Langmuir 1989,5, 741-753 dinated modes to be the most favored for all adsorbates. The long di-a-coordinated modes are less favored by about 0.17 eV. Besides, this latter mode requires that the adsorption is strong enough to initiate the transition of the Pt(ll0) surface from its (1x2) reconstructed state to the (1x1) unreconstructed state. The tetra-a-coordinated mode of 1,3-butadiene is as stable as the short di-a mode but also requires the (110) surface to be (1x1). The other possible adsorption modes investigated have been found to be less favorable, including the di-a coordination on the metal atoms of the second layer and the ?r-bonded state on the topmost layer. From a catalytic point of view, the understanding of the mechanism of hydrogenation of 1,3-butadiene requires a knowledge of the adsorption modes of the reactants as well as the monohydrogenated and dihydrogenated products. In addition, it is necessary to know the adsorption modes of the dehydrogenated species, which cause the poisoning of the reaction. The present study reports the investigation
741
of the adsorption modes of the reactant (C4H6)and of its dihydrogenated forms (C4H8). The differences found in their binding energies are weak but significant and understood. They imply that the hydrogenation should favor 1,3-butadiene, then 1-butene, and finally the 2-butenes, as these species compete for the same adsorption sites. The second part of our study, currently under investigation, is focused on the adsorption modes of the monohydrogenated and dehydrogenated species which may occur during the hydrogenation reaction.
Acknowledgment. This work has been motivated by fruitful discussions with C.M. Pradier, Y. Berthier, and Professor J. Oudar, whom we would like to acknowledge gratefully. We also thank the CIRCE for computational facilities on VP200. Registry No. Ethene, 74-85-1; 1,3-butadiene, 106-99-0; 1butene, 106-98-9; cis-2-butene,590-18-1; trans-2-butene,624-64-6; pT, 7440-06-4.
Specific Ion Effects in Electrical Double Layers: Selective Solubilization of Cations in Aerosol-OT Reversed Micelles E. B. Leodidis and T. A. Hatton* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received July 25, 1988. I n Final Form: January 5, 1989 Specific ion effecta in the solubilization of cations in an AOT reversed micellar solution in equilibrium with an excess aqueous phase are examined in terms of a phenomenological model of the electrical double layer formed inside the reversed micellar water pools. The model distinguishes between different cations via their charge, hydrated size, and electrostatic free energy of hydration. Other significant features of the model are the variation of the dielectric constant with the local electric field within the reversed micellar water pools and the smearing of the surfactant charge over a spherical shell of finite thickness. All the cations and water are allowed to penetrate this shell, which effectively accounts for the association of the cations with the surfactant head groups. The model accurately predicts the distribution of monovalent and divalent cations between the two phases of the system under consideration. A number of the implications of the model are discussed, including the possibility of treating similar cation effects important in other colloidal and biological systems. 1. Introduction Specific cation effects are encountered in a number of colloidal, macromolecular, and biological systems. Cations are seen to affect the activity ccefficients,l the second virial coefficients,2 and the salting-out behavior of protein^.^ They also affect the electrostatic potential, intermolecular interactions, and phase separation of polyelectrolytes4and (1) Joseph, N. R. J. Biol. Chem. 1938, 126, 389. (2) Bull, H. B.; Breeee, K. Arch. Biochem. Biophys. 1972,149, 164. (3) Arakawa, T.; Timasheff, S.N. Biochemistry 1984,23, 5912. (4) Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971. (5) Alberteson, P.-A. Partition of Cell Particles and Macromolecules; Wiley: New York, 1986. ( 6 ) Ohki, 5.;Sauve, R. Biochim. Biophys. Acta 1978,511, 377. (7) Goddard, E. D.; Kung, H. C. J. Colloid Interface Sci. 1971,37(3), 585. (8) McLaughlin, S.; Mulrine, N.; Gresalfi, T.; Vaio, G.; McLaughlin, A. J. Gen.Physiol. 1981, 77, 445. (9) Cevc, G.; Marsh, D. Phospholipid Bilayers: Physical Principles and Models; Wiley: New York, 1987. (10) Marra, J.; Israelachvili, J. Biochemistry 1985, 24, 4608.
0743-7463/89/2405-0741$01.50/0
the partition coefficients of biological solutes in biphasic aqueous polymer system^.^ The surface potential6 and the a-A isotherms of monolayers' and the t potential! phase transition properties: and interbilayer forces'O in phospholipid bilayers all show a strong dependence on the cation type present in the system. In surfactant solutions, significant specific cation effects have been observed in the aggregation and sphere-to-rod transition of SDS micelles," in the phase separation in anionic surfactant solutions,12 and in the water uptake by water-in-oil and middle-phase micrmmulsions.l2 While most of these cation effects have been partly understood in a qualitative way, there are currently no theoretical models able to explain or predict cation effects quantitatively. Recently, we have observed a new and interesting cation effect on water uptake by AOT reversed micellar solutions (11) Missel, P. J. T. 'Quasielastic Light Scattering Studies of Alkylsulfate Detergent Micelles";Ph.D. Thesis, MIT, June 1981. (12) Chou, S. I.; Shah,D. 0.J. Colloid Interface Sci. 1981,80(2), 311.
0 1989 American Chemical Society
742 Langmuir, Vol. 5, No. 3, 1989
Leodidis and Hatton
CAmodm = 0 1 M
Sodium
50
o Cesium
Rubidium Potassium
0 00
01
02
03
04
05
06
07
OB
j 09
10
30
o A
Calcium Stront~um Barium
20
0 00
I
1
,
I
J
0 05
0 10
0 15
0 eo
0 25
C M ~ + a q(l n M ) Figuw 1. Equilibrium water uptake (zoo) as a function of initial electrolyte concentration in excess aqueous phase for (a, top) monovalent and (b, bottom) divalent cations.
during the liquid-liquid extraction of proteins. Specifically, the water uptake by an AOT reversed micellar solution in equilibrium with a bulk aqueous electrolyte (in the absence of proteins) is a strong function of both cation concentration and type. As shown in Figure 1,the water uptake expressed in terms of wo, the molar ratio of water to surfactant in the reversed micellar phase, decreases significantly as the salt concentration increases, while the actual magnitude of wo at any given salt concentration depends on the salt type. Protein partitioning follows similar trends.13 Furthermore, it can be noted that cations of the same charge and of approximately the same hydrated size (e.g., Ca-Sr, K-Rb-Cs) produce significant differences in water uptake, indicating that cation charge and size are not the only factors influencing the equilibrium distribution of water between the two phases of the system. The primary goal of this paper is to gain insight into the reasons for the large differences in water solubilization by AOT W/O microemulsions noted for the different cations. The first important step toward the achievement of this goal is the prediction of the distribution of the various cations throughout the two-phase system. We do this through the development of a detailed model of the electrical double layer that forms inside reversed micelles. The model is presented in section 4 of this paper, and its va(13) Kelley, B,D.; Leodidis, E. B.; F a m a n , R. s.; Hatton, T. A., paper presented in the ACS 6Znd Collold and Surface Science Symposium, Pennsylvania State University, June 1988.
lidation, by comparison of predicted solubilization curves with the experimentally observed selective equilibrium distribution of different cations between the reversed micellar phase and an excess bulk aqueous solution, is presented in section 5. Additional theoretical resulh are also discussed here. Particular emphasis is placed on the more general implications of these results and on the possible extensions of the present work, which will further elucidate the behavior of cations in this and other systems. Our conclusions from this effort are summarized in the last section. 2. Previous Work The effeds of electrolytm on reversed micelles have been examined for the most part from a different perspective than that treated here. Kon-No and Kitahara investigated the effects of temperature, salt concentration, and type on the maximum water uptake by AOT reversed micellar phases before any second (excess) phase is formed.'@18 A similar study was undertaken by Chou and Shah.I2 In more recent years, Kunieda and Shinoda1*21 and Gosh and Millerz2focused on the influence of salinity on the phase behavior of AOT/water/oil systems. Studies more relevant to our own have been presented by Tosch et al.,23FletcherF4and Aveyard et al.25 These authors have measured the distribution of sodium salts between an electrolyte solution and a W/O microemulsion in equilibrium. They also measured water uptake by the reversed micellar solution and reported AOT distribution between the two phases, sizes of reversed micelles, and interfacial tensions. The first attempts to model such two-phase systems taking account of the electrostatics were reported by Adamson26and Levine and R~binson.~'Subsequent theoretical work on reversed micellar systemswM and on spherical has almost exclusively employed the Poisson-Boltzmann equation in either the linearized, truncated, or complete form. The Poisson-Boltzmann equation considers all ions as points, however, and can (14) Kitahara, A.; Watanabe, K.; Kon-No, K.; Ishikawa, T. J. Colloid Interface Sci. 1969,29(1), 48. (15) Kon-No, K.; Kitahara, A. J. Colloid Interface Sci. 1971, 35(3), 409. (16) Kon-No, K.; Kitahara, A. J. Colloid Interface Sci. 1971, 35(4), 636. (17) Kon-No, K.; Kitahara, A. J . Colloid Interface Sci. 1971, 37(2), 469. (18) Kon-No, K.; Kitahara, A. J. Colloid Interface Sci. 1972,41(1), 47. (19) Kunieda, H.; Shinoda, K. J. Colloid Interface Sci. 1979, 70(3), 517. (20) Kunieda, H.; Shinoda, K. J . Colloid Interface Sci. 1980, 75(2), 601. (21) Kunieda, H.; Shinoda, K. J. Colloid Interface Sci. 1987,118(2), 586. (22) Ghosh,0.;Miller, C. A. J . Phys. Chem. 1987, 91, 4528. (23) Tosch, W. C.; Jones, S. C.; Adamson, A. W. J. Colloid Interface Sci. 1969, 31(3), 297. (24) Fletcher, P. D. I. J. Chem. SOC.,Faraday Tram. 1 1986,82,2651. (25) Aveyard, R.; Binka, B. P.; Clark, S.; Mead, J. J. Chem. Soc., Faraday Trans. 1 1986,82, 125. (26) Adamson, A. W. J. Colloid Interface Sei. 1969,29(2), 261. (27) Levine, S.;Robinson, K. J. Phys. Chem. 1972, 76(6), 876. (28) Jonsson, B.; Wennerstrom, H. J. Colloid Interface Sci. 1981, 80(2), 482. (29) Jonsson, B.; Wennerstrom, H. J. Phys. Chem. 1987,91, 338. (30) Overbeek, J. Th. G. Faraday Discuss. Chem. SOC.1978, 65,7. (31) Overbeek, J. Th. G.; Verhoeckx, G. J.; De Bruyn, P. L.; Lekkerkerker, H. N. W. J . Colloid Interface Sci. 1987, 119(2), 422. (32) Ruckenstein, E.; Chi, J. C. J. Chen. Soc., Faraday Tram. 2 1975, 71, 1690. (33) Ruckenstein, E.; Krishnan, R. J. Colloid Interface Sci. 1980, 75(2), 476. (34) Akoum, F.; Parodi, 0. J. Phys. (Les Ulis, Fr.) 1985, 46, 1675. (35) Lampert, A.; Martinelli, R. U. Chem. Phys. 1984,88,399. (36) Tenchov, B. G.; Koynova, R. D.; Raytchev, B. D. J. Colloid Znterface Sci. 1984, 102(2), 337.
Langmuir, Vol. 5, No. 3, 1989 743
Specific Ion Effects in Electrical Double Layers
,'aN
M*',
Na'
\
CI-
I
Mixing
Phase Separation
Figure 2. Diagram of the experimental procedure.
distinguish them only in terms of their charge. Thus, it is quite inappropriate for our purpose, since we shall always deal with systems in which at least two different cations will be present simultaneously,one of them being sodium, which is the counterion of AOT. Statistical mechanics, too, is not particularly helpful in our case. In general, statistical mechanical theories of double layers are mathematically unreasonably complex, have not been extended to spherical geometries to date, and are strictly applicable to systems with relatively low salt concentrations and low electrostatic potential^.^'^^^ We have consequently developed a phenomenological electrostatic model in which the specific character of each cation (hydrated size and electronic properties) is introduced in an ad hoc manner and in which we utilize a number of ideas existing in the literature on phenomenological extensions of the Poisson-Boltzmann formalism for double layers.
3. Experimental Section 3.1. Materials. Aerosol-OT (bis(2-ethylhexyl)sodium sulfosuccinate) of 99% purity was obtained from Pfaltz and Bauer (Waterbury, CT) and spectrophotometric grade isooctane from Mallinckrodt (St Louis, MO); both were used without further purification. Chloride salts of alkali and alkaline earth metals were also purchased from Mallinckrodt and used as received. Atomic absorption standards were obtained from Aldrich (Milwaukee, WI). Karl Fischer solvent and titrant were obtained from Crescent Chemical Co. (Hauppauge, NY). The water used for preparation of electrolyte solutions was doubly distilled and deionized. 32. Methods. The phasetransfer experimentswere performed as shown schematidy in Figure 2. Aqueous electrolyte solution, 5 mL, containing the chloride salt of an alkali or alkaline earth metal [MCl,] was contacted with 5 mL of a 0.1 M solution of Aerosol-OT in isooctane. The phasecontacting experiment was carried out in carefully stoppered 16 x 75 mm glass test tubes. The phases were vigorously shaken for 15 s in a vibrating shaker and after that for 30 min in an orbital shaker at 120 rpm. Phase demixing was acceleratedby centrifugation at 3200 rpm for 5 min. It was observed that phase separation was rarely complete when the samples were removed from the centrifuge. The test tubes were subsequently placed in a temperature-controlled bath at 25 0.1 OC and left there for 48 h. Water uptake and cation distribution measurements over time showed that this 48-h equilibration period was adequate for complete phase separation and achievement of thermodynamic equilibrium. Of the two phases formed, the lower (denser) phase was an electrolyte solution containing a negligible amount of AOT26as well as the ions M*+,Na+,and Cl-. The upper phase was a reversed micellar solution under the conditions employed. During the course of this experiment it was observed that Li', Be2+,and
*
w'
(37) Levine, P. L. J . Colloid Interface Sci. 1975, 5I(1), 72.
(38)Carnie, S. L.;
Torrie, G. M. Adu. Chem. Phys. 1984, 56, 141.
Figure 3. schematic diagram of the assumed interfacial strudure of the reversed micelle.
did not form reversed micellar solutions in equilibrium with aqueous phases. The very large hydrated size and the electronic properties of these cations apparently favor the formation of different phases. After the test tubes were removed from the temperature-controlled bath, the two phases were carefully separated. The water content of the reversed micellar phase was measured by Karl Fischer titration. The cationic concentrationsof the finalaqueous phase ([Na+] and [M"]) were determined on a Perkin-Elmer Plasma 40 atomic emission spectrometer. 4. Theory 4.1. Our Picture of a Reversed Micelle: Structural Charge Smearing. Traditionally, reversed micelles have been viewed as rigid, spherical cavities containing an electrolyte solution. The surfactant heads have been assumed to be either fully ionized3l3or partly dissociated.% The interior of a reversed micelle has always been considered to be electrically neutral. In our work, we have retained the basic assumptions of electroneutrality, monodispersity, and spherical symmetry of the reversed micelles, and we have also assumed that the surfactant molecules are fully dissociated. This latter assumption is in contrast to the work of a number of authors who have interpreted experimental results on surfactant systems successfully by allowing for surfactant counterion binding or by using the Stern-Gouy-Chapman double-layer theory.6.828 Our picture of the spherically symmetric reversed micelle differs from the traditional view in one important aspect. Instead of assuming that the surfactant heads are placed on a rigid spherical surface, we allow for their distribution over a spherical shell having inner and outer radii Ri and R,, respectively, as shown in Figure 3. Ions in solution (and water as well) are permitted to penetrate this layer
144 Langmuir, Vol. 5, No. 3, 1989
Leodidis and Hutton 110
up to R,. In this way we simulate the well-known phenomenon of surfactant molecular motion and water penetration, using concepts employed successfully with normal micelle^.^^^^^ Furthermore, by permitting the cations to penetrate the interfacial layer of the surfactant molecules, we are effectively allowing for the close association of surfactant and cations, which can be interpretted as “binding”. The surfactant head distribution function is taken to depend only on the radial distance from the center of the water pool, so that the problem retain its spherical symmetry. A n i a n s ~ o nshowed ~~ that the distribution of surfactant heads away from the surface of a normal micelle of radius R, obeys the exponential decay law p d r ) = pW exp(-(r - R,)/O;
r 1 R,
80
(2)
This distribution has the convenient properties that pH(Ri) = 0 and (dpH/dr)rPR, = 0. The constant A is determined by the requirement that the total number of surfactant heads in each micelle must equal the aggregation number, i.e. (3) This results in the expression
A = N A G / [ ~ T ( ( R-: R ? ) / 5 - (R: - R,4)Ri/2 + (R: Ri3)Ri2/3)1 (4) The average charge displacement is not modified much when d (= R, - Ri, the thickness of the surfactant head shell) is varied.between 3 and I A. Consequently, neither were our numerical calculations, and we therefore set d equal to the constant value of 5 A for all values of R,. The (39) Aniansson, E. A. G.; Wall, S. N.; Almgren, M.; Hoffmann, H.; Kielmann, I.; Ulbricht, W.; h a , R.; Lang, J.; Tondre, C. J. Phys. Chem. 1976,80(9), 906. (40) Aniansson, E. A. G . J . Phys. Chem. 1978,82(26), 2805. (41) Cevc, G.; Svetina, S.;Zeks, B. J. Phys. Chem. 1981, 85, 1762. (42) Ohshima, H.; Ohki, S. Bioelectr. Bioenerg. 1986, 15, 173. (43) Ohshima, H.; Makino, K.; Kondo, T. J. Colloid Interface Sci. 1987,116(1), 196. (44) Nichols, A. L.; Pratt, L. R. Faraday Symp. Chem. Soc. 1982,17, 129.
iL ‘ ;Ii
’
l
’
l
’
,
l
,
0 1 M AOT
70 -
(1)
where 1 is the characteristic decay length, approximately equal to the length of a single C-C bond, i.e., 1.5-2 A. If 1 is taken to be equal to 1.5 A, it is found that more than 90% of the surfactant heads is within 5 A from the micellar “surface”. Cevc et al.4l have shown that the main effects of incorporating such an exponential law for the surfactant head distribution in bilayer lipid membranes are a significant reduction of the electrostatic potential at the surface of the membrane and the appearance of a maximum in the electric field intensity profile. Similar conclusions have been drawn by Ohshima et a1.42*43Furthermore, it appears that the electrostatic properties of the double layer are not very sensitive to the precise shape of the charge distribution function but depend strongly on the average displacement of the structural charge from the membrane surface.4lVu We have been able to verify these observations in the case of reversed micelles as well, by experimenting with constant, linear, binomial, and cubic distribution functions. On the basis of these considerations, we have elected to approximate Aniansson’s exponential decay law by the simpler distribution function
pH(r) = A(r - R J 2 ; Ri I r 5 R , = 0; r < Ri and r > R,
l o90 o
l
60
-
50
-
40
-
30 -
20 -
\ 35
0 1
M
AOT
-
30 -
-4 v
25-
20
-
15 -
lo^"'""""""""^
Langmuir, Vol. 5, No.3, 1989 745
Specific Ion Effects in Electrical Double Layers reversed micelle is obtained from the solution of Poisson’s equation in a dielectric medium (7) V.D = (pH(r) + p(r))/eo where D is the dielectric displacement vector, to is the vacuum permittivity, pH(r) is the surfactant head density given by eq 2 and 4, and p(r) is the local charge density in the solution given by m
p(r) = Czieci(r)
(8)
ill
The summation extends over all ionic species. We assume that the dielectric displacement is linearly related to the local electric field, the proportionality constant being identified as the local dielectric constant, i.e. D(r) = t(r) E(r) (9) Then, for spherically symmetric reversed micelles, Poisson’s equation reduces to (l/r2)d/dr[(r24r) EW)l = (pH(r) + p(r))/eo
(10)
or equivalently
In eq 11 we assume that the dielectric constant varies with radial position inside the water pools. This assumption, which we introduce mainly in order to enhance the water pool selectivity for different cations (as will be seen in section 4.3), is quite plausible for two reasons. First, the ionic concentrations vary with position, and it is well established that the dielectric constant of an electrolyte solution is a function of the solution ionic strength. Second, because strong electric field gradients are established near the micelle wall, dielectric saturation phenomena can also be important. Many authors have tried to account for one or both of these phenomena in double-layer theory. The most notable such efforts are those of Grahame,& Deviller et a1.,& Chernenko:’ Gur et a1.,& and Frahm et al.49 We have adopted the dielectric saturation formalism of Frahm et al.49 for its computational simplicity, using Booth’s correlationmfor the dependence of the dielectric constant on the local electric field strength. Thus, we have assumed t=n2+
+
’[
28No?r(n2 2)p 3(73ll2)E
+
731/2Ep(n2 2) 6kBT
]
(12)
with L ( x )= coth x - l / ~
(13)
where n is the refractive index of water, p its dipole moment, and No the number of water molecules per unit volume of solution. This equation has a statistical mechanical basis and is strictly applicable to pure water and not to electrolyte solutions. Nonetheless, we have been encouraged by the fact that this correlation has already been used previously with some success in treating electrical double-layer problem^.^^^^ (45) Grahame, D. C. J. Chem. Phys. 1950,18,903. (46) Deviller, C.; Sanfeld, A,; Steinchen, A. J. Colloid Interface Sci. 1967,25, 295. (47) Chernenko, A. A. Sou. Electrochem. 1981,17, 511. (48) Gur, Y.; Ravina, I.; Babchin, A. J. J. Colloid Interface Sci. 1978, 64(2), 326. (49) F’rahm, J.; Diekmann, S.J . Colloid Interface Sci. 1979, 70(3), 440. (50) Booth, F. J. Chem. Phys. 1951, 19(4), 391.
The electrostatic potential is defined in terms of the electric field by the equation d$/dr = -E (14) Equations 11and 14 can be solved simultaneously to yield the electric field and potential distributions within the reversed micelles. To complete the problem specification, however, two boundary conditions are required. One is provided by the spherical symmetry of the micelle, yielding E(r=O)= 0 (15) The second boundary condition is a “tentative” one, in which the potential at the micelle water pool outer surface is specified, i.e. $(r=R,) = $w (16) In the classical Gouy-Chapman theory of double layers, the “wall” boundary condition is known a priori if the charge or the potential at the charged surface has been specified. It is this boundary condition that guarantees the overall electroneutrality of the double-layer system. In our case, the structural charge of the surfactant head is not smeared over a surface but is distributed over a shell of finite volume. Thus, the value of $w in eq 16 is not known a priori but must be determined iteratively to ensure that the electroneutrality condition inside the micelle is satisfied. 4.3. Relation between the Ionic Concentrations and the Local Electrostatic Variables. The simultaneous integration of eq 11and 14 is feasible once the local ionic concentrations CFv(r) in the water pools are related to the local electrostatic variables e(r), E(r), and $(r). This connection is achieved by assuming that the water pools of the reversed micelles and the excess aqueous phase are in thermodynamic equilibrium, which implies equality of the electrochemical potentials in the different phases for the individual ionic species, i.e. pClt”w(r) (17) where the superscripts bw and mwp denote bulk water and micellar water pools, respectively, and pi is the electrochemical potential of any ionic species i. If we set the electrostatic potential in the bulk of the excess aqueous phase equal to zero, eq 17 can be written as p?(T,P) kBT In (rPwcpw) = P?(TS)+ ~ B InT (rF’(r)CY’V)) + ziei.4) (18) or equivalently kBT In (cPw/cyw(r)) kBT In (rPw/ryw(r)) zie$(r) = 0 (19) In the Gouy-Chapman theory, or the equivalent PoissonBoltzmann description for electrolyte solutions, the activity coefficient corrections are ignored to yield the simple relation pPW =
+
Crw(r) = CPw exp(-zie$(r)/(kBT)) (20) This neglect of the activity coefficient corrections is tantamount to ignoring a number of potentially important phenomena, the most significant being ionic size and polarizability, ion-ion interactions, variation of solvent compressibility and dielectric constant, and image interacmechanical calculation by Bell and t i o n ~ . ~ ’A , ~statistical ~ L e ~ i n for e ~ an ~ ionic double layer near a planar charged wall indicated that the ion concentration distributions should take on the more complex form (51) Cevc, G.; Marsh, D. J.Phys. Chem. 1983,87, 376. (52) Bolt, G. H. J. Colloid Sci. 1955,10, 206. (53) Levine, S.;Bell, G. M. J. Phys. Chem. 1960,64, 1188.
746 Langmuir, Vol. 5, No. 3, 1989
ci(x)= J i ( X ) CfefeXp(-Vi"(X)/kBT)
Leodidis and Hatton (21)
where Ji(x) depends on the hydrated volume of the ith ionic species, Cyf is a reference concentration, and Vio(x) is a "potential of the mean force", which contains contributions from all the interactions between the ith ionic species and all other components of the system. This approach has led to the development of the "Modified Poisson-Boltzmann (MPB)" theory of the double layer." Based on such considerations, we have approximated the statistical mechanical formalisms in our phenomenological model by making the assumption that the activity corrections can be expressed as a h e a r combination of a series of different interaction terms, viz.
[
~ B InT y ; G r ) ] = A P ~ W ~ $&~kL$TV + AP~KY% (22)
There is no rigorous justification for separating the various contributions to the activity coefficient ratio in this way, but it does simplify the problem formulation significantly. Equation 22 implies that only pair interactions are important, so that ion-solvent interactions are decoupled from ion-ion interactions. Entropic Term. The entropic contribution is a volume exclusion effect in recognition of the finite ionic size. I t is expected to be especially important close to the surfactant interface, especially in the surfactant head layer between Riand R,. In fact, a recent simulation by Wennerstrom et al.65shows that "hard core" effects can be very important close to a "discrete charge interface". A number of ways to account for the finite size of ions in a phenomenological context have been proposed. Glueckauf6B and Chernenkos7have proposed hydration number corrections, while Radic et aLMused hard-sphere theory. Numerous other theories have appeared in the past 50 years. The probabilistic treatment of the ionic size effect by Ruckenstein and Schiby72is appealing intuitively, and its simplicity is well suited to our purpose. The volume of the water pools is divided into layers, and each layer into identical sites, the volume of each site being equal to the volume of a water molecule. It is assumed that a hydrated ion of species i occupies T~ sites. The totalnumber of sites per unit volume at a distance r from the center of the water pool is rn
n, = n,(r)
+ jC= l ~ ~ n ~ ( r )
(23)
where nj(r)is the local number density of species j . The fraction of sites occupied by hydrated ions is
f ( r )=
1 m
C~jnj(r) ne j = l
(24)
The probability of finding T~ available neighboring vacant sites in which to introduce an ion of species i and generate a hydrated ion is (25) Pf"W(r) = [l - f(r)]" In the excess aqueous phase this probability is a constant
pp" (54) Levine, 1978, 74, 1670.
= [ l - p"]r,
These simple formulae show that in general the number of positions available locally to a hydrated ion is smaller than the number available to a nonhydrated species because of volume exclusion. We can now identify the entropic contribution to the activity coefficient ratio in eq 22 as being
Ion-Solvent Term. The difference in ion-solvent interactions between the excess aqueous phase and the micellar water pools is an additional factor for cation discrimination, and it is mostly due to modified electrical interactions and ionic polarizability. We assume that ApfGLTvcan be modeled as a difference in the electrostatic parts of the free energy of hydration of the ions, which can be treated according to a modified Born model. Although more complicated models have been proposed,"'SB1 it seems that electrostatic free energies of solvation correlate well with the inverse of the dielectric constant of the solvent,62 and the use of a modified Born model is justifiable. In the modified Born model the electrostatic free energy change upon moving an ion from vacuum to a solvent of dielectric constant t is AG&1 = -Ai(l - l / t )
(28)
Born originally postulated that
where zie is the ionic charge and ri the Pauling radius of the ion, but this identification of Ai significantly overestimated AG&. Reliable estimates for Ai can be obtained, however, by using the experimental data for the Acte1 tabulated by no ye^.^^ With these considerations in mind, we can express the ion-solvent part of the activity coefficient ratio in eq 34 as ApkL$Tv = Ai(l/tbw - l/t"W(r))
(30)
Gur et aL4*were the first to introduce this expression in phenomenological double-layer theory, and Cevc et a1.61 subsequently used it with considerable success in their double-layer analyses. Ion-Ion Term. Ion-ion interactions in the diffuse part of the double layer can be treated consistently by statistical mechanical theories only. In most phenomenological models these interactions are ignored. A notable exception is the model developed by Izumitani et al.,&iwho used the activity coefficient expressions of Harned and Owena to account for interionic interactions. This does not seem totally justified, since all available activity coefficient correlations presuppose a locally electroneutral solution, which is not the case in a double layer. An alternative approach is offered by Bolt.52 In view of the fact that a
(26)
S.;Outhwaite, C. W. J. Chem. Soc., Faraday Trans. 2
(55) Wennerstrom, H.; Jonsson, B. J. Phys. (Les Ulis, Fr.) 1988.49. 1033. (56) Glueckauf, E. Trans. Faraday Soc. 1966,51,1235. (57) Chernenko, A. A. Sov. Electrochem. 1986,22, 1478. (58) Radic, N.; Marcelja, S.Chem. Phys. Lett. 1978, 55(2), 377. (59) Ruckenstein, E.; Schiby, D. Langmuir 1985, I, 612.
(60) Abraham, M. H.; Liszi, J.; Meszaros, L. J. Chem. Phys. 1979, 70(5), 2491. (61) Kornyahev, A. A,; Volkov, A. G. J.Electroanal. Chem. 1984,180, 363. (62) Markin, V. S.;Volkov, A. G. Sou. Electrochem. 1987,23, 1042. (63) Noyes, R. M. J . Am. Chem. SOC.1962,84, 513. (64) Izumitani, Y.; Fukuoka, H.; Nakamura, T. J . Colloid Interface Sci. 1985, 108(2), 356. (65) Harned, H. S.;Owen, B. B. The Physical Chemistry of Electrolytic Solutions, 3rd ed.; Reinhold: New York, 1958.
Langmuir, Vol. 5, No. 3, 1989 141
Specific Ion Effects in Electrical Double Layers satisfactory phenomenological treatment of ion-ion interactions has not yet been proposed, we have chosen to neglect this term. Having identified all the terms in eq 22, we can return to eq 19 to obtain the required connection between the local ionic concentrations and the local electrostatic variables:
Table I. Variation of C R J / C bwJ with TN.+ for C kwsa= 0.1 M and C:$ = 0.1 M
50
This equation is implicit in CFW(r) because of the definition of f(r) (eq 24). Within the interfacial layer (between Ri and RJ,the preexponential correction term for ionic size effects must be modified to
[($k) - P V 9 ) / ( 1- Pw)lT~ With all contributions to eq 11and 14 now identified, we can carry out the integration to obtain concentration and potential profiles within the micellar water pools. The solution is tedious because of the requirements for micellar electroneutrality and material balance closures, described in Appendix C. The solution procedure is outlined in Appendix D. 4.4. Parameter Selection. A number of parameters must be assigned values before numerical solutions to the electrostatic problem can be obtained. The thickness of the surfactant head group shell has been fixed at 5 A, as discussed in section 4.1. The ionic parameters Ai in eq 31 were also uniquely determined by using eq 28, and the tabulated values of AG$] as given by no ye^.^^ The size parameters ~i reflect the hydrated sizes of the ions. There is, however, considerable ambiguity in the literature regarding both the size of hydrated ions and the hdyration number. In their original publication, Ruckenstein and Schiby59 used the value T = 5 for monovalent cations. Since T~ is a significant parameter in our model and since we would like to avoid adjustable parameters to the extent possible, we have assumed that, given a value for T N ~ +we , can calculate the other ~i values through the simple formula Ti
=
TNa+(ri,h/rNa+.h)
3
(32)
where ri,his the hydrated radius of ion i, as tabulated by Conway.ss For the chloride ion, which is largely excluded from the water pools, volume exclusion effects are unimportant, and the results are independent of the T C ~value used. We therefore arbitrarily set T C ~= 1. We have determined T N ~ by + fitting the model to a single experimental point, specifically, the case for which the surfactant concentration in the initial organic phase was 0.1 M and the initial aqueous phase contained 0.1 M KC1. The value of Qa+ thus obtained was used in all subsequent calculations, irrespective of AOT concentration or cation type and concentration in the initial aqueous phase. The criterion used for selecting T N ~ +was that the calculated ratio of sodium to potassium concentrations in the final excess aqueous phase agree with the experimentally observed value for this one case. From the results shown in Table I we conclude that the optimum value for T N ~ +is between 3.5 and 4. Since deviations of 2-3% were of the order of the experimentalerror, we simply selected the integer value (66) Conway, B. E. Ionic Hydration in Chemistry and Biophysics; Elsevier: Amsterdam, 1981.
45
*.
,
A
40 -
o
0.
+,
35 -
0
30
-
\
25
-
'?
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,
abs dev"
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TK+
TN.+
~
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~
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~
~
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-
Var of Salt Conc Var of AOT Conc Var o f Phase Volume Ratio
-
4 4
e
-
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-
-
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00
00
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e '
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'
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6
= N0*0T
09
'
12
'
'
15
1
'
18
/
1 ' 21
~
24
'
E7
30
NOKC1
Figure 5. Universal curve for potassium. Ratio of final con-
centrations of Na+ and K+ in the excess aqueous phase vs the with variation of ( 0 )[KCl], normalizing parameter 6 = pAOT/pK~I (0) phase volume ratio, and (A)surfactant concentration. T N ~ += 4 and used this value in the rest of this study. Given this choice of T N ~ +the , T~ for the other ionic species were calculated, and these are reported in Table 11. Also shown in this table are the values of the Ai.
5. Results and Discussion 5.1. Experimental Results. Comparison of Experiment and Theory. The model predictions were compared with experimental results on the solubilization of a range of different monovalent and divalent cations under varying conditions of salt and surfactant concentrations. The first interesting experimental finding is that the ratio 6 = pAOT/ZpMCl, is the most satisfactory choice for monitoring the behawor of om liquid-liquid extraction systems. pAoT is defined as the total number of AOT moles in the system, and pMc1, is the total number of moles of electrolyte MC1, introduced in the initial aqueous phase. In Figure 5 it is shown that when the ratio of sodium to potassium concentrations in the final aqueous solution is plotted against the parameter 6 the data of three different experiments collapse onto a single plot. In these experiments the value of 6 was changed in the following ways: (i) by changing the initial KC1 concentration in the aqueous phase, while keeping the AOT concentration in the organic phase constant, and setting = 1; (ii) by changing the AOT concentration in the organic phase, while keeping the initial KC1 concentration in the aqueous phase con/: & = 1; and (iii) by varying stant, and setting VLV V i / V&, while keeping the AOT concentration in the organic phase and initial KC1 concentration in the aqueous phase constant. The fact that we can collapse these three data sets onto one universal curve indicates that cation distribution is affected only by specific ionic parameters (e.g., charge,
eq/ erg
748 Langmuir, Vol. 5, No. 3, 1989 Nat 1,
A,
4 99.6
Leodidis and Hatton
Table 11. Values of ~i and Ai (kcal/mol) for Seven Cations K+ Rb+ cs+ Ca2+ 3.2 81.6
3.1 76.5
hydrated size, polarizability) and not by the micellar size, the surfactant concentration, or the ratio of the phase volumes. The universal distribution curves for the monovalent cations are shown in Figure 6a. The behavior of K+, Rb+, and Cs+ is quite similar, as anticipated, since these ions have almost the same hydrated size, and the dielectric exclusion is not very strong in the case of monovalent cations. A significant selectivity of the micellar pools toward these three cations with respect to sodium is observed, however. This must be attributed mostly to the size difference between the larger hydrated Na+ and the three other ions. The universal curves are almost linear, asymptotically approaching the "uniform distribution" or "no-selectivity" curve at low values of 6. Experimental results for potassium are also included in this plot. The agreement between theory and experiment is excellent (to within 1-5%). Similar agreement was obtained for rubidium and cesium, but the results are not presented in this figure for the sake of clarity. The behavior of divalent cations is compared in Figure 6b. Divalent cations are more strongly drawn toward the water pools because of the strong electrostatic interaction. There is a significant difference in selectivity between the three ions, which can only be understood in terms of the different hydration free energies, since the hydrated sizes of the ions are almost the same. The universal curves are quadratic, asymptotically approaching the no-selectivity curve at low values of 6. Experimental results for calcium only are presented in this plot, again for the sake of clarity. The agreement between theory and experiment is surprisingly good (to within 2-10%). Apparently, our neglect of ion-ion interactions in the double layer is partly compensated for by the dielectric constant variation. Typical ion concentration profiles inside the water pools are shown in Figure 7a for the particular case of 0.1 M AOT organic solution in contact with an aqueous phase initially containing 0.1 M KC1. The chloride ion is largely excluded from the water pools, because of the unfavorable electrostatic interaction owing to the negative potential inside the water pool. The combination of dielectric and volume exclusion leads to a distinct difference between the K+ and Na+ profiles. The potassium peak is much higher and is achieved closer to R, than is the sodium peak. Thus we see that potassium ions can penetrate the surfactant layer more effectively than sodium ions. The absolute magnitudes of the concentrations close to the micellar "wall" (2-4 M) are physically realistic, in contrast to the predictions afforded by the classical Gouy-Chapman theory. In the Poisson-Boltzmann approximation, ions are treated as points, and their calculated concentrations at the micellar wall can be unnaturally high, ranging from 10 to 100 M. The magnitudes of the various contributions to the Na+ concentration distribution for the same experimental conditions are shown in Figure 7b. The dielectric exclusion component is certainly not dominant but is nevertheless important for monovalent ions. For divalent ions, this term contributes significantly more to the overall ion concentration distribution functions, becoming as important as the volume exclusion term. The electrostatic potential profiles shown in Figure 7c indicate that the predicted electrostatic potential throughout the micellar pool is higher than that obtained
Sr2+
6
3.1
Ba2+
6
5.8
0 1 M AOT 225
-
/ /
Cesium
-
/
- - .... Rubidium
...
-_ 2a
175
-
u5:
150
-
\
125
-
+
-U.
+
Potassium
,,/
100 -
z c)
'
=
/
NoAOT
NoMCI
0 1 M AOT Barium 12
..-..- -.. Strontium
Unirorm distribution ~ u m e 0 ~ " 000
"
~
"
030
'
"
"
060
=
'OAOT
"
"
"
090
/
"
" " 120
'
1.50 1 50
180
2NoMC12
Figure 6. Universal curves (ratio of final concentration of Na+ and zM2+in excess aqueous phase vs the normalizing parameter 6 = NoAm/zNoMcl, for (a, top) monovalent and (b, bottom) divalent cations.
from the corresponding Poisson-Boltzmann solution. This is partly due to the fact that the dielectric constant is much lower than 78.3 in the last 10 A of the integration domain. Furthermore, the dielectric and volume exclusion of the cations from the micellar surface renders it impossible to randomly concentrate cations close to the micellar interface, as is allowed in the Poisson-Boltzmann description. This necessitates an increase in the potential to ensure that sufficient cations enter the micelle for electroneutrality to be attained. The electrostatic potential would have been even higher had we not permitted the cations to penetrate the surfactant head shell. If, in fact, we had assumed that all the surfactant heads are rigidly positioned on a spherical surface, and if we had used Booth's correlation
Langmuir, Vol. 5, No. 3, 1989 749
Specific Ion Effects in Electrical Double Layers
a
5000 4500
-7 3
C
300
7
Potassium
4000 3500
0
3000
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2500
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1000 500
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d e+/kLlT Diel. exclusion Volume exclusion
;
S u m of all c o m p o n e n t s
w
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-------lY-_-~~.-----_____
~
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~
right up to this surface, we would have found that the limiting value of the dielectric constant (c, discussed in detail below) drops to 5-10, and the limiting value of the potential ($,I takes on values as high as 400-600mV. The model predictions (with respect to cation distributions) would also be poorer because of the strong dielectric exclusion that is observed a t these low values of c,. Figure 7d shows a typical dielectric constant profile inside the micelle corresponding to the same conditions discussed above (0.1 M AOT/O.l M KCl). Dielectric saturation becomes important at a distance of 10 A from R,. The horizontal portion of the plot in the last 3 A can be explained as follows. Because the surfactant structural charge was assumed to be distributed over a spherical shell of finite thickness, and was treated as a volume charge density in the Poisson equation (eq l l ) , the electric field intensity was predicted to have a maximum somewhere between Ri and R,. If we were to use the Booth dielectric constant correlation right up to R,, the dielectric constant-being a function of the electric field only-would
exhibit a corresponding minimum. In addition, in the outermost 2 A of the micellar water pool the volume fraction #w(r)drops to below 50%,and the use of Booth's free water correlation in this region is totally unjustified. To circumvent these problems, Booth's correlation was used up to the point where the maximum in the electric field was observed, following which t was kept constant at the value obtained at this maximum. Figure 8 illustrates how the limiting value of the dielectric constant at the micellar wall (ew) behaves as a function of the micelle size, Ro. Two very encouraging characteristics are observed. The first is that the values o f t lie between 30 and 50 for Ro varying between 15 and 65 These values agree very well with dielectric constant values that have been "measured" at the surfaces of normal micelle^.^-'^ The second satisfactory characteristic is that
1.
~~
(67) Mukerjee, P.;Cardinal, J. R.; Desai, N. R. In Micellization, Solubilization and Microemulsiom; Mittal, K. L., Ed.; Plenum Press: New York, 1978 Vol. 1.
Leodidis and Hattan
750 Langmuir, Vol. 5, No. 3, 1989 50
35
I
, ,
~
1
,
,
,
,
1
,
,
,
,
/
/
,
,
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,
,
,
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,
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,
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e
30
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l
'
20
25
30
35
,
l
40
,
l
45
,
1
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,
I
55
60
65
15
00
01
02
03
04
OS
CKClaq,ln(
06
07
08
09
10
M )
Figure 8. Calculated limiting value of the dielectric constant at the micellar wall m a function of micellar size R, obtained with variation of ( 0 )KCl, (0) phase volume ratio, and (A)surfactant
Figure 9. Surfactant degree of dissociation given by eq 33 as a function of initial KC1 concentrationin the exceaa aqueous phase.
concentration.
Some experimentalresults of the present work offer insight into the optimum method for carrying out this extraction process in AOT systems. Since protein size exclusion from the micelles appears to be an important consideration, our experimental results indicate that for the forward extraction we should choose an accompanying electrolyte that promotes the formation of large water pools in the absence of protein, even at the sufficiently high salt concentrations required to ensure that we are far from critical points. Thus sodium and calcium salts seem to be much better accompanying electrolytes than do potassium, rubidium, cesium, and strontium salts. For the back-extraction or stripping step, high concentrations of a salt that produces small reversed micelles are desired. Thus potassium chloride should be much better for the back-extraction than is sodium chloride. (ii) The AOT/isooctane reversed micellar system does not have sufficient selectivity for the separation of monovalent cations. In the case of divalent cations, the system looks more promising. Divalent cations with smaller hydrated radii and smaller hydration free energies are favored for transfer to the water pools, especially at low divalent ion concentrations. In principle, it seems feasible to selectively extract one particular cation from a mixture, provided its hydrated radius and/or hydration free energy differs substantially from those of the other cations present. AOT or similar reversed micellar solutions could certainly be used as solvents for the removal of heavy metal ions from waste streams, the main problem being the necessity for regenerating the solvent after a certain number of extraction cycles. (iii) The success of the present model in treating a specific cation effect observed in reversed micellar solutions raises the interesting possibility that a similar treatment can explain cation effects in other systems. Missel's results" on the cmc values and hydrodynamic radii of SDS micelles in aqueous solutions with various counterions may possibly be explained quantitatively in terms of a thermodynamic model in which the electrostatic free energy Fd is calculated from our model of the diffuse double layer. Similar quantitative success might be obtained in the w e of the T-A isotherms of sodium nonadecylbenzenesdfonate in the presence of various electrolytes.' Cation effects observed in polyelectrolyte systems4in protein salting out:
eW appears to depend primarily on R, (or wo)and does not depend on external salt concentration, AOT concentration, or the ratio of the phase volumes. It is, however, a weak function of salt type, being 5 1 0 % higher for divalent cations at the same R,. Another interesting result is obtained when we determine the number of cations in the surfactant head layer between Ri and R,. In our model, we ignored the possible "covalent" binding of cations to the surfactant heads. However, if we consider all the cations in the surfactant head layer as being "bound", then the fraction of surfactant dissociated would be f d = 1 - ( N ' N ~+ + ZN'MZ+)/N,G (33)
where N : is the number of cations of species i in the surfactant head layer. In Figure 9, f d is shown as a function of initial KC1 concentration in the excess aqueous phase. The results obtained are encouraging. The calculated values of fd, which vary between 20% and 32%, agree well with experimental results reported in the literature72and with the theoretical predictions of Beunen et al.73and Evans et al.74 A binding selectivity toward the smaller cation, or the one with the smaller hydration free energy (in this case K+ vs Na'), is predicted. The most satisfying aspect of this calculation is that the degree of dissociation, or the number of bound counterions, is introduced in a natural way, and no additional length scale need be used to account for the binding, aside from the surfactant head shell thickness,
R,
- Ri.
5.2. Implications of Results of This Work. There are many significant implications of the theoretical and experimental results presented above. (i) The liquid-liquid extraction of proteins using reversed micellar organic solvents has been considered as a promising protein separation process in recent (68)Zachariasae, K. A.; Van Phuc, N.; Kozankiewicz, B. J. Phys. Chem. 1981,85,2676. (69)Lessard, J. G.; Fragata, M. J. Phys. Chem. 1986,90,811. (70)Drummond, C. J.; Grieser, F.; Healy, T. W. Faraday Discuss. Chem. SOC.1986,82,95. (71)Warr, G. G.; Evans, D. F. Langmui; 1988,4, 217. (72)Wong, M.;Thomas, J. K.; Nowak, T. J. Am. Chem. Soc. 1977, 99(14),4730. (73)Beunen, J. A.;Ruckenstein, E. J. Colloid Interface Sci. 1983, 96(2), 469. (74)Evans, D. F.; Mitchell, D. J.; Ninham, B. W. J.Phys. Chem. 1984, 88,6344.
(75)Goklen, K. E.; Hatton, T. A. Biotech. Prog. 1985, I, 69. (76)Dekker, M.;Van't Riet, K.; Weijers, S. R.; Baltussen, J. W. A.; Laane, C. Chem. Eng. J. (Lawanne) 1986,.33,B27. (77)Goklen, K.E.; Hatton, T. A. Sep. SCLTechnol. 1987,22(2,3),831.
Langmuir, Vol. 5, No. 3, 1989 751
Specific Zon Effects in Electrical Double Layers and in biphasic aqueous polymer systems6 can also be understood in principle both qualitatively and quantitatively in the same way. (iv) The present model can also be applied in two cases in which the initial aqueous phase contains at least two different cations. First, the system response when both sodium and potassium exist in the initial aqueous phase can be probed. A second dimension is introduced in this way, and the “universal curve” should become a “universal surface”. The second application would probe the selectivity of the AOT reversed micelles toward divalent cations. More than one divalent cation could be introduced in the initial aqueous phase, and it is important that we be able to predict their relative distributions between the phases. It is also necessary to check whether the present model is capable of treating divalent cations of transition elements with equal success or if specific chemical interactions need to be accounted for. 6. Conclusions
We have developed a phenomenological electrostatic model that has been successful in treating the distribution of cations between a reversed micellar solution of AOT in isooctane and an excess aqueous phase. Given information on the equilibrium water uptake by the reversed micellar solution, which determines the reversed micelle radius, the model can predict the distribution of alkali and alkaline earth cations between the two phases utilizing only one adjustable parameter, identified as a measure of the hydrated size of the Na+ cation. The model predictions are in excellent agreement with experimental results. Discrimination between different cations is on the basis of charge, hydrated size, and electrostatic free energy of hydration. The explicit inclusion of cation binding to the surfactant molecules (as frequently required in the Stern-Gouy-Chapman phenomenological theory) is avoided by distributing the micellar structural charge over a thin interfacial shell of 5-A thickness and permitting the cations and the solvent to penetrate this layer. Local variations in dielectric constant with electric field intensity are allowed in the model, as a means for achieving additional selectivity toward different cations, with predicted values of the dielectric constant at the micellar wall of between 30 and 50; these are in agreement with published experimental data on micellar systems. The success of the model in predicting the selectivity of reversed micellar salt solubilization suggests that similar electrostatic treatments might be capable of explaining qualitatively and quantitatively the specific cation effects observed in many colloidal and biological systems of importance. Acknowledgment. We are grateful to the NSF Biotechnology Process Engineering Center at MIT for support of this work (Cooperative Agreement CDR-85-00003). Appendix A A precise calculation of the volume fraction of water in the surfactant head layer is practically impossible. In view of the relative unimportance of the form of the head distribution function, we calculate #w(r)in the following approximate way. Each surfactant molecule is viewed as a rectangular block, with no distinctions being made between the head and tail regions of the molecule. The cross sectional area of each of these blocks is assigned a fictitious value fkoT, such that The volume that each surfactant molecule cuts from a
spherical shell of radius r and width dr is equal to f io?$r. The volume fraction occupied by the surfactant molecules at position r is then #s(r) =
fioTdr(number of heads at Ri Ix Ir ) 4rr2dr
- fiOTNAG 4rr2
ff(r)
4Ro)
642)
644)
On further simplifying, using eq Al, we obtain
We see from this result that our choice for fiOT (eq Al) ensures that &(R,) = 1. Since &,(r) = 1- &(r), we obtain eq 5. Appendix B For AOT systems, the water pool radii of reversed micelles have been determined under a range of conditions by using small-angle neutron scattering,7a81 small-angle X-ray scattering,82and time-resolved fluorescence metho d ~ Eicke . ~ ~and Rehaks4applied static light-scattering measurements to the water/AOT/isooctane system and compared results from static light scattering, ultracentrifugation, and vapor pressure osmometry. Their experimentally determined dependence of the interfacial area per AOT molecule (fAOT) on the amount of water added to the surfactant solution can be represented by the equation fAoT (A2/molecule) = 57 - (11exp[-0.09163(wo - lo)]) 031) With eq B1, one can also determine the radius of the water pools as a function of the water content of the organic phase. By assuming that all the surfactant is located at some fictitious radius R, (Ri < R, < R,) within the reversed micelle, we can write balance equations for the total water solubilized by these micelles and for the total surfactant in the system. These equations are ( N o T / N A G ) ( ~ / ~=) W~&RO T~V ~ ~ NAGfAOT(Wo)
= 4rRm2
(B2) 033)
respectively, where lVOA0T is the total number of surfactant molecules in the organic phase and vw is the molecular volume of water. Combination of these two equations gives Rm = 3wOvw/fAOT(wO)
034)
As noted before, the radius R, lies between Ri and R,. The outer radius, R,, can be calculated by equating the volume of the cavity in our micelles to (4/3)rRm3,giving the implicit equation (78) Kotlarchyk, M.; Chen, S.-H.;Huang, J. S. J.Phys. Chem. 1982, 86(17), 3273. (79) Kotlarchyk, M.; Stephens, R. B.; Huang, J. S. J. Phys. Chem. 1988, 92, 1533. (80) Cabos, C.; Delord, P. J. Appl. Crystallogr. 1979, 12, 502. (81) Robinson, B. H.; Toprakcioglu, C.; Dore, J. C.; Chieux, P. J. Chem. Soc., Faraday Trans. 1, 1984,80, 13. (82) Cabos, C.; Marignan, J. J. Phys. Lett. 1986, 46, L-267. (83) Lang, J.; Jada, A.; Malliaris, A. J. Phys. Chem. 1988, 92, 1946. (84) Eicke, H. F.; Rehak, J. Helu. Chim. Acta 1976, 59, 2883.
152 Langmuir, Vol. 5, No. 3, 1989
Leodidis and Hatton is a reasonable assumption for the salt conditions used in this work.
which must be solved numerically for R,. Here &(r) is the water volume fraction in the surfactant head layer, given by eq 5, and Ri = R, - 5 A. Thus, by assuming that Eicke's correlation given by eq B1 holds irrespective of surfactant concentration and salt type and concentration inside the micelles, we can transform simple experimental measurements of the water content into reasonably accurate water pool size information. The experimental results shown in Figure 1have been used to construct the curves shown in Figure 4, which relate the outer water pool radius, R,, to the salt concentration for the various salts used in this work. SANS data obtained by one of the present authors (T.A.H.) indicate that the micellar size calculation that we have presented in this section yields very satisfactory results.
Appendix D The simultaneous solution of the electrostatic equations (11) and (14) was achieved by using a fourth-order Runge-Kutta algorithm. A shooting method was employed with Newton-Raphson iteration on the necessary conditions. Because of the complexity of the system, three iteration loops were required: (1)satisfaction of the wall boundary condition, eq 16, for given It,in the integration of the electrostatic equations, which, in essence, required iteration on the potential at the center of the water pool; (2) iteration on rl, to satisfy the internal electroneutrality constraint in the water pool; and (3) closure of mass balances for all ionic species by iteration on C&$ to obtain the value of C@, which satisfies the electroneutrality condition for the excess aqueous phase (eq C3).
Appendix C A number of different charged species will be found in each micelle. These are (i) surfactant head groups, each with a charge of -e, their total number being the aggregation number N A G of the micelle; (ii) surfactant counterions (in our case, Na+);and (iii) Mz+and C1- ions which have been transferred to the water pools from the excess aqueous phase initially containing the electrolyte MC1,. The electroneutrality constraint inside each micelle can be expressed as N A G + Pel- = P N a + + z N m ~ * + (Cl)
Notation constant appearing in the surfactant head distribution function specific ionic constant in the expression of the electrostatic free energy of hydration concentration of ionic species i in the excess aqueous phase in the final equilibriumsituation concentration of ionic species i in the initial aqueous phase before contact local concentration of ionic species i in the reversed micellar water pools surfactant head layer width dielectric displacement vector electronic charge (=1.6 X C) electric field vector local fraction of sites occupied by hydrated ions in micellar water pools fraction of sites occupied by hydrated ions in excess aqueous phase interfacial area per AOT molecule dissociation fraction of surfactant electrostatic part of the Helmholtz free energy of the reversed micellar phase electrostatic free energy of hydration of ionic species i ionic size factor in the statistical mechanical theory of Bell and Levine Boltzmann constant (=1.38X J/K) surfactant head distribution decay length Langevin's function, given by eq 12 total number of ionic species in the reversed micelles total number of sites per unit volume of water local number density of ionic species i and of water, respectively aggregation number of the micelles Avogadro's number total number of AOT moles in the organic phase total number of electrolyte moles in the twophase system total number of reversed micelles in organic phase number density of water molecules in bulk water total number of ions of species i in one reversed micelle number of cations of species i in the surfactant head layer probability of generating a hydrated ion of species i in excess aqueous phase and reversed micellar water pools, respectively radial distance from the center of the micelle Pauling and hydrated radius of ionic species i internal and external radius of the surfactant head layer
or equivalently
A similar electroneutrality condition holds for the excess aqueous phase N&' = N&f (C3)
+ zmi
where the superscript bw denotes bulk water and the superscript f implies the final equililibrium situation. Material balance constraints must also be observed in the evaluation of the equilibrium distribution of ions between the two phases. Initially, an aqueous electrolyte solution of volume and ionic concentrations C!wF is contacted with a surfadant-rich organic solution of volume After equilibration, the aqueous phase volume is V&, and the final ionic concentrations are Cfwpf. Mass balance equations can be written for the various species as follows: water: v; = vaq + VmW (C4)
eq
c,.
ions
Mz+, C1-:
Na':
Vmwp is the total volume of water transferred to the organic phase, Ced is the concentration of surfactant in the initial organic phase, and Nmic is the total number of reversed micelles in the oil phase:
Equation C7 presupposes that the surfactant and oil are not soluble in the aqueous phase and that practically all the surfactant in the oil phase is found in the micellar interfaces. In light of the results of Aveyard et al.,25this
Langmuir 1989,5, 753-757 fictitious micellar radius given by eq B4 absolute temperature (K) potential of mean force used by Bell and Levine initial volume of organic phase before contact initial and final volume of the aqueous phase before and after contact total volume of water in organic phase after contact molar ratio of water to surfactant in organic phase after contact charge of ionic species i function appearing in eq 5 activity coefficients of ionic species i in excess aqueous phase and micellar water pools, respectively ratio of total surfactant to total electrolyte moles in the system C2N-' m-2) vacuum permittivity (r8.854 X dielectric constant of water at the micellar wall local dielectric constant inside the micellar water pool bulk dielectric constant of the excess aqueous phase dipole moment of water (=1.85 D)
753
chemical potential of ionic species i at a reference state, in the bulk of the excess aqueous phase, and inside the micellar water pools, respectively chemical potential difference of ionic species i between excess aqueous phase and micellar water pools due to ion-ion interactions, ionsolvent interactions, and volume exclusion, respectively distribution function of the surfactant heads and its value at the micellar wall local ionic charge density inside the water pools hydrated size parameter of ionic s ecies i molecular volume of water (-30 &) volume fraction occupied by water and by surfactant molecules, respectively, in the surfactant head layer local value of electrostatic potential in the water pools and its value at the micellar wall (i.e., at Ro)
Registry No. AOT, 577-11-7; Na, 7440-23-5; K, 7440-09-7; Rb, 7440-17-7; Cs, 7440-46-2; Ca, 7440-70-2; Sr, 7440-24-6; Ba, 744039-3; isooctane, 540-84-1.
Study of Aqueous Tetradecyltrimethylammonium Bromide-Brij 35 Solutions by Ion Activity Measurements E. B. Abuin,t E. A. Lissi,*it R. Niifiez,f and A. Oleat Departamento de Qdmica, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 5659, Santiago 2, Chile, and Departamento de Quimica, Facultad de Ciencias, Universidad de Chile, Casilla 3425, Santiago, Chile Received April 14, 1988. I n Final Form: October 25, 1988 Measurements of surfactant ion (tetradecyltrimethylammonium)and counterion (bromide) activities in binary tetradecyltrimethylammonium bromide ('I"I'AB)-Brij 35 surfactant mixtures have been performed by using ion-specific electrodes. From the data obtained in the TTAB solutions in the absence of Brij 35, a TTAB critical micelle concentration (cmc) of 3.6 (h0.2)mM was evaluated. The degree of counterion binding on TTAB micelles was calculated from the concentration dependence of bromide activity above the cmc, &, and by applying the charged phase separation model to the surfactant and counterion activity data obtained, jjcpe. Values found were &, = 0.75 and p+., = 0.8. From the data obtained in the surfactant mixtures, the micellar composition and the counterion binding degree (&) of the mixed micelles have been evaluated. It was found that (1) in dilute solutions (total surfactant concentration lower than 5 mM) mixed micelles having a very low fraction of associated counterions are formed until a critical situation is reached where the behavior of the binding changes sharply. This critical situation was found to be reached at micellar compositions ranging from 0.16 to 0.41 TTAB mole fraction depending on the experimental conditions. A rationale for thisfiiding is given in terms of micellar surface charge density requirements for the formation of a Stern layer. (2) A t total surfactant concentrations from 0.015 to 0.1 M, counterion binding was found to be mainly determined by the composition of the micelle. The binding dependence with micellar composition is accounted for by a simple binding model which considers site-specificadsorption promoted by the micellar surface potential. Surface potentials employed were evaluated from the measured micellar effect on the acid-base equilibrium of phenol red. The application of the charged phase separation model to account for the counterion binding on the mixed micelles is also discussed.
Introduction The study of ionic surfactant solutions by ion activity measurements provides valuable information regarding basic micellar properties such as the critical micelle concentration (cmc) and the degree of counterion binding @), as well as on the activities of free surfactant and counterions below and above the cmc. Singleion activity values can be obtained, if liquid junction potentials are disreUniversidad de Santiago de Chile. * Univeraidad de Chile. 0743-7463/89/2405-0753$01.50/0
garded, by using ion-specific electrodes. Intermicellar activities of surfactant ions and counterions have been measured in solutions of several ionic surfactants through the use of ion-specific electrodes.'-12 In all the systems investigated, comprising either anionic or cationic surfactants, it was found that the intermicellar counterion activity (a,) increases with increasing surfactant concentration above the cmc.l-l0 On the other hand, a decrease (1) Botr6, C.; Crescenzi, V. L.; Mele, A. J. Phys. Chem. 1969,63, 650. ( 2 ) Shedlovskt, L.; Jakob, C. W.; Epstein, M. B.J.Phys. Chern. 1963,
67,2075.
0 1989 American Chemical Society
