J. Phys. Ckem. 1986, 90, 538-541
538
is apparent if one calculates the ratio of the correct and approximate rate constants
k d / k b = 1 /(1 - r ~ ~ / a ~ )
(6)
Equation 4 underestimates the rate constant, and the error depends on the relative sizes of the reaction partners. For rA = rB the systematic error in kh is only -14%. However, in case of disparate sizes of the reactants the error becomes large. If rA/rB= 10, the approximate kh obtained from eq 4 is too small by a factor of 3, for rA/rB= lo2 by a factor of 25, and for rA/rB= 10" by a factor of 2.5 X lo3. Thus the use of the correct expression (5) is recommended for the calculation of diffusion-controlled dissociation rate constants, especially if macromolecules, enzymes, colloidal particles,"~'* or interfacial reactions' l , I 2 are involved. Naturally, the arguments presented in this Letter are equally valid for the corresponding theoretical calculations involving uncharged species, as well as for the calculation of the equilibrium constant of any reaction whose dissociation step is diffusion controlled.
/
'-' Figure 1. Schematic representation of the particle pair AB.
volume of particle A is not available for B in the complex AB, and this is analogous to the "excluded volume" of the van der Waals equation. Thus, one has to use the actual volume of the spherical shell AV, = 4 4 a 3 - rA3)/3in eq 3 resulting in the correct expression of kd 320
Acknowledgment. This work was partially supported by the R. A. Welch Foundation. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for additional support.
(5)
(11) (1984). (12)
The effects of the explicit consideration of the excluded volume
R. D. Astumian and Z. A. Schelly, J . Am. Chem. SOC.,106, 304 R. D. Astumian and P. B. Chock, J . Phys. Chem., 89, 3477 (1985).
Ionic Competition in Micellar Reactions: A Quantitative Treatment Clifford A. Bunton* and John R. Moffatt Department of Chemistry, University of California, Santa Barbara, California 931 06 (Received: September 23, 1985)
Distribution of reactive and inert hydrophilic ions about a spherical micelle has been estimated taking account of specific and Coulombic interactions by using the Poisson-Boltzmann equation over a range of [electrolyte] and [surfactant]. The treatment is applied to dephosphorylation and aromatic nucleophilic substitution by OH- and an SN2reaction of thiosulfate ion in micellized cetyltrimethylammonium chloride and bromide.
Inert counterions compete with reactive counterions for micellar surfaces, and reduce reaction rates, and an understanding of ion binding is an integral part of any description of reactions in such ionic colloids as micelles' and microemulsions.2 The effectiveness of the competition increases with decreasing charge density of the inert ion, and the variation of rate with concentrations of surfactant or added electrolyte can be treated quantitatively in terms of the pseudophase, ion-exchange, model.',2 This model treats water and micelles as distinct reaction regions so that rates will depend upon distribution of both reagents between these regions. For nonionic substrates distribution follows a simple (1) (a) Romsted, L. S. In 'Micellization, Solubilization and Microemulsions"; Mittal, K. L., Ed.; Plenum Press: New York, 1977; p 509. (b) Romsted, L. S. In "Surfactants in Solution"; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 2, p 1015. (c) Romsted, L. S. J . Phys. Chem. 1985.89, 5107, 5113. (d)Quina, F. H.;Chaimovich, H. Ibid. 1979, 83, 1844. (e) Almgren, M.; Rydholm, R. Ibid. 1979, 83, 360. (f) Abuin, E. B.; Lissi, E.; Araujo, P. S.; Aleixo, R. M. V.; Chaimovich, H.; Bianchi, N.; Miola, L.; Quina, F. H. J . Colloid Interface Sci. 1983, 96, 293. (g) Funasaki, N.; Murata, A. Chem. Phann. Bull. 1980,28,805. (h) Bunton, C. A. Caral. Reu. Sci. Eng. 1979, 20, 1. (i) Sudholter, E. J. R.; van der Langkruis, G. B.; Engberts, J. B. F. N. R e d . Truv. Chim. Pays-Bus 1980,99, 73. 6) Broxton, T. J.; Sango, D. B. Aust. J . Chem. 1983, 36, 711. (2) (a) Mackay, R. A. J. Phys. Chem. 1982,86,4756. (b) Bunton, C. A.; de Buzzaccarini, F. Ibid. 1982, 86, 5010.
Michaelis-Menten type of equation,3 but concentration of reactive ions in the micellar pseudophase is often calculated assuming that ionic competition is governed by eq 1, where Y and X are reactive
and inert ions respectively, and W and M denote aqueous and micellar pseudophases. The micelle is also assumed to be saturated with counterions; Le., the degree of fractional ionization, a, is constant.la This treatment has been strikingly successful in treating rate and equilibrium data, but there are some problems in its application. It depends upon reasonable, but unproven, assumptions, and it fails to account for reactions of some very hydrophilic ions, e.g., OH- and F,1*4 especially at high con~entration.~In addition widely different ion-exchange parameters have been used in fitting the kinetic data, especially for reactions of OH-.' Values of ~
~~
(3) Menger, F. M.; Portnoy, C. E. J . Am. Chem. SOC.1967, 89, 4698. (4) (a) Bunton, C. A.; Romsted, L.S.; Savelli, G. J . Am. Chem. Soc. 1979, 101, 1253. (b) Bunton, C. A.; Gan. L-H.; Moffatt, J. R.; Romsted, L. S.; Savelli, G. J . Phys. Chem. 1981, 85, 4118. (5) Nome, F.; Rubra, A. F.; Ionescu, L. G. J . Phys. Chem. 1982,86, 1881. Stadler, E.; Zanette, D.; Rezende, M. C.; Nome, F. Ibid. 1984, 88, 1892
0022-3654/86/2090-0538$01.50/00 1986 American Chemical Societv
Letters ion-exchange parameters for OH- or F,estimated by fluorescence quenching in cetyltrimethylammonium bromide (CTABr), are much larger than those generally used in fitting kinetic or equilibrium data.“ The simple kinetic model also appears to be unsatisfactory for dianions; for example, it was used to fit data for a reaction of thiosulfate ion in solutions of CTABr, but the ion-exchange parameter varied with [CTABr] .6 However, Lissi et al. fitted their fluorescence quenching data for mixtures of thiosulfate and univalent anions to an ion-exchange model.’ An alternative approach is to assume that ionic distribution about a micelle can be estimated from the ionic concentration in water and the micellar electrostatic surface potential, but the problem is estimation of this potential.2a*s However, the Poisson-Boltzmann equation (PBE) has been solved in spherical symmetry for a range of surfactant and electrolyte concentrations,”’ and reaction rates in CTAOH can be interpreted from the calculated distribution of OH-.12 Simpler, but less general, treatments of Coulombic binding have also been described and are applicable in very dilute surfactant, for e ~ a m p 1 e . I ~ A purely Coulombic model does not account for ion specificity and Rathman and Scamehorn have combined the Coulombic interactions, calculated from PBE, with a specific interaction term, but their calculation was for plane surfacesI4 rather than for spherical surfaces such as those of micelles. We have extended our earlier treatment of Coulombic, nonspecific, interaction of OH- with micelles of CTAOHIZto mixed ion systems on the assumption that such ions as C1- and Brinteract both Coulombically and specifically with cationic micelles. We apply it to reactions of OH- and thiosulfate ion for which rate and substrate binding constants are a ~ a i l a b l e . ~ ~ ~ ~ ’ ~ Results and Discussion The solution volume is divided into electrically uniform cells, each containing one micelle, radius, a, and aggregation number, N. The cell radius, R, is given by
where NA is Avogadro’s number, [DT] is total [surfactant], and cmc is the critical micelle concentration. The PBE, written in spherical symmetry for a solution containing univalent cations and various charged anions, gives the reduced potential, C$= eJ//kT, where $ is the electrostatic potential, in the form
where t is the dielectric constant,I6 T, the absolute temperature, e, the electrostatic charge (esu), and k Boltzmann’s constant. For a colloidal cation Z+ and Z- are the valencies of co- and coun~ )the cell terions and niRthe number concentration (ions ~ m - at wall. ( 6 ) Cuccovia, I. M.; Aleixo, R. M. V.; Erismann, N. E.; Van der Zee, N. T. E.; Schreier, S.; Chaimovich, H. J . Am. Chem. SOC.1982, 104, 4544. (7) Lissi, E. A.; Abuin, E. B.; Sepulveda, L.; Quina, F. H. J. Phys. Chem. 1984, 88, 8 1 . (8) Frahm, J.; Diekmann, S . J . Colloid Inferface Sci. 1979, 70, 440; In “Surfactants in Solution”; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 2, p 897. (9) Bell, C. M.; Dunning, A. J. Trans. Faraday SOC.1970, 66, 500. (10) Mille, M.; Vanderkooi, G. J . Colloid Interface Sci. 1977, 59, 21 1. ( 1 1) Gunnarsson, G.; Jonsson,B.; Wennerstrom, H. J . Phys. Chem. 1980, 84, 3114. (12) Bunton, C. A.; Moffatt, J. R. J . Phys. Chem. 1985, 89, 4166. (13) Mitchell, D. J.; Ninham, B. W. J. Phys. Chem. 1983, 87, 2996. Evans, D. F.; Ninham, B. W. Ibid. 1983, 87, 5025. (14) Rathman, J. F.; Scamehorn, J. F. J . Phys. Chem. 1984, 88,5807. (15) Bunton, C. A.; Robinson, L. J . Am. Chem. Soc. 1968, 90,5972. Bunton, C. A.; Mhala, M. M.; Moffatt, J. R.; Monarres, D.; Savelli, G . J. Org. Chem. 1984, 49, 426. Cipiciani, A.; Savelli, G.; Bunton, C. A.; Mhala, M. M.; Moffatt, J. R., submitted for publication in J . Chem. Sor., Perkin Trans. 2. (16) Values of c are 78.54 and 77.75 at 25 and 30 ‘C respectively.” (17) w e d ,H. S.; Owen, B. B. “The Physical Chemistry of Electrolyte Solutions , 3rd ed.; Reinhold: New York, 1958.
The Journal of Physical Chemistry, Vol. 90, No. 4, 1986 539 The boundary counditions are
4 = d4/dr = 0
at r = R
d$/dr = (1 -j)Ne2/ta2k T
at r = a
(4) (5)
The factor (1 -A wherefis the fractional coverage by counterions takes into account reduction of charge density at the micellar surface due to specific adsorption of counterions, because such a counterion will electrically neutralize one cationic head group at the micellar surface. Average concentrations, it, and those at the cell wall are related by
with
V = 4x(R3 - a 3 ) / 3
(7)
The cell is electrically neutral so that = Vit+ (1 - A N
+
We assume thatfis independent of and depends only upon the nature and concentration of counterions, and we estimated it using either a Langmuir or a Volmer isotherm (eq 9)18
f=
6 ~ x P ( - ~ / u- n ) [ x W - i
(9) 1 + 6 ex~(-f/(1 -n)[xW-i where 6 is the Volmer specificity constant. Equation 9 reduces to the Langmuir form asf- 0. Similar equations can be written for the reactive ion, Y-.I9 For mixtures of nucleophilic ions and inert halide ions we found that the Volmer isotherm (eq 9) gave the best fit for solutions of Br-, but with C1- either the Volmer or Langmuir isotherms were adequate. In the following discussion we consider only eq 9. The distributions of co- and counterions can be estimated by solving the PBE with the boundary conditions (eq 4 and 5) and taking into account the possibility of specific adsorption of counterions. The general procedure has been described.I2 Effects due to interactions between counter- and co-ions or to ion size are not included explicitly in the treatment, cf. ref 20. Kinetic Analysis. We applied the treatment to reactions of OHin solutions of CTABr or CTAC1I5 and of thiosulfate ion in CTABr. (The nonionic substrates bind readily to micelles.) For CTAOH we assumed that the micellar reaction took place in a 2.4-A shell around the micelle so that the first-order rate constants are given byI2
where kw and k2” are respectively second-order rate constants in water and the shell, Ks is the substrate binding constant, and [OH-]2,4~ is the average molarity within the 2.4-A shell. In the reactions discussed here the term kw[OHw-] can be neglected except in very dilute surfactant, and values of Ks have already been e ~ t i m a t e d . ~The . ~ values of the “kinetic” cmc were taken from the onset of the rate enhancement.’ These assumptions are generally made in treatments of micellar kinetics.’-6 We also assume that the micellar radius, a, is constant for a given surfactant, and that N for micelles of CTACl or CTABr is independent of [surfactant] but may increase on addition of electrolyte. This assumption is reasonably good for CTACI, but is probably less satisfactory for CTABr where increasing [surfactant] may (18) McLaughlin, S.In “Current Topics of Membranes and Transport’’; Bronner, F., Kleinzeller, A., Eds.; Academic Press: New York, 1977; Vol. 9, p 71. (19) Bratko and Lindman have accounted for micellar counterion selfdiffusion in terms of a modified PBE which takes into account ion size and excluded volume, but assumes that these parameters are the same for the ions under consideration.20 (20) Bratko, D.; Lindman, B. J. Phys. Chem. 1985,89, 1437.
540
The Journal of Physical Chemistry, Vol. 90, No. 4, 1986 [NoCl],
M
Letters TABLE I: Kinetic Fitting Parameters" [nucleophile], sursubstrate M factant 0.01 CTACI DNCN~
DNCN DNCN DNCN DNCBc
0.03 0.01 0.03
0.05 ~ N P D P P ~ 0.01 0.03 pNPDPP 0.01 pNPDPP
pNPDPP pNPDPP n-BuIe
0.03 0.05 0.00157
CTACI CTABr CTABr CTABr CTACI CTACl CTABr CTABr CTABr CTABr
lo4 cmc,
M
N 80 88 90 110
9 4 7 2 6 9 4 7 2 0 3.2
125 80 88 90
110 125 100
102k,", M-' s-I
1.05 1.15 1.05
1.05 0.035 6.0 6.0
6.0 6.0 6.0 32
"For reactions with OH- at 25.0 OC ref 4 and 15, unless specified; = 15, aBr = 120. * K , = 1600 M-', k , = 6.4 X M-' SKI. 'K, = 70 M-I, k , = 1.4 X M" s-l. K, = lo4 M-l, k , = 0.48 M-' s-l. eReaction with 0.00157 M sodium thiosulfate at 30.0 OC with 0, 0.00149, and 0.00975 M added NaBr; K , = 450 M-', k , = 0.146 M-' s-I. ref 6.
3 I
001
I
002 [CTACI],
I
0.03
I
004
005
M Figure 1. Predicted variations of k, for reaction of DNCN with OH- in CTACI. 0, 0.01 M OH-; 0,0.03 M OH-. Insert: 0 , 0.02 M CTAC1, 0.03 M OH-, and added NaCl. cause the micelle to grow and become spheroidal, with a considerable size dispersion.21 However, we assume that radius, a, will not be very much larger than the extended chain length, which is 21.2 A for a hexadecyl group.22 The rate data were fitted by computer simulation based on values of shell thickness, K,, and k2"' which were essentially invariant with changes in [surfactant] or [electrolyte]. Reactions of OH-. Reactions of nucleophilic anions with nonionic substrates are slower in CTABr than in CTACl and Bris a better inhibitor than C1-, and both ions displace OH- from micelles.' Therefore values of 6 (eq 9) should decrease in the sequence Br- > C1- > OH-, and in fitting data for CTAOH we assumed that OH- did not interact specifically with cationic micelles,I2 so that 6oH = 0. We illustrate our treatment for reactions of OH- with 2,4dinitrochloronaphthalene (DNCN) or p-nitrophenyldiphenyl phosphate (pNPDPP) but it has also been applied successfully to reaction with 2,4-dinitrochlorobenzene (DNCB). Figure 1 shows the observed and calculated values of k+ for reaction of DNCN with OH- in CTACl based on the parameters given in Table I. The model also satisfactorily accounts for rate retardation by NaCI. Equally good fits were obtained for reactions of OH- with pNPDPP (Table I). The value of a of 21 A is slightly smaller than the value of the hydrodynamic radius which is ca. 27 A2Idand N is similar to literature values.2' The fits are only slightly affected by a change in 6 from 15 to 30. Figure 2 shows observed and calculated values of k, for reaction of pNPDPP with OH- in CTABr based on the parameters in Table I, and reasonably good fits were obtained under other conditions. The value of jlBr is much larger than that of bel, and the radius, a 24 A, is slightly larger than the extended chain length but is smaller than the hydrodynamic radius of ca. 32 A, estimated
-
(21) (a) Porte, G.; Appel, J. In 'Surfactants in Solution"; Mittal, K. L., Lindman, B., Us.; Plenum Press: New York, 1984; Vol. 2, p 805; (b) Ikeda, S. Ibid. p 825; (c) Dorshow, R.; Briggs, J.; Bunton, C. A,; Nicoli, D. F. J . Phys. Chem. 1982,86,2388. (d) Dorshow, R.B.; Bunton, C. A.; Nicoli, D. F. Ibid. 1983, 87, 1409. (22) Tanford, C. 'The Hydrophobic Effect"; Wiley: New York, 1973. Leibner, J. E.; Jacobus, J. J . Phys. Chem. 1977, 81, 130.
I
.,
I
I
I
0 01
I
0.02
[CTABr], M
Figure 2. Predicted variations of k, for reaction of pNPDPP with OHin CTABr. 0.01 M OH-; W, 0.03 M OH-; +, 0.05 M OH-.
by dynamic light scattering.21cValues of N are similar to those in the literature, although there is a spread of quoted values.21 The fit of calculated to experimental values of k, is poorer than with CTACl, probably because if addition of electrolyte causes growth of the micelle it will become ellipsoidal, and eq 2-6 are derived for spherical particles. In fitting the data we take values of the cmc lower than those in water,*j because added electrolytes and hydrophobic solutes reduce the cmc. "Kinetic" cmc values are typically lower than those in water.' There is a problem with both the pseudophase and our present model because they both neglect the possible existence of submicellar aggregates in very dilute surfactant and the extent to which a hydrophobic substrate may perturb the aggregate. Thus both models are least satisfactory with very dilute surfactant. The difference between values of 6 = 15 and 120 for CTACl and CTABr (Table I) is consistent with the relative affinities of the ions toward cationic micelles and with the generalization that weakly hydrophilic, polarizable, monoanions interact most strongly with mice1les.l.' Reactions of Thiosulfate Zon. The data for reaction of thiosulfate ion with n-butyl iodide in CTABr at 30.0 OC are from ref 6,which also gives Ks = 150 M-' and kw = 0.146M-' s-'. The cmc of cetyltrimethylammonium thiosulfate (CTA2T) is 1.6 X M, (3.2 X M for CTAT1l2)and we used that value consistently, because thiosulfate ion is in excess in dilute CTABr and divalent counterions give lower cmc than otherwise similar univalent ions.".23 The rate-surfactant profiles (Figure 3) were (23) Mukerjee, P.; Mysels, K. J. 'Critical Micelle Concentrations of Aqueous Surfactant Systems''; Natl Bur. Stand. 1970.
J . Phys. Chem. 1986, 90, 541-542
541
eH
0 05 [CTABr], M
01
Figure 3. Predicted variations of k , for reaction of n-butyl iodide with 0.00157 M thiosulfateion, ref 6. 0,no added NaBr; 0,00149 M NaBr; A, 0.00975 M NaBr. fitted assuming that thiosulfate ion does not bind specifically to CTA+ micelles. This assumption is consistent with evidence that Br- binds more strongly than thiosulfate ion to cationic micelles,6s7 and the fit is satisfactory for reactions in solutions containing NaBr. Our value of kzm= 0.32 M-' s-' is lower than that estimated by Chaimovich and co-workers: in part because they took a relatively large molar phase volume element of reaction of 0.37 L in their calculations with a variable ion-exchange parameter.24 Both treatments give a higher second-order rate constant in the micelles than in water. Comparison with the Ion-Exchange Model. The ion-exchange, pseudophase model is based on the assumption that the micellar reaction takes place in a region generally identified with the micellar Stern layer,' although recently reaction across the Stern layer boundary has been p o s t ~ l a t e d .Interionic ~ competition is described in terms of eq 1, although it is hard to rationalize some (24) The molar phase volume calculated for a 2.4-A shell is 0.1 15 L for This value is lower than those generally used in calculations based on the pseudophase model which vary from 0.14 to 0.37 L.'ses u = 24 A.
kinetically estimated values of with those estimated by fluorescence quenching." In addition fractional ionization of micelles is assumed to be independent of the nature or concentration of counterions. The ion-exchange model is essentially an empirical description of interionic competition for an ionic micelle, but it gives values of k2mwhich are similar to those calculated by solving the PBE with inclusion of a specific adsorption term for halide ion.2s Most workers have assumed that the volume element of reaction is independent of the nature of the reactants, just as we assume that reaction occurs in a 2.4-A shell.12 (An increase of shell thickness to 3 A, for example, does not affect the qualtiy of the fits, but it modestly increases k,"). These assumptions are suspect,26and variations in k2"/k, (Table I and ref 1, 2, 4-6) may simply be due to dependence of the volume element of reaction upon the reactants. But, despite imperfections, the simplicity of the ionexchange model makes it a convenient way to rationalize ratesurfactant profiles. There is a major different between the ion-exchange model and our present model, which describes interionic competition as a combination of Coulombic and specific interactions, so that a simple equation, such as eq 1, should be satisfactory only over a limited concentration range. Recent experiments show the failure of the ion-exchange model at high [OH-] or [H30+],sv27 and there is kinetic, and other evidence showing that ion-exchange parameters may depend upon the method of estimation and ionic concentration.~~6~7~Zs
Acknowledgment. Support of this work by the National Science Foundation (Chemical Dynamics Program) and the US.Army of Research (DAA9 29-83-9-0098 and DAAG 29-85-K-0016) is gratefully acknowledged. We are grateful to Drs. L. S. Romsted and F. H. Quina for very interesting discussions. (25) The agreement depends upon the choice of the volume element of reaction or the thickness of the reactive shell, eq (26) Hicks, J. R.; Reinsborough, V. C. Ausl. J . Chem. 1982, 35, 15. (27) Gonsalves, M.; Probst, S.; Rezende, M. C.; Nome, F.; Zucco, C.; Zanette, D. J . Phys. Chem. 1985, 89, 1127. (28) Reactions were followed spectrophotometrically in solutions of NaOH or KOH in CO, free water with M substrate as de~cribed.~.'~
The Si-0-Si Force Field: Ab Initio MO Calculations M. O'Keeffe* and P. F. McMillan Department of Chemistry, Arizona State University, Tempe, Arizona 85287 (Received: September 24, 1985)
The energy of H6Si207has been calculated as a function of bridging bond length and bond angle for a number of structures close to the minimum energy configuration, and harmonic stretch, bend, and stretch-bend force constants are derived. These are in accord with observed systematics of bond length-bond angle correlations in silicates. Fair agreement is also found between the observed bulk modulus of a-quartz and that calculated by a simple rigid tetrahedron model.
There is considerable current interest in the dynamics of silicate structures' and the nature of the Si-O-Si force field in silicas and silicates has been the subject of innumerable discussionsa2 It is now well established that molecular orbital calculations on suitably chosen molecular fragments accurately model local geometries in silicates and related materials3 and, in particular, that calcu-
TABLE I: F~~~~Constants ~ ~ f iin, ,the d Text for the si-o-si Moiety of H&i20, in Units of N ,-I 4 K rl r2
0
643 30.4, 616 23.0, 23.0, 17.2
basis sets accurately reproduce (1) seee.g. ~ i to ~ ~ ~ ~ ~~~~i~~~ ~ ~~ in ~ ~ i ~~ ~vel,i ~ ~ ~ ~lations ~ i~ with~l sufficiently ~* ~ , flexible ~ , bond length-bond angle variations experimentally observed in 14, Kieffer. S. W., Navrotsky, A., Eds.; Mineralogical Society of America: Washington, DC, 1985; and references therein. crystal^.^ It is also well established that SCF MO calculations 121 OKeeffe. M.: Newton. M. D.: Gibbs. G. V. Phvs. Chem. Miner. 1980. 6, 305. References to previous theoretical studies ofihe lattice dynamics of (4) O'Keeffe, M.; DomengEs, B.; Gibbs, G. V. J . Phys. Chem. 1985,89, a-quartz are given here. (3) Gibbs, G. V. Amer. Minerul. 1982, 67, 421. 2304. 0022-3654/86/2090-0541$01.50/0
0 1986 American Chemical Society
