The Journal of
Physical Chemistry
0 Copyright, 1991, by the American Chemical Society
VOLUME 95, NUMBER 18 SEPTEMBER 5,1991
LETTERS Acid Hydrolyses of Hydrophobic Dioxoianes in Cationic Micelles: A Quantitative Treatment Based on the Poisson-Boitzmann Equation Andre Blasko, Clifford A. Bunton,* Department of Chemistry, University of California-Santa Barbara, Santa Barbara, California 93106
Cathy Armstrong, Walter Cotham, Zhen-Min He, Jackie Nikles, and Laurence S. Romsted* Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick, New Jersey 08903 (Received: May 24, 1991; In Final Form: July 18, 1991)
Acid hydrolyses of hydrophobic dioxolanes, la and lb, are inhibited by cationic micelles of myristyl- and cetyltrimethylammonium halides, MTAX and CTAX, respectively (X = CI,Br), but at surfactant concentrations sufficient to bind all the substrate, firstsrder rate constants, at constant [HX], increase with increasing [surfactant] and on addition of NaX. In the absence of added NaX k increases linearly with [MTAX] and [CTACI], but at constant [surfactant] there are breaks in plots of k, against [Nadl. These results can be explained quantitatively in terms of a pseudophase model in which hydrogen ion concentrations at micellar surfaces are calculated by solution of the Poisson-Boltzmann equation.
Introduction Ionic micelles inhibit, but do not completely suppress, reactions of co-ions and bound nonionic substrates.'*2 Residual rates increase with addition of surfactant or inert electrolyte to an extent that depends upon counterion type. The pseudophase ion-exchange (PIE)model explains these residual rates as incursions of both counterions and -ions into the reaction region, the micellar Stem layer, and specific counterion effects on surface co-ion concentrations are expressed in terms of a Donnan equilibrium.'.* An (1) Chimovich, H.; AI&, R. M. V.; Cuccovia, 1. M.; Zanettc, D.; Quina, F. H. In Sdufion Behoulor ojSur/oelants; Mittal, K. L., Fendler, E. J., Eds.; Plenum Preas: New York, 1982; Vol. 2, p 949. (2) (a) Annatrong, C.; Gotham, W.;Jennings, P.;Nikles, J.; Romsted, L. S.;Venace, M.; Waidlich, J. In Surfuctcmrs in Solution; Mittal, K. L., Ed.; Plenum h New York, 1989; Vol. 9, p 197. (b) He,Z.-M.; Loughlin, J. A.; Romsted, L. S. Bol. Soc. Chil. Quim. 1990,35,43.
0022-3654/91/2095-6747$02.50/0
SCHEME I
:%+ n
0
0
H20
Me0
-
n
H+
Me0
Me
la, R = n-CIsH2,;
Ib, R
Me
CI,H,I
alterative Coulombic model3 postulates that added ions reduce the surface potential, allowing co-ions to enter the reaction region to an extent that can be calculated from classical electrostatics by solving the Poisson-Boltzmann equation (PBE)in appropriate ~ymmetry.~ Rate constants of acid hydrolyses of hydrophobic dioxolanes (Scheme I) increase with increasing concentrations of cationic Q 1991 American Chemical Society
6748 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991
Letters
6r
5-
a
*i4 -
*
z
321-
0.2 1 .o
0.5 [NaCI], M
Figure 1. Comparison of observed rate constants for the hydrolysis of la in 0.02 M CTACl 0.01 M HC12 ( 0 ,right-hand scale) and predicted values of [H+], (left-hand scale) obtained with parameters in Table I. Solid line, A = 2.4 A; with constant N a n d a, -- - and are for A = 2.4 and 3 A, respectively, and N inmasing with NaCl (see text). The dashed line through the rate data is to guide the eye.
+
surfactant, myristyl- or cetyltrimethylammonium halide (MTAX and CTAX, respectively; X = CI, Br), or of NaX at constant [surfa~tant].~ The PIE model predicts linear relations between overall fint-order rate constants, k , and [surfactant] or [NaX]. The rate increase on addition of MTAX or CTACl to aqueous HX is linear, but slopes of initially linear plots of &.+ against [NaX] decrease at higher [NaX]? We examined the possibility that the PBE model would predict the nonlinearity of plots of k, against [NaCI] at constant [surfactant]. The current treatment is only applicable to rate effects of spherical micelle^,^-^ so we did not attempt to treat data quantitatively for CTABr micelles that undergo a sphere-to-rod transition.68
Results Observed rate constants for co-ion reactions with hydrophobic substrates decrease with added surfactant as substrate becomes micellar bound but then increase with additional surfactant or We consider only conditions in which our very hydrophobic substrates are fully bound and k, increases? The reaction region is a shell of thickness A at the micellar surface?) and k, is proportional to the molar concentration of hydrogen ions, [H+IA,in this region
k, = kz”[H+I*
(1)
(3) Bunton, C. A,; Mhala, M. M.; Moffatt, J. R. J . fhys. Chem. 1989,93, 7851. (4) (a) Bunton, C. A.; Moffatt, J. R. J . fhys. Chem. 1986,90,538. (b) Bunton, C. A,; Moffatt, J. R. J. fhys. Chem. 1988,92, 2896. ( 5 ) Rodenas, E.; Ortega, F. J . fhys. Chem. 1987, 91,837. (6) (a) Fendler, J. H. Membrune Mfmerfc Chcmfsrry; Wiley-Intedence: New York, 1982. (b) Zana, R., Ed. Sur/oerunr Solurfons; New Methods of Inuesffgulfon; Marcel Dckker: New York, 1987. (7) (a) Donhow, R.;Brig& J.; Bunton, C. A.; N i d i , D. F. J. fhys. Chcm. 1982,86,2388. (b) Donhow, R. B.; Bunton, C. A,; Nicoli, D. F. J . fhys. Chem. 1983,87, 1409. (8) Imae, T.; Kamiya, R.;Ikcda, S. J . Collofd Inrerfuce Scf. 1985, 108, 215. (9) The observed rate constanta for la and l b in 0.01 M MTAX and CTACl in the a b ” of added u l t arc lower by factom of a.10, than those of short-chain analogua.’ The ssmd-order rate corutant in water, k., is 5.9 M-’ I-I W on the acid hydro1 si8 of 1, R = CaHs, in 1.0 M NaCl at 30.0 OC. There is a p i t i v e salt on acetal hydrolysa in aqueous acetic medial0 so valua of kam/kwwill be larger if comparison is b a d on reaction without added ult, but counterion concentrations are high at micellar surfaced1 so aqueous salt is a reasonable medium for com b n pu-. (10) (a) Lon F,A.; Mclntyre, D. J. Am, Chem,Soc. 76,3243. (b) Bunton, C, A,; kdnheimer, J. D.J. Phys. Chem. 1970, 74,4457. (11) (a) Bunton, C. A,; Savelli, 0. Adu. fhys. Org. Chem. 1986,22,213. (b) Romrted, L. S. In Surfucrunrs In Solurfon; Mittal, K. L., Lindman, B., Edr.; Plenum Preu: New York, 1984; Vol. 2, p 1015. (c) Quina, F. H.; Chaimovich, H. J . fhys. Chem. 1979,83, 1844.
efL
lk,
0.4 0.6 [NaCI], M.
0.8
1 .o
Figure 2. Variations of &$ with [NaCl] in 0.02 M CTACl + 0.01 M HCI; ( 0 , O ) reactions of l a and lb, respectively. The lines are predicted; see text.
TABLE I: Calculated Second-Order Rate Constants in the Micellar PseudoDha&
substrate la la la la la lb lb
medium MTACI MTABr MTABr CTACl 0.02 M CTACl CTACI 0.02 M CTACI
+ NaCl + NaCl
a, A
N 70 90 80 8W
806 8W 806
+
17 19 19 20 20 20 20
klm,
M-I s-’ curveb 0.23 I 0.21 I1 0.18 I11 0.19 0.30 0.16 IV 0.24
+
*At 30.0 f 0.1 OC, with 0.02 M H X MTAX or 0.01 M HCI CTAC1: cmc = 2, 1 , and 1 mM for MTACI, MTABr, and CTACl, respectively. bFigure 3. CWithN increasing linearly to 95 in 0.5 M CTACI. With N constant up to 0.2 M NaCl and increasing linearly with [NaCI] to 90 in 1 M NaCI.
where kZm (M-l s-l) is the second-order rate constant at the micellar surface. We predict [H+IAas a function of [HX], [NaX], and [surfactant] following our approach applied to reactions of OH- in anionic micelles of sodium dodecyl sulfate.’ Distributions of counterions and co-ions are estimated for a spherical micelle by using the cell model and solving the PBE.h5J2 We assume that Na+ and H+ interact only Coulombically with the micelle but that CI- and Br- interact both Coulombically and ~ ~ i c a l l y . ~ ~ ~ This specific interaction between anions and the surface neutralizes the charge of an equivalent number of cationic micellar head groups. The fractional coverage by anions,f, is written in terms of a Volmer isotherm
6 expl(-1)(1 -A[xV-lt (2) = 1 + 6 cxp((-j)(l -j)[&-]) where 6 is a specificity parameter whose values, 15 and 120 M-I for Cl- and Br-, respectively, were estimated for reactions of anions in micellized CTAX.4 Surface electrical potential and counterion and co-ion concentrations are calculated by numerical integrat i ~ n , ” *within ~ J ~ the shell of width A, at the micellar surface and with values for the micellar aggregation number, N,and the radius of the charged surface, Coulombic models predict that an increase in N,at constant a, will increase the surface charge density and counterion concentration but decrease co-ion concentration at the surface. We made calculations for both constant and variable N.13 (12) (a) Mille, M.; Vanderlrooi, G. J . Collofd Inre@ce Scf. lSn, 59,211. (b) Gunnarsson, G.; Jonson, 8.;Wennerstrom, H. J. fhys. Chcm. 19M),84, 3114. ( I 3) Valuea of u for CTACI arc similar to those used in fitting data for reactions of nucleophilic anions, with the assumption that a will be lower than the hydrodpamic radius which enwmpaesg aa8cciated counterion, and water molecules. (14) Modest increases in N with increasing [electrolyte] were a b assumed in fitting rate data for anionic reactions in micellized cationic surface?
The Journal of Physical Chemistry, Vol. 95. NO.18, 1991 6749
Letters
+
(Figure 3, curve IV), and 0.02 M CTACl 1 M NaCl (Figure 2) to be about 1%, lo%, and 25% of that in bulk solvent, respectively. However, the volume of the micelles is much less than that of water, so the fraction of bound Htis usually less than ca. 0.1%. Our PBE treatment is limited to spherical micelles and is inap licable as micelles become rodlike, as for CTABr + NaHowever, classical electrostatics predicts that, for a given charge density, surface electrical potential will increase with a sphere-to-rod transition and will decrease. Thus, the downward curvatures of plots of k, against [CTABr] or [NaBr] are readily understandable? although we cannot fit the data quantitatively. The pseudophase model predicts the minima observed in plots of k, against [anionic surfactant] for reactions of OH-,',' but we cannot examine this question because substrates l a and l b are very hydrophobic and water insoluble. Consequently, we plot k, - ko in Figure 3. By assuming that the reaction only occurs in the micellar pseudophase (eq l), we neglect contributions from reaction in water or reaction induced by nonmicellar aggregationh*" which may account for the small discrepancy in the fits at low [NaCI] or [CTACI] (Figures 1-3). All our calculations are based on values of a, 6, A, and k2mthat do not change with [surfactant] and [salt].'.' The assumption of constant kzmis common to pseudophase treatments of micellar rate effects." Changes in the values of u, N,and A do not affect forms of predicted variations of [H+IA(and of k,) with either [surfactant] or [NaCl]. Figure 1 illustrates similarities of plots of k, vs [NaCl] and [H+IAvs [NaCI] in CTACI, regardless of assumptions made in the calculations. For a given reactive shell width, A = 2.4 A, predicted curvatures of plots of [H+IAvs [NaCl] increase if we allow N to increase with [NaCI], but the extent of curvature is not markedly affected by an increase of A to 3 A. However, these changes affect [H+IAand therefore k2min the micellar pseudophase. Figure 2 shows predicted variation of k, with [NaCl] in 0.02 M CTACl + 0.01 M HCI for substrates l a and l b based on the parameters given in Table I. Fits are good, except in dilute NaCl as noted above. Figure 3 illustrates the effect of a change of N of MTABr from 80 to 90 for reaction of la (curves I1 and 111). Similar results are obtained for reactions in MTACl (data not shown). An increase in N decreases [H+IAand increases k2"', but the quality of the fits is unaffected. In fitting the data for reaction of l b in CTACl solutions, we assume that N increases with increasing [CTACI]. There is a slight downward curvature of the plot if N is kept constant, but an increase of N increases the charge density of the cationic head groups and therefore decreases [H+]& Similar results are obtained for reactions of l a in CTACl (data not shown). We assume that MTGX and CTACI micelles remain spherical even with a modest increase in N? because they show no sphere-to-rod transition under our Values of k2'" (Table I) are similar in magnitude for reactions of l a and l b in the presence or absence of NaCI. As for acid hydrolysis of acetals in anionic micelles, k2"/kW< I , l 7 where k, (M-I PI) is the second-order rate constant in water. Comparison of second-order rate constants in micellar and aqueous pseudophases requires choice of concentration units, and following convention," we must specify the thickness, A, of the reactive shell (or its molar volume") to write concentrations as molarities. We took A = 2.4 A. Note that if A = 3 A, values of kzmare reduced by ca. 20% (Figure 1). The slightly smaller values of k p for reactions of l b compared to l a (Figure 1) may be due to the more hydrophobic substrate, lb, being immersed more deeply in the micelle." They key point of our treatment is prediction of the downward curvature of plots of k, against [NaCI] at constant [CTACl] and of the magnitude of kzmat cationic micellar surfaces. The PBE
Br.dI6
10s[H+],,
M
Figure 3. Variations of corrected first-order rate constants with [H+],. Reaction of la, left-hand scale: I, 0.02 M HCl, MTACI; I1 and III,0.02 M HBr, MTABr (see text). Reaction of I b in 0.01 M HCI, CTACl right-hand scale, IV. The fitting parameters are in Table I, and k0 is the value of k+ extrapolated to zero surfactant.t
Data Simulations. For reaction wholly in the micellar pseudophase, kl is proportional to [H+IA(eq l), and plots of [H+], and against [NaX] or [surfactant] should have similar forms. Figure 1 shows the dependence of k, on [NaCI] for reaction of l b (right-hand scale) and predicted values of [H'], (left-hand scale). We calculate [H+JAwith constant Nand u and also with N increasing with increasing [NaCl]. The broken lines are based on N = 80 up to 0.2 M NaCl and a linear increase with increasing [NaCI] to N = 90 at 1 M NaC1.I' Note the similarity in the breaks in plots for k* and [H+] and also the increase in [H+IA Figure 2 shows the predicted as A is increased from 2.4 to 3 dependence of k for reactions of l a and l b in 0.02 M CTACI and 0.01 M HCrwith added NaCI, based on parameters given in Table I and with an increase of N on addition of NaCI. In the absence of salt k, increases linearly with [MTACI], (MTABr], and [CTACI]? and in Figure 3 we plot k, - k,,against [H+],. As noted below, plots of k, versus [surfactant] do not extrapolate to the origin? and ko is the extrapolated value. We considered various assumptions regarding values of N. We assumed constant values of N for MTACl and MTABr, but for CTACI we made calculations with a constant N = 80 and with a linear increase of N from 80 up to 95 in 0.5 M CTACI." Values of N for MTABr with no added salt are 60-70,15and dynamic light scattering data in up to 0.1 M NaBr and 0.05 MTABr were fitted with N = 90.7b
A.
Dimmion The PIE and PBE models predict finite H+ concentrations at surfaces of cationic micelles that increase with added surfactant and electrolyteand are larger with B r than with CI-as counterion. In the PBE approach, counterions and co-ions are respectively in negative and p i t i v e concentration gradients extending radially from the s u r f a ~ e .An ~ increase of ionic concentration markedly reduces both concentration gradients. We estimate at the micellar surface in 0.02 M CTACI (Figure 2), 0.5 M CTACl ~
(15) (a) Jona,M.N.;Reed, D.A. Kollold Z . Z . Polym. 1969,235, 1196. (b) Dorranca, R, C.; Hunter, T. F. J. Chcm. Soc., Faraday Tram. 1 1974, 70, 1572. (c) Lianol, P.;a n a , R. J . Collold Intcr/occ Scl. 1981,81, 100.
( 16) The PIE model does not explicitly consider changes in micellar aize and shape. (17) (a) Bunton, C. A.; Romstcd, L. S.;Smith, H.J. J. Org. Chcm. 1978, 13,4299. (b) Bunton, C. A,; Romsted, L. S.;Savclli, G. J . Am. Chcm. Soc. 1979, 101, 1253. (c) Gonsalves, M.;Probst, S.;Rezende, M.C.; Nome, F.; Zucco, C.; Zanette, D. J . Phys. Chem. 1985, 89, 1127.
J. Phys. Chem. 1991, 95, 6750-6751
6750
approach alp0 predicts that with no added salt 4 increases linearly with increasing [surfactant] provided that reaction occurs wholly in the micellar pseudophase. Experimental Section The rate constants are from ref 2. The simulationswere carried out as described in ref 3, and values of k2"' are from plots of k, against calculated values of [H+],.
Acknowledgment. We are graW to a number of organizations for financial support: both C.A.B. and L.S.R.to the NSF U. S.-Latin American Cooperative Program-Brazil; C.A.B. to the Organic and Molecular Chemistry Program of NSF;and L.S.R. to the Busch and Biological Sciences Research Fund of Rutgers University, the donors of the Petroleum Research Fund, administered by the American Chemical Society, Research Corporation, and National Institutes of Health (GM32972).
Molecular Dynamics Simulations of SdM Buckminsterfulleretnes Ailan Cbeng and Michael L.Klein* Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6323 (Received: June 5, 1991; In Final Form: July 1 1 , 1991)
The results of constant-pressure molecular dynamics (MD) calculations are presented for facecentered-cubic solid buckminsterfullerene based on an atom-atom intermolecular potential acting between rigid C, molecules. At room temperature, potential parameters fitted to the oaxis compressibility of graphite yield a lattice constant in agreement with the measured value. The molecular dynamics trajectories are analyzed to obtain estimates of the reorientational relaxation time and the lattice vibrational frequency distribution. Below 200 K,orientational freezing is observed on the MD time scale.
The carbon cluster Cm (Buckminsterfullerene) can now be prepared more or less routinely in macroscopic quantitie~l-~This possibility has lead to a flurry of research activity directed at characterizing the properties of the pure solid and studying its chemistry.c6 For example, it is now established that under ambient conditions solid Cbois a facaomtered-cubic crystal7$ with lattice parameters a = 14.17 A and that the individual 'Bucky balls" are rotating.'-" However, on cooling to only 249 K, there appears to be a phase transition, as evidenced by the appearance of additional X-ray diffraction peaks.* Calorimetric measurements* and more recent NMR data" suggest a phase transition at 260 K. The effect of applying pressure has also been investigated, and rudimentary information is now available conceming the compressibility of the ~ o l i d . ~ . ' ~ The purpose of the present Letter is to see to what extent classical molecular dynamics calculations based on a simple ( I ) Kroto, H. Science 1988, 242, 1139-1145. (2) KrHtschmer, W.; Lamb, L. D.; Foatiropoulos, K.; Huffman, D. R. Nature. 1990. 347. 354-358. . --(3) Krlltnchmer, W.; Fastiropoulos, K.: Huffman, D. R. Chem. Phys. Lett. 1990,170,167-170. (4) Hawkim. J. M.:M e w .. A,:. Lewis. T. A.: Loren.. S.:. Hollander. F. J. Sciitice 1991,2S2.31i-313. (5) Holczer, K.; Klein, 0.;Gruner, 0 . ; Thompson, J. D.; Diedcrich, F.; Whetten, L. R. Submitted for publication in Phys. Rm. b i t . (6) H?+, K.;Klein, 0.;Hung, S. M.:Kaner, R. E.;Fu, K. J.; Whetten, R. L.: Dledench. F. Science 1991. 252. IlS&l157. (7) Fischer, J: E.; Heiney, P. A.;Mkhie, A. R.; Romanow, W. J.; Denenstein, A. M.; McCauley, J. P., Jr.; Smith, A. B., 111; Cox, D. E. Science 1991.252. -.---.1288-1 - -~290. (8) Heiney, P. A.; Fischer, J. E.; McGhie, A. R.; Romanow, W. J.; Denenstein, A. M.:McCaulcy, J. P., Jr.: Smith, A. B., 111; Cox, D. E. Phys. Rev. Lett. 1991,66, 291 1-2914. (9) Yannoni, C. S.;Johnson, R. D.; Meijer, G.; Bethune, D. S.;Salem, J. R.J. Phys. Chcm. 1991, 95,9-10. (IO) Tycko, R.: Haddon, R. C.; Dabbagh, G.; Glarum, S.H.; Douglas, D. C.; Mujm, A. M. J . Phys. Chem. 1991, 95, 518-520. ( I I ) Tycko, R.; Dabbagh, G.; Fleming, R. M.; Haddon, R. C.; Makhija, A. V.; Zahurak, S. M. Submitted for publication in Phys. Reu. Lett. (12) Duclos, S.;Brister, K,; Haddon, R. C.; Kortan, A. R.; Theil, F. A. Nature 1991, 351, 380-382.
-.- .--
~
~
0022-3654/91/2095-6750$02SO/O
atomatom potential can account for the physical properties of Buckminsterfullerene. Anticipating our results, we will see that this atomatom potential, with parameters fitted to the c-axis compressibility of graphite, gives a remarkably good account of the existing experimental data.7 Indeed, the agreement is sufficiently good that we have been motivated to present some information on the dynamical properties of the crystal. The simulations were carried out using the constant-pressure Parrinello-Rahman equations of The translational degrees of freedom were solved by using a third-order Gear predictor-corrector algorithm, and the rotational degrees of freedom, expressed by quaternions, were generated by using a fourth-order Gear a1g0rithm.I~ Each Cm molecule is treated as a rigid molecule with bond lengths lI = 1.37 A and 1, = 1.448 A, obtained from high-level quantum chemistry calculation^.^^*^^ This choice yields a 7. I-A diameter for the nuclear framework of the C,~molecule. Pairs of carbon atoms on different molecules are considered to interact via a Lennard-Jones (12-6) potential, with parameters e = 28.0 K and u = 3.4 A taken from early work on graphite.'* These parameters agree quite well with a recent fit19to the sublimation energym and lattice c o n ~ t a n t .Although ~ the molecule is nearly spherical, the interaction between two molecules depends on their relative orientation. The dimer energy minimum occurs between 9.7 and 10.3 A, with energy ranging from -2200 to -2900 K. The potential energy barrier for rotational motion can be estimated (13) Parrinello, M.;Rahman, A. Phys. Rev. Lett. 19so,45,1196; J. Appl. Phys. 1981,52,7182. (14)No&, S.: Klein, M.L. Mol. Phys. 1983,50, 1055. Impey, R. W.; Sprik, M.;Klein, M.L. J. Chem. Phys. 1905.83, 3638. (1 5) Gear, C. W. Numerical Initial Value Problems In OrdlnOrY D U f b ential Equations; PrenticuHall: Englewood Cliffa, NJ, 1971. (I 6) Scuseria, G. E. Chem. Phys. Lett. 1991, 176, 423-427. ( 17) Fowler, P.W.; Lazmeth, P.: Zanasi, R. Chem. Phys. Lett. 1998,165, 19. (1 8) Steclc, W. A. The Interaction of Gases wlth Solid Surfaces; Pergamon: New York, 1974. (19) Giriflco, L. A. Submitted for publication in J. Phys. Chem. (20) Pan, C.; Sampson. M.P.; Chai, Y.; Hauge, R. H.; Margrave, J. L. J . Phys. Chem. 1991, 95, 2944-2946.
Q 1991 American Chemical Society
