This type of derivative is not directly accessible from experimental data, and it makes the estimation of individual activity coefficients through eq 16 difficult. However, for the calculation of mean ionic properties, we can rewrite the operators in terms of neutral salt concentrations, as is shown in Appendix A, so that (41) Reed, T. M.; Gubbins, K. E. Applied Statistical Mechanics. Thermodynamic and Transport Properties of Fluids; McGraw-Hill: New York, 1973. (42) Ebeling, W.; Scherwinski, K. Z. Phys. Chem. (Leipzig) 1983, 264, 1. (43) Humffray, A. A. J. Phys. Chem. 1983, 87, 5521. (44) Corti, H. R. J. Phys. Chem. 1987, 91, 686. (45) Lu, J.-F.; Yu, Y.-X.; Li, Y.-G. Fluid Phase Equilib. 1993, 85, 81. (46) Sun, T.; Le´nard, J.-L.; Teja, A. M. J. Phys. Chem. 1994, 98, 6870. (47) Tikanen, A. C.; Fawcett, W. R. J. Electroanal. Chem. 1997, 439, 107.
D(Ps) )
( ) ∂Ps ∂FMX
FNY
FMX +
( ) ∂Ps ∂FNY
FMX
FNY
(34)
Now, the two differential coefficients can be obtained from tabulated experimental data. (c) Conversion between Theoretical and Experimental Quantities. The previous theory has been developed in the McMillan-Mayer framework, that is, at constant volume.48,49 Nevertheless, the experiments are usually performed in the laboratory at constant pressure. To compare experimental and theoretical results, it is necessary to perform the conversion from the McMillanMayer (MM) to the Lewis-Randall (LR) activity and osmotic coefficients, according to the approximated relationships.33,34,50
∑s Csvjs) ) φMMCwvjw
φLR ) φMM (1 -
(35)
where the s and w subscripts refer to salt and solvent, respectively, and vj s is the partial molar volume of the s salt. Each vj s value can be evaluated from known values of the solution densities. Alternatively, vj w can be calculated, through
10-3 Mw
vj w ) d-
)
( ) ∂d
∑s Cs ∂C
10-3 Mw
( )
d - Caj
s Ct(t*s)
(36)
∂d
∂Caj
where Mw is the molecular weight of w, d is the density of the solution, the t subscript refers to any other salt in the solution, different from s, and Caj is any linear combination of the salt concentrations appropriated to fit the density data. The second equality is more convenient for practical purposes, because it requires less information about the exact solution composition. Thus, the Lewis-Randall activity coefficient in the molar scale is defined by33,34 MM + (φLR - φMM) ) ln yLR i ) yi
ln yMM + (Cwvj w - 1)φMM (37) i and it is directly related to the Lewis-Randall activity coefficient in the molal scale, according to LR ln γLR i ) ln yi + ln
Ci mid0
(38)
where d0 is the pure solvent density. The i subscript in eqs 37 and 38 may refer to either ions or salts. Comparison between experimental and theoretical osmotic coefficients needs a further conversion. Apparent values, φap, of experimental osmotic coefficients are usually reported:
φ
ap
)-
103 ln aw
∑i
Mw
(39)
mi
A given type of electrolyte dissociation must be assumed (48) Poirier, J. C. J. Chem. Phys. 1953, 21, 965; 1953, 21, 972. (49) Friedman, H. L. J. Solution Chem. 1972, 1, 387; 1972, 1, 418. (50) Pailthorpe, B. A.; Mitchell, D.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1984, 80, 115.
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Molero et al.
to compute φap from ln aw values. In the case of micellar systems and for practical purposes, it is usually assumed that surfactant electrolyte (BA) is fully dissociated into monomers, and the aggregation process is neglected when computing the number of species in solution, so that
ap
φ
)-
103 ln aw MwνBAmBA
)
∑i mi
φLR
νBAmBA
(40)
where ∑imi now corresponds to a BA-BnA electrolyte mixture, that is, with explicit consideration of the micellar aggregation number. 3. Results and Discussion Aqueous solutions of SDS at 25 °C have been selected to illustrate a practical application of the present theory. This system is known to be monodisperse for low surfactant concentrations3 where it consists of micelles of spherical shape from CT ) Ccmc ) 8.3 mM up to about CT ) 60 mM, where a phase transition to cylindrical micelles is observed.51 Although experiments to obtain the aggregation number are scarce, it has been estimated by different techniques (SANS,52-55 fluorescence,56-63 and positron annihilation lifetime spectroscopy64). The published values depend on temperature and the technique used, neutron scattering usually giving higher n values than fluorescence methods.58 This can be the consequence of the use of different theories to extract the geometrical parameters from the raw experimental data. Although a dependence of the aggregation number with the 1/4 power of the surfactant concentration has been proposed,53,57,58 in the small concentration range considered in this work, the experimental values of n55,60 can be fitted to the following linear relationship:
n ) 62 + 165CT
Ccmc < CT < 60 mM
(41)
D(σNa), and D(σM)), three accounting for the charge variations (D(zDS), D(zNa), and D(zM)), and D(). As the charge of Na+ and DS- is independent of CT, D(zDS) ) D(zNa) ) 0. Further, we will assume that σNA+ and σDS- are concentration independent, so that only D(zM), D(σM), and D() need to be calculated. The value for the Na+ diameter (0.34 nm) was obtained from a fitting of the experimental activity coefficients of simple sodium salts, such as NaCl.42 No significant variations have been obtained in the calculations when σNa+ changes in (0.1 nm. Theoretical results are nearly independent of the choice for σDS-, because of its low concentration, and we have set σDS- ) 0.92 nm, based on geometrical considerations. The low sensitivity of the results to the values of σNa+ and σDS- prevents us from considering concentration dependent radii. Otherwise, the fitting procedure would be overparametrized. On the other hand, the micellar diameter gives an important contribution to the osmotic coefficient. Its value and its variation with the surfactant concentration will be considered as the only adjustable parameters of the theory. For the determination of D(zM) and D(), eqs 41 and 42 can be rewritten in a more general form as follows:
Ccmc < CT < 60 mM (42)
Equations 41 and 42 suffice to estimate the operators D(Ps) appearing in eqs 24-26. In principle, seven of these operators are required: three related to the sizes (D(σDS), (51) Barchini, R.; Pottel, R. J. Phys. Chem. 1994, 98, 7899. (52) Cabane, B.; Duplessix, R.; Zemb, T. J. Phys. 1985, 46, 2161. (53) Bezzobotnov, V. Yu.; Borbe´ly, S.; Cser, L.; Farago´, B.; Gladkih, I. A.; Ostanevich, Yu. M.; Vass, Sz. J. Phys. Chem. 1988, 92, 5738. (54) Kabir-ud-Din; Kumar, S.; Aswal, V. K.; Goyal, P. S. J. Chem. Soc., Faraday Trans. 1996, 92, 2413. (55) Kumar, S.; David, S. L.; Aswal, V. K.; Goyal, P. S.; Kabir-ud-Din Langmuir 1997, 13, 6461. (56) Quina, F. H.; Nassar, P. M.; Bonilha, J. B. S.; Bales, B. L. J. Phys. Chem. 1995, 99, 17028. (57) Bales, B. L.; Almgren, M. J. Phys. Chem. 1995, 99, 15153. (58) Bales, B. L.; Stenland, C. J. Phys. Chem. 1993, 97, 3418. (59) Siemiarczuk, A.; Ware, W. R.; Liu, Y. S. J. Phys. Chem. 1993, 97, 8082. (60) Pe´rez-Benito, E.; Ro´denas, E. J. Colloid Interface Sci. 1990, 139, 87. (61) Gehlen, M. H.; De Schryver, F. C. J. Phys. Chem. 1993, 97, 11242. (62) Croonen, Y.; Gelade´, E.; Van der Zegel, M.; Van der Auweraer, M.; Vandendrlessche, H.; De Schryver, F. C.; Amgren, M. J. Phys. Chem. 1983, 87, 1426. (63) Almgren, M.; Linse, P.; Van der Auweraer, M.; De Schryver, F. C.; Gelade´, E.; Croonen, Y. J. Phys. Chem. 1984, 88, 289. (64) Bockstahl, F.; Duplaˆtre, G. Phys. Chem. Chem. Phys. 1999, 1, 2767; 2000, 2, 2379.
CT > Ccmc
(43)
) 0 + 1CT
CT > Ccmc
(44)
Formally, the solution can be described as a mixture of two strong electrolytes: NaDS and NanM. The concentration CNaDS ) Ccmc ) 8.3 mM is fixed. From the mass balance, the total surfactant concentration can be related to the concentrations of the two salts as
CT ) CNaDS + nCNanM
(45)
Because zM ) -n, substituting eq 45 in eq 43 and taking derivatives with respect to CNanM gives
( )
Permittivity measurements of micellar systems are rather scarce in the literature. The results of Barchini and Pottel51 for SDS solutions can be expressed as
) 84.73 + 336CT
n ) n0 + n1CT
∂zM ∂CNanM
)-
CNaDS
n1n 1 - n1CNanM
(46)
No information is available for the variation of the aggregation number with monomer concentration, which is considered to be fixed. We will assume that (∂zM/ ∂CNaDS)CNanMCNaDS , (∂zM/∂CNanM)CNaDSCNanM, whose validity rests on the success of the theory to reproduce the experimental behavior. Then, eqs 34 and 46 lead to
D(zM) ≈
( ) ∂zM ∂CNanM
CNanM )
CNaDS
n1 n(CT - Ccmc) (47) n0 + n1Ccmc An expression for the D() operator can be derived in a similar way, to obtain
D() ≈
(
)
∂ ∂CNanM
CNanM ) 1(CT - Ccmc) (48)
CNaDS
Analogously, we have chosen a linear functionality to describe the variation of the micellar diameter (σMn-) with CT:
σMn- ) (σMn-)0 + (σMn-)1CT
(49)
Isotropic Model for Micellar Systems
Langmuir, Vol. 17, No. 2, 2001 319
φ0 ) ∆φMSA + ∆φCS 0 0
(53)
φ1 includes the contributions due to the derivatives of the free energy with respect to the size, charge, and permittivity:
+ ∆φCS φ1 ) ∆φMSA 1 1
(54)
) ∆φMSA + ∆φMSA + ∆φMSA ∆φMSA 1 σ z
(55)
CS ∆φCS 1 ) ∆φσ
(56)
with
and
Figure 1. Osmotic coefficients of SDS solutions. (b) Experimental values (from ref 65). The solid line shows the theoretical osmotic coefficients corresponding to σMn-/nm ) 3.8 + 8CT, σNa+ ) 0.34 nm, and σDS- ) 0.92 nm. The following contributions to the osmotic coefficients are indicated: hairlines correspond to φ0 and φ1 and dotted hairlines correspond to ∆φMSA , ∆φMSA , z MSA CS ∆φσ , and ∆φσ . See text for explanation.
so that the operator D(σM) can be approximated as in eqs 47 and 48:
( )
∂σMnD(σMn-) ≈ ∂CNanM
It is interesting to point out that an extensive numerical cancellation occurs in φ1, because of the opposite signs of and ∆φMSA . Thus, it its two most important terms, ∆φMSA z is necessary to consider simultaneously both the aggregation number and dielectric permittivity dependence with surfactant concentration to explain the experimental values. Otherwise, the simpler alternative is to assume φ1 ) 0, though this approach appears to be unsound from the theoretical point of view. and ∆φCS The ∆φMSA σ σ contributions are always positive, increasing with surfactant concentration. Their sum, ∆φσ MSA ) ∆φMSA + ∆φCS , the other positive σ σ , is smaller than ∆φ CS contribution. The ∆φσ term has the same qualitative MSA , although ∆φCS , so that it can behavior as ∆φMSA σ σ , ∆φσ MSA be assumed that ∆φσ ) ∆φσ to a good approximation. According to our fit, the diameter of the SDS micelles is given by
σMn- ) 3.8 + 8CT CNanM ) (σMn-)1(CT - Ccmc)
CNaDS
(50)
To obtain the apparent osmotic coefficient (φap), molar concentrations of SDS (CT) were first computed from reported molal values65 and solution densities66 which were found to obey a simple quadratic dependence on CT:
d ) 0.997097 + 0.039938CT - 0.00422283CT2
(51)
Theoretical osmotic coefficients φMM were obtained from eqs 7, 29, and 32. Then, φLR values were derived from eq 35, which requires the previous determination of solvent partial molar volume, according to eqs 36 and 51, with Caj ) CT. Figure 1 shows a comparison between the experimental and theoretical osmotic coefficients. Separate contributions to the theoretical values are also included. Formally, these contributions are expressed as follows:
φ ) φ0 + φ1
(52)
where φ0 represents the zero order osmotic coefficient calculated using eqs 41, 42, and 49 where variations of n, σMn-, and with surfactant concentration are taken into account, but the corresponding free energy derivatives are neglected: (65) Crisantino, R.; De Lisi, R.; Milioto, S. J. Solution Chem. 1994, 23, 639. (66) De Lisi, R.; Genova, C.; Testa, R.; Turco Liveri, V. J. Solution Chem. 1984, 13, 121.
(57)
with σMn- in nm and CT in mol dm-3. The positive value of (σMn-)1 means that the micelles are growing with surfactant concentration, which seems physically reasonable, considering the increase of the aggregation number. The theoretical mean activity coefficients corresponding to the parameters of Figure 1 are represented in Figure 2. Calculations were extended to concentrations smaller than the cmc, where the typical behavior of a 1:1 electrolyte is predicted. The sharp decrease of the mean activity coefficient, and thus of the osmotic coefficient, is brought about by the formation of micelles. To get more insight into the origin of this behavior, we have also analyzed the activity coefficients of the two salts, NaDS and NanM, into which the solution can be formally divided. As can be seen from the comparison of the different curves, the activity coefficient of the micelle is the responsible factor for the sudden decrease of the mean activity coefficient above the cmc. This sharp dependency disappears when the activity coefficients are plotted against the ionic strength, I ) (1/2)∑Ci zi2, suggesting that the aggregation process is mainly controlled by electrostatic factors. On the other hand, the low sensitivity of the results to the value of σNa+ is illustrated with the dotted lines. Next, the sensitivity of the fit to (σMn-)0 and (σMn-)1 values was investigated. Figure 3 illustrates the consequences of a variation of (0.2 nm in (σMn-)0, keeping constant all other parameters. When (σMn-)0 is increased, φ0 also increases, whereas φ1 decreases, so the final result depends both on the concentration range and on the values of all other parameters. In the SDS micellar system, an increase of (σMn-)0 leads to higher values of φ. The influence of (σMn-)1 is shown in Figure 4. When (σMn-)1 increases, higher values
320
Langmuir, Vol. 17, No. 2, 2001
Figure 2. Activity coefficients corresponding to the same parameter values as in Figure 1. For the mean activity coefficient, the upper dashed line corresponds to σNA+ ) 0.64 nm, the solid line to σNA+ ) 0.34 nm, and the lower dashed line to σNA+ ) 0.04 nm. The dotted line corresponds to the plot of the NanM activity coefficient against the ionic strength.
Molero et al.
Figure 4. Influence of (σMn-)1 on theoretically calculated osmotic coefficients. The results correspond to the same parameter values as in Figure 1, except for (σMn-)1 ) (1 nm/M. The upper dotted line corresponds to (σMn-)1 ) 9 nm/M and the lower dotted line corresponds to (σMn-)1 ) 7 nm/M. The solid line is as in Figure 1. (b) Experimental osmotic coefficients (from ref 65).
value is expected for the solvated radius of the micelle. Also, our values are in good agreement with those calculated by E. Ro´denas and E. Pe´rez-Benito67 using the Tanford equation,68 (σMn-)0 ) 3.5 nm and (σMn-)1 ) 4.2 nm /M. An alternative estimate of the micellar size can be derived from the solution density values in the following way. Starting from eqs 43 and 45, the derivative of the solution density with respect to the micellar concentration is
∂d ∂d n ) ∂CNanM 1 - n1CNanM ∂CT
(58)
and considering that the molecular weight of NanM is n times that of NaDS,
MNanM ) nMNaDS
(59)
we may obtain vj NanM, that can be split in its individual ionic contributions as
vj NanM ) nvj Na+ + vj MnFigure 3. Influence of (σMn-)0 on theoretically calculated osmotic coefficients. The results correspond to the same parameter values as in Figure 1, except for (σMn-)0. Upper dotted line, (σMn-)0 ) 4 nm; solid line, (σMn-)0 ) 3.8 nm; lower dotted line, (σMn-)0 ) 3.6 nm; solid line as in Figure 1. (b) Experimental osmotic coefficients (from ref 65).
of φ0 and φ1 are obtained, and therefore the osmotic coefficient also increases. Within the 0.008 M e CT e 0.06 M range, the micellar radii lie in the 1.93 nm e rMn- e 2.14 nm interval. These values are in good agreement with the radius of 1.84 nm derived from neutron scattering for a 2% SDS solution53 (ca. 0.07 M). The value of 1.84 nm refers to the radius of the hydrocarbon core of the micelle, so a somewhat higher
(60)
0 Subtracting the contribution of the sodium ion69 (vj Na + ) 3 mol-1) and assuming a spherical shape, the -6.7 cm micellar radius can be estimated as
rMn- )
(4π3 (vj
NanM
)
- nvj Na+)
1/3
(61)
The obtained values (1.85 nm e rMn- e 1.93 nm) agree reasonably with those derived from the fit of the osmotic (67) Ro´denas, E.; Pe´rez-Benito, E. An. Quim. 1993, 89, 674. (68) Tanford, C. J. Phys. Chem. 1974, 78, 2469. (69) Marcus, Y. Ion Properties; Marcel Dekker: New York, 1997; p 137.
Isotropic Model for Micellar Systems
Langmuir, Vol. 17, No. 2, 2001 321
coefficients by the present model. The Na+ contribution in eq 61 can also be estimated more consistently from its assigned radius of 0.17 nm, but this last choice gives essentially the same micellar radii. In the concentration range studied in this work, differences between φMM and φLR remain e1.5% and are comparable to the experimental accuracy. At higher concentrations, such as those studied by Amos et al.,30 conversion between the MM and LR scales should be necessary. Under these circumstances, the repulsive effect arising from the finite sizes of the species may be the dominant contribution to the osmotic coefficient, thus explaining the type of results obtained in ref 30. In the range of concentrations considered in the present work, the attractive electrostatic forces prevail and determine the observed decrease of the osmotic coefficient with surfactant concentration. Experimental values of in a wider concentration range would be required to extend this type of analysis to higher surfactant concentrations; however, micelles are likely to lose the spherical shape at higher concentrations, thus breaking a basic assumption of the model. The aim of the present work is mainly to present the formulation of the theory. Its application to other surfactants will be considered in future work.
where Ps stands for a solution property such as ionic size, ionic charge, or dielectric permittivity. Individual ionic concentrations are related to salt concentrations according to
FM ) νMFMX FX ) νXFMX FN ) νNFNY FY ) νYFNY
Under these conditions, eq A3 can be rewritten as
[ ( ) ( ) ] ( ) ] [( ) ∂Ps ∂FM
D(Ps) ) νM
Appendices A. Reformulation of Operator D(Ps). Let us consider two strong electrolytes MX and NY that dissociate completely according to
MνMXνX a νMM+νX + νXX-νM
(A1)
NνMYνY a νNN+νY + νYY-νN
(A2)
∑i
Fi
∂Ps ∂Fi
( ) ( )
) FM FN
∂Ps
∂FM
FX,FN,FY
∂Ps
∂FN
FM,FX,FY
( ) ( )
+ FX + FY
∂Ps
∂FX
+
FM,FN,FY
∂Ps
∂FY
FM,FX,FY
∂Ps ∂FX
FM,FN,FY
∂Ps ∂FY
+ νY
FMX +
FM,FX,FN
FNY (A5)
dPs )
( ) ∂Ps ∂FM
FX,FN,FY
dFM +
( ) ∂Ps ∂FN
( ) ∂Ps ∂FX
FM,FX,FY
FM,FN,FY
dFN +
dFX +
( ) ∂Ps ∂FY
FM,FX,FN
dFY (A6)
Equation A4 can be differentiated to give
dFM ) νM dFMX dFX ) νX dFMX dFN ) νN dFNY dFY ) νY dFNY
(A7)
which can be substituted in eq A6 to obtain
[( )
dPs ) νM
( ) ] ( ) ] [( )
∂Ps ∂FM
FX,FN,FY
∂Ps ∂FN
∂Ps ∂FX
+ νX
FM,FX,FY
dFMX +
FM,FN,FY
+ νY
∂Ps ∂FY
FM,FX,FN
dFNY (A8)
Differentiating with respect to the concentration of each electrolyte and keeping constant the concentration of the other, it follows that
( ) ( ) ( ) ( ) ∂Ps ∂FMX
∂Ps ∂FNY
FNY
FMX
) νM
) νN
∂Ps ∂FM
∂Ps ∂FN
FX,FN,FY
FM,FX,FY
( ) ( )
+ νX
+ νY
∂Ps ∂FX
∂Ps ∂FY
(A9)
FM,FN,FY
(A10)
FM,FX,FN
Replacing eqs A9 and A10 into eq A5, we have
with concentrations FMX and FNY and stoichiometric numbers νMX ) νM + νX and νNY ) νN + νY, respectively. Then, we define
D(Ps) )
+ νX
and
νN
Acknowledgment. This work has been supported by the Spanish DGESIC under Grant PB98-1123.
FX,FN,FY
∂Ps ∂FN
νN
4. Conclusions A model that considers micellar solutions as an electrolyte mixture with a common ion is proposed. The model is an extension of the primitive model for electrolyte solutions, incorporating changes of the ionic sizes and of the solution permittivity with the surfactant concentration. This model is solved with the aid of the MSA theory and the Carnahan-Starling approximation. Micellar solutions of SDS have been analyzed, and a good agreement between experimental and theoretical osmotic coefficients was found. The micellar radii obtained from the fit agree very well with other independent estimations of the micellar size in the literature. The success of the theory implies that only the consideration of electrostatic and hard-sphere interactions is able to explain the thermodynamic behavior of diluted SDS solutions.
(A4)
FM,FX,FN
(A3)
D(Ps) )
( ) ∂Ps ∂FMX
FNY
FMX +
( ) ∂Ps ∂FNY
FMX
FNY
(A11)
where Ps stands for the selected property, as indicated before. B. Carnahan-Starling (CS) Expressions for the Properties of a Multicomponent Mixture of Hard Spheres. For the sake of simplicity, we have rewritten the thermodynamic properties of a multicomponent mixture of hard spheres in matrix form, as follows:
322
[
Langmuir, Vol. 17, No. 2, 2001
ln
yCS i,0
(
β ∆ACS ) MA‚MY F
(B1)
∆φCS 0 ) Mφ‚MY
(B2)
ln yCS m,0 ) My‚MY
(B3)
(
)
∂MS ∂β ∆ACS ) ‚M4‚MY )F ∂Fi ∂Fi
∂β ∆A ∂σi
)
CS
∂MS )F ‚M4‚MY ∂σi
MA ) [(1 - b) (3a + b) b 0]
(B6)
Mφ ) [0 (3a + 1) (3a + 3b) 2b]
(B7)
My ) [(1 - b) (6a + b + 1) (3a + 4b) 2b]
(B8)
MS ) [ln(S0) ln(S1) ln(S2) ln(S3)]
(B9)
[
(B10)
]
σi σi2 σi3 ∂MS xi ) 0 2 3 ∂σi σi S1 S2 S3
[ ]
ln(1 + Y) Y MY ) 2 Y Y3
0 0 3b (3a + b)
0 0 0 2b
(B13)
∑k xk σk
n
a)
S1S2 S3
b)
S23 2
Y)
S3
η 1-η (B14)
where xk is the molar fraction of species k in the mixture and η is the packing fraction:
η)
π
π
∑Fk σk3 ) 6FS3 6 k
(B15)
Explicit multiplication of the matrices in eqs B1-B3 gives
β ∆ACS ) (1 - b) ln(1 + Y) + (3a + b)Y + bY2 (B16) F 2 3 (B17) ∆φCS 0 ) (3a + 1)Y + 3(a + b)Y + 2bY
ln yCS m,0 ) (1 - b) ln(1 + Y) + (6a + b + 1)Y + (3a + 4b)Y2 + 2bY3 (B18)
3
σi σi σi ∂MS 1 ) 1 ∂Fi F S1 S2 S3
[
]
0 3a (3a + 3b) (1 - 2b)
with Sn, a, b, and Y defined as
Sn )
where MA, Mφ, My, MS, and its derivatives are 4 × 1 arrays. These arrays are defined in eqs B6-B11 and contain information about the sizes and relative composition of the species in solution. MY is a 1 × 4 array that depends on the packing fraction only, and it is defined in eq B12. M4 is a 4 × 4 array given by eq B13. In some interesting cases, such as single electrolyte solutions in the primitive model, M4, MA, Mφ, My, and the products F(∂MS/∂Fi)‚M4 and F(∂MS/∂σi)‚M4 are concentration independent.
2
1 0 M4 ) -3b 2b
(B4) (B5)
]
Molero et al.
(B11)
(B12)
In eqs B4 and B5, the matrix products can be done using the associative rule:
MR‚Mβ‚Mγ ) (MR‚Mβ)‚Mγ ) MR‚(Mβ‚Mγ)
(B19)
The use of the variable Y, rather than the usual η, simplifies the equations and helps to identify the lowdensity limit, where Y f η f 0, and the Y2 and Y3 terms can be neglected in eqs B16-B18. LA0010267
