837
J . Phys. Chem. 1987, 91, 837-840
nmaxat which fd
and R peak for a given laser intensity. For the H20system, we have compared in Table I the effect of the laser on the dissociation dynamics of the states (3, 9) and ( 6 , 6 ) . In the absence of a laser,fd for the states (3, 9) and ( 6 , 6 ) equals 0.71 and 0.09, respectively, indicating that (3, 9) is a highly dissociative state. In the presence of a 0.8 TW/cm2 laser, fd for the (3, 9) and the ( 6 , 6 ) states becomes 0.72 and 0.20, respectively. Thus, the coupling with a laser field has very little effect on the dissociation dynamics of the state (3,9), but enhances the dissociation by about a factor of two for the long-lived complex. a
4. Conclusions We have found that the effect of laser-induced dissociation of highly excited vibrational states of triatomic molecules depends on the nature of classical dynamics of the state. In the case where the excited state is a short-lived complex, the laser has very little effect on the dissociation dynamics or lifetime. However, if the
molecule is in a state which exhibits mostly quasibound motion, the presence of a laser can enhance the dissociation probabilities and rates. Not surprisingly, such enhancement increases with laser intensity, and for each intensity there is a resonant driving frequency at which the maximum enhancement occurs. A possible experimental way to probe for such states would be to investigate which prepared states can have their lifetimes significantly altered. Systems which exhibit non-RRKM behavior would be good candidates for demonstrating such effects.
Acknowledgment. This work was supported by NSERC of Canada and U.S. DOE under contract DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc. The use of the facilities of the University of Tennessee Computing Center and the University of Waterloo Science VAX under the WATDEC project S34 is acknowledged. Registry No. HzO, 7732-18-5; D,O,7789-20-0.
An Electrostatic Approach for Explaining the Kinetic Results in the Reactive Counterion Surfactants CTAOH and CTACN Francisco Ortega and Elvira Rodenas* Departamento de Qdmica Fjsica, Universidad de Alcalii de Henares, Alcalii de Henares, Madrid, Spain (Received: March 13, 1986; In Final Form: September 23, 1986)
The association of hydroxide and cyanide ions to spherical micelles is analyzed by using the nonlinearized Poisson-Boltzmann equation, on the basis of the cell model. The calculated ion distribution around the micelle was used to compute the concentration of reactive ions and then the fraction of micellar head group neutralized, 8. Comparison of the theoretical results with estimates of counterion binding from kinetic results in micellized cetyltrimethylammoniumhydroxide and cyanide, CTAOH and CTACN, shows that the aggregation number increases with surfactant and ion concentration and that it is always larger for CTACN than for CTAOH. .
Introduction
It is known that cationic micelles affect the rates of bimolecular reactions that involve the attack of anions on hydrophobic substrates.'q2 In the case of reactive counterion surfactants, most of the kinetic results can be explained by the pseudophase mass-action model that describes the micelles as a phase separate from the aqueous phase and counterion distribution between both phases with a binding constant, Khu(Nu- = co~nterion).~When the counterion is hydrophobic, e.g. Br-, it appears that the micellar surface is essentially saturated with counterions with a constant value of the fraction of micellar head group neutralized, 0, while for hydrophilic counterions, the mass-action model predicts that (3 increases with surfactant and ion concentration. The electrostatic ion-micelle interaction involved in the formation of micelles is analyzed in the literature on the basis of the nonlinearized Poisson-Boltzmann e q ~ a t i o n . ~ In - ~ this paper we (1) (a) Romsted, L. Micellization, Solubilization and Microemulsions, Vol. 2, Mittal, K. L., Ed.; Plenum: New York, 1977; p 509. (b) Romsted, L. S. In Surfactants in Solution, Vol. 2, Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; p 1015. (2) (a) Fendler, J. H.; Fendler, E. J. Catalysis in Micellar and Macromolecular Systems; Academic: New York, 1975. (b) Fendler, J. H. Membrane Mimetic Chemistry; Wiley-Interscience: New York, 1982. (3) (a) Bunton, C. A,; Gan, L. H.; Moffat, J. R.; Romsted, L. S.;Savelli, G. J. Phys. Chem. 1981, 85, 3114. (b) Bunton, C. A.; Romsted, L. S. In Solution Behavior of Surfactants. Theoretical and Applied Aspects, Vol. 2, Mittal, K. L., Fendler, E. J., Eds.; Plenum: New York, 1981; p 975. (4) Overbeek, J. Th. G.; Stigter, D. R e d . Trau. Chim. Pays-Bas. 1956, 75, 1263. (5) Stigter, D. J. Phys. Chem. 1974, 78, 2428. 1975, 79, 1008. 1975, 79, 1015.
0022-3654/87/2091-0837$01.50/0
discuss the validity of this electrostatic approach for explaining the reaction of some low-spin diimine iron(I1) complexes with CTAOH and CTACN micelles. Theoretical Description and Computation Procedure
The electrostatic treatment we used is based on the cell model, which is frequently used in the l i t e r a t ~ r e . ~The - ~ total volume of micellar solution is divided into cells, each of them containing a micellar aggregate and the amount of water and electrolyte given by the whole concentration of the particular system. The cells and the micelles are considered spherical with the micelle, of radius rm, located at the center of thecell, of radius rc. The volume of the cell is given by
4ifr,3/3 = N / 1 O 3 N ~ ( [ D-]CmC)
(1)
where NA is Avogadro's number, [D]the surfactant concentration, cmc the critical micelle concentration, and N the aggregation number. The micellar charge is assumed to be uniformly distributed over the micellar surface with a surface charge per m2, u, given by u =
Ne/4ifrm2
(2)
where e is the electron charge, and the counterions are located (6) Bell, G. M.; Dunning, A. J. Trans. Faraday SOC.1970, 66, 500. (7) Mille, M.; Vanderkooi, G. J . Colloid Inrerface Sci. 1977, 59, 211. (8) Gunnarsson, G.; Jonsson, B.; Wennerstrom, H. J . Phys. Chem. 1980, 84, 3114. (9) Linse, P.; Gunnarsson, G.; Jonsson, B. J. Phys. Chem. 1982, 86, 413.
0 1987 American Chemical Society
Ortega and Rodenas
838 The Journal of Physical Chemistry, Vol. 91, No. 4, 1987 in the aqueous region of the cell. In the region rm< r < rc the distribution of ions around the micelles is assumed to be given by the nonlinearized PoissonBoltzmann equation (NLPB) which for spherical symmetry is expressed as crcO(l/r2)d / d r (r2 d$/dr) = -p = -&Fci
i
ic
1
(3)
Equation 3 reduces to eq 4, if only monovalent ions are present 2
in solution, where tr is the relative permitivity, assumed to be constant in the cell, to is the vacuum permitivity, Tis the absolute temperature, k is the Boltzmann constant, 4 is the electrostatic potential, p is the charge density, zi is the valency of charged ~ c - ~are the concentrations (mol m-3) of species i, and c + and positive and negative ions, respectively, at the point of the solution where 4 = 0. For a micellar solution at equilibrium 4(r=rc) = 0
/ 0010
0020
0030
Oblb
0060
ICTAOHI(K)
Figure 1. Variation of the pseudo-first-order rate constant, k y , with CTAOH concentration for reaction of l a (0,0) and l b (0,U) with OH-. 0, 0 in the absence of additional NaOH; a. U, [NaOH] = 0.060 .M. Solid lines are predicted values.
(5)
and from the electroneutrality of the cell: (d4/dr)(,+
= 4’(r=rm) = -g/tOt,
(7)
The parameters c + and ~ c4 are related to the whole number of positive and negative ions in the cell, n+ and n-, by the normalization conditions
By guessing values of c+,, and c4, one can solve eq 4 by a fourth-order Runge-Kutta method using initial values of 4 and $’ at rc, 4 = 4’ = 0, eq 5 and 6. The solution was then checked against the additional condition, eq 7, and the normalization condition, eq 8 and 9, and error minimization gave new values ~ c-~,in accord with an algorithm based on the Powto c + and ell-Zagwill method, until convergence was reached. The main problem in this calculation is that rather good initial values for c + and ~ c0are needed to avoid divergence. With this procedure the ion distribution around the spherical micelle was obtained, and from these values, the average concentration of reactive ion in a layer of thickness Ar was computed from [Nu-],, = N A I Y A r c - oexp(e4/kT)(4?ir2) d r
(10)
and the fraction of micellar head group neutralized will be Plr
= [Nu-l~r/N
( 1 1)
where Ar is the Stern-layer thickness of micelles where reaction occurs. More details about the numerical method are given in ref 10. Results and Discussion We applied the above electrosta.tic treatment to the reaction of low-spin diimine iron(I1) complexes with reactive counterion surfactants CTAOH and CTACN.” The experimental pseudo-first-order rate constants for the reaction of tris(3,4,7,8tetramethyl-1 ,10-phenanthroline)iron(II) (la) and tris(4,7-diphenyl-1,IO-phenanthroline)iron(II) ( l b ) ions with OH- and CN-, in CTAOH and CTACN, can be explained with the macroscopic (10) Ortega. F. Ph.D. Thesis, Universidad de Alcall de Henares, Alcall de Henares, Madrid, 1985. ( 1 1) Ortega, F.; Rodenas, E. J . Phys. Chem. 1986,90,2408.
Figure 2. Variation of the pseudo-first-order rate constant, k,, with CTACN concentration for reaction of la (0,0) and 1: (m) with CN-. 0 , in the absence of additional NaCN; 0, [NaCN] = 0.040 OM. Solid lines are predicted values.
pseudophase mass-action kinetic model,3 Figures 1 and 2, equations:
where kw and kM are the rate constants in aqueous and micellar pseudophases, kMis expressed in terms of the mole ratio of micellar nucleophile bound to micellar head groups, [NUT:] = [NUM-] + [Nuw-], mNu= [NuM-]/[Dn], and Ks is the binding constant of the substrate to the micelle written in terms of micellized surfactant:
where [Dn] = [D] - cmc. The parameters kM,Khu,and Ks have been taken as adjustables parameters and the values that best fit the experimental results, already discussed,” are given in Table I, k,”’ = 0.14kM,’’and in Figures 1 and 2, solid lines representing the calculated values of the pseudo-first-order rate constant obtqined by using these parameters. The values of KfOHand KhN, 80 and 300 M-l, are the same for these two substrates with similar structure but different hydrophobicity. With these values of the equilibrium distribution constants of ions between the micellar and aqueous phases, the fractions of micellar head group neutralized at different CTAOH and CTACN concentrations have been calculated with eq 13, p = mNu,and the values are represented in Figure 3. We also estimated the reactive ions concentrations in the Stern layer of micelles by using the nonlinearized Poisson-Boltzmann equation,
Kinetics of CTAOH and CTACN Reactants
The Journal of Physical Chemistry, Vol. 91, No. 4, 1987
839
TABLE I: Mass-Action Parameters That Best Fit the Experimental Results for Reactions of l a and l b with OH- and CN- in CTAOH and CTACN substrate l a substrate l b surfactant [NaNu]/M k2"'/M-' s-I Ks/M-' K",,/M-' surfactant [NaNu]/M k2"'lM-l s-' Ks/M-' K'NJM-' CTAOH 0.000 9.52 x 10-3 1.3 80 CTAOH 0.000 1.10 X 300 80 3.1 80 0.040 1.10 x 10-2 1000 80 0.060 9.52 X lo-' 0.000 0.040
CTACN
1.04 X lo-* 1.04 X lo-*
9 10
300 300
CTACN
8.54 x 10-3
0.000
1200
300
and Fraction of Micellar Head Group within 2-, 3-, and 4-A Shells for Different Surfactant Concentrations and TABLE 11: Values of e-,,, dm Aggregation Numbers of 50 and 100' -1 0-94; b 0.568 0.568 0.568 0.568
P7A
B dh
100 100
do, V 0.292 0.239 0.222 0.199
h h
0.403 3.248 6.255 15.468
0.548 0.560 0.565 0.569
0.609 0.625 0.627 0.638
0.644 0.664 0.670 0.680
50 50 50 50
0.7 15 5.909 11.579 29.674
0.233 0.181 0.164 0.141
0.284 0.284 0.284 0.284
0.303 0.335 0.345 0.361
0.361 0.400 0.410 0.435
0.404 0.448 0.460 0.485
[CTANu], M 0.002 0.020 0.040 0.100
N 100
0.002 0.020 0.040 0.100
100
c4, mol m-)
'Corresponding to a surface charge density of u(N=100) = 0.394 C m-2 and u(N=50) = 0.197 C m-2.
TABLE 111: Values of NThat Fit the Experimental Results for Reaction of l a and l b with CTAOH and CTACN, and Corresponding Values of e-,,, $0, and 8 ICTANul.6 M N c4, mol m-3 4 0 %v a: C m-2 P3A 0.002 0.004 0.006 0.008 0.010 0.016 0.024 0.030 0.040 0.060
25
(45) (60)
37
1.1687 2.4363
(78) 45 (104)
(0.224) (0.224)
0.129
0.145
0.270 (0.551)
0.177
0.352
(0.247)
(0.409)
0.204
(0.650)
0.271 (0.261) (0.229)
0.212 0.213
(0.329) (0.441)
(0.307)
0.188 (3.2988) (3.8545)
(0.177) (0.236)
(0.240)
(2.551 1) 7.0229 9.2752
0.098
0.181
5.3416 (150) (170)
0.165
(1.7320) 3.3560
69 88 100
(0.77614) (1.1504)
0.499 (0.590) (0.669)
0.346 0.394
(0.726) (0.757)
0.580 0.639
" r m= 18 A; t = 25 "C; er = 78.3. bValues in parentheses correspond to CTACN. 'Calculate from @ ',, with eq 7.
zoo}
1
1
0.010
0.020
0.030
0.bLO
'0.060
[CTANuI(M)
Figure 3. Variation of (3 with [CTANu] and [NaNu] for the reaction of l a and lb with OH- ( 0 , O ) and CN- ( 0 ,m). 0 , O in the absence of added NaNu; 0 for CTAOH, [NaOH] = 0.060 M; for CTACN, [NaCN] = 0.040 M.
I
//
0010
002c
0030
0040
1 0360
:CTANuI(M)
assuming spherical micelles under all conditions. This assumption is supported by the rekults in t h e literature which show t h a t for CTABri2 a n d CTAC113 micelles, with counterions of hydrophobicity similar to CN-, are spherical under similar conditions a n d
(12) (a) Lindblom, G.; Lindman, B.; Mandell, L. J . Colloid Interface Sci. 1973, 42, 400. (b) Henriksson, U. Chem. Phys. Left. 1977,52, 554. (c) Reiss-Husson, F.; Luzzati, V. J . Phys. Chem. 1964,68, 3504. (d) Ulmius, J.; Wennerstrom, H. J . Magn. Reson. 1977, 28, 309. ( e ) Johanson, L.; Lindblom, G.; Norddn, B. Chem. Phys. Lett. 1976,39, 128. (13) (a) Ekwall, P.;Mandell, L.; Solyom, P.J. Colloid Interface Sci. 1971, 35,519. (b) Danielsson, I.; Rosenholm, J. B.; Stenius, P.; Backlund, S. Prog. Colloid Polym. Sei. 1976,61, 1.
Figure 4. Variation of the aggregation number (N) with [CTANu] and [NaNu]. 0 for CTAOH in the absence of additional NaOH; 0 for CTACN in the absence of additional NaCN; 0 for CTAOH, [NaOH] = 0.060 M; for CTACN, [NaCN] = 0.040 M. recent results show t h e s a m e is t r u e for CTAOH.14.'5 To calculate t h e ion concentration in t h e S t e r n layer with eq 3, we fixed the values of some of the parameters: the aggregation number, N , Stern-layer thickness, Ar, a n d micellar radius, rm. ~~
(14)Flory, P.J. Statiscal Mechanics of Chain Molecules; Wiley: New York, 1969. A size similar to a sixteen-bond segment of polyethylene was assumed for the micellar group CI6H3+ (15) Lianos, P.; Zana, R. J . Phys. Chem. 1983,87, 1289.
Ortega and Rodenas
840 The Journal of Physical Chemistry, Vol. 91. No. 4 , 1987 TABLE I V Values of N That Fit the Experimental Results for Reaction of la and l b with CTAOH,Using Tanford’s Equation, and Corresponding Values of c4 dotand u‘ ICTANul. M N CA mol m-3 bo, V u, C m-* &A 0.179 0.149 0.129 17 1.2615 0.002 0.190 0.199 0.352 40 3.4631 0.010 0.469 0.196 0.294 4.8403 0.020 68 0.204 0.294 0.580 130 6.3745 0.040 0 210 0.340 0.639 200 7.3324 0.060
30c N
20c
5 r = 25 OC; cr = 78.3.
Initial1 we assumed a constant value for the micellar radius, rm = 18 in accord with ref 14. Different values of the aggregation number are used in the literature to fit the experimental results: for CTAOH values between 46 and 5 1 i 5 and 65.16 In our first calculation, two different values for N were used, 50 and 100, and we calculated the fraction of micellar head group neutralized at Ar = 2 , 3, and 4 A. The values for oh, the electrostatic surface potential, &, and &,’ are in Table 11, and from these results we concluded that the fraction of micellar head group neutralized depends on the aggregation number and on the Stern-layer thickness. Micelles grow with increasing surfactant and counterion concentration,12-18and, assuming that the Stern-layer thickness is independent of surand used factant concentration, we fixed this value at Ar = 3 the aggregation number as an adjustable parameter in order to fit the experimental kinetic data. The obtained values for the aggregation number are showed in Figure 4, and the values of &A, u, &, and c4 are given in Table 111. These results are consistent with an increase in the aggregation number of the micelles with increasing surfactant concentration, and, in all cases, the aggregation number for CTACN is larger than the aggregation number for CTAOH under equivalent conditions, indicating that the more hydrophobic ion induces larger micelles, as often noted in the l i t e r a t ~ r e , ~ although ~ ~ ~ ~ , ”the high aggregation number found in the case of CN- as the reactive ion indicates that, for this ion, not only electrostatic but also specific interactions should be c o n ~ i d e r e d . ’ ~ ~ ~ ~ We tried to improve our calculations using Tanford’s theory which relates the micellar volume to the aggregation number and the micellar radius, rmZ1
1,
4 a r m 3 / 3= N(27.4 + 2 6 . 9 ~ )
(15)
where n, is one fewer than the total number of carbon atoms of the alkyl chain, n, = 15 for the CTANu surfactant. By considering (16) Athanassakis, V.; Moffat, J. R.; Bunton, C. A.; Dorshow, R. B.; Savelli, G.; Nicoli, D. F. Chem. Phys. Lett. 1985, 115, 467. (17) Lindman, B.; Wennerstrom, H. In Micelles; Springer-verlag: Berlin, 1980 Topics in Current Chemistry Series, Vol. 87, p 45 and references therein. (18) (a) Rhode, A,; Sackmann, E. J . Colloid Interface Sri. 1979, 70, 494. (b) Ibid. J . Phys. Chem. 1980, 84, 1598. (19) Rathman, J. F.; Scamehorn, J. F. J . Phys. Chem. 1984, 88, 5807 (20) Burton, C. A,; Moffat, J. R. J . Phys. Chem. 1986, 90, 538. (21) (a) Tanford, C. J . Phys. Chem. 1974, 78, 2469. (b) Leibner, J. E.; Jacobus, J. J . Phys. Chem. 1977. 81, 130.
1oc
0.020
O.OL0
0.060
[ C T A O H I (M) Figure 5. Variation of the aggregation number (N)with [CTAOH] with (0)rm = 18 A and ( 0 )rm calculated with eq 15.
this variation in the micellar radius, r,, eq 15, we repeated the calculation and the new values of the aggregation number that fit the experimental results are in Figure 5 . Table IV lists values of &, u, &A, and c4. Our results for CTAOH show that small aggregation numbers are needed at low surfactant concentrations, but at high surfactant concentrations higher aggregation numbers are needed in order to fit the kinetic data. For CTACN the values of the aggregation number, obtained by using Tanford’s equation, are too high to have any physical sense. We note that using Tanford’s equation the surface charge per mz, u, and the aggregation number are different than those obtained by using a fixed value for r, = 18 A, while similar values of surface potential are obtained. From all of these results it can be concluded that the nonlinearized Poisson-Boltsmann equation for spherical symmetry can partially explain the macroscopic kinetic results for the reaction of la and l b with OH- and CN- in the reactive counterion surfactant CTAOH and CTACN, using the aggregation number as an adjustable parameter. The calculated results show that N increases with surfactant concentration, according to the results in the literature for other reactions,22and that, without considering specific interactions, the aggregation number of CTACN micelles is always larger than that for CTAOH micelles, although the high aggregation numbers obtained for CTACN indicate that some kind of specific interaction may occur between the cyanide ion and the head group of the micelle.
Acknowledgment. F.O. gratefully acknowledges financial support from the “Plan de Formacion de Personal Investigador” of the Spanish Government. Registry No. la, 17378-70-0; l b , 21412-03-3; CTAOH, 505-86-2; CTACN, 74784-26-2 (22) Bunton, C A., Moffat, J. R. J Phys. Chem 1985, 89, 4166.
